Fast Solution of Boundary Integral Equations by Using Multigrid Methods and Multipole Evaluation Techniques C

Fast solution of boundary integral equations by using multigrid methods and multipole evaluation techniques C. Caspar

Szechenyi htvdn College, Department of Mathematics P.O.Boz 70), #-2007 Oyo'r, #tm#an/

Abstract

The standard Boundary Integral Equation Method generally results in dense and nonsymmetric algebraic equations. To speed up the computations, special techniques are needed. In this paper a multigrid method is presented applied to boundary integral equations. The main idea of the method is to convert a mixed boundary value problem to a sequence of pure Dirichlet and Neumann subproblems. To evaluate the appearing boundary integral operators, a special panel clustering method based on the fast multipole evaluation technique is applied. A completely different multigrid approach for solving the scattered data interpolation problem arising in the dual reciprocity method is also presented.

1 Introduction

The usual discretisation techniques applied to boundary integral equations lead to an algebraic system with dense and non-self adjoint matrix even if the original problem was self adjoint in some Sobolev space. Therefore tra- ditional direct solvers (e.g. Gaussian elimination) are usually applied. The computational cost of these methods is proportional to the third power of the number of boundary nodes. In fact, the computational cost may exceed that of some advanced techniques applied to the original partial differential equation. For instance, if a standard multigrid method is applied to a 2D elliptic problem, the number of the necessary algebraic operations is pro-

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portional to the second power of the number of boundary gridpoints only, and can be reduced further by applying non-uniform grids. To speed up the solution of the boundary integral equations, several methods have been proposed. A natural attempt is the use of multigrid tools in boundary integral equation context. This technique is well known if the resulting boundary integral equation is of the second kind (Hackbusch [8]). However, if the original problem is supplied with mixed boundary conditions, this is not the case and the applicability of the multigrid method is not straightforward. A remedy is to convert the original (mixed) problem to a sequence of pure Dirichlet and Neumann subproblems, the solutions of which converging rapidly to the solution of the original problem as proposed by Caspar [3]. Here we present this method combined with the fast multipole technique of Rokhlin [13] to evaluate the appearing singular and hypersingular boundary integral operators. Using complex potentials, we derive the multipole expansions of these boundary integral operators. The technique uses a quadtree cell system controlled by the boundary and makes it possible to significantly reduce the computational cost of the appearing matrix-vector multiplications. The memory requirement is reduced as well, since it is not necessary to store the boundary element matrices. In many cases, the boundary integral equations contain also domain integrals to be evaluated. The usual trick to evaluate these integrals is a transformation to boundary integrals. This can be carried out by using a (fast) Fourier transform (Tang [14]) or the multiple reciprocity method (Nowak and Brebbia [11]) or the dual reciprocity method of Nardini and Brebbia (see e.g. Partridge and Brebbia [12]). In this latter technique, using also some internal points, the function to be integrated is approximated by simple radial basis functions, the integral of which can be transformed to the boundary in an easy way by applying Green's formulas. In fact, a scattered data interpolation problem is hidden in the background based on radial basis functions as pointed out by Golberg and Chen [6,7]. From many points of view the best choices for solving general scattered data interpolation problems have been proved Hardy's multiquadric method and the so-called thin plate spline method (see Franke [2] and Kansa [10]). Both methods are closely related with the biharmonic interpolation, where the interpolation function is assumed to satisfy the biharmonic equation except the interpolation points. However, these interpolation techniques generally produce dense, nonsymmetric and ill-conditioned algebraic systems. This causes numerical problems similar to the boundary integral equations. In this paper we present also a multigrid-based solution technique. The essen- tial idea is to solve the associated biharmonic problem, whose discretisation is completely independent of the original boundary integral equations and can be multigridded in a particularly efficient way using quadtree-generated nonuniform grids as proposed by Caspar and Simbierowicz [5].

