Computer methods in applied

mechanics and englneerlng

EISEWIER Comput. Methods Appl. Mech. Engrg. 162 (1998) l-17

Adaptive H- and H-R methods for Symm’s integral equation

Thang Cao School of Mathematics, University of New South Wales, Kensington, NSW 2052, Australia

Received 26 August 1996; revised 12 September 1997

Abstract

In this paper we present the adaptive h-, r- and h-r methods for the Galerkin approximation of Symm’s integral equation. The a posteriori error estimate depends on a localized a priori error estimate and local finite differences of the computed solution. The optimal mesh for these

three methods is obtained by using a mesh grading function. The numerical results for a polygonal problem, and a comparison between our

error estimate and the residual error estimate, are also presented. Moreover, we briefly describe how to apply our method to a

two-dimensional elasticity problem. 0 1998 Elsevier Science S.A. Ail rights reserved.

1. Introduction

A large class of both exterior and interior elliptic boundary value problems can be reduced to boundary integral equations using Green’s identities. Such integral equations are found in many applications such as acoustic scattering [lo], elasticity and potential theory [ 161 and fluid mechanics [19], etc. The various methods of discretization of these integral equations by finite elements on the boundary manifold are called boundary element methods. The main advantages of the boundary element methods are that they reduce the computational dimensions by one, and give a simple discretization of the exterior problems. In the latter case, boundary element methods are clearly more advanced than finite element methods because the relevant domain is unbounded and so a direct finite element discretization is problematic (see [27,16,22]). There are many books and papers about applications of boundary element methods to engineering problems, among which we may mention [7,6], etc; for a more complete bibliography of engineering works see [l].

At the heart of adaptivity is knowledge about the size and distribution of the error for a given finite element method (FEM) or boundary element method (BEM). Usually, this knowledge is extracted a posteriori from the approximate solution, in the form of an error indicator for the local error and an error estimator for the global error in some norm. Let 7 be the error estimator for the norm llell of the error e = u - u,,, where u and u,, are the exact and approximate solutions, respectively. The estimate 8 should depend only on u,, and satisfy

where C, and C, are positive constants independent of uh. The error indicator Aj, j = 1, . _ . , N, is computed for the jth element, and is related to 7 by

Several authors have studied adaptive BEM that use a posteriori error indicators depending on local residuals [21,28,14], etc.). However, due to the nature of boundary integral equations, the cost of computing the local residuals is high (see Section 5). Therefore, we propose a different adaptive BEM that is computationally cheap, and is effective in refining the mesh. For simplicity, we only consider the Gale&in BEM for Symm’s equation

0045-7825/98/$19.00 0 1998 Elsevier Science S.A. All rights reserved. PII: SO0457825(97)00326-5

2 T. Cao I Comput. Methods Appl. Mech. Engrg. 162 (1998) 1-17

(25) on a closed smooth curve (for our theoretical analysis only) with logarithmic kernel using piecewise- constant trial functions. The error control will be based on the localized a priori error estimate (13) and Cea’s Lemma, which yield the following error estimate

(3)

where

hj=tj-t,_, and J’=<tj_,,tj) forj= l,..., N,

and where t,, t,, . . . , t,, are the values of parameter t that correspond to the nodal points, and Du is the derivative of u with respect to t. The local quantities IIDu~~~,+~, can be approximated by using finite differences

of the computed solution u,,. Hence, our adaptive algorithm is based on an a posteriori error estimate of the form

(3) with /(Du((,o,~~ replaced by an approximation obtained through the computed solution u,,. This error

estimator is attractive because it is composed from local quantities defined for individual elements. Any change in the mesh affects the local error on the elements where the changes have been made. The constant c (resulting from Cea’s Lemma and the parameterization) is bounded but is very difficult to calculate in a practical problem and therefore can only be ignored. The local error indicators A, are defined by

Aj = h,?‘z~(Du~~H~~I;~ for j = 1, . . . , N, (4)

so that the local error indicators and the error estimator are related by (6). A good mesh will be shown to be one which has a similar value of the local error indicator on each element.

There are three versions of the adaptive FEM, each involving a different strategy for reducing the error: l The h-version which achieves the accuracy by refining the mesh while using a low fixed degree p for the

piecewise polynomials, usually p = 0, 1,2. l The p-version which keeps the mesh fixed. The accuracy is achieved by increasing the degree p for the

piecewise polynomials. l The r-version keeps the number of nodal points and the degree for the piecewise polynomials fixed, and

adjusts their locations to achieve higher accuracy. Following Babuska and Rheinboldt [4] we use a mesh grading function to obtain the optimal mesh. Both h-

and r-mesh strategies are used for this optimal mesh analysis; cf. the approach used in [8] and [13] for the FEM. We are also able to devise a combined h-r mesh strategy.

