Moving Mesh (Dynamic mesh beyond fluent)

8/12/2019 Moving Mesh (Dynamic mesh beyond fluent)

1/45

www.torvergata-karting.it

Dynamic mesh: beyond FluentDynamic mesh: beyond Fluent

Marco Evangelos Biancolini

Department of Mechanical Engineering

University of Rome Tor Vergata

Marco Evangelos Biancolini

Department of Mechanical Engineering

University of Rome Tor Vergata


2/45


OutlineOutline

Why use smoothing

Review of methods for mesh smoothingPseudo solid approach, FEM solver

Radial basis approach

Example 2D

Example 3D


3/45


Why?Why?

Smoothing means that mesh topology is

preserved: no new mesh required!Shape optimisation

Moving or deforming boundaries (Fluid

structure interaction, rigid bodymovements, morphing)

Parametric analysis


4/45


Review of scientific contributionsReview of scientific contributions

198 relevant publications in the last 7 years(keyword moving mesh method) scanned

25 fully examined

Strategies:

Laplacian smoothing Spring models (rotational spring correction)

Pseudo solid (variable moduli, fictitious loadscorrection)

Boundary methods Others


5/45


Selected papersSelected papers

AUTOMATIC MESH MOTION FOR THE UNSTRUCTURED FINITEVOLUME METHOD Hrvoje Jasak,Zeljko TukovicISSN 13331124An adaptive mesh rezoning scheme for moving boundary flows and

fluidstructure interaction Computers and FluidsVolume: 36, Issue: 1,January, 2007, pp. 77-91 Masud, Arif; Bhanabhagvanwala, Manish; Khurram,Rooh A.A three-dimensional torsional spring analogy method forunstructured dynamic meshes Computers and StructuresVolume: 80,Issue: 3-4, February, 2002, pp. 305-316 Degand, Christoph; Farhat, CharbelMesh deformation using radial basis functions for gradient-basedaerodynamic shape optimization Computers and FluidsVolume: 36,Issue: 6, July, 2007, pp. 1119-1136 Jakobsson, S.;Amoignon, O.Mesh deformation based on radial basis function interpolationComputers and StructuresVolume: 85, Issue: 11-14, June - July, 2007, pp.784-795 de Boer, A.; van der Schoot, M.S.; Bijl, H.Finite element mesh update methods for fluidstructure interaction

simulations Finite Elements in Analysis and DesignVolume: 40, Issue: 9-10, June, 2004, pp. 1259-1269 Xu, Zhenlong;Accorsi, Michael


6/45


Pseudo solid approachPseudo solid approach

The mesh is supposed to be a deformablestructure

Elastic moduli can be chosen on an element byelement basis to minimise distortion

Corrective loads are added to improve element

quality after mesh deformationOne step method: Masud strategy in whichstiffness is proportional to element area (volume)

Two step methods: stiffness correction and extraloads added to preserve elements shape


7/45


Pseudo solid approachPseudo solid approach

Element distortion estimator: ratio between innerand outer circle radii (for triangles)

Advantages Very good results optimising element stiffness Physical pseudo solid guarantees mesh compatibility Large motions can be handled by means of non linear

FEM modelling

Pitfalls: Large problem solution Large memory usage (can be alleviated by means of

explicit solver) Non conformal meshes, connectors and polyedra requires

special treatment


8/45


9/45


Implementation details (2D)Implementation details (2D)

Material stiffnessmatrix

Element area

Interpolation matrix

Stiffness matrix

Q E ,( ) E

1

2

1

0

1

0

0

0

1

2

:=

At x1 y1, x2, y2, x3, y3,( ) x2 x1( ) y3 y1( ) x3 x1( ) y2 y1( )

2:=

B x1 y1, x2, y2, x3, y3,( )1

2 At x1 y1, x2, y2, x3, y3,( )

y2 y3

0

x3 x2

0

x3 x2

y2 y3

y3 y1

0

x1 x3

0

x1 x3

y3 y1

y1 y2

0

x2 x1

0

x2 x1

y1 y2

:=

kt x1 y1, x2, y2, x3, y3, E, , t,( ) t At x1 y1, x2, y2, x3, y3,( ) B x1 y1, x2, y2, x3, y3,( )T Q E ,( ) B x1 y1, x2, y2, x3, y3,( ):=


