Research Article Unstructured Grids and the...

Hindawi Publishing CorporationScience and Technology of Nuclear InstallationsVolume 2013 Article ID 641863 26 pageshttpdxdoiorg1011552013641863

Research ArticleUnstructured Grids and the Multigroup NeutronDiffusion Equation

German Theler

TECNA Estudios y Proyectos de Ingenierıa SA Encarnacion Ezcurra 365 C1107CLA Buenos Aires Argentina

Correspondence should be addressed to GermanTheler gthelertecnacom

Received 22 May 2013 Revised 20 July 2013 Accepted 20 July 2013

Academic Editor Arkady Serikov

Copyright copy 2013 GermanTheler This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The neutron diffusion equation is often used to perform core-level neutronic calculations It consists of a set of second-orderpartial differential equations over the spatial coordinates that are both in the academia and in the industry usually solved bydiscretizing the neutron leakage term using a structured grid This work introduces the alternatives that unstructured grids canprovide to aid the engineers to solve the neutron diffusion problem and gives a brief overview of the variety of possibilities theyoffer It is by understanding the basic mathematics that lie beneath the equations that model real physical systems better technicaldecisions can be made It is in this spirit that this paper is written giving a first introduction to the basic concepts which can beincorporated into core-level neutron flux computations A simple two-dimensional homogeneous circular reactor is solved usinga coarse unstructured grid in order to illustrate some basic differences between the finite volumes and the finite elements methodAlso the classic 2D IAEA PWR benchmark problem is solved for eighty combinations of symmetries meshing algorithms basicgeometric entities discretization schemes and characteristic grid lengths giving even more insight into the peculiarities that arisewhen solving the neutron diffusion equation using unstructured grids

1 Introduction

The better we engineers are able to solve the equationsthat model the real physical plants we design and buildthe better services we can provide to our customers andthus general people can be benefited with better nuclearfacilities and installations The Boltzmann neutron transportequation describes how neutrons move and interact withmatter It involves continuous energy and space-dependentmacroscopic cross-sections that should be knownbeforehandand gives an integrodifferential equation for the vectorial fluxas a function of seven independent scalar variables namelythree spatial coordinates two angular directions energy andtime It represents a balance that holds at every point in spaceand at every instant in time Such an equationmay be tackledusing a variety of approaches one of them is a simplificationthat leads to the so-called neutron diffusion approxima-tion that states that the neutron current is proportional tothe gradient of the neutron flux by means of a diffusion

coefficient which is a function of the macroscopic transportcross-sectionWhen this approximationmdashwhich is analogousto Fickrsquos law in species diffusion and to the Fourier expressionof the heat fluxmdashis replaced into the transport equation apartial differential equation of second order on the spatialcoordinates is obtained Formally the neutron diffusionequation may be derived from the transport equation byexpanding the angular dependance of the vectorial neutronflux in a spherical harmonics series and retaining both thezero and one-moment terms neglecting the contributions ofhigher moments [1 2] As crude as it may seem this diffusionequation gives fairly accurate results when applied under theconditions in which thermal nuclear reactors usually operateIndeed it can be shown that in a homogeneous bare criticalreactor the neutron current is proportional to the neutrongradient for every neutron energy [3]

The energy domain is usually divided into a finite numberof groups thus transforming one partial differential equationover space and energy into several coupled equationsmdashone

2 Science and Technology of Nuclear Installations

for each groupmdashcontaining differential operators appliedonly over the spatial coordinatesThis resulting set of second-order PDEs is known as the multigroup neutron diffusionequation and is usually used to model design and analyzenuclear reactor cores by the so-called core-level calculationcodesThese programs take homogenizedmacroscopic cross-sections (which may depend on the spatial coordinatesthrough changes of fuel burnup materials temperature orother properties) computed by lattice-level codes as an inputand solve the diffusion equation to obtain the flux (and itsrelated quantities such as power xenon etc) distributionwithin the core

Given a certain spatial distribution of materials and itsproperties inside a reactor core chances are that the resultingreactor will not be critical That is to say in general therate of absorptions and leakages will not exactly overcomethe neutrons born by fissions sources and some kind offeedbackmdasheither through an external control system or bymeans of an inherent stability mechanism of the core [4]mdashis needed to keep the reactor power within a certain intervalMathematically this means that the transportmdashand thus thediffusionmdashequation does not have a steady-state solution Inpractice given a certain reactor configuration its steady-stateflux distribution is computed by setting all the time deriva-tives to zero as usual but also by dividing the fission sources bya positive value 119896eff called the effective multiplication factorwhich becomes also one unknown and turns the formulationinto a eigenvalue problemThe largest (or smallest dependingon the formulation) eigenvalue is therefore the effectivemultiplication factor If 119896eff lt 1 the original configurationwas subcritical and conversely As successive configurationsmake 119896eff rarr 1 the associated eigenfunctions approach thesteady-state critical flux distribution [5]

Core-level codes traditionally use regular grids to dis-cretize the differential operators over the space Dependingon the characteristics and symmetry of the reactor coreeither squares or hexagons are used as the basic shapeof the mesh Usual discretization schemes involve cell-centered finite differences or two-step coarse-meshcouplingcoefficient methods [6ndash8] which are accurate and efficientenough formost applicationsThere are nevertheless certainlimitations that are not inherently related to structured gridsbut to the way their coarseness is utilized and how they arecomputed and applied to the reactor geometry These issuescan be overcome by discretizing the spatial operators usinga scheme which could be applied to nonstructured gridsnamely finite volumes or finite elements

This way unstructured grids may be used to studyanalyze and understand the numerical errors introduced bythe discretization of the leakage operator with a difference-based scheme over a coarse structured grid by successivelyrefining the mesh whilst comparing the solutions with thestructured one as depicted in Figure 1 Another example ofapplication may be the improvement of the response matrixof boundary conditions over cylindrical surfaces which isthe case for most of the nuclear power plants cores Whena coarse structured mesh is applied to a curved geometrythere appears a geometric condition known as the staircaseeffect Even though arbitrary shapes cannot be perfectly

tessellated with unstructured grids for the same number ofunknowns they better represent the geometry than struc-tured meshes as Figure 2 illustrates Again by refining themesh and comparing solutions the matrix responses usedto set the boundary conditions on the coarse meshes canbe optimized Finally other complex geometries such asslanted control rods (Figure 3) or irradiation chambers can bedirectly taken into account by unstructured grids possibly byrefining the mesh in the locations around said complexities

2 The Steady-State Multigroup NeutronDiffusion Equation

In the present work we take the steady-state multigroupneutron diffusion equation for granted That is to say wefocus on the mathematical aspects of the eigenvalue problemand make no further reference to its derivation from thetransport equation nor to the validity of its applicationto reactor problems as these subjects that are extensivelydiscussed in the classic literature [1 5 11]

