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Invited Paper

Dual reciprocity boundary element based

on complete set global shape functions

A. H-D. Cheng", S. Grilli*, 0. Lafe'

"'Department of Civil Engineering, University of

Delaware, Newark, Delaware 19716, USA

Department of Ocean Engineering, University of

Rhode Island, Kingston, Rhode Island 02881, USA

"OLTech Corporation, Beachwood, Ohio 44122,

USA

Abstract

A dual reciprocity boundary element method based on global shapesfunctions belonging to a complete set to transform domain integralsinto boundary ones is presented. Governing equations of Poissonand Helmholtz type are tested. It is demonstrated that the completeset shape functions are more efficient and accurate in approximatingdomain integrals than those based on incomplete set.

INTRODUCTION

The application of Boundary Element Method to non-standard linear par-tial differential operators, or to nonlinear and time-dependent ones, oftenleads to domain integrals that are cumbersome to handle. The dual reci-procity method pioneered by Nardini and Brebbia* is a powerful numericaltechnique which converts domain integrals into boundary ones using a set ofglobal shape functions to approximate the integrand. It has been success-fully applied in many problems involving dynamic, diffusion and nonlineartype operators. See Partridge, et al. for a comprehensive bibliographicalreview.

The success of the dual reciprocity hinges on the ability to express theglobal shape functions into the Laplacian of a set of known functions suchthat the divergence theorem can be applied to resolve the domain integralinto a boundary one. Such a constraint has so far limited the selection ofshape functions to a set of radial functions of the form 1 + ar -f br^ -f • • ••

Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

344 Boundary Elements

These radial functions do not form a complete set hence may lack conver-gence properties. Although practical evidences have indicated that thoseshape functions work quite satisfactorily, it is still desirable to use shapefunctions from a set that is complete in the mathematical sense. This pa-per present a dual reciprocity BEM based on global shape functions derivedfrom a polynomial series. Perhaps due to its mathematical completeness,we are able to demonstrate that the numerical results are more efficient andaccurate than the traditional radial set.

DUAL RECIPROCITY

For many non-standard, transient, or nonlinear partial differential equa-tions it is possible to extract a standard, time-independent, linear partialdifferential operator C and lump the remainder to the right-hand-side:

jC% = 6(z, ?/, Z, f , %, %\ . . . , %,., %„ , %,, %,, %=\ . • •) (1)

For the purpose of BEM implementation, these "non-standard" portionscan be formally regarded as body forces. By the application of generalizedGreen's theorem

= /Jr

(2)

where v is a conjugate variable, £* is the adjoint operator of £, B» andB* are operators resulting from integration by parts, which may be viewedas generalized normal derivatives, F is the solution boundary, and fi thesolution domain, a boundary integral equation can be formed as:

cu = / (u*q - uq*) dT - j bu* dSl (3)

where quantities denoted by asterisk are free-space Green's functions satis-fying £*u* = —6, with 6 a Dirac delta function, c is a constant dependent onthe Cauchy principal value integration of the singularity at the base point,q and q* are generalized normal derivatives of u and u*.

To circumvent the domain integration, the function b can be approx-imated by an expansion based on a set of global shape functions /, suchthat

&*fX£ (4)j=l

in which a, are coefficients determined by collocation, and n^ is the numberof collocation nodes. The shape functions are chosen such that particularsolutions of the set of inhomogeneous equations

Ciij = /,• (5)


Boundary Elements 345

are explicitly known. The domain integral in (3) can be transformed as

[ V" /j^bu da = i y /ju

(6)j=l L ^1 J

The last expression was a consequence of the application of (2). Substitutionof the above into (3) yields

- / (u*q - uq"} dY = Y a, \cu, - I (%*& - titf) dT\ (7)•/r fr(. L 7r J

cu

We hence in principal have achieved the elimination of the domain integral.However, the detail of the implementation is somewhat dependent on thetype of body force terms. We shall illustrate below the implementation byexamples.

EXAMPLES

Operator of V^u = b(x, y) type

For the governing equation of Poisson type,

q in (7) is du/dn, and u* is — lnr/2%- for 2-D or l/4%r for 3-D geometries.For simplicity, only two-dimensional problems will be studied herein.

If the right-hand-side 6(x, y) is a mathematical expression whose partic-ular solution is known as Up, the solution of (8) can be expressed as

% = %& + %? (9)

where u^ is the homogeneous solution and

V2%& = 0 (10)

It is apparent that Uh = u — Up satisfies the boundary integral equation

c(u - «„) = / [u(q - q,} -(u- u,)q'] dT (11)

If we approximate the right-hand-side b(x,y) as (4), and find the particularsolution term by term as

V - = /,- (12)



it is clear that

j=l

By the above interpretation, we have provided an alternative way for de-riving (7), which is rewritten below with explicit reference to coordinatesas

c(x)u(x)- f(u*(x',x)q(x')-u(x')q-(x',x)] dT(x')j r

(14)

in which x denotes the base point where the singularity is located, x' isthe field point for integration, and Xj marks the collocation points for theapproximation defined in (4).