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2 Multigrid solution of boundary integral equations

As a model problem, consider the simplest 2D Laplace equation in a bounded and piecewise smooth domain ft supplied with mixed boundary conditions:

A(/ = 0 (1)

U\ri = uolri, -Q^lr* = fob (2) where F = <9Q, the boundary of H and PI, ^ form a disjoint decomposition of F. We assume that neither F% nor ^ is empty. UQ, VQ are given functions of the Sobolev space Jf^F) and #-*/2(F), respectively. Then (l)-(2) has a unique solution in the Sobolev space //*(H,A) := {U £ H*(tt) : A[/ £

1/2(0)}. Equation (1) is equivalent to the corresponding boundary integral equation: TTU + Ku - Rv = 0 (3) and also to the normal derivative boundary integral equation:

(4)

Here u = t/|r, v — |^|r, and K, R denote the usual double-layer potential and the single-layer potential, respectively:

'= /r while K* is the adjoint of K and Q is a hypersingular boundary integral operator defined as the normal derivative of the double layer potential. The coefficient of u in Equation (3) becomes the solid angle of the boundary where F is not smooth. The main idea of the multigrid approach mentioned in the introduction is to convert (l)-(2) to a sequence of pure Dirichlet and Neumann subprob- lem. To do this, let P be a (not necessarily orthogonal) projector of the closed subspace of the functions of H*/*(T) vanishing along PI. Then the operators PI := I — P, PI \— P* are also projectors in the spaces //*/^(F) and //~*^(P), respectively, and can be interpreted as certain extensions from FI to F and from F2 to F, respectively. Define the following iteration

At/n+l/2 = 0, £/n+l/2|r =%" + ^l(%0 ~ %n) (5)

or r 1 = 0, "** |r =

where Un := t/n|r, Vn+i/2 := ^%T^~lr- Equation (5) is a pure Dirichlet, while Equation (6) is a pure Neumann problem. The iteration can be interpreted

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as follows: in the first half-step (Equation (5)) the Dirichlet boundary condition along PI is exactly satisfied, while in the second half-step (Equation (6)) it is the Neumann condition along 1^2 that is exactly fulfilled. Intro- ducing the Dirichlet-to-Neumann operator J by Ju := -^, where At/ = 0 in 0 and f/|r •= u, the iteration (5)-(6) can be written in a simpler form:

%n+i := J-X/ - P')JP^ + 6, 6 := J-'[?2i;o + (/ - ^2)^1^0], (7)

Based on the concept of quasi-orthogonality, it can be shown that, under some conditions, the operator «/"*(/ — P*)JP is a contraction in the space //*/^(F), thus the iteration (7) is convergent. For details, see Caspar [4]. In the practice, PI, P% can be defined e.g. by the coarse-grid approximation of the special mixed problems P^u := f/|r, where A/7 = 0, in ft, [/|r. - u|r,, f|r, = 0 and P,v := f |r, where At/ = 0, in 0, [/[PI — 0? §^|r2 = Hr2 The steps of the iteration (7) can be realized by solving pure Dirichlet and pure Neumann subproblems, or, equivalently, the corresponding boundary integral equations (3) and (4), which are of the second kind. This allows the use of standard multigrid tools. In this case, it is clear that the whole algorithm needs O(N*) operations only, where TV is the number of boundary nodes. The computational cost can be reduced further, if the appearing matrix-vector multiplications (after some usual discretisation) are performed in a "fast" way (requiring less than O(N^) operations). Such a general technique is the panel clustering method (Hack- busch and Nowak [9]). In the next section we show a special version of this method based on the multipole expansions.

3 Fast multipole method for evaluating boundary integrals

In the basic version of the fast multipole method (for details, see Rokhlin [13]; van Dommelen and Rundensteiner [1]) the function to be evaluated is expressed in terms of multipole series centered at the points z\, ..., ZN G C:

*(*):= XX$,-(z) (8) .7 = 1 where the functions 3>j are of the form:

™ j) (9) * Z with given coefficients a,j and a^ (j = 1 , . . . , TV, r = 1,2,...). The evaluation is based on the shifting theorem which states that a multipole series centered at Zi can be transformed to another one centered at another point z^ in the following way: OO OO 1

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where the new coefficients are defined by

, , ao(zi - 22)" , ^ /r - 60 := GO, br := +

Moreover, if the original multipole series is convergent outside a circle centered at z\ with radius R\, the shifted one is also convergent outside the circle centered at z^ with radius R^ = R\ + \z\ — Zz\.