The differential formulation of the steady-state multi-group neutron diffusion equation over an 119898-dimensionaldomain 119880 with 119866 groups of energy is the set of 119866 coupleddifferential equations

div [119863119892 (x) sdot grad 120601119892 (x)] minus Σ119886119892 (x) sdot 120601119892 (x)

+

119866

sum

1198921015840 = 119892

Σ1199041198921015840rarr119892 (x) sdot 1206011198921015840 (x)

+ 120594119892 sdot

119866

sum

1198921015840=1

]Σ1198911198921015840 (x)119896eff

sdot 1206011198921015840 (x) = 0

(1)

where 120601119892 are the 119892 = 1 119866 unknown flux distributionsand the Σs and the 119863s are the macroscopic cross-sectionsand diffusion coefficients which ought to be computed by alattice-level code and taken as an input to core-level codesHowever for the purposes of fulfilling the objectives of thiswork we take the macroscopic cross-sections as functions ofthe spatial coordinate x isin R119898 which is known beforehandThe coefficient 120594119892 represents the fission spectrum and is thefraction of the fission neutrons that are born into group 119892 Asstated above the ]-fissions are divided by a positive effectivemultiplication factor 119896eff and all the time derivatives that isthe right hand of (1) is set to zero We note that in order todefine a consistent nomenclature we use the absorption crosssection Σ119886119892 of the group 119892 which is equal to the total crosssection Σ119905119892 minus the self-scattering cross section Σ119904119892rarr119892Therefore the sum over the scattering sources excludes theterm corresponding to 119892

1015840= 119892 If we had used the total

cross section in the second term of (1) we would have hadto include the self-scattering term as a source

If the cross sections depend only in an explicit wayon the spatial coordinate x then (1) is linear If as is thegeneral case the cross sections depend on x through the flux120601119892(x) itselfmdashsuch as by means of the xenon concentrationor by local temperature distributionsmdashthen (1) is nonlinearNevertheless this general case can be solved by successive

Science and Technology of Nuclear Installations 3

(a) Coarse structured grid commonly used in diffu-sion codes such as in [9] Each lattice cell is divided intoa 2 times 2 grid for solving the neutron diffusion equation

(b) Discretization of the lattice cell using a fineunstructured grid as proposed in this work

(c) Further refinement of the unstructured grid

Figure 1 Discretization of a homogenized PHWR array of fuel channels for a core-level diffusion computation Each square is a lattice-levelcell comprising one fuel channel and the surrounding moderator

linear iterations so the basic problem can be regarded as beingpurely linear

It should be noted that if at least one of the diffu-sion coefficients 119863119892(x) is discontinuous over space thenthe divergence operator is not defined at the discontinuitypoints Therefore the differential formulationmdashalso knownas the strong formulationmdashis not complete when there existmaterial discontinuities that involve the diffusion coefficientsAt thesematerial interfaces the differential equation has to bereplaced by a neutron current conservation condition

119863+

119892(x) sdot grad 120601

+

119892(x) = 119863

minus

119892(x) sdot grad 120601

minus

119892(x) (2)

where the plus and minus sign denote both sides of the inter-face As the diffusion coefficients are different the resultingflux distribution120601119892(x) ought to have a discontinuous gradientat the interface in order to conserve the current

When transforming the strong formulation into a weakformulationmdashnot just into an integral formulationmdashboth (1)and (2) can be taken into account by a single expression Ineffect let 120593119892(x) be arbitrary functions of x for 119892 = 1 119866Multiplying each of the 119866 equations (1) by 120601119892(x) integrating

over the domain 119880 isin R119898 and applying Greenrsquos formula [12]to the leakage term we obtain

int119880

119863119892 (x) sdot [grad 120593119892 (x) sdot grad 120601119892 (x)] 119889119898x

+ int120597119880

120593119892 (x) sdot 119863119892 (x) sdot [grad 120601119892 (x) sdot n] 119889119878

+ int119880

120593119892 (x) sdot Σ119886119892 (x) sdot 120601119892 (x) 119889119898x

+ int119880

120593119892 (x) sdot119866

sum

1198921015840 = 119892

Σ1199041198921015840rarr119892 (x) sdot 1206011198921015840 (x) 119889119898x

+ 120594119892 sdot int119880

120593119892 (x) sdot119866

sum

1198921015840=1

]Σ1198911198921015840 (x)119896eff

sdot 1206011198921015840 (x) 119889119898x

(3)

These 119866 coupled equations should hold for any arbitraryset of functions120593119892(x)Making an analogy between (3) and theprinciple of virtual work for structural problems we call thesefunctions virtual fluxes This formulation does not involveany differential operator over the diffusion coefficient and yet


(a) Continuous two-dimensional domain (b) Structured grid

(c) Unstructured grid

Figure 2 When a continuous domain (a) is meshed with an unstructured grid there appears a geometric condition known as the staircaseeffect (b) For the same number of nodes unstructured grids reproduce the original geometry better (c)

Figure 3 Cross section of a hypothetical reactor in which thecontrol rods enter into the core from above with a certain attackangle with respect to the vertical direction

at the same time can be shown to be equivalent to the strongformulation given by (1)

In any case both formulations involve the computationof 119866 unknown functions of x and one unknown real value119896eff Except for homogeneous cross sections in canonicaldomains 119880 the multigroup diffusion equation needs to besolved numerically As discussed below any nodal-baseddiscretization scheme replaces a continuous unknown func-tion 120601119892(x) by 119873 discrete values 120601119892(119894) for 119894 = 1 119873

Figure 4 The finite volumes method computes the unknown fluxin the cell centers (squares) whilst the finite elements methodcomputes the fluxes at the nodes (circles)

If we arrange these unknowns into a vector 120601 isin R119873119866

such as

120601 =

[[[[[[[[[[[[[[[[

[

1206011 (1)

1206012 (1)

120601119866 (1)

1206011 (2)

120601119892 (119894)

120601119866 (119873)

]]]]]]]]]]]]]]]]

]

(4)


(a) Fast flux distribution

minus100minus50 0 50 100minus100

minus500

50100

0

06

xy

(b) Fast flux unknowns

(c) Matrix 119877 structure (d) Matrix 119865 structure

Figure 5 A bare homogeneous circle solved with finite volumes (184 unknowns)

then the continuous eigenvalue problemmdashin either for-mulationmdashcan be transformed into a generalized matrixeigenvalueeigenvector problem casted in either of the follow-ing forms