The paper is organized as follows. In Section 2, we introduce the well-known results and basic notations, which are used throughout the paper. In Section 3, we introduce the mesh grading function. In Section 4, we present our adaptive algorithms for the h-, I- and h-r versions. In Section 5, we describe our numerical experiments. In Section 6, we obtained a similar local error estimate for elasticity as (3) then briefly describe how to apply our method to a two dimensional elasticity problem.

We will denote C as a generic constant throughout the paper.

2. Preliminaries

2.1. Sobolev and spline spaces

The definitions of the Sobolev spaces to be used throughout the paper are as follows. For s E [w, the Sobolev space H”(rW*) is defined as a space of temperate distributions u ES’, such that

where li is the Fourier transform of U. It is well know that

T. Cao I Comput. Methods Appl. Mech. Engrg. 162 (1998) l-17

and

Let fi be a bounded domain with Lipschitz boundary i! Let r C r be a closed or open curve such that r = f if r is closed and r # r if r is open. As in [17] and [26], we define

H”(~)={u~~:uEZ-Z’+“~(R~)} fors>O

Ho(r) = L2(1;)

H’(r) = (K”(F))’ for s < 0 .

where (.)’ denotes the dual space with respect to L2-inner product. Then, we define for all s E R

w(r) = fuIr : u E ~“(1;)) I?‘(r) = iu E H”(F) : supp u G ij .

For s = l/2, f’(r) = Hi,(T) as defined in [17]. The duality properties are as follows (see [26]):

(w(r))’ = KS(r) and @‘(r)y = H-“(r) For s > 0, the norms in H”(F), H”(T) and I!?(r) are defined by

I]u]IHSC~, = inf{llull,S+~,@2) : ul~ = u}

IbllHlcr, = infMles~~ : UL- = ~1

II4Zi.yr, = lMf~(~ .

(5)

For s < 0, the norms are defined by duality. Their interpolation relationships for 0 <s < 1 are as follows:

w(r) = [f?(r), &J(r)] F (6)

f?‘(r) = [ii I(r), fP(r)l s (7)

where [., .],s denotes complex interpolation spaces as defined in [ 17,261 and (6), (7) are from Theorems 2.10.1 and 2.10.4 of [26]. The duality theory for interpolation spaces [26] ([A,, A ,I,: = [A& A :],y) and (5) imply that

H-‘(r) = [fi’(ry, fP(r)ys = [P(r),HO(r)l,y (8) HP’(r) = p?(r)‘, fP(r)~],y = [I?-‘(r), ffo(r)l,y . (9)

The following lemma is due to Von Petersdorff [20, Lemma 3.2, p. 491 and it is a main tool for deriving an a priori error estimate in Lemma 2.3.

LEMMA 2:l. Let r be decomposed into non-intersecting Lipschitz open subarcs I;, j = 1,. . , M, such that r = U ,” , q. We have the following decomposition inequalities

Ilull (10)

for s E R.

(11)

The spline spaces used throughout the paper are chosen in the sense of Aziz and Babuska [2]. Let y[O, l] + r be the parameterization of K and let r be a partition of [0, 11 with mesh points tj for j = 0, . . . , N. The mesh sizes are denoted by h, = tj - tj_, and we put h = maxlSjGN hj for j = 1, . . . , N. We define the space of all

4 T. Cao I Comput. Methods Appl. Mech. Engrg. 162 (1998) 1-17

l-periodic smoothest splines of degree Y subordinate to the partition r, i.e. each element of this space is l-periodic on R and its restriction on (tj_, , t,) is a polynomial of degree equal r - 1. With the parameterization y, we transplant these splines onto r then denote those spline spaces by SL. The order r is chosen to satisfy the conformity condition Si C H”(r), which is satisfied if (Y < r - l/2. We note that Si C H-“2(L-‘) is piecewise constant space. This spline space will be used as both test and trial spaces for the Gale&in approximation of the weakly singular integral equations introduced in the next sections. The following approximation properties of spline spaces Sk play an important role for the error analysis of the Galerkin approximation.

