SAND92-2762 Distribution Unlimited Release Category UC-814 Printed July 1993 Ii Use of an Iterative Solution Method for Coupled Finite Element and Boundary Element Modeling J. Richard Koteras Engineering Mechanics and Material Modeling Department Sandia National Laboratories Albuquerque, NM 87185 Abstract Tunnels buried deep within the earth constitute an important class of geome- chanics problems. Two numerical techniques used for the analysis of geome- chanics problems, the finite element method and the boundary element method, have complementary characteristics for applications to problems of this type. The usefulness of combining these two methods for use as a geome- chanics analysis tool has been recognized for some time, and a number of cou- pling techniques have been proposed. However, not ali of them lend themselves to efficient computational implementations for large-scale prob- lems. This report examines a coupling technique that can form the basis for an efficient analysis tool for large scale geomechanics problems through the use of an iterative equation solver. OIST'RIB'(.,ITICNOF THIS DOCUMENT IS UNLIMITED I_ i ii ,
Transcript
SAND92-2762 DistributionUnlimited Release Category UC-814Printed July 1993
Ii
Use of an Iterative Solution Method for CoupledFinite Element and Boundary Element Modeling
J. Richard Koteras
Engineering Mechanics and Material Modeling DepartmentSandia National Laboratories
Albuquerque, NM 87185
Abstract
Tunnels buried deep within the earth constitute an important class of geome-chanics problems. Two numerical techniques used for the analysis of geome-chanics problems, the finite element method and the boundary elementmethod, have complementary characteristics for applications to problems ofthis type. The usefulness of combining these two methods for use as a geome-chanics analysis tool has been recognized for some time, and a number of cou-pling techniques have been proposed. However, not ali of them lendthemselves to efficient computational implementations for large-scale prob-lems. This report examines a coupling technique that can form the basis for anefficient analysis tool for large scale geomechanics problems through the useof an iterative equation solver.
OIST'RIB'(.,ITICNOF THIS DOCUMENT IS UNLIMITED
I_ i ii ,
Acknowledgment
The initial phase of this research was done in conjunctionwith Steven L. Crouch and You Tian, both of Geologic Re-search, Inc. in Minneapolis, MN. Their knowledge of the "boundary element method was instrumental in the develop-ment of a method to couple finite elements and boundary el- ,ements in a manner that would lend itself to solution with an
iterative solver. They provided a basic version of the bound-ary element code HOBE1 that was used in the research forthis report. Marc Loken, also of Geologic Research, Inc.,verified that the basic approach for the coupling techniquewas correct by the use of direct solutions methods. Thishelped resolve some difficulties and allow the research withiterative solvers to move forward.
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Contents
Acknowledgment ....................................................................................................... iv
Use of an Rerative Solution Method for Coupled Finite Elementand Boundary Element Modeling
Coupling the Finite Element and Boundary. Element Methods ................................. 2Implementation of the Coupled Technique in JAC2D .............................................. 13Test Problems ............................................................................................................ 15
Cylindrical Cavity Problem .................................................................................. 16Tunnel Problem .................................................................................................... 20
Multicavity Problem ............................................................................................ 24Conclusion ................................................................................................................. 28References ................................................................................................................... 30
Figures
Figure 1. Coupled Finite Element/Boundary Element Problem ............................. 4Figure 2. Conversion of Boundary Element Tractions to Nodal Forces ................ 6Figure 3. Geometry for Cylindrical Cavity Problem ............................................... 17Figure 4. Coarse Mesh for Cylindrical Cavity Problem .......................................... 19Figure 5. Fine Mesh for Cylindrical Cavity Problem .............................................. 19Figure 6. Geometry for Tunnel Problem ................................................................. 21Figure 7. Mesh for Tunnel Surrounded by a Circular Boundary Element Contour 22Figure 8. Mesh for Tunnel Surrounded by a Square Boundary Element Contour.. 22Figure 9. Geometry for Multicavity Problem .......................................................... 25Figure 10. Geometry for Modeling Multicavity Problem with Boundary Elements
and Finite Elements ................................................................................. 25
Figure 11.Mesh for Multicavity Problem for Combined Finite Element/BoundaryElement Solution ...................................................................................... 27
Figure 12. Geometry for Multicavity Problem for Finite Element Solution ............. 27
Tables
• Table 1" Cylindrical Cavity Coarse Mesh Results ................................................. 18Table 2: Cylindrical Cavity Fine Mesh Results ..................................................... 20Table 3" Tunnel Results from Coupled Models 23Table 4: Tunnel Results from Finite Element Model ............................................. 24Table 5' Multicavity Results for Various Modeling Techniques .......................... 26
SAND92-2762 was prepared under Yucca Mountain Project WBS number 1.2.4.2.3.1.
The data in this report was developed subject to QA controls in QAGR S 12423 IB Revision0, PCA 1.0 Task 1.1; the data is not qualified and is not to be used for licensing.
. Use of an Iterative Solution Method forCoupled Finite Element and Boundary
• Element Modeling
Introduction
The prediction of stresses and displacements around tunnels buried deep within the earthrepresents an important class of problenas in geomechanics. For many tunnel problems, thematerial distant from the tunnel can often be assumed to behave, when viewed from pointsnear the tunnel, as a simple linear-elastic material. The material near the excavation canexhibit, however, signit:'icant nonlinear behavior such as cracking, slip on joint planes, orcreep. To analyze such problems accurately, it is desirable to capture the nonlinearbehavior near the tunnel while incorporating an accurate representation of the effects of thesurrounding rock mass.
Two numerical tools that have been used to analyze geomechanics problems are the finiteelement method and the boundary element method. The finite element method isparticularly well suited for modeling problems with rapidly varying properties and highlynonlinear material behavior, lt is not particularly well suited for very large-scale problems,because the entire domain must be discretized, lt can become quite expensivecomputationally to model large volumes solely with finite elements. The boundary elementmethod, on the other hand, offers a means to model large volumes o!"material efficientlybecause only the boundary has to be discretized. If material properties vary significantlyover a portion of the domain or if the material behaves nonlinearly, then use of theboundary element approach becomes cumbersome relative to the finite element method.
lt is apparent ftore the above discussion that a computational tool combining the finiteelement and boundary element methods would be useful fl_r modeling geomechanicsproblems involving tunnels buried deep within the earth. The finite element method couldbe used to account for complex nonlinear behavior near the tunnels, and the boundaryelement method could be used to model the rock mass remote from the tunnels and to
provide accurate boundary conditions for the outer boundaries of the finite element region.The boundary element region would be modeled as elastic material, which is a reasonable
" assumption for volumes away ftore the lunncls.
,, The benefits of coupling the finite element method and boundary element method forgeomechanics problems have been recognized for some time, and a variety of schemes
have been proposed for coupling the two techniques. !-8 Although there is a significantbody of literature on coupling the finite element and boundary element methods in general,little attention has been paid to the problem of developing an efficient coupling scheme for
large-scale geomechanics problems. The ability to efficiently analyze large-scale
geomechanics problems will become increasingly important for the design and assessment
of nuclear waste repositories and for other classes of geomechanics problems.
Initial work to develop an analysis approach that couples boundary elements and finite °
elements for large-scale problems is discussed in Reference 9. This approach produces a
nonsymmetric operator. A primary focus of the initial research concerning the coupling
technique has been to determine if an iterative scheme for nonsymmetric operators is a
viable solution method for the nonsymmetric operator used in the coupling technique. The
use of iterative solvers makes possible the modeling of large-scale problems, which is one
of the key goals of this research. Althougil the engineering community has a great deal of
experience with iterative solution techniques for symmetric operators, it has much less
experience with iterative solution techniques for nonsymmetric operators. The research
presented here is focused on determining if an iterative solution technique for
nonsymmetric operators will work well with the coupling scheme discussed in Reference9. If it does not, then it will be necessary to examine alternative coupling schemes.