119877 sdot 120601 =1

119896effsdot 119865 sdot 120601

119865 sdot 120601 = 119896eff sdot 119877 sdot 120601

119877minus1sdot 119865 sdot 120601 = 119896eff sdot 120601

(5)

where 119877 and 119865 are square 119873119866 times 119873119866 matrices We call119865 the fission matrix as it contains all the ]-fission termswhich are the ones that we artificially divided by 119896eff Therest of the terms are grouped into the removal or transportmatrix 119877 which includes the rest of the neutron-matterinteraction mechanisms It can be shown [11] that for anyreal set of cross sections 119877minus1 exists and that the119873119866 pairs ofeigenvalueeigenvector solutions of (5) satisfy that

(1) there is a unique real positive eigenvalue greater inmagnitude than any other eigenvalue

(2) all the elements of the eigenvector corresponding tothat eigenvalue are real and positive

(3) all other eigenvectors either have some elements thatare zero or have elements that differ in sign from eachother

21 Boundary Conditions Being a differential equation overspace the neutron diffusion equation needs proper boundaryconditions to conform a properly definedmathematical prob-lem These can be imposed flux (Dirichlet) imposed current(Neumann) or a linear combination (Robin) However dueto the fact that in the linear problem in absence of externalsourcesmdashsuch as (1) or (3)mdashthe problem is homogeneousif 120601119892(x) is a solution then any multiple 120572120601119892(x) is also asolution That is to say the eigenvectors are defined up to amultiplicative constant whose value is usually chosen as toobtain a certain total thermal power Thus the prescribedboundary conditions should also be homogeneous and bedefined up to amultiplicative constantTherefore the allowedDirichlet conditions are zero flux at the boundary theNeumann conditions should prescribe that the derivative in




minus500

50100

0

06

xy



Figure 6 A bare homogeneous circle solved with finite elements (218 unknowns)

the normal direction should be zero (ie symmetry condi-tion) and the Robin conditions are restricted to the followingform

grad 120601119892 (x) sdot n + 119886119892 (x) sdot 120601119892 (x) = 0 (6)

n being the outward unit normal to the boundary 120597119880 of thedomain 119880

22 Grids and Schemes One way of solving the neutrondiffusion equationmdashand in general any partial differentialequation over spacemdashis by discretizing the differential oper-ators with some kind of scheme that is applied over a certainspatial grid Given an 119898-dimensional domain a grid definesa partition composed of a finite number of simple geometricentities In structured grids these elementary entities arearranged following a well-defined structure in such a waythat each entity can be identified without needing furtherinformation than the one provided by the intrinsic structureOn the other hand the geometric entities that composean unstructured grid are arranged in an irregular patternand the identification of the elementary entities needs to

be separately specified for example by means of a sortedlist For instance Figure 2(b) shows a structured grid thatapproximates the continuous domain of Figure 2(a) Eachsquare may be uniquely identified by means of two integerindexes that indicate its relative position in each of thehorizontal and vertical directions In the case of Figure 2(c)that shows an unstructured grid there is no way to system-atically make a reference to a particular quadrangle withoutany further information

Almost all of the grid-based schemesmdashwhich are knownas nodal schemes which are to be differentiated from modalschemes based on series expansionsmdashare based in either thefinite differences volumes or elements method Finite differ-ences schemes provide themost simple and basic approach toreplace differential (ie continuous) operators by difference(ie discrete) approximations However they are not suitablefor unstructured meshes and may introduce convergenceproblems with parameters that are discontinuous in spacewhich is the case for any reactor core composed of atleast two different materials Moreover boundary conditionsare hard to incorporate and usually give rise to incorrectresults


20 40 600995

09975

1

Analytical solutionFinite volumesFinite elements

keff

a985747c

Figure 7 The effective multiplication factor of a two-group barecircular reactor of radius 119886 as a function of the quotient 119886ℓ

119888between

the radius and the characteristic length of the cellelement

20 40 60

Finite volumesFinite elements

Erro

r ink

eff

10minus3

10minus4

10minus5

a985747c

Figure 8 Absolute value of the error committed in the computationof 119896eff versus 119886ℓ119888

Methods of the finite volumes family involve the inte-gration of the differential equation over each elementaryentity applying the divergence theorem to transform volumeintegrals into surface integrals and providing a mean to esti-mate the fluxes through the entityrsquos surface using informationcontained in its neighbors In this context each elementaryentity is called a cell and finite volumes methods give themean value of each of the group fluxes 120601119892(119894) at the 119894th cellwhich may be associated to the value of 120601119892(x119894) where x119894 isthe location of the cell barycenter Being an integral-basedmethod spatial-discontinuous parameters are handled moreefficiently than in finite differences and boundary conditionscan be easily incorporated as forced cell fluxes Nonethelesseven though these methods may be applied to unstructuredgrids the estimation of the fluxes on the cellsrsquo surfaces isusually performed by using some geometric approximationsthat may lose validity as the quality of the grid is worsened

10

70

90

130

150

170

10 30 50 70 90 110 130 150 170

4

1 38

36 37353

3 32 33 34312

26 27 28 29 3025

19 20 21 22 23 2418

10 11 12 13 14 15 16 17

2 331 4 5 6 7 8 9

x (cm)

y (cm

)

Figure 9 The 2D IAEA PWR benchmark geometry

Moreover the simple integral approach cannot take intoaccount discontinuous diffusion coefficients so when twoneighbors pertain to different materials the flux has to becomputed in a certain particular way to conserve the neutroncurrent according to (2)

Finally finite elements methods rely on a weak formula-tion of the differential problem similar to (3) that maintainsall the mathematical characteristics of the original strongformulation plus its boundary conditions The method isbased on shape functions that are local to each elementaryentitymdashnow called element defined by nodes as cornersmdashand on finding a set of nodal values such that a certaincondition is met which is usually that the residual of thesolution has to be orthogonal to each of the shape functionsThis condition is known as the Galerkin method and impliesthat the error committed in the approximate solution ofthe continuous problem is confined into a small subset ofthe original vector space of the continuous functions Thesemathematical properties make finite elements schemes veryattractive However these features depend on a large numberof integrations that ought to be performed numerically soa computational effortdesired accuracy tradeoff has to beconsidered Not only does the finite elements method givethe values of the flux 120601119892(x119894) at the 119894th node but also theshape functions provide explicit expressions to interpolateand to evaluate the unknown functions at any location x ofthe domain119880 Boundary conditions are divided into essentialand naturalThefirst group comprises theDirichlet boundaryconditions which are satisfied exactlymdashwithin the precisionof the eigenvalue problem solvermdashby the obtained solutionThe latter include the Neumann and the Robin conditionsthat are satisfied only approximately by the derivatives of theinterpolated solution through the shape functions with finermeshes giving better agreement with the prescribed values

The same unstructured grid may be used either for thefinite volumes or for the finite elements method In the first