LEMMA 2.2 (Approximation property). Assume that t, < r - l/2. Then, for any u E H”(T) with t, G s G r

there exists 5 E SL such that

((u - &lcr, G ChS-‘lluI(HS(,-J for all t G t, . (12)

PROOF. See [2]. q

The local a priori L*-projection error estimate for functions in El(T) is expressed in the following lemma

LEMMA 2.3. For any u E I? ‘(f ), there exists 5 E Sl such that

where Osscl, J=(tj_l,tj), j= l,..., N.

PROOF. Let Ph denote the L*-projection onto Si. For each u E Z?‘(T), we define [ E SE by

[(x):=Phu(x)=h,Y1/ru(x)dx forxEq,j=l,...,N. /

For any x E q, we now use Cauchy-Schwarz inequality to obtain

It&> - &)I2 = h,-’ I,. (44 - U(Y)) dy / * G 1 h,-’ I, b(x) - 4~11 dy 1 * I I

s h,Y* )U > r 1 dy I

= h,-’ I

r /u(x) - u(y)[* dy . I

Also, for any x, y E 4, we have

I44 - 4Y)12 = / 1; u’(t)dti’=z[x-y$u’(t)/*dt

< h, lu’(t)[* dt c hjllu#,ocr;, .

We then integrate both sides of inequality (16) on 4 to obtain

I r b(x) - U(Y)/* dy 6 h/211u’ll;~(I;) . I

It then follows from (15) and (17) that

/U(X) - 5(x)(‘s hjIIU’IIio(q) 9 x E c *

Integrating both sides of (18) on q, we have

(13)

(14)

(15)

(16)

(17)

(18)

T. Cao I Comput. Methods Appl. Mech. Engrg. 162 (1998) l-17

lb - P,&~,r, 6 q&(l;)

By the duality property (5), we have

Using the interpolation property (9), it follows from ( 19) and (20) that

\l” - phuIlj?s(I;, s ch;+s\b'\\H"(I;) '

Therefore, (13) is a consequence of Lemma 2.1 and inequality (21), as follows:

5

(19)

(20)

(21)

1124 - P,ull&, s 5 [IL4 - P,ull;-5cl;, 6 c E h)(‘+awII&/~ . 0 j=l j=l 2.2. Boundary integral equations

Throughout the paper we study the weakly singular integral equations of the first kind. These equations are particular examples of pseudodifferential operator equations (see [27]), and can be expressed abstractly as follows.

Let X be a Hilbert space, X’ its dual space and A : X +X’ a bounded linear operator. The general equation is

Au=f for f EX’ (22)

The existence of a solution u is based on the well-known Lax-Milgram theorem. The theory of boundary integral equations is vast and complex. We only attempt to introduce in this paper some prototype boundary integral equations of the first kind reformulated from Dirichlet problems without any technical proof. The

technical details of reformulation of elliptic boundary value problems as boundary integral equations can be found in [18,22] and the references in [22]. It is well known that the Dirichlet boundary value problem

-AU=0 inR, (23)

U=g onr, (24)

can be converted to Symm’s integral equation via a direct or indirect method.

Direct method for the _Di;shlet problem

Find u = au/an E H (@J) such that

“u(x):= -$ I (25) r u(y)loglx-ylds,=f(x) forxEf,

where Ix - y/ is Euclidean distance between x and y, & is the element of arc length, and f(x) is defined by

f(x) := u + m(x)

with

Mx):= -; r g(Y)-$loglx-y~dsy. I Y

6 T. Cao I Comput. Methods Appl. Mech. Engrg. 162 (1998) 1-17

The solution U is then obtained by

U(Y) log]x - YI - g(y) $ loglx - Y] >

ds, 3 xE0. Y

The solution u of the integral equation (25) is actually the ‘jump’ of W/an across the boundary K

Indirect method for the Dirichlet problem

Find u EE?-l’*(r) such that

Vu(x):= -1 I

for x E I’ rr F

u(y) loglx - yI ds, = 2g(x) =f(x) (27)

where the solution U is obtained by

U(x) = - & I F U(Y) hlx - Yl by 7 XE0. (28)

The weakly singular operator V and the double layer operator K define continuous linear operators

V: H-Ii2 (r) + H1’2(F) , K: H”2(F)+H”2(F) ; (29)

See [ll].