Direct solution techniques are not being considered for this work because they become very
cumbersome in terms of storage and efficiency for large-scale problems, especially for
three-dimensional problems, lt would be easy to i'nplement the coupling scheme proposedin Reference 9 with a direct solution technique but not particularly useful.
The puq_ose ot" this report is to present the work done to date in determining if an iterative
scheme for nonsymmetric operators is a viable solution technique for the coupling
technique presented in Reference 9. The current research addresses the viability of the
solution technique for linear elastic example problems. The real benefit of a coupled
scheme will be lk_rmodeling of problems with nonlinear material behavior. Problems with
nonlinear material behavior will be examined in the next phase of research. In the next
section, the coupled finite element/boundary element technique is described. Then the
implementation of the method in the JAC2D finite element code is described. The iterative
solver in conjunction with the coupling scheme is used to solve several sarnple problems
that allow the accuracy of the method to be assessed. Comments are also made regarding
issues of efficiency.
Coupling the Finite Element and Boundary ElementMethods
Before discussing techniques for coupling the finite element and boundary elernent
methods, a brief discussion of the equations characterizing the two numerical methods will
be presented. In general, the finite element method yields a system of equations of the form
[K] {u} = {f}, (EQ1)
where K denotes a stiffness matrix, u represents nodal point displacements, and l
represents nodal point forces. The boundary element equations have the general form
, ii, ,ill I , ,,,_ip, i,,, ,, Ii' _ ' ' ' ' '' " II' " '' II "''ll['Irll II"'" '= = n,,,, , ,' ' '" ' , li ' " 'II 'tl" '
''' Pl'' = , _ii, ii ,,, I'I "lilHlrl=_,, ,,_ ' '" =ill' ,il, il ....
[7] {u} = [U] (t}, (EQ2)
where u represents nodal displacements and t represents tractions, l°'ll The matrices
• designated by T and U contain influence coefficients associated with the displacementsand tractions, respectively. The boundary element problem is often posed as a mixedproblem, i. e., there are both unknown displacements and tractions. For a mixed problem,
" the boundary element equations are rewritten so that ali of the unknowns are on the left-hand side and ali of the known quantities are on the right-hand side. The system ofequations would appear as
ITl U 1 [ tl] = I-Tr , (EQ 3)Lt J
where the r subscript denotes known quantities and their associated influence coefficientsand the I subscript denotes unknown quantities and their associated influence coefficients.
Since the quantities ur and tr are known, the matrix multiplication can be carried out on
the right-hand side to yield a set of equations of the form
[AI {x} = {b}, (EQ4)
where the A matrix contains the T/ and U / influence coefficients and the x vector contains
the unknowns u I and t/. The b vector is the product of the right-hand side in Equation 3.
It is important to note that the T and U influence coefficients do not have the same units asthe terms in the K matrix, and b does not have the same units as [. Even though the systemof boundary element equations in Equation 2 can be reduced to a form resembling the finiteelement equations, one cannot simply add terms from the two different systems to producea combined finite element/boundary element technique.
For the development of a coupled finite eleinent/boundary element technique, consider theproblem shown in Figure 1. Part of the domain is modeled with finite elements, and theremainder is modeled with a boundary element region. The finite element and boundaryelement regions do not overlap, but they do share a common interface. The problem shownin Figure 1 represents a general coupling problem in that some of the finite elem,':_t nodesare strictly within the finite element domain, and some of the finite elements noaes are onthe interface. Likewise, some of the boundary element nodes are strictly on a boundaryelement surface, and some of the boundary element nodes are on the interface. A morespecialized case of the problem shown in Figure 1 would be one where ali of the boundary
" element nodes were interface nodes.
•, As the first step in developing a combined finite element-boundary element technique, theboundary element equations (Equation 2) need to be written in terms of nodal point forcesrather than tractions. This can be done in a straightforward manner. For the two boundaryelements shown in Figure 2, a nodal point force at node i can be computed based on thetractions prescribed for boundary' elements n and n + 1. The global coordinate system for
i, i , i i i lr r,i ii i t I • '
Region 2Boundary Element Region
_:> Boundary element nodes
Interface nodes
,4e o Interface
.................._.... Region 1Finite Element Region
!
i.
o node point (not ali node points shown)
Figure 1. Coupled Finite Element/Boundary Element Problem ,,a
"' ii' .............. II ll',ll_.......... ,,,. _r,_r.... ,r..... ,.......... ' "_' ....IIIIq_'_'''''r'"lln"' 1,,l_qrrallrmim"'",'IIIIli li ,l_lr'r'l_........ _r H' '_"lir ' i
the boundary elements in Figure 2 is defined to be an xy cartesian system. The local
coordinate system associated with each boundary element is denoted by _, with _ ranging
from -1 to + 1 over the length of an element. If the tractions are assumed to vary linearly
over the boundary elements, the shape functions approximating the tractions over atb
boundary element are
. _ N 1 (_) = ( 1 - _)/2 (EQ 5)
and
N2(_) = (1 +_)/2. (EQ6)
- The traction component tx over element n written in terms of the shape functions is
• fxi = t.,i[_l-, (1 +_)/2(l,,/2)d_+II_ I (l-_)/2(l,,+l/2)d_], (EQll)
" which simply reduces to
fi_i = t,i(l,,+ l,t+ I)/2' (EQ 12)
For the y component of force at node i,
, ' ' ' ' II ' '' '
node i+ 1
txiNl (_)element n+ 1
boundary•"-- element n+ 1
frjlxi
node itxiN2 (_ )
element n "-
boundary element n
node i- 1
Figure 2. Conversion of Boundary Element Tractions to Nodal Forces
fyi = tyi(In + ln+ 1)/2. (EQ 13)
In general, it is possible to write at node i
txi = fx./ai (EQ 14)
" and
tyi = fyi/ai , (EQ 15)
where
a i = (I,, + In+ I)/2. (EQ 16)
Equation 2 can now be written in terms of forces rather than tractions by using the relations
[U] {t} = [U] [I/a] {f} = [Uf] {.f}. (EQ17)
In Equation 17, the matrix represented by 1/a is a diagonal matrix, and the matrix Uf is
simply the matrix U with its columns divided by appropriate a i terms, where i ranges from
one to the number of nodes in the boundary element model. Calculation of the Uf matrixis actually a simple process and can be done without the construction of the 1/a matrix.
In general, a moment will result from replacing the tractions by nodal forces. For boundaryelements, this does not result in significant errors if the ratios of lengths of adjacent
elements are less than or equal to 3.12 If the boundary elements are only used to give anaccurate far-field representation for the finite element regions, this should not be a difficultcriterion to meet. In most cases, the analyst can probably construct the boundary elementsso that the ratios of adjacent elements are close to 1 for the entire boundary element model.This scheme for converting the tractions to nodal forces should, therefore, introducenegligible inaccuracies if reasonable guidelines are followed, lt has the attractive featureof being easy to implement from a computational standpoint.
Now that the boundary element equations have been written in terms of nodal point forcesrather than nodal point tractions, it is possible to derive a very simple scheme for couplingthe finite element and boundary element methods. For the sake of simplicity in thefollowing derivations, it will be assumed that ali of the unknowns of the boundary element
" region not on the interface are displacements only. This is not a restriction on the couplingtechniques presented; it is merely a special case that is used to simplify the equations that
, arise during the derivations.
Consider a boundary element formulation of the form
[T] {u} = [Ufl {f}. (EQ18)
, iT , i , , , , , , ,11 i iii i
If the matrix Ut is inverted and both sides of Equation 18 are multiplied by the inverse, then
[uf] -I IT] {u} = [KBE] {U} = {f}, (EQ19)
where KtjF_represents a stiffness matrix derived from the influence coefficients of the
boundary element formulation. The terms in the KBE matrix could be added directly to theterms from a finite element formulation. A combined finite element-boundary elementformulation would have the form
[KFE+KBE ] {U} = {f}, (EQ20)
where the coefficients derived from the finite element formulation are denoted by Kt_t;.