Largest eigenvalue 1029690 (288336 pcm)1135 (3066 3079)1306 (5171 13076)

Number of unknowns 15740Outer iterations 3Linear iterations 32Inner iterations 1967Residual normRelative errorError estimateMemory used 49824 kBSoft page faults 13401Hard page faults 0Total CPU time 074 seconds

Max 1206012(x y) coreMax 1206012(x y) reflector

2483 times 10minus8

1223 times 10minus8

403 times 10minus9

keff

quarter-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 002

(a)

123456789

1011121314151617181920212223242526272829303132333435363738

k Pk 1206011k 1206012k075 3282 555133 4242 984147 4645 1090123 3921 911061 2677 453094 2995 694093 2927 689074 2011 549mdash 322 756146 4597 1078150 4730 1110133 4199 985107 3405 789104 3271 767095 2971 700072 1956 534mdash 306 733149 4695 1102136 4289 1007118 3733 876107 3367 792097 2892 718066 1628 491mdash 234 549120 3797 891097 3096 718090 2842 669084 2221 619mdash 568 1215mdash 067 257047 2042 348068 2055 504058 1414 430mdash 222 562057 1374 425mdash 388 820mdash 053 203mdash 060 229

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)

Figure 10 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution

120601119892(119909 119909) along the diagonal

case the unknowns are themean value of the fluxes over eachcell whilst in the latter the unknowns are the fluxes evaluatedat each node Therefore the number of unknowns 119873119866 isdifferent for each method even when using the same gridFigure 4 shows this situation and in Section 31 we further

illustrate these differences A fully detailed mathematicaldescription of the actual algorithms for both volumes andelements-based proposed discretizations can be found in anacademic monograph written by the author of this paper[13]





1394 times 10minus8

6865 times 10minus9

3425 times 10minus9

keff

quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 013

(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



3 Results

We now proceed to show two illustrative results that are tobe taken as an overview of the possibilities that unstructuredgrids can provide in order to tackle the multigroup neutron

diffusion problem The examples are two-dimensional prob-lems as they contain some of the complexities a real three-dimensional reactor posse yet the reported results are notso complicated as to be easily understood and analyzed Inparticular we state some basic differences between the finite


Largest eigenvalue 1029695(288388 pcm)1112 (3076 3032)838 (5000 13000)



1056 times 10minus8

5202 times 10minus9

5122 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



volumes and the finite elements methods by solving a two-group homogeneous bare reactor with the Robin boundaryconditions over the very same grid although a rather coarseone so the differences can be observed directly into theresulting figuresWe then solve the classical two-dimensionalLWR problem also known as the 2D IAEA PWR benchmark

Not only do we show again the differences between the finitevolumes and elements formulation but also we solve theproblemusing different combinations ofmeshing algorithmsbasic shapes and characteristic lengths of the mesh

To solve the two examples shown below we employed themilonga code which was written from scratch by the author





5408 times 10minus8

2665 times 10minus8

8424 times 10minus9

keff

Milongarsquos 2D LWR IAEA benchmark problem case no 031quarter-symmetry core meshed using Delquad (quads 985747c = 4) solved with finite volumes

(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



of this paper and is currently still under development withinhis ongoing PhD thesis There exists a first public release[14] under the terms of the GNU General Public Licensemdashthat is it is a free softwaremdashthat can only handle structuredgrids There is a second release being prepared whose

main relevance is that it can work with nonstructured gridsas well which is the main feature of the code It uses ageneral mathematical frameworkmdashcoded from scratch aswellmdashwhich provides input file parsing algebraic expres-sions evaluation one- and multidimensional interpolation of





2296 times 10minus12



keff

Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes

(a)

1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779

10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244

k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which

features are expected and what to do with the obtainedresults

It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies





2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff

Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes

(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into

materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense



Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error





keff

Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds

9993 times 10


Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes

(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be

evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome



Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti


26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies

were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and



Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm


566 times 10minus12



keff

Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds

566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case

with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is


500 1000 2000 5000 10000 20000 50000Number of unknowns

2840

2860

2880

2900

2920

ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes

Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes

Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements

120588(p

cm)

105 2 times 105 5 times 105

Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference

twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices

In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions

appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators

As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is

119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)

being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)

Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold


2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105

Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements

Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements

Figure 20 Total wall time versus number of unknowns

for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors

We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids

32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861

2

119911119892= 08times10

minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as

120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)

The expected results are as follows

(1) Maximum eigenvalue(2) Fundamental flux distributions

(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)



105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)

Figure 21 Time needed to mesh the geometry versus number of unknowns

Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)

1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912

Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector

Note the fluxes shall be normalized such that

1

119881coreint119881core

sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)

(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given

(3) Average subassembly powers 119875119896

119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)

where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9

(4) Number of unknowns in the problem number ofiterations and total and outer

(5) Total computing time iteration time IO-time andcomputer used

(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid

of 20 times 20 cm(8) Dependence of results on mesh spacing

Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry


Quarter Delaunay triangles elements

Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105

Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes

Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements


01

003

001

03

1

3

10

30

100

300

Figure 22 Time needed to read the mesh versus number of unknowns

which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves

Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to

milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied

As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps

(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh

(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results

The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes


Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm

basicshapediscretization

schemeandcharacteris

ticelem

entc

elllengthTh

ereference

solutio

nis120588=28749times10minus5w

hich

correspo

ndsto119896eff=10296

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105


Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements

01

003

001

03

10

30

100

300

1

3

Figure 23 Time needed to build the matrices versus number of unknowns

was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm

1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817

119877 + (1119896eff) sdot 119865lt 10minus8 (11)

The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus

1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)

respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy

When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information

about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873

2) task which is

unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem

The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a


Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3

Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns

desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid

Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets

are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid

It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements


4 Conclusions

Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem

Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better

References

[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970

[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976

[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006

[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010

[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984

[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008

[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992

[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008

[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011

[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977

[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975

[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005

[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013

[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011

[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010

[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009

[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013

[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997

[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005

[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009

[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005

[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004

[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference


on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997

[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011

[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871

infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013

[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002

[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975

TribologyAdvances in

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Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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Submit your manuscripts athttpwwwhindawicom


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Journal ofEngineeringVolume 2014

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International Journal ofPhotoenergy


Nuclear InstallationsScience and Technology of


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High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


for each groupmdashcontaining differential operators appliedonly over the spatial coordinatesThis resulting set of second-order PDEs is known as the multigroup neutron diffusionequation and is usually used to model design and analyzenuclear reactor cores by the so-called core-level calculationcodesThese programs take homogenizedmacroscopic cross-sections (which may depend on the spatial coordinatesthrough changes of fuel burnup materials temperature orother properties) computed by lattice-level codes as an inputand solve the diffusion equation to obtain the flux (and itsrelated quantities such as power xenon etc) distributionwithin the core