We have so far not discussed the selection of the global shape functionsfj. It was proposed^ that a series of locally distributed radial functions beused. In the general form,

COr { \ "^^ k 1 i i 2 i /i r \fj(x, Xj) = 2^r=l + r + r + - - - (15)

k=Q

where r(x,Xj) = |x — Xj|. In that case, Uj may be solved from (12) as

&=o v* ~ )

However, in actual implementation a much simplified form such as

f.(~ y .\ — 1 j_ „ (17}M^^jJ — i -h r (it)

and

has been used. The coefficients OLJ needed in (14) are determined by collo-cation. Applying (4) to the n^ collocation nodes, which include boundaryas well as domain nodes, the matrix system is formed

FOL = b (19)

where

•••6(XnJJ^ (20a)

j (20b)


Boundary Elements

1x y

xy y*

347

level 012

34

Table 1: Pascal triangle for polynomial functions.

and

(21)

The coefBcients otj are then solved from (19).Although the above shape functions have generally produced satisfactory

results in approximating domain integrals of various types/ they suffer fromthe theoretical deficiency that they do not form a truncated subset of acomplete set. Convergence of the approximation may not be guaranteed.Considering the above shortcoming, it was propounded^ that global shapefunctions which are a part of a complete set be used. In particular, we usethe polynomial series which has components of

fj(x) = fj(x,y) = l,x,y,x*,xy,y*,x*,x*y,--- for ; = 1,2,3,---(22)

The above can be arranged into a Pascal triangle as shown in Table 1. Wenote that this set of functions is independent of the collocation nodes Xj.To perform collocation, we take the same procedure as the above, exceptthat the F matrix is defined as

F =

1

2/2(23)

The functions Uj(x) are found through (12). The first score of those arelisted in Table 2. For the purpose of computer implementation, it is moreconvenient to have explicit formulae. For fj = x™?/", the function Uj canbe expressed as

T t m+2* -.n-

r-*+i./ k-\

> n (24a)

for m < n (24b)



fi •14,•

xyy**

2/3/6

3,4/12

891011121314

3/5/20

15161718192021

- 1/7/420

Table 2: Polynomial shape functions and particular solutions.

where Int[ ] denotes the operation of taking integer part of the argument.Its normal derivative is

n

2A)!(n-2Jfcfor m > n (25a)

= | -'n-f-2^-1

(m-26 + 2)!(n + 26-l)!

in which n^ and Uy are the directional cosine components of the unit out-ward normal. To reflect the fact that Uj(x) and (jj(x) are independent ofcollocation nodes, equation (14) is repeated as follows

c(x)u(x) - I [u*(x',x)q(x') - u(x')q*(x',x)] dT(x')

"•d

(26)

For the purpose of computer programming, it is necessary to associatethe index j with the exponents m and n. The hierarchial location of elementfj in the Pascal triangle can be identified by its level I and element numbere of the Pascal triangle

e = int1 + v/8 * J - 7

2-1

e - j - Mint

(27a)

(27b)



7.

u = 1

8

1

q = 06

1

. 9

1

2q = 0

Figure 1: A box geometry.

where Nint[ ] denotes the nearest integer. The exponents are then evaluated

as

m = l-e+l

n — t — m

(28a)

(28b)

We thus have defined the complete set global shape functions in place of

the incomplete set.

node solutionnumber type

exact 4-node 4-node 9-nodecomplete incomplete incomplete

du/dn

.463542 .463542 .457607 .459959-.750000 -.750112 -.734468 -.7362851.08333 1.08345 1.11448 1.09485

Table 3: Comparison of numerical results for box problem, b = x\

We shall examine a few numerical examples below. The first test prob-lem involves one of the simplest geometry as depicted in Figure 1. Withthe boundary conditions prescribed and a body force term of b = x*, theproblem is one-dimensional

S = (29)

For the BEM solution, the geometry is discretized using one quadratic ele-ment per side (8 nodes total). For the complete set case, using 4 boundarynodes (2, 4, 6 and 8) as collocation points, the BEM solution produces



u = 0

0.6m

0.4mu = 0

Figure 2: A benchmark problem.

exact result at nodes 2 and 6. Numerical solutions at several points arepresented in Table 3 together with exact solutions. The same problem issolved using the incomplete set shape functions defined in (17). For thesame 4 collocation nodes used, the result as presented in Table 3 is clearlyinferior. When the collocation nodes are increased to 9, that is using allthe boundary nodes plus one interior node (node 9), the accuracy improves,but the convergence to exact solution appears to be slow.