Throughout the paper we assume that

cap(f) < 1 (30)

where cap(r) denotes capacity (or conforr_nal radius or transfinite diameter) of K Hence, the operator V defines a positive definite bi_li_n,e,a: form on H -“2(lJ [23]. Furthermore, it was shown in [24,25] that V is strictly coercive on the subspace H (F), i.e. there exist a positive constant c such that

(Vu, V) 2 cIIuII~~~~~~~, for all u EE?-“2(r). (31)

Therefore, by the Lax-Milgram lemma, each of the equations (25) and (27) has a unique solution u E I? -1’2(r)

for any given right-hand side f E H “2(F). The following lemma will provide very important properties of the

weakly singular operator V which imply the convergence of the Galerkin boundary element method for Symm’s

integral equation defined in the next section.

LEMMA 2.4. The weakly singular operator V : I?“-’ (f) + H?(F) is linear and bounded for any 7 E [0, 11, and

bijective for q E (0, 1) when F is open. When F is closed, it is linear, bounded and bijective for 17 E [0, 11.

PROOF. See [ 1 l] for linear and bounded properties of V for q E [0, l] and bijectivity when r is closed. For f

is open, see [24]. 0

2.3. Galerkin boundary element method

The classical Galerkin method for approximately solving the abstract equation (22) uses a finite dimensional subspace X, C X for the trial as well as the test space. Thus, the method is: Find uh E X,, such that

(Au,, 4) = (.J 4) for all #J E X,, . (32)

The existence of the numerical solution uh is based on the following lemma.

LEMMA 2.5 (Cea ‘s lemma (271). If A : X +X’ is bijective and coercive then (32) has a unique solution u,,. Furthermore,

lb - %llx s c ,‘:f,ll” - 4x * (33)

The Galerkin approximations for equations (25) and (27) are as follows:

T. Cao I Compur. Methods Appl. Mech. Engrg. 162 (1998) I-I 7

Direct method for Lht?, flirichlet problem Find uh E SL C H (J’) such that

(Vu,, 4) = (5 4) for all 4 E SL

where f = (Z + K)g. The approximation U,, of U is defined by

1 U,(x)= -2n r

J( uI(Y)loglx-Yl-~(Y)~log/~-Y/ d+

> XEn.

Y

Indirect method for Dirichlet problem

Find u,, E Sk C I?-“‘(T) such that

(Vu,, 4) = (i 4) for all #J E Si

where f = 2g. The approximation U,, of U is defined by

u,cd = - & r %(Y) loglx - Yl d.-y” 9 i

XE:n.

(34)

(35)

(36)

(37)

From Lemma 2.4, the weakly singular operator V : H - -l”(r) -+ H”2(r’) is bijective and coercive. Therefore,

due to Cea’s lemma, Eqs. (34) and (36) have unique solutions. It is well known that V is bijective strongly

elliptic pseudodifferential operators of order - 1 when r is closed and smooth. The asymptotic error estimates

then follow from the next theorem.

THEOREM 2.1 (271. Let A be a bijective strongly elliptic pseudodifferential operator of order (Y and let r be a

smooth closed curve. Let u,, E Si is a Galerkin solution and (Y < 2r - 1 and LY - r < r S (u/2 6 p s r, then we

have the asymptotic error estimate of optimal order

(38)

REMARK. For problems with singularities, such as occur for open curves or polygons, a modified form of (38)

is required. The reference for open curve problems is [22]. In polygonal domains, similar results for weakly

singular integral equations were presented in [ 121. In the practical computation of the Galerkin solutions, we have to choose bases for the spline spaces Si. The

standard basis for Si is {4j; j = 1, . . . , N}, where 4j equals 1 on the interval [tj_ 1, tj) and is zero everywhere

else. Using this basis, the Galerkin equations (34) and (36) can be expressed as the systems of linear equations

VUh=J, (39)

where the entries of the stiffness matrix v are given by

ek=(V$,+kk)t j,k=l,...,N,

and 7 is a vector with components 7, = (J 4j) in (39), for j = 1,. . . , N.

3. Mesh grading function

The optimal mesh is obtained by minimizing the error estimator on the boundary r We assume that r is a closed smooth curve. Therefore, the weakly singular operator V is a pseudodifferential operator of order - 1, and

the error estimate (3) holds. As presented in Section 2, the boundary r can be parameterized into I = [O, 11. We

approximate the derivative Du by D,,uhr defined by

T. Cao I Comput. Methods Appl. Mech. Engrg. 162 (1998) 1-17

U&l > - %&) h,

tE(t,,t,)t

D/p/#) = Uh(tj-1/2) - uh(tj-3/2) 'hCtj+112) - 'h('j-112)

h, + h,_, +

hj + hj+, tE(tj-,*tj),

U#,) - Uh&+l) h,

tE(t,-,,t,).