10/45



Elementalstiffnessmatrixescalculation

Material

topologyextracted formdatabase readin the nastran

file

Kelementi

isez elementiiel 2,

imat sezioniisez 2,

E materialiimat 2,

G materialiimat 3,

materialiimat 4,

t sezioniisez 3,

x1

y1

submatrix nodi elementiiel 3,

, elementiiel 3,

, 2, 3,( )T

x2

y2


, elementiiel 4,

, 2, 3,( )T

x3

y3


, elementiiel 5,

, 2, 3,( )T

Kelementiiel

kt x1 y1, x2, y2, x3, y3, E, , t,( )

iel 1 Nelementi..for:=


11/45



Linear system isassembled

partitioned

Kaug KaugNdof Ndof,

0

IDrow LMiel irow,

IDcol LMiel icol,

KaugIDrow IDcol,

KaugIDrow IDcol,

Kelementiiel( )

irow icol,+

icol 1 6..for

irow 1 6..for

iel 1 Nelementi..for

out Kaug

:=

Ktt submatrix Kaug 1, NdofA, 1, NdofA,( ):=

Kut submatrix Kaug NdofA 1+, Ndof, 1, NdofA,( ):=

Kuu submatrix Kaug NdofA 1+, Ndof, NdofA 1+, Ndof,( ):=

Ktu submatrix Kaug 1, NdofA, NdofA 1+, Ndof,( ):=


12/45



Linear systemsolution

Complete solution isbuilt

t lsolve Ktt Pt Ktuu,( ):=

Ru Kutt Kuuu+:=

P stack Pt Ru,( ):=

stack t u,( ):=


13/45


Radial basis approachRadial basis approach

Born as a data interpolation procedure for scattered dataMeshless field functions are defined

A set of sources point is defined (may be mappedeverywhere in the domain)A radial function is defined (compact support, or definedeverywhere)Polynomial corrector is added to guarantee compatibility for

rigid modesA linear system (order equal to the number of source pointintroduced) is solved for coefficients calculationThe motion of an arbitrary point inside or outside the domain(interpolation/extrapolation) is expressed as the summation

of the radial contribution of each source point (if the pointfalls inside the influence domain)


14/45



We look foran

interpolationfunctioncomposed bya radial basis

and apolynomial

Typical radialfunctions are

reported inthe table


15/45



Basiscoefficients are

obtainedimposing thedesired functionvalues at sourcepoints

Furthermorethe polynomialterms has togive 0contributions at

source points


16/45


17/45



Applying the

method for eachof the threedisplacementsfunctions weobtain theinterpolationfield


18/45



Node positions of the original mesh read from anastran data file (only GRID, SPC and SPCD entries

are processed)Processing

Constrained nodes are used as source points

Imposed displacements are used for mesh deformation

Basis generation

Calculation of displacements

Field functions fro x and y displacements can bevisualized by a contour or a surface plot

Deformed grid is represented by a scatter plot


19/45



Source points are firstextracted

Displacementfunctions values at

source points areextracted

Radial function isdefined

x.k

INOD vincolii 1,

X

i

submatrix nodi INOD, INOD, 2, 3,( )T

i 1 rows vincoli( )..for:=

g.k ID 1

GID 1,

SPCalli 3,

GID 2,

SPCalli 4,

ID ID 1+

SPCalli 1,

1 SPCalli 2,

1if

i 1 rows SPCall( )..for

OUT G

:=

r( ) 1

1 r2

+

:=


20/45



Interpolation matrixis calculated

Constraint matrix iscalculated

M

Mi j, x.ki

x.kj

j 1 rows x.k( )..for

i 1 rows x.k( )..for:=

P

Pi 1,

1

Pi j 1+,

x.ki

j

j 1 rows x.k1

..for

i 1 rows x.k( )..for:=


21/45



Linear system isassembled and solved

Interpolationcoefficients are

extracted

Interpolation functionis defined

MP augment stack M PT,( ) stack P zero,( ),( ):=

ic lsolve MP stack g.kic zero1 ,,:=

ic submatrix ic 1, rows x.k( ), 1, 1,( ):=

ic submatrix ic rows x.k( ) 1+, rows ic( ), 1, 1,( ):=

s.xyx( )

1

rows x.k( )

i

1( )i

2( )i

x x.k

i

=

1stack 1 x,( )

2stack 1 x,( )

+:=


22/45



Node positions of the original mesh read from anastran data file

Processing

Constrained nodes are used as source points

Imposed displacements are used for mesh deformation

Basis generation

Calculation of displacements

Nodal displacements are written in standardnastran file *.f06

Post processing by FEMAP


23/45



Advantages

Meshless method only grid points are moved regardless of

element connected Suitable for parallel implementation

Small systems have to be solved (sparse if the compactsupport is chosen)

Fast

Pitfalls

Reversed elements can arise

Boundary tracking requires special effort if not allboundary grids are included as source points