The manner in which the terms of Kt_E are added to the terms of K_.t_:depends on the
pattern of shared nodes for the combined problem. The formulation just outlined results in
a nonsymmetric KBI:. matrix, which means that the sum of the two stiffness matrices in
Equation 20 is nonsymmetric.
Although the coupling technique just outlined is straightforward, it is not practical from acomputational standpoint for large-scale geomechanics problems. When coupling thefinite element and boundary element methods, it is necessary to use some technique thatcan handle large-scale problems in a computationally efficient manner. The resultingcomputational tool needs to be efficient enough so that it can be used routinely for two-dimensional problems and makes the solution of three-dimensional problems feasible.
The coupling technique just outlined requires that the U/ matrix be assembled and inverted,
and then a product of two full matrices (U f and T) be computed. Unless the boundaryelement portion of the problem is relatively small, this method has large memoryrequirements. The memory requirements for this coupling technique may becomeimpractical for some two-dimensional problems and are impractical for ali but the smallestof three-dimensional problems.
If the coupled finite element-boundary element approach is constrained to handle largeproblems efficiently, then some other coupling technique must be considered. For theproblem in Figure 1, the finite element equations can be partitioned so that thedisplacements and forces associated only with the finite element model are groupedseparately from the displacements and forces associated with the interface. The results ofthe partitioning are given in Equation 21. In Equation 21, those stiffness coefficientsrelating displacements at nodes in the finite element region to forces in the finite element
region are designated as Kil, and those stiffness coefficients relating displacements at
interface nodes to interface forces are designated as Kit. Those coefficients relating
quantities in the finite element region to those on the interface are designated as K I1 and
Ktl .
' " 'm r
• A similar partitioning for the boundary element equations in Equation 18 leads to
Lrt2 rtd 2 u{2 '
where the subscript notation is similar to that used in the partitioning of the finite elementequations. Both compatibility and equilibrium conditions must be satisfied at the interface.The compatibility aod equilibrium equations can be written simply as
u li = u21 = u !, (EQ 23)
and
--fi! = f2! = f/' (EQ 24)
respectively. Both the displacements and the forces at the interface are treated asunknowns. Equations 21 and 22 can now be written so that the unknowns are consolidatedon the left-hand side of the equations and the known quantities are consolidated on theright-hand side. The resulting equations are
Kll Kit u t = , (EQ 25)I1 Ktt
and
'2 TII-I_IJ u, U_I _2]" (EQ26)
The Y matrix is a sparse matrix containing l's and O's and is constructed so that, if node j• is an interface node, then
where the notation (K) j is used to denote the row of matrix K associated with the node jon the interface. An expression for interface node j that involves only the influence
9
ii i i , i i i i ii , ii
coefficients in Equation 26 arises from the boundary element part of the problem (Equation28).
+ Eul- .Equations 25 and 26 caa now be combined to give
KII Kll 0 0- Ul I 0
K/] K// 0 Y u/ = 0 _¢_0 T21 T22 -U/2/ u 2 U/22 _2_1' (EQ29)
0 r,, r,_-_,
The matrix multiplication on the right-hand side of Equation 29 ,'an be carried out to yield
a vector. If the matrix Ac is defined as
-Kll KIt 0 0 -
[A_] = Kt, Kt/ 0 Y0 T2/ T22 -uf2/' (EQ 30)
o r. T,_-vi,,
the vector xc is defined as
IIu 1
u i= , (EQ 31)Ex ]LI,J
and the matrix multiplication on the right-hand side of Equation 29 is carried out, the resultis
,m
The quantities u I, u/, u 2, and [t in Equation 31 are unknowns. Equation 32 is the same as
the general form given in Equation 4.4
Consider the determination of the unknowns in Equation 32 by an iterative scheme that usesrepeated products on the left-hand side of Equation 32. For such iterative solution schemes,
0 is usually made for the unknowns, and repeated products are computeda starting guess x c
i i+1
using the left-hand side ::atrix Ac and subsequent iterations for x c (x c, xc ,...) todetermine the solution, lt is important to note that the matrix product on the left-hand sidecan be computed on an element-by-element basis. The element-by-element calculations
• can be done for both the finite element and boundary element portions of the problemdescribed by Equation 29. For the boundary element portion, one could construct a singlerow of the boundary element matrices (which involves the computation of influence
• coefficients for one boundary element in relation to all other boundary elements) andcompute the appropriate products. Since the matrix product on the left-hand side ofEquation 29 can be calculated on an element-by-element basis, there is no need to construct
the matrix A c in order to compute the product of A c and xc .
From the preceding discussion, it is obvious that an iterative solution technique andelement-by-element calculations in conjunction with the coupling technique outlined byEquation 29 would allow for the solution of large problems with minimal memoryrequirements. The coupling technique in Equation 29 is, therefore, a viable scheme forlarge-scale geomechanics problems.
lt is important at this point to consider computational efficiency for the coupling techniquein Equation 29. In an iterative solution method, the A matrix in Equation 4 is typically notstored but recomputed for each iteration. For nonlinear problems, the coefficients in the Amatrix can change at each iteration, which means that the use of iterative solvers is a highlyreasonable approach to nonlinear problems. S_.lce the boundary element region of thecoupled problem is assumed to remain elastic, the influence coefficients fc_ the boundaryelement remain unchanged. If the boundary element portion of a coupled problem is smallenough, the boundary element matrices can be stored and reused for each iteration. If thestorage of the boundary element matrices is not feasible, then they must be recomputed ateach iteration. Since both the T and U matrices are full, it is important that these twomatrices be computed efficiently if they must be recomputed at each iteration. Onetechnique for greatly reducing the computation time for the T and the U matrices is analyticintegration for the integral expressions for the boundary element influence coefficients asopposed to numerical integration. If the boundary element geometry and the assumeddisplacement and traction functions over an element are simple enough, it is possible tocarry out analytic integration of the integral expressions for the boundary element influencecoefficients. Computation of the influence coefficients with expressions resulting fromanalytic integration is much more efficient than computation of the integral expressionswith numerical methods.
The computation of the T and U matrices can also be simplified by limiting the boundary" element contour to a very simple shape with boundary elements of uniform length. If the
boundary element contour is a circle with each boundary element having the same length,, for example, the rows of the T and U matrices are simple permutations of a single pattern.
Restricting the boundary element contour for a problem to a simple shape is not difficultwhen considering coupled problems. The finite element region models complex interiorgeometry and is extended to some simple contour where the boundary element region is
defined. With some of the newer mesh generation techniques being developed, 13'14
11
moving the finite element mesh from a complex interior geometry to a simple contour is aneasy process. From the preceding considerations, it appears that the coupling technique inEquation 29 can be implemented so that it is computationally efficient even for large-scaleproblems.
o
Before discussing a prototype implementation of the coupling technique described byEquation 29, the equations for a special class of coupled problems will be considered. Thisspecial class of problems will be those where ali of the boundary element nodes are alsointerface nodes. For this special class of problems, Equation 29 reduces to
! u/ _(1_" (EQ 33)
Til -U f
In Equation 33, ali of the coefficients in the U matrix appear in the Ac matrix, which is
different from the more general problem outlined in Equation 29. The equations for thisspecial class of problems is presented because ali of the example problems considered infollowing sections fall into this class.