(40)

where tj_,,2 =+(tj_, +tj) forj= l,..., N. The error indicator Aj is approximated by Aj,h, defined by

A:,=h~llDh~hll~~~~,, j=l,...,N. (41)

This replacement of Du by D,u, to obtain the approximation is purely heuristic. It will be shown to be a good and effective error estimator in our numerical experiments, but further mathematical analysis is still required.

We say that 5 E C’[O, l] is a mesh grading function for the mesh if

sCt,)=i forj=O,...,N, (42)

and if the function 5 is strictly increasing. Thus, 5 has an inverse function 4, and

tj=+ 5 , (3

j=O,...,N. (43)

For hj sufficiently small we have

The error estimator 11 defined by (2)-(4) can be defined as a functional of 5 by using (42) and (44) as follows:

where 5 satisfies the condition

I

I [‘(t)dt=&l)-&O)=l.

0

Using the Lagrange multiplier method, the optimal error estimator can be found by minimizing

with the result

3 @(t)12 _-- N3 5’(t)4 + CL = 0.

We then have

bWi2 5’o”=N3i 9 vtE(O, 11,

where ii = p/3. Hence, the optimal local error indicator is given by

(45)

(46)

(47)

(48)

1 IDU(tj-i,2)1* Aj’ = h~llDu&,,I;, = ~4 5,(t,_ l4 = ,ii k as h +O .

, 112

(49)

T. Cao I Comput. Methods Appl. Mech. Engrg. 162 (1998) I-17 9

Since fi is a constant independent of the mesh and N is fixed, the local error indicators are in equilibrium for an optimal partition.

4. The adaptive h- and h-r algorithms

The h-algorithm has been proposed by Eriksson and Johnson [13] for the FEM. Making the appropriate changes for the BEM the h-algorithm is based on the equilibrium principle for the local error indicators, which holds for BEM as shown in (49), i.e. the partition is optimal if

A: = h;lID,u,ll;o”;, = g, j=l,...,N, (50)

where b is a positive constant independent of the elements. Hence, the h-algorithm can be formulated as follows:

(1) (2)

(3)

(4)

Specify an initial boundary element mesh and tolerance cr > 0. Solve (25) approximately by the Galerkin BEM, i.e. solve (34), and calculate ~~~]Dh~&~~r;,, j = 1 ..,N. If we have

j=l,...,N (51)

then accept u,,. Else go to 4. Refine the mesh for those elements that violate (5 1). Then go to 2.

We now show how the mesh grading function can be used to yield the h-r algorithm. It follows from (48) and the boundary conditions ((0) = 0 and [( 1) = 1, that

$; \Du\“~ dr 60) = sd IDUli,2 d7 for 6 (0, 1) .

Approximating Du by D,u,, then substituting it into (52) one obtains

J; ID,u,1"2 dr &Q lD,u,ll/2dT fortE(O,l).

Hence, the asymptotically optimal mesh is characterized by

IDhuhl112 dt = ((t,) I, I I

lDhu,1112 dt = $ I, ID,u,l"2 dt

forj=l,..., N - 1. Then, we have

Since D,u, is piecewise constant, the mesh sizes h, = tj - t,_ , , j = 1, . . . , N - 1, are approximated by

A $; ID,u,,~"~ dt

hj= hi = N(D,u,(tj_,,,)l”2 ifD,u,(tj_1,2)>0,

(52)

(53)

(54)

(56)

and by heuristic

h,=i, =i if D,,uh(tj_,,2)=0.

The nodal points tj, j = 1, . . , N - 1, are then determined by

tj=tj_, +aj,

(57)

(58)

10 T. Cao I Comput. Methods Appl. Mech. Engrg. 162 (1998) I - 17

where e is a scaled positive constant, which is usually calculated by

(59)

The mesh redistribution (58) is used to achieve the optimal error estimator 77min with the optimal local error

indicator hj is given by Aj = u/N as in (49). By (2), we then have

Replacing Du by D,u,, then the function t’(t) in (48) and vmi, are related by

(61)

Hence, the optimal error estimator rlmin is obtained by substituting t’(t) in (61) into (45) with Du replaced by

DtPtP

1 712” = 2 (i,,’ hd”2 dt)l

We combine the h- and r-versions to obtain the following h-r algorithms:

(1) Specify an initial boundary element mesh and tolerance u > 0.