Given a certain spatial distribution of materials and itsproperties inside a reactor core chances are that the resultingreactor will not be critical That is to say in general therate of absorptions and leakages will not exactly overcomethe neutrons born by fissions sources and some kind offeedbackmdasheither through an external control system or bymeans of an inherent stability mechanism of the core [4]mdashis needed to keep the reactor power within a certain intervalMathematically this means that the transportmdashand thus thediffusionmdashequation does not have a steady-state solution Inpractice given a certain reactor configuration its steady-stateflux distribution is computed by setting all the time deriva-tives to zero as usual but also by dividing the fission sources bya positive value 119896eff called the effective multiplication factorwhich becomes also one unknown and turns the formulationinto a eigenvalue problemThe largest (or smallest dependingon the formulation) eigenvalue is therefore the effectivemultiplication factor If 119896eff lt 1 the original configurationwas subcritical and conversely As successive configurationsmake 119896eff rarr 1 the associated eigenfunctions approach thesteady-state critical flux distribution [5]

Core-level codes traditionally use regular grids to dis-cretize the differential operators over the space Dependingon the characteristics and symmetry of the reactor coreeither squares or hexagons are used as the basic shapeof the mesh Usual discretization schemes involve cell-centered finite differences or two-step coarse-meshcouplingcoefficient methods [6ndash8] which are accurate and efficientenough formost applicationsThere are nevertheless certainlimitations that are not inherently related to structured gridsbut to the way their coarseness is utilized and how they arecomputed and applied to the reactor geometry These issuescan be overcome by discretizing the spatial operators usinga scheme which could be applied to nonstructured gridsnamely finite volumes or finite elements

This way unstructured grids may be used to studyanalyze and understand the numerical errors introduced bythe discretization of the leakage operator with a difference-based scheme over a coarse structured grid by successivelyrefining the mesh whilst comparing the solutions with thestructured one as depicted in Figure 1 Another example ofapplication may be the improvement of the response matrixof boundary conditions over cylindrical surfaces which isthe case for most of the nuclear power plants cores Whena coarse structured mesh is applied to a curved geometrythere appears a geometric condition known as the staircaseeffect Even though arbitrary shapes cannot be perfectly

tessellated with unstructured grids for the same number ofunknowns they better represent the geometry than struc-tured meshes as Figure 2 illustrates Again by refining themesh and comparing solutions the matrix responses usedto set the boundary conditions on the coarse meshes canbe optimized Finally other complex geometries such asslanted control rods (Figure 3) or irradiation chambers can bedirectly taken into account by unstructured grids possibly byrefining the mesh in the locations around said complexities

2 The Steady-State Multigroup NeutronDiffusion Equation

In the present work we take the steady-state multigroupneutron diffusion equation for granted That is to say wefocus on the mathematical aspects of the eigenvalue problemand make no further reference to its derivation from thetransport equation nor to the validity of its applicationto reactor problems as these subjects that are extensivelydiscussed in the classic literature [1 5 11]

The differential formulation of the steady-state multi-group neutron diffusion equation over an 119898-dimensionaldomain 119880 with 119866 groups of energy is the set of 119866 coupleddifferential equations

div [119863119892 (x) sdot grad 120601119892 (x)] minus Σ119886119892 (x) sdot 120601119892 (x)

+

119866

sum

1198921015840 = 119892

Σ1199041198921015840rarr119892 (x) sdot 1206011198921015840 (x)

+ 120594119892 sdot

119866

sum

1198921015840=1

]Σ1198911198921015840 (x)119896eff

sdot 1206011198921015840 (x) = 0

(1)

where 120601119892 are the 119892 = 1 119866 unknown flux distributionsand the Σs and the 119863s are the macroscopic cross-sectionsand diffusion coefficients which ought to be computed by alattice-level code and taken as an input to core-level codesHowever for the purposes of fulfilling the objectives of thiswork we take the macroscopic cross-sections as functions ofthe spatial coordinate x isin R119898 which is known beforehandThe coefficient 120594119892 represents the fission spectrum and is thefraction of the fission neutrons that are born into group 119892 Asstated above the ]-fissions are divided by a positive effectivemultiplication factor 119896eff and all the time derivatives that isthe right hand of (1) is set to zero We note that in order todefine a consistent nomenclature we use the absorption crosssection Σ119886119892 of the group 119892 which is equal to the total crosssection Σ119905119892 minus the self-scattering cross section Σ119904119892rarr119892Therefore the sum over the scattering sources excludes theterm corresponding to 119892

1015840= 119892 If we had used the total

cross section in the second term of (1) we would have hadto include the self-scattering term as a source

If the cross sections depend only in an explicit wayon the spatial coordinate x then (1) is linear If as is thegeneral case the cross sections depend on x through the flux120601119892(x) itselfmdashsuch as by means of the xenon concentrationor by local temperature distributionsmdashthen (1) is nonlinearNevertheless this general case can be solved by successive








119863+

119892(x) sdot grad 120601

+

119892(x) = 119863

minus

119892(x) sdot grad 120601

minus

119892(x) (2)




int119880


+ int120597119880


+ int119880

120593119892 (x) sdot Σ119886119892 (x) sdot 120601119892 (x) 119889119898x

+ int119880

120593119892 (x) sdot119866

sum

1198921015840 = 119892

Σ1199041198921015840rarr119892 (x) sdot 1206011198921015840 (x) 119889119898x

+ 120594119892 sdot int119880

120593119892 (x) sdot119866

sum

1198921015840=1

]Σ1198911198921015840 (x)119896eff

sdot 1206011198921015840 (x) 119889119898x

(3)











such as

120601 =

[[[[[[[[[[[[[[[[

[

1206011 (1)

1206012 (1)

120601119866 (1)

1206011 (2)

120601119892 (119894)

120601119866 (119873)

]]]]]]]]]]]]]]]]

]

(4)




minus500

50100

0

06

xy





119877 sdot 120601 =1

119896effsdot 119865 sdot 120601



(5)









minus500

50100

0

06

xy











20 40 600995

09975

1


keff

a985747c


119888between


20 40 60


Erro

r ink

eff

10minus3

10minus4

10minus5

a985747c



10

70

90

130

150

170

10 30 50 70 90 110 130 150 170

4

1 38

36 37353

3 32 33 34312

26 27 28 29 3025

19 20 21 22 23 2418

10 11 12 13 14 15 16 17

2 331 4 5 6 7 8 9

x (cm)

y (cm

)









2483 times 10minus8

1223 times 10minus8

403 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









1394 times 10minus8

6865 times 10minus9

3425 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



3 Results







1056 times 10minus8

5202 times 10minus9

5122 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










5408 times 10minus8

2665 times 10minus8

8424 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


(a)