lt is interesting to examine the Ac matrix in Equation 33 in terms of symmetry and diagonal
dominance. Some generalizations will be made about the submatrices in the A,, matrix
based on the matrices produced by one of the example problems in the following sections(a cylindrical cavity problem). The finite element terms (for linear-elastic problems) are
symmetric about the diagonal and are diagonally dominant. The submatrix -uf t from the
boundary element portion of the problem is not symmetric although it shows some veryapproximate symmetry. Neither is it diagonally dominant, although the diagonal terms in
-uf/tend to be larger than most of the off diagonal terms. Therefore, the introduction of
-Uftr into the coefficient matrix destroys the symmetry and diagonal dominance in thecoefficient matrix. We are left, however, with a coefficient matrix that shows biases toward
symmetry and diagonal dominance. Next, compare the Y and Tr/ matrices to determine
their impact on the symmetry properties of the coefficient matrix. The interface forces canbe ordered so that Y is a diagonal matrix. This means that the Y matrix is symmetric and
diagonally dominant. The 7"// matrix is very nearly symmetric, but it is not diagonally
dominant. As in the case of the -U_t/ matrix, however, it appears that the diagonal terms
are larger than most of the off diagonal terms. The diagonal terms in Y and Ttr are on the
same order of magnitude. This means that the Y and Tn matrices form an approximately
symmetric system about the diagonal of the coefficient matrix, lt appears, at least for someinstances, that the coefficient matrix for the coupled problem is not symmetric ordiagonally dominant, but it does have biases toward these properties (symmetry anddiagonal dominance). If the number of interface nodes is much smaller than the number offinite element nodes, which will probably be the usual case for geomechanics models
employing the coupled technology, then the largest portion of the coefficient will besymmetric and diagonally dominant.
If we go to the more general case where some of the boundary element nodes are not
• interface nodes, then the asymmetry in the A c matrix increases. Notice in Equation 29, for
example, that the terms that are the mirror image of the T2/submatrix about the diagonal" form a zero submatrix.
Implementation of the Coupled Technique inJAC2D
An experimental version of a coupling technique based on Equation 26 has been
implemented in JAC2D. 15 JAC2D is a two-dimensional finite element code for nonlinearquasistatic problems that uses the conjugate gradient method as a solution technique.JAC2D, because of its architecture, provides an ideal basis for rapidly developing anexperimental version of the proposed coupling technique. This section describes some ofthe details of the implementation of the coupling technique in JAC2D.
The boundary element portion of the coupled problem can easily be defined within the
existing input/output data structure (EXODUS 16)utilized by JAC2D. EXODUS includesentities called nodal point sets that are lists of nodes with a unique identifier for each list.Nodal point sets are used to define the set of ali nodes in the boundary element model andali nodes that are interface nodes between the finite element and boundary elementdomains. Multiple boundary element contours can be described within the framework ofnodal point sets. The unknowns associated with the finite element portion of the mesh areaccounted tor by default with the nodes used to define the finite elernent mesh. By addingnodes to the mesh that are not associated with any finite elements, one can account forunknown displacements for points associated only with the boundary element model andthe unknown interface forces.
The nodal point set definition of the boundary element problem is used to set up a data
structure that can be used by routines from the boundary element code HOBEI. 10 HOBEIuses two-noded boundary elements with linear interpolation of the tractions anddisplacements. The terms in the T and U matrices are calculated from expressions obtainedfrom the analytic integration of the integral expressions for the influence coefficients. Theroutines from HOBE1 process the data structure and generate the T and U matrices. The
• matrices are saved for the coupled calculations.
In the experimental version of JAC2D with the coupled technique, the standard solver (aa,.
modified conjugate gradient method) has been replaced with the Bi-CGSTAB 17 method.The coefficient matrix on the left hand side of Equation 29 is nonsymmetric. Any iterativesolver used to solve the system of equations given by Equation 29 must be able to handle t,.nonsymmetric coefficient matrix. The use of conjugate gradient type iterative methods in
conjunction with boundary element problems is discussed in Reference 18. The studies
indicate that a conjugate gradient squared (CGS) method 19 is an accurate and robustiterative solver tor boundary element problems with mixed unknowns. The studies alsosuggest that the selection of the Bi-CGSTAB method is the most appropriate solutionmethod for the coupling technique in Equation 29. The Bi-CGSTAB method is a variation "
of the CGS method, and both fall in the category of K,ylov iterative methods. 2°'21 Thereare other iterative schemes that could be used, but only the Bi-CGSTAB method is •investigated in this study. Additional references discuss iterative solvers for problems with
nonsymmetric operators. 22-25
When the T and U matrices are calculated in HOBE 1, the influence coefficients are scaled
so that ali of the terms in the A c matrix are ali approximately the same order of magnitude.
In the formulation for the influence coefficients given in Reference 9, the influence
coefficients in the U/ matrix ali have a factor of 1/ (2G), where G is the shear modulus,and they tend to be several orders of magnitude smaller than the influence coefficients inthe T matrix. The influence coefficients in T depend only on geometry, and they tend tobe several orders of magnitude smaller than the stiffness terms in K. The stiffness terms in
K tend to be on the order of magnitude of Young's modulus of the material being modeledby the finite elements. If both sides of Equation 18 are multiplied by 2G, then the influence
coefficients of the form 2GTij are approximately the same order of magnitude as the
stiffness terms in K. Equation 34 shows Equation 18 with both of the coefficient matricesmultiplied by 2G.
The influence coefficients of the form 2G g./_)a,'e denoted by Ufi/in Equation 34. The U{)'coefficients arc several orders of magnitude smaller than the stiffness terms in K and the
influence coefficients of the lbrm 2GT_/. If ali influence coefficients of the lk_rm Ui/i'al'e
again multiplied by 2G, then ali of the influence coefficients in Equation 34 will have the
same order of magnitucie. If the Uil)' coefficients are multiplied by 2G, then the vector onthe right hand side of Equation 34 must be written in terms of //(2G) rather than /.Equation 34 becomes, therefore,
, ( .i,.
2G[T_{U} = 2GIUI 1 {.17(2G)}. (EQ35)
- " Equation 35 represents a boundary element problem where ali of the influence coefficientsare on the same order of magnitude. In a coupled problem, the coefficients in Equation 35
•- would be on the same order of magnitude as the coefficients derived fl'om the finite elementportion of the problem. Apply the above scaling to Equation 33 to obtain a coefficientmatrix for the coupled problem where ali terms are on the same order of magnitude. When
' . _3this scaling process is applied to Equation ._,, the unit terms in the Y matrix must be
multiplied by 2G since we solving for ])/(2G) as opposed to f/" The scaled version of
Equation 33 is
• Kll KI! 0 u 1
IKII K,/ 2GY u/ = _1_' (EQ 36)
• 2GT/I-2GUf_ /(2G)
When development of the coupled technique was start@, it was assumed that fixed [50hats
would have to be included in the boundary element model to eliminate rigid body motions
in ,he A c matrix. Experience has shown that this is not necessary. Provisions have been
made for including fixed points in the boundary element model, but this feature has notbeen used.
The boundary element method does not preclude the exploitation of symmetry in problemsto simplify modeling. A boundary element code can be set up to handle symmetryconditions to reduce problem size. The boundary element code incorporated into JAC2Dis very basic and does not have a data structure set up to exploit symmetry. Since theprimary focus of research so far has been to determine if iterative solvers are viable solutionmethods for the proposed coupling technique, effort has not been expended to addcapabilities to the boundary element code to handle symmetry conditions. Some of theproblems in the following sections have one or two planes of symmetry. Since theboundary element code has not been set up to handle symmetry conditions, the symmetryconditions in various sample problems has not been exploited.
Test Problems
The experimental version of the coupled technique has been used to solve three testproblems. All of these test problems share common characteristics. First, the behavior inthe finite element regions problems is linear elastic for ali three. The phenomenon ofnonlinear behavior in the finite element region will be studied in future research. Second,ali test cases are problems where ali of the boundary element nodes are interface nodes. Sofar, it has not been possible to solve coupled problems where some of the boundary elementnodes are not interface nodes. The reason for this is not readily apparent. Careful
,, inspection of intermediate code results indicate that the experimental version of the coupled.t_!chnique has been implemented properly to handle this more general class of problems.When attempts are made to solve problems where some of the boundary element nodes arenot interface nodes, the Bi-CGSTAB method will not converge to a solution. More
comments will be made on this problem in the concluding remarks of this report. Third,none qf the test problems include fixed points in the boundary element model, because thefixed points represent boundary element points that are not interface nodes.