Although the polynomial shape functions are demonstrated to be supe-rior in approximating the body force, it is important to point out a defi-ciency. With an improper selection of collocation nodes, the matrix (23)can become singular. For example, the use of the 4 corner nodes 1,3,5 and7 causes such a plight. The rules under which singular matrix will resultare not fully understood. In general, any regular mesh laid on a square gridis problematic. For the case of 4 corner nodes, the addition of a domainnode 9 can break the unison and resolve the problem. The rotation of thegeometry also eliminates the difficulty.

The second example adopted here is referred to as NAFEMS Ther-mal Analysis Benchmarks #9(ii) ref 2D/PO/CNSR.^ The geometry andboundary conditionsC are depicted in Figure 2. The problem is governedby the Poisson equation

v.~ises!» w

Using the mesh reported in Partridge, et al./ i.e. 0.1 m per node on eachside, the solution at the center is found to be 310.0 using only one collocationnode by the complete set method. The exact solution is 310.1. The excellentperformance with such low collocation node is hardly surprising—using oneterm in the series for the current problem, and 4 terms in the precedingproblem, we actually achieved exact representation of the body force term!Using the incomplete set method, Partridge et al. reported the solution atthe center to be 314.6. It is not clear what were the collocation nodes used.Using our own program, we apply the incomplete set method employing allboundary nodes as collocation points. The solution at the center is found



Figure 3: Relative error of approximating 6 = -1000000/52 by incomplete

set, 20 boundary points.

Figure 4: Relative error of approximating b = -1000000/52 by incompleteset, 20 boundary and 15 interior points.

to be 303.2. To improve result, 15 additional interior collocation nodes aredeployed in a regular grid. The solution does converge well and gives thevalue 309.8 at the center.

To gain idea of the performance of collocation by the incomplete set,we plot in Figure 3 the relative error of the approximation of the constantfield b = -1000000/52 using the 20 boundary nodes. We observe that al-though the error is zero at boundary nodes, the interior is generally not wellrepresented. But the addition of the 15 interior nodes ties down the errorsignificantly. The result is demonstrated as Figure 4. Here it is remindedthat the complete set gives the exact representation everywhere using only

one collocation point.It may be argued that for body forces of polynomial type, the polynomial



Figure 5: Absolute error of approximating b — sin2z sin2y by incompleteset, 20 boundary points.

Figure 6: Absolute error of approximating b = sin2z s'm2y by completeset, 12 boundary points.

shape functions possess an unfair advantage. A test is then conducted forthe function

b(x) = sin2x sin2% (31)

defined in the same region as Figure 2. Using 20 boundary nodes, theabsolute error of approximation using the incomplete set is presented inFigure 5. For the complete set, 12 of the 20 nodes are selected to form anirregular pattern. The result of approximation is shown as Figure 6. Therepresentation is apparently superior to that of the incomplete set, despitethat fewer nodes were used.



node solution exact 4-node 4-node 9-nodenumber type complete incomplete incomplete

2 u .443409 .443396 .442781 .4432694 <9%/<9n -.850318 -.852337 -.842975 -.8534858 du/dn 1.31303 1.31460 1.30849 1.31579

Table 4: Comparison of numerical results for box problem, b = u

Operator of V^u = (3(x, y}u type

We next examine governing equations of Helmholtz type with variable co-efficient

For b = /3u, the matrix equation (19) is now written as

Fa = Bu

where

(33)

u = [H(XI) u(

and

B =

0

0 £(*,)

(34)

(35)

0 0 ..- 0(*,J J

In the current case a cannot be explicitly solved. Rather, it is represented

asa = F-^BU = Au (36)

where A — F~^B can be pre-computed. Substitution into (14) becomes

)Uj(x, Xj)

(37)

in which ajh are the matrix elements of A.To test the algorithm, the same box geometry and boundary condition as

sketched in Figure 1 is studied. The governing equation under considerationin one-dimensional geometry is

= u (38)



1 cP-

0.8

0.6

0.4

0.2

n n

0 0.2

!./

ac

0.4oqo

u>

cuc

1

.6p3

C

00

5

.8

r

13

s3

1.

-

°1 0

0.8

0.6

0.4

0.2

n nD.O 0.2 0.4 0.6 0.8 1.CT

x

Figure 7: Potential field for b — — (1 + 2y)u.

With the same 8-node discretization and 4-node collocation, the completeset solution is presented in Table 4, together with the exact solution andincomplete set solutions. The complete set result again outperforms theincomplete set.