1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff


(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of
























119863+

119892(x) sdot grad 120601

+

119892(x) = 119863

minus

119892(x) sdot grad 120601

minus

119892(x) (2)




int119880


+ int120597119880


+ int119880

120593119892 (x) sdot Σ119886119892 (x) sdot 120601119892 (x) 119889119898x

+ int119880

120593119892 (x) sdot119866

sum

1198921015840 = 119892

Σ1199041198921015840rarr119892 (x) sdot 1206011198921015840 (x) 119889119898x

+ 120594119892 sdot int119880

120593119892 (x) sdot119866

sum

1198921015840=1

]Σ1198911198921015840 (x)119896eff

sdot 1206011198921015840 (x) 119889119898x

(3)











such as

120601 =

[[[[[[[[[[[[[[[[

[

1206011 (1)

1206012 (1)

120601119866 (1)

1206011 (2)

120601119892 (119894)

120601119866 (119873)

]]]]]]]]]]]]]]]]

]

(4)




minus500

50100

0

06

xy





119877 sdot 120601 =1

119896effsdot 119865 sdot 120601



(5)









minus500

50100

0

06

xy











20 40 600995

09975

1


keff

a985747c


119888between


20 40 60


Erro

r ink

eff

10minus3

10minus4

10minus5

a985747c



10

70

90

130

150

170

10 30 50 70 90 110 130 150 170

4

1 38

36 37353

3 32 33 34312

26 27 28 29 3025

19 20 21 22 23 2418

10 11 12 13 14 15 16 17

2 331 4 5 6 7 8 9

x (cm)

y (cm

)









2483 times 10minus8

1223 times 10minus8

403 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









1394 times 10minus8

6865 times 10minus9

3425 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



3 Results







1056 times 10minus8

5202 times 10minus9

5122 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










5408 times 10minus8

2665 times 10minus8

8424 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


(a)

1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff


(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


























such as

120601 =

[[[[[[[[[[[[[[[[

[

1206011 (1)

1206012 (1)

120601119866 (1)

1206011 (2)

120601119892 (119894)

120601119866 (119873)

]]]]]]]]]]]]]]]]

]

(4)




minus500

50100

0

06

xy





119877 sdot 120601 =1

119896effsdot 119865 sdot 120601



(5)









minus500

50100

0

06

xy











20 40 600995

09975

1


keff

a985747c


119888between


20 40 60


Erro

r ink

eff

10minus3

10minus4

10minus5

a985747c



10

70

90

130

150

170

10 30 50 70 90 110 130 150 170

4

1 38

36 37353

3 32 33 34312

26 27 28 29 3025

19 20 21 22 23 2418

10 11 12 13 14 15 16 17

2 331 4 5 6 7 8 9

x (cm)

y (cm

)









2483 times 10minus8

1223 times 10minus8

403 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









1394 times 10minus8

6865 times 10minus9

3425 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



3 Results







1056 times 10minus8

5202 times 10minus9

5122 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










5408 times 10minus8

2665 times 10minus8

8424 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


(a)

1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff


(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of




















minus500

50100

0

06

xy





119877 sdot 120601 =1

119896effsdot 119865 sdot 120601



(5)









minus500

50100

0

06

xy











20 40 600995

09975

1


keff

a985747c


119888between


20 40 60


Erro

r ink

eff

10minus3

10minus4

10minus5

a985747c



10

70

90

130

150

170

10 30 50 70 90 110 130 150 170

4

1 38

36 37353

3 32 33 34312

26 27 28 29 3025

19 20 21 22 23 2418

10 11 12 13 14 15 16 17

2 331 4 5 6 7 8 9

x (cm)

y (cm

)









2483 times 10minus8

1223 times 10minus8

403 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









1394 times 10minus8

6865 times 10minus9

3425 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



3 Results







1056 times 10minus8

5202 times 10minus9

5122 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










5408 times 10minus8

2665 times 10minus8

8424 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


(a)

1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff


(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of




















minus500

50100

0

06

xy











20 40 600995

09975

1


keff

a985747c


119888between


20 40 60


Erro

r ink

eff

10minus3

10minus4

10minus5

a985747c



10

70

90

130

150

170

10 30 50 70 90 110 130 150 170

4

1 38

36 37353

3 32 33 34312

26 27 28 29 3025

19 20 21 22 23 2418

10 11 12 13 14 15 16 17

2 331 4 5 6 7 8 9

x (cm)

y (cm

)









2483 times 10minus8

1223 times 10minus8

403 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









1394 times 10minus8

6865 times 10minus9

3425 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



3 Results







1056 times 10minus8

5202 times 10minus9

5122 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










5408 times 10minus8

2665 times 10minus8

8424 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


(a)

1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff


(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















20 40 600995

09975

1


keff

a985747c


119888between


20 40 60


Erro

r ink

eff

10minus3

10minus4

10minus5

a985747c



10

70

90

130

150

170

10 30 50 70 90 110 130 150 170

4

1 38

36 37353

3 32 33 34312

26 27 28 29 3025

19 20 21 22 23 2418

10 11 12 13 14 15 16 17

2 331 4 5 6 7 8 9

x (cm)

y (cm

)









2483 times 10minus8

1223 times 10minus8

403 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









1394 times 10minus8

6865 times 10minus9

3425 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



3 Results







1056 times 10minus8

5202 times 10minus9

5122 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










5408 times 10minus8

2665 times 10minus8

8424 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


(a)

1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff


(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















2483 times 10minus8

1223 times 10minus8

403 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









1394 times 10minus8

6865 times 10minus9

3425 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



3 Results







1056 times 10minus8

5202 times 10minus9

5122 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










5408 times 10minus8

2665 times 10minus8

8424 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


(a)

1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff


(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















1394 times 10minus8

6865 times 10minus9

3425 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)



3 Results







1056 times 10minus8

5202 times 10minus9

5122 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










5408 times 10minus8

2665 times 10minus8

8424 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


(a)

1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff


(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















1056 times 10minus8

5202 times 10minus9

5122 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










5408 times 10minus8

2665 times 10minus8

8424 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


(a)

1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff


(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















5408 times 10minus8

2665 times 10minus8

8424 times 10minus9

keff


(a)

123456789

1011121314151617181920212223242526272829303132333435363738


(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


(a)

1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff


(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of
























keff


(a)

1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)










2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff


(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















2019 times 10minus8

9946 times 10minus9

7138 times 10minus9

keff


(a)

1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)












keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of
























keff


9993 times 10



(a)

1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















26 times 10minus8

1281 times 10minus8

4996 times 10minus9

keff


1281 times 10 8

4996 times 10minus9


(a)

1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)