(2) Solve (25) by the Gale&in BEM, i.e. solve (34).

(3) Calculate vii” by (62); if vii” < g2 then stop, otherwise continue.

(4) The number of elements needed to achieve v,,,~” 6 u is calculated by

1 N=,,

u (I,’ ID,u,/1’2 dt)4’3 , (63)

obtained from (62).

(5) Construct a uniform mesh with the number of elements N calculated in 4, and solve (34) with this mesh.

(6) Construct the optimal mesh given by (58) for j = 1, . . . , N - 1.

(7) Go to 2.

The r-algorithm is similar to h-r algorithm but with steps 4-5 excluded. The required accuracy can be

achieved by subdividing them redistributing the mesh.

(62)

5. Numerical results

In practice, the adaptive methods are very important for problems with singularities. Therefore, we perform the numerical experiment on a polygon. The following numerical results indicate that our method is effective for

solving (34) on the L-shaped polygon, even though our error analysis is only for closed smooth curves. We

consider the following problem:

Let g = Im z2’3

(l,O). The E?-1’2

and let 0 be the L-shaped polygon with vertices (0, 0), (0, l), (- 1, l), (- 1, -l), (1, - 1) and

norm is approximated using

Mii- I’+-) -(Vu, uy2 = IlUll” 1

where - denotes equivalence of norms. A full description of the numerical implementation of the Gale&in

scheme for this problem was presented in [14]. The optimal mesh (58) was obtained by applying the r- and h-r algorithms presented in the last section to obtain the optimal error estimator on each edge separately, then summing these quantities to obtain an approximation to the global optimal error estimator rlmin and T,,,~,, should be kept below a given tolerance. The quantity D,u, is also calculated separately on each edge.

In general, there are two components in any adaptive BEM: l The a posteriori error estimate. l The adaptive strategy.

T. Cao 1 Comput. Methods Appl. Mech. Engrg. 162 (1998) I-17 11

The only available a posteriori error estimator for the BEM with a complete mathematical foundation is the

one based on residuals (see [21,28]). In that approach, the local error indicator on jth element was shown to be

the local residual e,(q) in He”*, which is approximated by using the interpolation property between L* and H ’

norms (see [14]), as follows:

e,(Z,)’ G e~(Z,)(e~(Z,)* + et(Z,)‘)“* ,

where

Our h-method depends on the local L*-projection error estimate (13) with s = - l/2, and we use the

abbreviations l hL for h-method with the local difference error indicator (41) - l hR for h-method with the local residual error indicator,

l r- for r-method with the local difference error indicator (41),

l h-r for combined h- and r-methods with the local difference error indicator (41),

where the strategy in Section 4 is used for both & and B. We use this equilibrium strategy for M in order to

have a fair comparison between CPU times for hr, and !zJ. In practice, a better mesh strategy has been devised

by Ervin et al. 1141 (see Tables 4 and 5 and Fig. 1).

The goals of this numerical experiment are to verify that our method is efficient, reliable and inexpensive.

To measure efficiency, we introduce the global convergence rate constants for both the error in the solution of