As a final demonstration, the problem

is solved. This type of equation may correspond to harmonic wave propa-gation in a heterogeneous medium. With the same geometry and boundarycondition as in Figure 1, a finer discretization of 2 quadratic elements perside is adopted. The resultant distribution of u is plotted in contour linesas Figure 7.

POTENTIAL APPLICATIONS

The potential applications of dual reciprocity cover a wide area. We discussbelow only a few problems in flow through porous media that are of interestto us. One possible application involves Darcy's flow in inhomogeneousmedium, with the governing equation

where h is the piezometric head and K the hydraulic conductivity. DefiningY = InK, the above equation can be rewritten as

The body force falls into the category of CL(X)UX, b(x}iiy, etc. Algorithm fordealing with this type of gradient terms has been developed in Partridge,et al.



An alternative to the above algorithm is to apply the transformation^

% = VY/t (42)

Equation (40) can be expressed as

V2% = /?% (43)

where

Another problem of interest is the solution of non-Darcy flow. Thegoverning equations can be expressed as^

- Vu (45)

(46)- - iot

where v is the seepage velocity, v = |i?|, s is the added mass coefficient, ctiand c%2 are empirical coefficients for Forchheimer's friction formula, and His the total head given by

with p the pressure, p the density and g the gravitational constant. Theabove equations are apparently time-dependent and highly nonlinear. Themost efficient way to solve them is by explicit time stepping. The followingfinite difference formulae in time may be utilized

v,+i = fl - — (ai + <wOl *„ - — Vff, (48)L «s J «5

V'ffn+1 = -a2«n+l ' Vtfn+i (49)

in which the subscripts n and n -f 1 denote time level, and At is the timeincrement. Equation (49) is amenable to a dual reciprocity treatment.

CONCLUSION

The findings of the present work are summarized as follows:

1. The polynomial series based complete set global shape functions arenot only theoretically more appealing, but also computationally moreefficient in representing and converting domain integrals.

2. For smoothly behaved functions, the complete set can often achieveexcellent representation using only boundary nodes. The incompleteset normally requires a significant number of domain nodes.



3. The cost of adding the domain nodes is small for Poisson type equa-tions, as the sub-matrix for the boundary unknowns can be solvedindependently of the domain ones. The domain unknowns are foundone by one by matrix multiplication. This is however not true if theright-hand-side contains dependent variables, such as the Helmhoitztype equation studied here. In those cases the full matrix of domainand boundary nodes must be solved. The computational cost can bequite heavy for a problem studded with domain nodes.

4. The polynomial based shape functions however suffer from the defi-ciency that the collocation matrix can become singular for a meshlaid on a regular grid. The exact rules under which the malady de-velops are not yet known. Simple remedies at this moment includethe selection of collocation nodes that form irregular geometries, theaddition of interior nodes that are offset from the boundary ones, andthe rotation of the coordinate system.

Although only the polynomial series are presented here, other completesets, especially those possessing orthogonality properties, should also beinvestigated. The key element, however, is the existence in closed forms theparticular solutions of Poisson equations defined in (12).

Acknowledgment

A.C. and O.L. are supported in part by the grant "Microcomputer based re-gional groundwater modeling system,"DHR-5544-G-00-1036-00, U.S.-IsraelCooperative Development Research Program, Office of the Science Advisor,U.S. Agency for International Development.

References

[1] Nardini, D. and Brebbia, C.A., "A new approach to free vibrationanalysis using boundary elements", In: Boundary Element Methods inEngineering, Proc. J^th Int. Sem., Southampton, (ed.) C.A. Brebbia,Springer-Verlag, 312-326, 1982.

[2] Partridge, P.W., Brebbia, C.A. and Wrobel, L.C., The Dual ReciprocityBoundary Element Method, CMP/Elsevier, 1992.

[3] Cheng, A.H-D. and Ouazar, D., "Groundwater," to appear as Chap. 9in Boundary Element Techniques in Geomechanics, eds. G.D. Manolisand T.G. Davies, CMP/Elsevier, 1993.

[4] Cameron, A.D., Casey, J.A. and Simpson, G.B., Benchmark Tests forThermal Analysis, NAFEMS Publ.,Glasgow, 1986.



[5] Cheng, A.H-D., "Darcy's flow with variable permeability—a boundaryintegral solution," Water Resour. Res., 20, 980-984, 1984.

[6] Harrouni, K. El, Ouazar, D., Wrobel, L.C. and Brebbia, C.A., "Dualreciprocity boundary element method for heterogeneous porous me-dia," Boundary Element Technology VII, eds. C.A. Brebbia and M.S.Ingber, CMP/Elsevier, 151-159, 1992.

[7] Grilli, S., Unpublished notes, 1993.


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