The first problem is an internally pressurized cylindrical cavity in an infinite medium(Figure 3). Analytic expressions are available for both the stresses and displacements inthe surrounding medium. If the radius of the cylindrical cavity is a, the internal pressure is •p, and the surrounding medium has a Young's modulus of E and a Poisson's ratio of v, then
the radial displacement ur as a function of r, the radial distance from the center of the
cavity, is
pa 2 ( 1 + V)u,. = Er (EQ 37)
and the radial stress component is
2_ -pa
(Yrr 2 (EQ 38)F
Equations 37 and 38 are for plane strain conditions.
This problem has been modeled with a coupled finite element/boundary element model fora geometry where the radius a is 10 na and the pressure p is 6.895 MPa (1000 psi). The
value for E is 15200 MPa ( 1.04804 x 108 psi), and the value for v is 0.22. These values
for E and v are the elastic properties used in benchmarks of codes for the Yucca Mountain
Project (YMP). 26
The first mesh used for the cylindrical cavity problem is shown in Figure 4. The cylindricalcavity is completely encircled by finite elements, and the finite element region is encircledby the boundary element model. The circle defining the boundary element contour hasbeen placed at a radius r of 16 m. There are 96 quadrilateral elements in the finite elementmesh and a total of 128 geometric nodes. The geometric nodes define the geometry for thefinite and boundary elements. There are 32 nodes at the interface between the finiteelements and boundary elements. Since we must solve tk_rthe unknown interface forces,32 dummy nodes must be added to the EXODUS database. These dummy nodes create theextra unknowns in JAC2D to account for the unknown interface forces. The problem ismodeled, therefore, with a total of 160 nodes (320 degrees of freedom).
Table 1 shows displacement results from the finite element/boundary element model versusanalytic results. These displacements are along a radial line that extends from the surface ,.of the cavity to the finite element/boundary element interface. As can be seen, thenumerical results are ali within 1.12% of the analytic results. The small errors in the finiteelement/boundary element results are due in part to the piece-wise linear geometric "approximation of the circular geometry of the cavity. For a circular cavity with a radius of
10 na and a pressure of 6.985 m, the total load at the surface is 4.3323 x 102 N. The piece-
wise linear approximation gives a total load at the surface of 4.3253 x 102 N. The error in
16
I'IllHqP'IlPllIIIIIqllllI l!_lll!II_llilllqHIl'l'llNllll'lllllHIIllllII_IIIIqPlflllqll_lqlllHIlI'1qllllIIIIIllllIlqlflIlliIIIIlqWlllq_'"l''1' IIIMJIHUnlIIIHIt_nllqIIIqllllmliIlllIllPl,,,,PlIInlllllUllllSllllllllllllaallPlqqnYII,llpll,lqmq_lmmMmm,M,an,r,,,mnwp,avm,,la,_wn,,llmlmp.I,,mimm,,v,,Hmw_m,nmn,=.nx Mi ,,mm.,.,m.x
r
v
,, Figure 3. Geometry for Cylindrical Cavity Problem
the total load due to the piece-wise linear approximation of the cavity is 0.16%. At thesurface of the cavity, the numerical displacements differ from the analytical results by0.61%, which is larger than the error that can be attributed solely to the piece-wise linear
approximation of the cavity. Some of the error arises from round-off in the calculationsand errors allowed within the tolerance set on the residual norm.
Table 1: Cylindrical Cavity Coarse Mesh Results
Radial Distance (m) Analytic ur (in) JAC2D u,. (m)
l0 0.005534 0.00550
12 0,004612 0.00458
14 0.003953 0,00392
16 0.003459 0.00342
As indicated in the previous section, the solution technique chosen for these studies of thecoupled technique is the Bi-CGSTAB method. This method requires some starting vectorfor the iterative process. As part of the pr_ tess of verifying the code, the starting vectorwas chosen to be the exact solution as calculated by Equations 37 and 38, which varies
slightly ft'ore the actual computed solution due to approximations in the circular geometry.Under these circumstances, the first residual calculated for the iterative process should havecontained small terms, which it did. This served as a means to check the implementation
of the coupling technique. Interestingly enough, it took 109 iterations to converge to a
solution with a residual norm tolerance of 1.0 x 10 -7 with the initial vector computed by
Equations 37 and 38. By using a zero vector as the starting vector (i.e., ali unknowns wereassumed to be zero), it took 104 iterations to converge to a solution with a residual norm
tolerance of 1.0 x 10-7 .
The same cylindrical cavity problem described above was solved by using a finer mesh thanthe one shown in Figure 4. The finer mesh for the cylindrical cavity problem is shown inFigure 5. This mesh has 864 finite elements. There are 960 geometric nodes in thecombined finite and boundary element mesh. Since there are 96 nodes at the finite element/
bounda_'y element interface, a total of 1056 nodes (2112 degrees of fi'eedom) are requiredin JAC2D. This problem converged to a solution with a residual norm tolerance of
1.0 X 10 -.7 in 1804 iterations. Displacement results for the mesh in Figure 5 versus theanalytic solution are shown in Table 2. There is very close agreement between the analyticresults and the results fl)r the finer mesh. The higher degree of accuracy for the fine mesh
is due to two phenomena. First, the finer mesh gives a more accurate approximation of thecircular geometry of the cavity than the fine mesh. Second, because of the definition of theresidual norm tolerance in JAC2D, the fine mesh is actually subjected to a tighterconvergence tolerance than the coarse mesh. This probably accounts for the relatively
18
i ' ' ' ' ,, i
I I I I I I I
13.5
1 I I I I-13.5 -9.0 -4.5 .0 4.5 9.0 13.5
X
Figure 4. Coarse Mesh for Cylindrical Cavity Problem
I I I I _ I I
13.5
9.0
4.5
.0
-4.5
-9.0
-13,5
" I I I I-13.5 -9.0 -4.5 .0 4.5 9.0 13.5
X
" Figure 5. Fine Mesh for Cylindrical Cavity Problem
19
ml , I II i,i I I _ , I I rl '11
greater number of iterations needed for the fine mesh, and it has some bearing on theaccuracy of the results.
Table 2: Cylindrical Cavity Fine Mesh Results
Radial Distance (m) Analytic u r (m) JAC2D ur (m)...
10.0000 0.005534 0.00553
10.6667 0.005188 0.00518
11.3333 0.004883 0.00488,.,
12.0000 0.004612 0.00461
12.6667 0.004369 0.00437
13.3333 0.004151 0.00415
14.0000 0.003953 0.00395
14.6667 0.003773 0.00377
15.3333 0.003609 0.00361
16.0000 0.003459 0.00346
The two sets of results from the cylindrical cavity problem suggest that the couplingtechnique proposed in Equation 26 is valid and has been correctly implemented in JAC2D.The combined finite element/boundary element technique shows a high degree of accuracy.
Tunnel Problem
To further test the coupling technique, a problem with a noncylindrical opening wasexamined. The opening is a tunnel with the geometry shown in Figure 6. As in the previousexample, a pressure of 6.895 MPa is used to pressurize the tunnel. The surroundingmedium has a Young's modulus of 15200 MPA and a Poisson's ratio of 0.22. For oneversion of this problem, the surface defined by the boundary elements is a circle (centeredat (0,0)) with a radius of 16.0 m (Figure 7). For the second version of this problem, thesurface defined by the boundary elements is a square (centered at (0,0)) with sides of length30 m (Figure 8). Both versions contain 1152 finite elements in the region surrounding thetunnel, and 1248 geometric nodes to define the geometry of the finite elements andboundary elements. There are 96 interface nodes, which means that the problem is definedby a total of 1344 nodes (2688 degrees of freedom) in JAC2D.