the integral equation (25) and the error in the potential at the point P = (0, -0.5):

~~~~ll~~~ll~lll~~,II~~ log(~Nk(P)IVe,(PN ffNk =

log&, 1%) and fiNk =

log(N, IN,)

where Nk is the number of elements and eNk is the error at kth refinement.

To measure reliability, we introduce the effectivity index y = ~/l]e]], t o investigate the asymptotic behaviour

of our error estimator. This effectivity index is expected to be bounded, or better converge to 1 as h +O. It is

unlikely that the latter can occur as the constant in the local error estimates for BEM depends on the

parameterization as well as Cea’s Lemma.

The residual error estimator depends very much on the governing operator of the problem. This property

means that a good effectivity index can be achieved but also a large amount of numerical integration is required

for the local error indicators (see Table 5). This can be seen by the CPU time in Tables 4 and 5, which is probably too large to allow the method to have any practical use in adaptive BEM.

The error estimator (3) depends only on the solution u and the local estimates of its derivative and so is

independent of the governing operator of our problem. As a result its calculation is very simple and fast (see

Table 1) but we can expect to sacrifice the quality of effectivity index. Happily, the index in Table 1 appears to

Initial h-grid h-grid after 7 refinements

Fig. I Initial and final hl-adaptive grids with (+ = 0.001.

12 T. Cao I Comput. Methods Appl. Mech. Engrg. 162 (1998) 1-17

Table I Adaptive &method, (T = 0.001

N Index CPU time

12 O.l14E-00 5.89 0.761E-02

24 0.710E-01 0.68 3.23 0.337E-02 1.18 0.36

44 0.445E-01 0.72 2.80 O.l31E-02 1.56 1.67

68 0.278E-01 0.81 2.61 0.525E-03 1.54 5.82

80 O.l70E-01 1.00 2.70 0.207E-03 1.90 15.48

90 0.994E-02 1.21 3.18 0.813E-04 2.25 28.72

96 0.479E-02 1.52 3.64 0.3168-04 2.64 45.21

Table 2

Adaptive h-r method. The horizontal lines indicate separate experiments with different tolerances, and the initial mesh is the same for three

cases

Tolerance CPU time Index PM N

12 O.l14E-00 5.89 0.76lE-02

22 0.311E-01 2.14 27.57 0.81 lE-03 3.69 0.005 1.81

27 0.266E-01 1.79 8.91 0.253B02 1.36 0.005 4.76

40 O.l34E-01 1.78 5.15 0.369E-03 2.51 0.001 4.39

42 0.106E-01 1.89 8.11 O.l69E-03 3.03 0.001 12.30

50 0.8lOE-02 1.85 6.31 0.6378-04 3.35 0.0005 6.54

Table 3

Adaptive r-method

N 11% - 4"

ll4” ffN Index lb%% - W)I

Iwp,l CPU time

12 O.l14E-00 5.89 0.761 E-02

24 0.439E-01 1.37 5.71 O.l12E-02 2.76 1.70

48 O.l94E-01 1.28 3.67 0.310E-03 2.31 8.26

96 0.863E-02 1.24 2.80 0.782E-04 2.20 34.20

Table 4

Adaptive @-method, u = 0.00 1

Index N

12 O.l14E-00 5.89

22 0.7llE-01 0.78 2.23

32 0.452E-01 0.94 2.67

44 0.287E-01 1.06 3.56

54 O.l88E-01 1.20 5.02

64 O.l24E-01 1.33 7.30

72 0.905E-02 1.41 9.88

80 0.730E-02 1.45 12.20

Iw, - uW)l lVu(P)I

CPU time

0.76lE-02

0.341&02 1.79

O.l32E-02 2.13

0.48lE-03 2.56

0.1638-03 2.46

O.l23E-03 2.59

0.734E-04 2.60 0.539E-04 1.96

5.50

23.79

62.17

135.57

244.63

396.96

591.25

T. Cao I Comput. Methods Appl. Mech. Engrg. 162 (1998) 1-17 13

be stable and bounded. It always keeps close to 3 as N increases, indicating that our error estimator is reliable

and effective. However, a mathematical foundation for this error estimate is lacking at the moment.

The r-method is easy to implement but can never reduce the error below a fixed limit. This limitation can be

overcome by the combined h-r method introduced in Section 4. However, these two methods can converge

erratically due to crude change of mesh size, leading to poor values of the effectivity indices (see Tables 2 and

3). The optimal meshes observed with hL, hR and h-r adaptive procedures are presented in Figs. 1-3. Near the

re-entrant corner we observe strong domination of mesh refinement for these three methods, due to the strength of the singularity at this comer. This example illustrates the importance of adaptive methods for problems

exhibiting strong singularities. An advantage of the h-r method over the other two methods is that it does not

over-refine the mesh.

A comparison of accuracy of the three adaptive strategies hL, hR, h-r and of the results obtained by uniform

h- and r-refinement are presented in Figs. 4 and 5. There the superiority of the h-r approach is clear for relative