566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















566 times 10minus12



keff


566 times 10




(a)

1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885


k Pk 1206011k 1206012k

(b)

0 40 80 120 1600

15

30

45

1206012(x 0)1206011(x 0)

(c)

0 40 80 120 1600

15

30

45

1206012(x x)1206011(x x)

(d)







2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















2840

2860

2880

2900

2920




120588(p

cm)

105 2 times 105 5 times 105






119896eff =]Σ1198912 sdot Σ1199041rarr2

[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)

2]

(7)




2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















2000 3000 6000 10000 20000 30000 60000

01

003

001

03

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 times 105 3 times 105







2

119911119892= 08times10


120597120601119892

120597119899= minus

04692

119863119892

sdot 120601119892 (8)






105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















105 2 times 105 3 times 105



01

003

001

03

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)



1198631

1198632

Σ1rarr2

Σ1198861

Σ1198862

]Σ1198912



1


sum

119892

]Σ119891119892 sdot 120601119892119889119881 = 1 (9)



119875119896 =1

119881119896

int119881119896

sum

119892

]Σ119891119892 sdot 120601119892119889119881 (10)









Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 times 105 3 times 105




01

003

001

03

1

3

10

30

100

300










Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















Table2

Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetrym

eshing

algorithm



ticelem

entc

elllengthTh

ereference

solutio


hich

correspo


Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

114

Delaunay

998779Vo

lumes

40

minus120

1145

8264

332

1033

6119890minus09

3119890minus09

1119890minus09

349470

214

Delaunay

998779Vo

lumes

30

85

1135

15740

332

1967

2119890minus08

1119890minus08

4119890minus09

4813401

314

Delaunay

998779Vo

lumes

20

197

1124

31188

332

3898

1119890minus08

6119890minus09

3119890minus09

7621932

414

Delaunay

998779Vo

lumes

10153

1123

127476

332

15934

7119890minus09

3119890minus09

3119890minus09

273

81886

514

Delaunay

998779Vo

lumes

05

187

1119

510676

332

63834

2119890minus09

9119890minus10

1119890minus09

1148

352261

614

Delaunay

998779Elem

ents

40

173

1100

4308

332

538

2119890minus08

8119890minus09

4119890minus09

318794

714

Delaunay

998779Elem

ents

30

117

1107

8108

332

1013

5119890minus09

2119890minus09

2119890minus09

4712946

814

Delaunay

998779Elem

ents

20

84

1112

15936

332

1992

6119890minus09

3119890minus09

2119890minus09

7321347

914

Delaunay

998779Elem

ents

1062

1116

64478

332

8059

6119890minus09

3119890minus09

4119890minus09

262

77105

1014

Delaunay

998779Elem

ents

05

58

1117

256700

332

32087

1119890minus09

5119890minus10

1119890minus09

1091

331969

1114

Delaunay

◻Vo

lumes

40

53

1149

5042

332

630

2119890minus08

1119890minus08

5119890minus09

288007

1214

Delaunay

◻Vo

lumes

30

346

1133

8560

332

1070

7119890minus09

3119890minus09

2119890minus09

3910895

1314

Delaunay

◻Vo

lumes

20

308

1131

15576

332

1947

1119890minus08

7119890minus09

3119890minus09

541564

514

14Delaunay

◻Vo

lumes

10177

1122

60774

332

7596

8119890minus09

4119890minus09

3119890minus09

167

50115

1514

Delaunay

◻Vo

lumes

05

199

1120

244936

332

30617

4119890minus09

2119890minus09

2119890minus09

698

215678

1614

Delaunay

◻Elem

ents

40

196

1100

5222

332

652

3119890minus08

1119890minus08

8119890minus09

4713098

1714

Delaunay

◻Elem

ents

30

123

1107

8810

332

1101

2119890minus08

9119890minus09

7119890minus09

6918598

1814

Delaunay

◻Elem

ents

20

901112

15922

332

1990

1119890minus08

5119890minus09

5119890minus09

107

30591

1914

Delaunay

◻Elem

ents

1063

1116

61456

332

7682

5119890minus09

2119890minus09

4119890minus09

389

113237

2014

Delaunay

◻Elem

ents

05

58

1117

246332

332

30791

8119890minus10

4119890minus10

1119890minus09

1676

498192

2114

Delq

uad

998779Vo

lumes

40

minus45

1144

6228

332

778

4119890minus08

2119890minus08

7119890minus09

308495

2214

Delq

uad

998779Vo

lumes

30

570

1153

11820

332

1477

3119890minus08

1119890minus08

4119890minus09

4010879

2314

Delq

uad

998779Vo

lumes

20

459

1103

24100

332

3012

2119890minus08

1119890minus08

5119890minus09

6217715

2414

Delq

uad

998779Vo

lumes

10512

1104

9640

03

3212050

2119890minus08

1119890minus08

9119890minus09

207

61714

2514

Delq

uad

998779Vo

lumes

05

528

1106

385600

332

48200

2119890minus09

1119890minus09

2119890minus09

838

255735

2614

Delqu

ad998779

Elem

ents

40

220

1093

3288

332

411

3119890minus08

2119890minus08

6119890minus09

308495

2714

Delq

uad

998779Elem

ents

30

140

1104

6148

332

768

4119890minus08

2119890minus08

8119890minus09

39110

0028

14Delq

uad

998779Elem

ents

20

82

1110

12392

332

1549

2119890minus08

9119890minus09

6119890minus09

6117067

2914

Delqu

ad998779

Elem

ents

1063

1115

48882

332

6110

2119890minus09

1119890minus09

1119890minus09

197

5804

630

14Delq

uad

998779Elem

ents

05

58

1116

194162

332

24270

2119890minus09

9119890minus10

2119890minus09

790

238769

3114

Delq

uad

◻Vo

lumes

40

minus416

1174

3132

332

391

5119890minus08

3119890minus08

8119890minus09

267322

3214

Delq

uad

◻Vo

lumes

30

minus248

1151

5922

332

740

9119890minus09

4119890minus09

2119890minus09

318779

3314

Delq

uad

◻Vo

lumes

20

minus132

1139

12050

332

1506

1119890minus08

7119890minus09

4119890minus09

4412239

3414

Delq

uad

◻Vo

lumes

10minus49

1126

48200

332

6025

6119890minus09

3119890minus09

2119890minus09

122

36094

3514

Delq

uad

◻Vo

lumes

05

minus23

1123

192800

332

24100

5119890minus09

2119890minus09

3119890minus09

464

140228

3614

Delq

uad

◻Elem

ents

40

224

1093

3294

332

411

3119890minus08

1119890minus08

7119890minus09

359852

3714

Delq

uad

◻Elem

ents

30

141

1104

6144

332

768

3119890minus08

2119890minus08

9119890minus09

5114018

3814

Delq

uad

◻Elem

ents

20

921111

12392

332

1549

1119890minus08

6119890minus09

5119890minus09

8122526

3914

Delq

uad

◻Elem

ents

1065

1115

48882

332

6110

5119890minus09

3119890minus09

4119890minus09

276

78431


Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















Table2Con

tinued

Case

no

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ119888

(cm)