24/45


Example 2D: moving holeExample 2D: moving hole

FEMreferencesolution by

NastranDisplacementBC on the

boundariesConstantstrain CTRIAelements


25/45



Deformed

meshStrainenergy

density


26/45



Mathcad solution

FEM

Max difference (Nastranproposed FEM)

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

nodi3

nodid fscala( )3

nodi2

nodid fscala( )2

,

max NASTRANT

TT( ) 2.7217 10 6=


27/45



Mathcadsolution

Radial basis

0 0.2 0.4 0.6 0.8 1

0

0.5

1

mplot


28/45



Mathcadsolution

FEM (blue)

Radial basis(green)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

nodi3

nodid fscala( ) 3

nodid2 fscala( ) 3

nodi2

nodid fscala( ) 2

, nodid2 fscala( ) 2

,


29/45



Mathcad solution

Mesh quality comparison changing deformation extent

0 0.5 10.25

0.3

0.35

FEMRadial BasisFEMRadial Basis

Minimum quality factor

0 0.5 10.44

0.46

0.48

0.5

Mean quality factor


30/45


Example 2D: moving hole, boundary

points


pointsNastransolution

4 boundarypoints on theholeExternal

boundaryNote thesharpdeformationof the

boundary


31/45



points

Example 2D: moving hole, boundarypoints

Radial basissolution

4 sourcepoints on thehole

All the

externalboundaryNote how theboundaryremains

smooth 0 0.2 0.4 0.6 0.8 1

0

0.5

1

nodi3

nodid2 fscala( ) 3

nodi2

nodid2 fscala( ) 2

,


32/45



points


Nastransolution

12boundary

points


33/45



points


Radial basissolution

12 sourcepoints

0 0.2 0.4 0.6 0.8 1

0

0.5

1

nodi3

nodid2 fscala( ) 3

nodi2

nodid2 fscala( ) 2

,


34/45


Example 3D: spherical shellExample 3D: spherical shell

Undeformedshape


35/45


Example 3D: spherical shellExample 3D: spherical shell

Deformed shape

Radial basis 3D

Post processing ofdeformation usingnastran format

4 internal sourcepoints

Boundary sourcepoints

Smooth shapeeven with 4

internal controlpoints!


36/45



Solid mesh

Externalwalls fixed

Various

movementof holesurface


37/45



Undeformedshape

Radial basis3DPostprocessing ofdeformationusing nastranformatRigid rotationof hole

surface

Front

Rear


38/45



Deformedshape

Radial basis3DPostprocessing ofdeformationusing nastranformatRigid rotationand

translation ofhole surface

Front

Rear


39/45


Fluent implementationFluent implementation

Pseudosolid External solver that interact with *.cas file (see

Sculptor) External solver (Nastran) linked via udf (variable

stiffness not easy to implement)

External FEM library linked via UDF Internal FEM solver in UDF (faster)

Key points Element decomposition (polyedron handling)

Non conformal interfaces


40/45


Fluent implementationFluent implementation

Radial basis

External tool that interact with *.cas file

External tool that interact via UDF

Internal tool via UDF (faster, GUI difficult to implement)

Key points

Choosing of the basis functions (compact support) Location of source points

Extra Laplacian smoothing or iterative stage in themethod to guarantee shape preservation


41/45


Computational effortComputational effort

Simple test case with MathCAD

Description 3D case previously presented (moving hole)

About 20000 nodes

About 500 source pointsResults

Total computation time (including I/O) 90s

Input and basis generation 10s Mesh motion and output 80s


42/45


Constrained boundaries strategyConstrained boundaries strategy

There are many practical application inwhich part of the boundary nodes have tomove onto a prescribed surface

The method have to preserve a goodquality of the mesh on the deformableboundary

The method have to propagate all

boundaries motion (prescribed orconstrained) in the interior nodes


43/45


44/45


Overall strategyOverall strategy

Boundary nodes are first partitioned:

Set A Prescribed motion nodes

Set B Constrained nodes (linked onto surfaces)

A first radial basis step is performed generating thefield using set A to move set B

The correction is applied projecting transformedset B onto the constraint, nodes of set B can nowbe handled ad prescribed motion nodes

A second radial basis step is performed generatingthe field using set A and B to move interior nodes


45/45


Example: A cylinder with a holeExample: A cylinder with a hole

Two step strategy is applied to move a holein a cylinder, preserving the shape of the

cylindrical surface

Date post:	03-Jun-2018
Category:	Documents
Upload:	hari-kirupakar
View:	264 times
Download:	2 times

Moving Mesh (Dynamic mesh beyond fluent)

Documents