In the practice, of course, the infinite series are approximated by trun- cated series (up to a prescribed maximal index p). The shifting theorem makes it possible to simplify the summation in (8) in the following way. For a fixed z, collect the terms for which the corresponding multipole centers lie "sufficiently near" to z and calculate these terms directly. Otherwise, several "far" terms can be unified by shifting them in a common center: their effect can be replaced with a single multipole series. The near-far classification can be conveniently defined by the quadtree algorithm. This is a systematic, recursively defined subdivision of an initial square (see e.g. van Dommelen and Rundensteiner [1]) controlled by the points Z\,...,ZN. The algorithm results in a non-equidistant, non-uniform but Cartesian cell system whose spatial resolution follows the spatial density distribution of the controlling points. The cell system has a tree-like structure: the root element corresponds to the initial square while the edges represent the sub- divisions. In order to apply the above technique to boundary integral operators, they should be discretised and expanded in terms of multipole series. Now we derive the expansions in the case of the simple 2D Laplace equation and the corresponding boundary integral operators (potentials) K,R,Q. As- sume that F is (approximated by) a polygonal with sides Fi,...,Fyv and vertices W\,...,WN. Using piecewise constant approximations of u,v and a collocation method with collocation points Zj := (wj -f itfj_i)/2 (where WQ := WN), in each step of iteration (7) (or other iteration technique as well) one has to evaluate finite linear combinations of the functions Kijjj, Ri/jj, K*il>j,Qi/>j at the collocation points Z\,...,ZN. Here t/y denotes the piecewise constant basis function which equals to 1 on Tj and zero else- where. Thus, it is sufficient to express these functions in terms of multipole series. Identifying R^ with the complex plane C it is easy to see that R^j is the real part of the complex single layer potential:

f log(z-CK (10) JT where e*^' := a,jl\dj\, a,j := Wj — Wj-\. Using the series of log(l — z), we have:

= -e-™> log(z - ,,) rfC + .„ •((- *;)X (11 ~

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The integrals in the right-hand side can be calculated analytically and we obtain the multipole series of the function Rj around the center Zji

R,(z) = r=l \* Zj) where a.® = — o^e "*"•*; a^ is zero for the odd indices and

for the even indices.

The function Ktyj is the real part of the complex double layer potential

dC (13)

Observing that Rj(z] = —ie~*"'Kj(z) and using the above multipole series of RJ, we obtain the multipole series of the function Kj around the center

where the coefficients /?,. vanish for the even indices, and

if r is odd. The functions A'*0j, Q^j can be expressed in a similar way. Since these functions are the normal derivatives of the single and double layer potential, respectively, their multipole series can be obtained immediately from the derivative series of A^j? R^j .

Summarizing the above considerations, the fast mulipole evaluation of the discretised boundary integral operators consists of the following steps. 1. Generate a regular quadtree grid controlled by the boundary. Deter- mine the coefficients of the multipole series of R^ Kj.

2. Shift each multipole series to the smallest cell center containing of the multipole center. Unify the multipole series belonging to the same cell. 3. Travelling up in the graph structure, unify the multipole series belonging to the same parent cell by shifting them to the center of the parent.

Repeat this procedure until the initial cell is reached. 4. For each value of z, evaluate the discretized boundary integral operators. The terms for which z and the multipole centers lie in the same or a neighbouring leaf-cell should be calculated directly: otherwise, they should be approximated by the previously unified multipole series. The overall computational cost of the algorithm is typically 0(/Vlog N). Note that the memory requirement is also reduced since the discretised

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boundary element matrices need not be stored. It is necessary to store the multipole coefficients, which requires 0(N\og N) bytes instead of 0(N*)

4 Multigrid in the dual reciprocity method

When the boundary integral equation contains domain integrals (the right- hand side of the model problem (1) is not zero), the advantages of the BEM can be preserved by transforming the domain integrals to the boundary.