errors in the energy norm. It is not quite convincing for the errors in the potential because our error estimator is

based on the energy norm.

In Fig. 6 we observe that the r-optimal mesh is actually the graded mesh defined by

P

t= I for j = 0, . . . , N ,

with p = 1.6 exponent. This exponent and the rate of convergence in Table 3 are in agreement with the theoretical result in Satz 3.7 of [20].

Table 5

Adaptive h-method, local residual error estimator, Ervin et al.‘s mesh strategy

N II% - UIIV

II4 Index

IV& W)l -

IWP)I CPU time

8 O.l88E-00 2.01 0.329E-01

IO O.l15E-00 2.21 1.94 0.716E-02 6.85 1.62

12 0.761E-01 2.23 2.04 0.502E-03 10.32 4.20

14 0.525E-01 2.28 2.24 O.l72E-02 5.27 7.84

18 0.350E-01 1.83 2.69 0.278E-02 3.03 12.69

22 0.295E-03 1.61 2.99 0.302E-02 2.36 20.40

26 0.281E-01 1.62 3.08 0.308E-02 2.01 31.47

32 O.l20E-01 1.98 2.99 0.435E-03 3.12 46.40

, E Initial h-grid h-grid after 8 refinements

Fig. 2. Initial and final hR-adaptive grids with (T = 0.001.

[Ii Initial h-grid h-r grid after 2 refinements

Fig. 3. Initial and final h-r adaptive grids with (r = 0.001.

14 T. Cao I Comput. Methods Appl. Mech. Engrg. 162 (1998) l-17

I

0.1

“h-uniform” + “h-r adaptive” +

“r-adaptive” 8 “hR-adaptive” CX.

“h-adaptive” &

0.001

ElKor

0.0001

100 I” Number of Unknowns

100

Fig. 4. Relative errors in energy norm (log-scale graph)

Fig. 5. Relative pointwise errors at (0, -0.5) (log-scale graph)

lOg(tj) -2

-5 -3 -2.5 -2 -1.5 -1 -0.5 0

log(jlN)

Fig. 6. loglt,l vs. 1ogljlMI on 4th edge.

T. Cao / Comput. Methods Appl. Mech. Engrg. 162 (1998) l-17 15

6. Adaptive H- and H-R boundary element method for elasticity

We only consider planar elasticity problems here, hence i, j always denote two possible values 1, 2 or

specified otherwise. The bold-face characters are used to denote the vectors, for instance when we write ‘u’ we

mean a vector whose components are u,, u2. The following notations will be used throughout this section.

Cauchy stress tensor

Cartesian coordinate (x, , x2) Displacement vector (u,, u2) Strain tensor

Kronecker delta symbol

Young modulus of elasticity; Poisson’s ratio

Shear modulus; 2G = E/( 1 - V)

Domain occupied by load body

Boundary surface of ti

Outward normal unit vector to r (n,, n2)

traction boundary conditions vector

Derivatives with respect to xi

Boundary point (5,) 6,)

v’(x, - 5, )2 + (x* - 62)’ (xj - ti)ir H”(T) x H”(T), a E R

H”(T) x I?“(T), a E R

(u,,_v,> + (5 4

Il4l&r, II4liPm + II~211&-,

ItI&-, II4&(r, + II4w-,

v, s; x s; We consider the following elasticity problem

with the following Hooke’s law

(65)

where

1 ‘in = 7 C”,,j + ‘j,i> 9 (66)

with the boundary condition

uI=gi onr, i-1,2. (67)

It had been shown in [27,9] that (64)-(67) can be converted to the following system of boundary integral equations of the first kind:

Find t EH-“‘(r) such that

w4 := I r w, 5x 5) b, =f(x) , x E r ,

where t is the traction data of (64)-(67), U is a 2 X 2 matrix whose components are

(68)

uij<x* 5) =

-(3 - 4~) ln(r)S,:, + r,ir,j

8lTG(l-v) ’ (69)

16 T. Cao I Comput. Methods Appl. Mech. Engrg. 162 (1998) I-l 7

and f is defined by

where T is a 2 X 2 matrix whose components are

Tij(x, 5) =

ar/an{(l - 2V)6;j + 2r,,r,j) - (1 - 2v)(nir,j - njr,i>

4n(l - V)I

The displacement vector now can be calculated by Betti-Somigliana formula as follows:

(70)

(71)

(72)

Now, the Galerkin equation is:

Find t, E V,, such that

(m,, UJ,) = ti u,,) t v,, E v,, ’

The approximated displacement vector uh is obtained by

(73)

u,(x) := r (u(X, 4?th(8) - T(x, &(8) dSg > I X E fi . (74)

It has been shown in [27] the approximated property (12) also hold for this elasticity problem, we then using Lemmas 2.1 and 2.3 to obtain the following local error estimtes

Now, the arguments in Sections 3 and 4 can be applied straightforward to derive the H- and H-R adaptive BEM

algorithms for elasticity problems.

Acknowledgments

This work is a part of the author’s Ph.D. Thesis at School of Mathematics, The University of NSW. The

author is supported by APRA and an ARC grant held by Professor I. Sloan and Dr W. McLean. I would like to

thank Professor I. Sloan, Dr W. McLean and Professor E. Stephan for many valuable advises and discussions.

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