4
Table 3 shows a comparison of the results from the two different meshes. Displacementsat six points on the tunnel ((2.5,0), (0,2.5), (-2.5,0), (-2.5,-2.0), (0,-2.0), (2.5,-2.0)) fromthe two different meshes are compared. Both problems converged to a solution with a
2O
Y
C A x
E
Figure 6. Geometry for Tunnel Problem
21
, ' i i II ' i ,, i i I i I i ,,, i I
22
residual norm less than 1.0 x 10 -7. The results from the two different meshes show verygood agreement. The results are symmetric about the line x = 0.
E (0, -2.0) -1.97e-11 -2.14e-3 -6.27e-11 -2.14e-3..........
F (2.5,-2.0) 4.24e-4 -5.40e-4 4.38e-4 -5.54e-4
The results can be checked to some extent by treating the tunnel as a cylindrical cavity witha radius of 2.5 m. If the tunnel were really a cylindrical cavity with a radius of 2.5 m, one
would predict a radial displacement of 2.1618 × 10-4 _ f-r r= 16 m. For the node at (0,16)on the mesh with the circular boundary element interface, ti_c r94ial displacement is
2.59 × 10-4 m; for the node at (0,-16) on the mesh with the circular boundary element
interface, the radial displacement is 2.98 x 10-4 m. Both of these radial displacements arefairly close to the value predicted by the circular cavity problem, whkh serves as averification that _he numerical results are reasonable.
Another check of the accuracy of the solutions is to simply embed the tunnel in a large finiteelement mesh that should reasonably approximate the behavior of the tunnel in an infinitedomain. This has been done with a model of the tunnel embedded in a large mesh with anouter circular boundary. The geometry for the finite element mesh resembles the geometryfor Mesh 1 except for the fact that the outer circular radius is at 100 m as opposed to 16 m.The relation of the tunnel to the coordinate system remains the same. This finite element
mesh has 6912 elements and 7008 nodes. At the points (0, 100) and (0,-100), u., = 0. At
the points (100, 0) and (-100, 0), u>.= 0. These two latter boundary conditions introducesome approximation into the problem since the x-axis is really not a plane of symmetrybecause of the tunnel _.eometry. The degree of approximation is probably very slight,however. The results from the finite element calculations are given in Table 4. The resultsfrom the coupled calculations agree with the result: from the finite element calculations to
" within 2.49%.
Table 4: Tunnel Results from Finite Element Model
Location ux Uy
A (2.5, 0) 0.156e-2 0.274e-3
B (0, 2.5) 0.203e-9 0.145e-2
C (-2.5, 0) -0.156e-2 0.274e-3
D (-2.5, -2.0) -0.425e-3 -0.548e-3.....
E (0, -2.0) -0.144e-8 -0.215e-2
F (2.5,-2.0) 0.425e-3 -0.548e-3
Multicavity Problem
The last problem consists of two circular cavities in an elastic material. One cavity ispressurized, as shown in Figure 9. Both cavities are circular, and both have a radius of twometers. The distance from the center of one circular cavity to the other circular cavity istwelve meters. For the material surrounding the cavities, Young's modulus is 15.2 GPa
( !.04804 × 108 psi), and Poisson's ratio is 0.22. The cavity centered at x, y = (0, 0) is
pressurized to a pressure of 6.895 MPa ( 1000 psi).
There is no known analytic solution to this particular problem. Results from boundaryelement and finite element calculations are used to verify results from the coupled finite
element/boundary element technique. Displacements along the x-axis are used as a basisof comparison.
For the boundary element calculations, each cavity is modeled with ninety-six boundaryelements so that there is a total of 192 elements for the entire problem. Table 5 gives
displacements for various locations along the x-axis. For the cavity centered at(x, y) = (0, 0), the radial deflection for the point located at x = 2.0 m is slightly greater
than the radial deflection for the point located at x = -2.0 m. This is to be expected because
of the influence of the cavity centered at (x, y) = ( 12, 0). For the cavity centered at
(x, y) = ( 12, 0), the point located at x = 10.0 m displaces a greater distance than thepoint located at x = 14.0 m. Again, this is the expected result. The point (x = 10.0) is closerto the pressurized cavity and should experience a greater deflection than the point (x =14.0). Finally, the deflections for the cavity centered at (x, y) = ( 10, 0) are smaller than
those for the pressurized cavity at (x, y) = (0, O) as expected.
For the coupled solution, a finite element/boundary element interface is established for •each cavity at a radius of four meters (Figure 10). The region outside the two interfaces ismodeled with boundary elements. The annular regions between cavities and interfaces aremodeled with finite elements. For each annular finite element region, there are ninety-sixelements around the circumference and fifteen elements through the radial depth. A plot
24
+1
• x
Figure 9. Geometry for Multicavity Problem
Y
finite element/b19,undary element interfacesr=am
JX
finite element regionsq
" Figure 10. Geometry for Modeling Multicavity Problem with Boundary Elements andFinite Elements
25
of the undeformed mesh is shown in Figure 11. As can be seen in Table 5, the results from
the boundary element and coupled finite element/boundary element calculations agree towithin 0.03%. For those points on the cavity surfaces (x = -2, 2, 10, 14), the boundary
element and coupled finite element/boundary element calculations agree exactly to within
the accuracy shown. •
For the finite element model, a large region surrounding the cavities has been modeled in
order to minimize the effect of approximating the original problem (two cavities in an
infinite medium) with a finite mesh. The outer dimensions of the mesh and the applied
boundary conditions are shown in Figure 12. At the two vertical boundaries, the x
displacement (ux) is constrained to be zero; at the horizontal boundaries, the y displacement
(Uy) is constrained to be zero. The mesh has a total of 6624 elements. The finite element
Table 5: Multieavity Results for Various Modeling Techniques
x, (y=0) (m) ux (m) b. e. method ux (m) coupledmethod ux (m) f. e. method
-4.0 -0.543e-3 -0.544e-3 -0.542e-3
-2.0 -0.1 !0e-2 -0.110e-2 -0.110e-2
2.0 0. I!3e-2 0.1 !3e-2 0. I 13e-2
4.0 0.581 e-3 0.582e-3 0.583e-3
8.0 0.328e-3 0.327e-3 0.328e-3
I0.0 0.301e-3 0.301e-3 0.301e-3
14.0 0. !03e-3 0. !03e-3 0. 102e-3
16.0 0.990e-4 0.987e-4 0.980e-4......
results agree very closely with the boundary element and coupled finite element/boundaryelement results.
lt is significant that the proposed coupling technique in conjunction with the iterative solveris able to handle a problem involving multiple boundary element contours. The ability to
use multiple contours could be extremely useful for modeling large tunnel complexes. If
there are groups of tunnels somewhat distant from other groups but there is a concern with
interaction among the groups, then a multi-contour modeling technique could be quite
useful. This particular problem is one that could not be modeled in an analogous manner
using infinite elements. 27 Infinite elements involve extending the domain of a finiteelement so that it is unbounded. This requires appropriate shape functions that are defined
up to infinity and tend to a finite value in a suitable way. When modeling with infinite
elements, it is not possible to have two separate meshes with each one being surrounded by
a ring of infinite elements. The typical infinite element formulation does not recognize nearor distant cavities. The shape functions in infinite elements do not account for effects that
would arise from some perturbation in the distant field. To model the multicavity problem
26
, |i
i i ii i ,i i i i ,,
Figure 11. Mesh for Multicavity Problem for Combined Finite Element/Bound_ ry Ele-ment Solution
x -- -100 m y x - 106 m
Uy-" O [y= lOOm
Ul t. -- 0
Ux=O
y = -100 m,
Uv=O
Figure 12. Geometry for Multicavity Problem for Finite Element Solution
27
with infinite elements, one would have to enclose the two cavities in a single contour ofinfinite elements. Obviously, the infinite element scheme presents much more limitedmodeling possibilities than the coupled scheme.