The domain integral arising in the boundary integral equations are of the form: '' f(y)Lo(x - y)dy (15) L where LQ(X) := ^ log ||x||, the fundamental solution of the Laplacian. The function / is assumed to be known at the discrete points #1, ...,%#, where some of these points are located at the boundary, while the remaining points are inside the domain. In the dual reciprocity method, the function / is approximated by the following expression:

where the function $ is a simple radial basis function, typically (but not necessarily) <&(x) = 1 + ||z||. Introducing the function \P by A\B = 0 (then

# has the form *(x) = \\\x\\* + |||z|H, Green's theorem implies that (15) can be expressed by boundary integrals, since

The unknown coefficients QI, ...c%# should satisfy the interpolation equations N

(17)

(k = 1,..., N). However, this system is dense and generally ill-conditioned. Therefore usually a direct solver (e.g. Gaussian elimination) is applied: the computational cost is then O(N^). Now we outline a multigrid approach for the above problem. Remark: The problem is a special case of the scattered data interpolation problem investigated by a lot of authors e.g. Franke [2], Kansa [10]. The "best" methods seem to be the Hardy's method of multiquadrics (with the basis functions v/||x||^ -f c^, where the constants cj play some opti- mization role) and the "thin plate spline" method (with the basis function

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Instead of the radial basis function 1 4- ||z||, we use the thin plate spline functions in an implicit way. We do not solve Equation (17) directly: instead, we look for an interpolation function as a solution of the biharmonic equation AA/ = 0 in %-{%!,. ..,%#} (18) where DO is an arbitrary domain containing the points x\, ..., XTV • Equation (18) should be supplied with boundary conditions which are regular in the biharmonic problem and also with the interpolation conditions

/W = A (6 =!,...,#) (19)

Since the biharmonic equation possesses continuous fundamental solution, the above problem does have a unique solution in some closed subspace of the Sobolev space //^(Ho). To solve this problem, standard multigrid tools can be applied, using a (sufficiently fine)gri d and finite differences indepen- dently of the original boundary integral equations. The computational cost of this procedure is typically O(TV^), but using a quadtree grid generated by the scattered points, this amount can be reduced to O( N log N) as pointed out by Caspar and Simbierowicz [5]. The biharmonic interpolant can be expressed in the form

/(4 = E&Wz-zj)-W4 (20) J=l where L\(x] := ^||z||^(log ||z|| — 1) is the fundamental solution of the biharmonic operator, and w is a function satisfying the biharmonic equation everywhere in HO (including also the points z%, ...%#). From (20) it is clear that AA/(z) = ^Af(z-z,) (21)

j=i where 6 denotes the 2D Dirac distribution. Hence the coefficients /?i, ...,/?/v can be determined without solving any additional system of equations. Note that the numerical evaluation of (21) requires the same schemes for the biharmonic operator as in the solution of (18)-(19). Once the coefficients /?i, ...,,3/v have been determined, the values as well as the derivatives of the biharmonic function w can be computed by direct applications of Green's formulas on the boundary of HQ. On the other hand, the representation (20) allows also a dual reciprocity formulation. Indeed, introducing the fundamental solutions of the iterated Laplacian by

Lo, AZ/2 = Li (22)

(which can be calculated analytically), a substitution into (15) yields:

f f(y)'Lo(x - y)dy =

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r / i^(i/)AAZ/2(j; - z/)ck/ j=i -in and each integral in the right-hand side can be transformed to the boundary without difficulty by using Green's formulas. Thus we have avoided the direct solution of the interpolation equations. In fact, this formula a special combination of the dual and multiple reciprocity formulas, since the first N integrals (containing the function LI) can be calculated by the dual reciprocity, while the last integral (containing the biharmonic function w) can be evaluated by the multiple reciprocity: however, since w is biharmonic, the multiple reciprocity method consists of finite number of steps only. The approach can be obviously generalized to higher powers of the Laplacian used in constructing the interpolation function /.

Remarks:

(a) The procedure described in this section can be generalized to 3D in

a natural way: in this case, the fundamental solutions of the iterated Laplacian have even simpler forms. Moreover, the fundamental solution of the biharmonic equation is const. • \\x\\, which shows the

similarity to both the classical basis function of the dual reciprocity method and the multiquadric interpolation method.

(6) Without going into details we note that, instead of the thin plate splines, the multiquadric interpolation can be used also in 2D, ex-

ploiting the fact that the 2D multiquadric interpolation is equivalent to a 3D biharmonic interpolation.

Acknowledgement: The research was partly supported by the Hungarian National Scientific Research Fund under the contract T17323.

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