Δ120588

(pcm

)max1206012

(mdash)

Total

unkn

owns

Outer

iter

Linear

iter

Inner

iter

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

4014

Delq

uad

◻Elem

ents

05

59

1117

194162

332

24270

1119890minus09

5119890minus10

1119890minus09

1104

318967

4118

Delaunay

998779Vo

lumes

40

minus99

1139

4228

332

528

4119890minus12

2119890minus12

1119890minus12

359630

4218

Delaunay

998779Vo

lumes

30

50

1132

7712

332

964

1119890minus11

6119890minus12

2119890minus12

4211578

4318

Delaunay

998779Vo

lumes

20

177

1120

15776

332

1972

2119890minus12

1119890minus12

7119890minus13

5515436

4418

Delaunay

998779Vo

lumes

10142

1118

63902

332

7987

3119890minus12

1119890minus12

1119890minus12

160

46691

4518

Delaunay

998779Vo

lumes

05

176

1115

254612

332

31826

2119890minus12

8119890minus13

1119890minus12

555

168526

4618

Delaunay

998779Elem

ents

40

183

1096

2252

332

281

6119890minus12

3119890minus12

2119890minus12

359589

4718

Delaunay

998779Elem

ents

30

119

1104

4040

224

505

2119890minus08

1119890minus08

7119890minus09

41114

3748

18Delaunay

998779Elem

ents

20

85

1108

8156

332

1019

2119890minus12

8119890minus13

6119890minus13

5515053

4918

Delaunay

998779Elem

ents

1063

1112

32480

332

4060

1119890minus12

6119890minus13

8119890minus13

151

43671

5018

Delaunay

998779Elem

ents

05

58

1113

128358

332

1604

47119890minus13

3119890minus13

7119890minus13

527

157581

5118

Delaunay

◻Vo

lumes

40

37

1144

2380

224

297

5119890minus08

2119890minus08

1119890minus08

5314173

5218

Delaunay

◻Vo

lumes

30

218

1120

4194

332

524

7119890minus12

4119890minus12

2119890minus12

3610034

5318

Delaunay

◻Vo

lumes

20

345

1110

7960

332

995

5119890minus12

3119890minus12

1119890minus12

4712879

5418

Delaunay

◻Vo

lumes

10165

1117

30652

332

3831

2119890minus12

1119890minus12

7119890minus13

100

28851

5518

Delaunay

◻Vo

lumes

05

191

1117

122066

224

15258

2119890minus08

8119890minus09

9119890minus09

343

103825

5618

Delaunay

◻Elem

ents

40

175

1097

2526

332

315

4119890minus12

2119890minus12

1119890minus12

6016159

5718

Delaunay

◻Elem

ents

30

114

1104

4386

332

548

4119890minus12

2119890minus12

2119890minus12

5013768

5818

Delaunay

◻Elem

ents

20

901108

8234

332

1029

2119890minus12

1119890minus12

1119890minus12

7320094

5918

Delaunay

◻Elem

ents

1062

1112

31186

224

3898

2119890minus08

7119890minus09

9119890minus09

208

59589

6018

Delaunay

◻Elem

ents

05

58

1113

123122

224

15390

9119890minus09

5119890minus09

1119890minus08

807

2366

7061

18Delq

uad

998779Vo

lumes

40

minus210

1168

3224

332

403

1119890minus11

5119890minus12

2119890minus12

43117

8462

18Delq

uad

998779Vo

lumes

30

369

1150

6000

332

750

1119890minus11

6119890minus12

3119890minus12

5013502

6318

Delq

uad

998779Vo

lumes

20

312

1104

12316

332

1539

2119890minus11

8119890minus12

5119890minus12

6016395

6418

Delq

uad

998779Vo

lumes

1044

01094

48714

332

6089

9119890minus12

4119890minus12

4119890minus12

1183444

165

18Delq

uad

998779Vo

lumes

05

534

1081

193980

332

24247

5119890minus12

2119890minus12

4119890minus12

423

126768

6618

Delq

uad

998779Elem

ents

40

240

1090

1750

332

218

9119890minus12

4119890minus12

2119890minus12

43117

8667

18Delqu

ad998779

Elem

ents

30

150

1100

3184

332

398

1119890minus11

5119890minus12

2119890minus12

5013509

6818

Delq

uad

998779Elem

ents

20

89

1106

6426

332

803

5119890minus12

3119890minus12

2119890minus12

5916047

6918

Delq

uad

998779Elem

ents

1064

1111

24886

332

3110

2119890minus12

1119890minus12

1119890minus12

113

32788

7018

Delq

uad

998779Elem

ents

05

59

1113

98052

224

12256

1119890minus08

5119890minus09

8119890minus09

398

118153

7118

Delq

uad

◻Vo

lumes

40

minus398

1172

1650

332

206

1119890minus11

5119890minus12

2119890minus12

369907

7218

Delq

uad

◻Vo

lumes

30

minus249

1146

3058

332

382

4119890minus12

2119890minus12

1119890minus12

349349

7318

Delq

uad

◻Vo

lumes

20

minus131

1136

6228

224

778

3119890minus08

1119890minus08

5119890minus09

4010930

7418

Delq

uad

◻Vo

lumes

10minus51

1123

24536

224

3067

3119890minus08

2119890minus08

1119890minus08

7822716

7618

Delq

uad

◻Elem

ents

40

237

1090

1752

332

219

6119890minus12

3119890minus12

2119890minus12

4111213

7718

Delq

uad

◻Elem

ents

30

145

1101

3184

332

398

6119890minus12

3119890minus12

2119890minus12

43118

1178

18Delq

uad

◻Elem

ents

20

941107

6426

332

803

3119890minus12

1119890minus12

1119890minus12

6016220

7918

Delqu

ad◻

Elem

ents

1065

1111

24874

332

3109

1119890minus12

6119890minus13

8119890minus13

156

4400

080

18Delq

uad

◻Elem

ents

05

59

1113

9804

22

2412255

8119890minus09

4119890minus09

8119890minus09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3




119877 + (1119896eff) sdot 119865lt 10minus8 (11)


1

119896effsdot 119865 sdot 120601

10038171003817100381710038171003817100381710038171003817


1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817

(12)




2) task which is




Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 times 105 3 times 105



01

003

001

03

10

30

100

300

1

3







4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















4 Conclusions



References



































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of




























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















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