6
Conclusion
The finite element method and boundary element method, when characterized in terms ofgeomechanics problems involving tunnels buried within the earth, have complementaryproperties. The finite element method is particularly well suited for modeling problemswith rapidly varying properties and highly nonlinear material behavior, lt is notparticularly well suited for very large-scale problems, because the entire domain must bediscretized, lt can become quite expensive computationatly to model large volumes solelywith finite elements. The boundary element method, on the other hand, offers a means tomodel large volumes of material efficiently because only the boundary has to bediscretized. If material properties vary significantly over a portion of the domain or if thematerial behaves nonlinearly, then use of the boundary element approach becomescumbersome relative to the finite element method. As a result, a variety of schemes hasbeen proposed for coupling them for use with geomechanics problems. However, not alicoupling methods lend themselves to practical computational implementations. This reporthas examined one scheme that is practical from a computational viewpoint for large-scalegeomechanics problems. The proposed coupling technique (Equation 29) combined withan iterative solution method and element by element calculations produces a viable schemefor large-scale problems. An experimental version of the coupling technique used inconjunction with an iterative solver has been implemented in JAC2D. This experimentalversion has been used to solve several linear elastic problems. The experimentalimplementation in JAC2D verifies that the coupling technique works for problems whereali of the boundary element nodes are interface nodes and that the coupled technique isquite accurate. Future research will address the issues of efficiency, especially for three-dimensional problems. The next phase of study will also examine problems wherenonlinear behavior occurs in the finite element region of a problem. This is the particularclass of problems where an efficient coupled technique capable of handling large-scaleproblems would be extremely useful.
So far, it has not been possible to use the experimental version of the coupling technique tosolve problems where some of the boundary element nodes are not interface nodes. Thisis not fully understood, although this phenomenon may be related to the degree of
asymmetry in the A c matrix. Recent studies 21 indicate that Krylov iterative methods showmore robustness and better convergence for systems that show an approximate symmetry.As the asymmetry increases, the Krylov iterative methods are not as robust. This mayexplain why the more general problem (some of the boundary element nodes are notinterface nodes) will not converge while the simpler problem will. The coupled techniquein Equation 29 may work for the general problem if other iterative solution methods areused. One candidate for an alternative solution scheme is the Generalized Minimum
Residual Method (GMRES). 25 Although GMRES is also a Krylov iterative method, it has
28
ii , i
some fundamental characteristics that are significantly different from those of the Bi-CGSTAB method. These differences might make it a viable solution method for the moregeneral coupling problem.
• The current inability to solve the more general problem with the Bi-CGSTAB method maynot be a serious issue. Consider a problem with two tunnels. The region immediately
• surrounding one of the tunnels exhibits nonlinear behavior, and a region around the othertunnel tlp to the tunnel surface exhibits linear elastic behavior. The tunnel with the regionexhibiting nonlinear behavior is modeled with finite elements surrounded by a boundaryelement contour. The other tunnel is modeled with boundary elements alone. Thisrepresents a problem where not ali of the boundary element nodes are interface nodes,which is the type of problem that has so far not been solved successfully with the Bi-CGSTAB method. The problem could be solved, however, by modeling the tunnelexhibiting linear elastic behavior only with a layer or several layers of finite elements plusa boundary element contour. (One impact of this change is to introduce more degrees offreedom into the problem.) This latter modeling approach converts the problem to one thatcan be handled by the coupling technique in conjunction with the Bi-CGSTAB method. Ingeneral, then, there is a way to work around the present inability to solve problems wherenot ali of the boundary element nodes are interface nodes.
Even if it were possible to solve the more general problem with the Bi-CGSTAB method,it may not be desirable to use the more general modeling approach. As indicated earlier,the more general problem has a higher degree of asymmetry than the problem where ali ofthe boundary element nodes are interface nodes. Since it is known that the Bi-CGSTABmethod shows faster convergence and greater robustness as the asymmetry in a problemdecreases, it may be best to always model problems so that the amount of asymmetry in theoperator for the coupling problem is minimized even it means increasing the degrees offreedom in the problem.
It is premature at this point to look at timing information in detail. First, the implementationused for the sample problems is experimental and certain portions of the code wereimplemented without consideration for efficiency. Some of the boundary elementcalculations need to be closely studied in terms of efficiency, especially if the calculationsare done on a vector machine. Second, the problems presented here are relatively smallcompared to a large-scale geomechanics problems. Extrapolating timing information fromthe current example problems to larger problems may not give an accurate indication of theefficiency of the scheme for large-scale problems. This latter consideration is complicatedby the fact that the details of the boundary element computations could be varied dependingon problem size, as indicated earlier. Several computational implementations would have
' to be studied to get an idea of efficiency for a full range of problems.
, The issue of efficiency is complicated by accuracy considerations. If a large-scalegeomechanics problem were modeled only with finite elements, it might take severalmodels to assess accuracy. If the same problem were modeled with a coupled approach,confidence in the accuracy of a coupled approach might be such that only one model isused. Even if a single finite element model required less solution time than the coupled
finite element model, determining the accuracy of the finite element models might be amore expensive computational process than simply using the coupled model.
tl
In conclusion, the research presented here shows that an iterative solution scheme for thecoupled finite element/boundary element boundary element problems is viable. The °coupling scheme can be implemented in a reasonably efficient manner. Efficiencyconsiderations have yet to be identified that rule out further study of the coupling scheme ,in conjunction with the Bi-CGSTAB method or some other iterative solution technique.
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27Bettes, E, 1977, "Infinite Elements," International Journal for Numerical Meth-
ods in Engineering, Vol. 11, pp. 53-64. (NWA.930330.0008)
32
" APPENDIX
Information fr,_m the Reference Information Base
Used in this Report
This report contains no information from the Reference Information Base.
J
Candidate Informationfor the
Reference Information Base
This report contains no candidate information for the Reference Information Base.
Candidate Informationfor the
Geographic Nodal Information Studyand Evaluation System
This report contains no candidate information for the Geographic Nodal InformationStudy and Evaluation System.
YUCCA MOUNTAIN SITE CHARACTERIZATION PROJECT
DISTRIBUTION LIST
1 Director (RW-I) 1 C.P. Gertz (RW-20)OCRWM Office of Geologic Disposal
US Department of Energy OCRWM1000 Independence Avenue SW US Department of EnergyWashington, DC 20585 1000 Independence Avenue, SW
Washington, DC 205851 F.G. Peters (RW-2)
Deputy Director 1 S.J. Brocoum (RW-22)OCRWM Analysis and Verification Division
US Department of Energy OCRWMI000 Independence Avenue SW US Department of EnergyWashington, DC 20585 I000 Independence Avenue SW
Washington, DC 25851 T.H. Isaacs (RW-4)
Office of Strategic Planning and 1 J. Roberts, Acting Associate DirectorInternational Programs (RW-30)
OCRWM Office of Systems and ComplianceUS Department of Energy OCRWMI000 Independence Avenue SW US Department of EnergyWashington, DC 20585 I000 Independence Avenue, SW
Washington, DC 205851 J.D. Saltzman (RW-5)
Office of External Relations 1 J. Roberts (RW-33)OCRWM Director, Regulatory Compliance
US Department of Energy DivisionI000 Independence Avenue SW OCRWMWashington, DC 20585 US Department of Energy
I000 Independence Avenue, SW1 Samuel Rausso (RW-10) Washington, DC 20585
Office of Program and Resource Mgt.OCRWM 1 G.J. Parker (RW-332)
US Department of Energy OCRWML
I000 Independence Avenue SW US Department of EnergyWashington, DC 20585 _' I000 Independence Avenue, SW
Washington, DC 205851 J.C. Bresee (RW-10)
OCRWM 1 R.A. Milner (RW-40)
US Department of Energy Office of Storage andI000 Independence Avenue SW TransporationWashington, DC 20585 OCRWM i
US Department of Energy1000 Independence Avenue, SWWashington, DC 20585 *
Distribution - 1
1 S. Rousso, Associate Director 1 D.R. Elie, Director(RW-50) Environmental Protection and Division
Office of Contract Business DOE Nevada Field Office
Management US Department of Energy
, OCRWM P.O. Box 98518US Department of Energy Las Vegas, NV 89193-85181000 Independence Avenue, SWWashington, DC 20585 1 Repository Licensing & Quality
Assurance
1 T. Wood (RW-52) Project Directorate
Director, M&O Management Division Division of Waste ManagementOCRWM US NRC
US Department of Energy Washington, DC 205551000 Independence Avenue, SWWashington, DC 20585 1 Senior Project Manager for Yucca
Mountain
4 Victoria F. Reich, Librarian Repository Project BranchNuclear Waste Technical Review Board Division of Waste Management
1100 Wilson Bird, Suite 910 US NRC
Arlington, VA 22209 Washington, DC 20555
5 C.P. Gertz, Project Manager 1 NRC Document Control DeskYucca Mountain Site Division of Waste Management
Characterization Project Office US NRCUS Department of Energy Washington, DC 20555P.O. Box 98608--MS 523
Las Vegas, NV 89193-8608 1 Philip S. JustusNRC Site Representf_ve
1 C.L. West, Director 301 E Stewart Avenue, Room 203Office of External Affairs Las Vegas, NV 89101DOE Nevada Field Office
US Department of Energy 1 E.P. BinnallP.O. Box 98518 Field Systems Group Leader
Las Vegas, NV 89193-8518 Building 50B/4235Lawrence Berkeley Laboratory
12 Technical Information Officer Berkeley, CA 94720DOE Nevada Field Office
US Department of Energy 1 Center for Nuclear WasteP.O. Box 98518 Regulatory Analyses
Las Vegas, NV 89193-8518 6220 Culebra RoadDrawer 28510
1 P.K. Fitzsimmons, Technical Advisor San Antonio, TX 78284Office of Assistant Manager for
Organization 6302 US Geological SurveyP.O. Box 5800 101 Convention Center Drive
Albuquerque, NM 87185 Suite 860, MS 509Las Vegas, NV 89109
1 J.F. DevineAsst Director of Engineering Geology 1 D. ZesigerUS Geological Survey US Geological Survey106 National Center 101 Conventional Center Drive
12201 Sunrise Valley Drive Suite 860, MS 509Rest#n, VA 22092 Las Vegas, NV 89109
US Geological Survey 421 Federal CenterP.O. Box 25046 Denver, CO 80225 q
Denver, CO 80225
Distribution - 3
]1 i ,I '' ' III
1 A.L. Flint 2 L.D. Foust
US Geot'ogi_ li Survey Nevada Site ManagerMS 721 TRW Environmental SafetyP.O. Box 327 Systems
Mercury, _qV 89023 101 Convention Center DriveSuite 540, MS 423
1 D.A. Beck Las Vegas, NV 89109
Water Resources Division, USGS6770 S Paradise Road 1 C.E. Ezra
Las Vegas, NV 89119 YMP Support Office ManagerEG&G Energy Measurements lnc
1 P.A. Glancy MS V-02US Geological Survey P.O. Box 1912Federal Building, Room 224 Las Vegas, NV 89125Carson City, NV 89701
1 E.L. Snow, Program Manager
1 Shern_,a S.C. Wu Roy F. Weston lhcBranch of Astrogeology 955 L'Enfant Plaza SW
US Geological Survey Washington, DC 200242255 N Gemin;_ Drive
Flagstaff, AZ 86001 1 Technical Information CenterRoy F. Weston lnc
1 J.H. Sass - USGS 955 L'Enfant Plaza SWBranch of Tectonophysics Washington, DC 200242255 N Gemini Drive.
Flagstaff, AZ 86001 1 D. Hedges, Vice President, QARoy F. Weston Jnc
1 DeWayne CampbeV.: 4425 Spring Mountain RoadTechnical Preject Officer - YMP Suite 300US Bureau of Recla_ation Las Vegas, NV 89102Code D-3790P.O. Box 25007 1 D.L. Fraser, General Manager
Denver, CO 80225 Reynolds Electrical &Engineering Co, lnc
1 J.M. LaMonaca MS 408
Records Specialist P.O. Box 98521US Geological Su.-vey Las Vegas, NV 89193-8521421 Federal CenterP.O. Box 25046 1 B.W. Colston, President and
Denver, CO 80225 General ManagerLas Vegas Branch
1 W.R. Keefer - USGS Raytheon Services Nevada913 Federal Center MS 416P.O. Box 25046 P.O. Box 95487
SAIC Raytheon Services Nevada101 Convention Center Urive Suite P-250, MS 403Suite 407 101 Convention Center Drive
Las Vegas, NV 89109 Las Vegas, NV 89109
Distribution - 4
i I i i ilr
lP
1 Paul Eslinger, Manager 1 C.H. JohnsonPASS Program Technical Program ManagerPacific Northwest Laboratories Agency for Nuclear ProjectsP.O. Box 999 State of Nevada
Richland, WA 99352 Evergreen Center, Suite 252 *1802 N. Carson Street
1 A.T. Tamura Carson City, NV 89710 r
Science and Technology DivisionOSTi 1 John Fordham
US Department of Energy Water Resources CenterP.O. Box 62 Desert Research Institute
Oak Ridge, TN 37831 P.O. Box 60220Reno, NV 89506
1 Carlos G. Bell Jr
Professor of Civil Engineering 1 David RhodeCivil and Mechanical Engineering Dept. Desert Research Institute
University of Nevada, Las Vegas P.O. Box 602204505 S Maryland Parkway Reno, NV 89506Las Vegas, NV 89154
1 Eric Anderson
1 P.J. Weeden, Acting Director Mountain West Research-Nuclear Radiation Assessment Div. Southwest IncUS EPA 2901 N Central Avenue #1000
Environmental Monitoring Phoenix, AZ 85012-2730Systems Lab
P.O. Box 93478 1 The Honorable Cyril Schank
Las Vegas, NV 89193-3478 ChairmanChurchill County Board of
1 ONW! Library CommissionersBattelle Columbus Laboratory 190 W First StreetOffice of Nuclear Waste Isolation Fallon, NV 89406
505 King AvenueColumbus, OH 43201 1 Dennis Bechtel, Coordinator
Nuclear Waste Division
1 T. Hay, Executive Assistant Clark County Department ofOffice of the Governor Comprehensive PlanningState of Nevada 301 E Clark Avenue, Suite 570
Capitol Complex Las Vegas, NV 89101Carson City, NV 89710
1 Juanita D. Hayes
3 R.R. Loux Nuclear Waste RepositoryExecutive Director Oversight Program
Agency for Nuclear Projects Esmeralda CountyState of Nevada P.O. Box 490
Evergreen Center, Suite 252 Goldfield, NV 890131802 N. Carson Street
Carson City, NV 89710 1 Yucca Mountain InformationOffice q
Eureka CountyP.O. Box 714
Eureka, NV 89316
Distribution - $
1 Brad Mettam 1 Jason Pitts
inyo County Yucca Mountain Lincoln County Nuclear WasteRepository Assessment Office Project Office
Drawer L Lincoln County Courthouse
Independence, CA 93526 Pioche, NV 89043
1 The Honorable Gloria Derby 1 Economic Development Dept.Chairman City of Las VegasLander County Board of 400 E. Stewart Avenue
Commissioners Las Vegas, NV 89101315 South Humbolt
