Transactions on Modelling and Simulation vol 18, 1997 WIT Press, ISSN X

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1 Application of the panel clustering method to the three-dimensional elastostatic problem K. Hayami* and S. A. Sauter** ^Department of Mathematical Engineering and Information Physics, School ofengineering, University of Tokyo, 113 Tokyo, Japan t. u-tokyo. ac.jp **Lehrstuhlfur Praktische Mathematik, Mathematisches Seminar Bereichll, Christian-Albrechts-Universitat zu Kiel, D Kiel, Germany Abstract A formulation of the panel clustering method for the three-dimensional elastostatic problem is presented, in order to reduce the computational work and memory required for the boundary element method. In order to make the necessary polynomial expansions for the clustering simple, the fundamental solution for the displacement and traction components can be expressed as a linear combination of partial derivatives of the distance r between the observation point and thefieldpoint. However, this requires a higher order expansion compared to directly expanding the usual expression for the kernels. 1 Introduction Although the Boundary Element Method (BEM) enjoys the advantage of the boundary only discretization, a serious computational difficulty arises due to its dense matrix formulation, particularly for large scale threedimensional problems arising in engineering. This is because the method requires O(A^) memory and O(A^) computational work using the conventional approach, where TV is the number of unknowns. The situation is even worse for the three-dimensional elastostatic problem, where the number of unknowns is three times that of the potential problem, since the unknowns at each node is a vector instead of a scalar. Rokhlin[8] and Hackbusch and Nowak[5] independently proposed the

2 626 Boundary Elements multipole method and the panel clustering method, respectively, in order to overcome this difficulty. The main idea is to approximate the far field using multipole or polynomial expansions around a centre of a cluster of panels or boundary elements, thus reducing the O(A^) dense matrix vector multiplication to a 0(7V(log AT)^) sparse matrix vector multiplication for each iteration of the iterative linear solver, where d is the dimension of the space. Yam ad a and Hayami[ll] proposed a multipole boundary element method for two-dimensional elastostatics. In this paper, we will present a formulation of the panel clustering method for the three-dimensional elastostatic problem [6]. 2 The boundary integral equation for 3-D elastostatics The boundary integral equation for the three-dimensional (linear, isotropic) elastostatic problem is given by [1] (x, yx(y)df(y) = (/== 1,2,3), (1) where we have used Einstein's convention for the summation over the repeated index k 1,2,3, F is the boundary of the domain under consideration, Uk,pk are the displacement and traction components, respectively, c/jk(x) = ^Sik when F is smooth at x, and the body force term has been neglected. u*j.(x,y) is the fundamental solution corresponding to the A:-th component of the displacement at y due to a unit point load in the /-direction at x, which is given by where JJL is the shear modulus, v is the Poisson's ratio, r = \ y x and r,& = <9r/<%, where y = (2/1,2/2,2/3)^. p*j.(x,y) is the traction component at y corresponding to u*^,(x,y), which is given by k - «, + 3r,r k} + (1-2i/)(n/r,* - OTT (1 V ' JT (3) where n\ is the component of the unit outward normal vector at y G F. (2)

3 3 The boundary element discretization Next, the boundary integral equation (1) is discretized by discretizing the boundary F into boundary elements F^, (a = l,...,n). If constant elements are used, and x* is a point representing F^, we obtain t4(x,y)dr(y), (a = l,...,n, ( = 1,2,3) (4) or where iad 3 a 8 8 / t o o\ /_\ %<=^%, (a=l,...,^, / = 1,2,3), (5) 2 ik t4(x,y)dr(y), T/3 and Einstein's convention is now also applied for the summation over the elements f3 = 1,..., n. Equation (5) can be rearranged as where JI IK k J Ik i k ' \ ) ' * * 5 7 "> ") } i \ ) for u], u^ : unknown boundary displacement component, for uf = uf : given boundary displacement component, ~a8 _ a8 -aft _ ^ 9ik = 9ik > 9ik = v for pi pi : unknown boundary traction component, and -a/3 _ n -a/3 _ a/3 9 ik = u, 9ik = 9 ik for pf = pf : given boundary traction component. Equation (6) is a system of linear equations for the unknown boundary displacement and traction components uf and %. Since its matrix is dense and nonsymmetric, it is usually solved using LU-decomposition, which leads to O(A^) computational work and O(A^) memory, where TV = 3n is the number of unknowns. Alternatively, one could use Krylov subspace iterative solvers for nonsymmetric matrices, such as the GMRES(Generalized Minimal Residual)

4 628 Boundary Elements method[9]. The iterates of this method is guaranteed to converge to the true solution within TV iterations if exact arithmetic is used. More important, the system of equations (6) arising from boundary integral equations such as equation (1) is better conditioned compared to its finite difference and finite element counterparts. That is, for the same mesh (element) size A, the condition number for BEM is 0(1) or 0(ft~*), whereas for the FEM, it is 0(/i~^)[3]. Hence, if a suitable preconditioner is used, the method should converge within M «N iterations for the BEM. Note also that it makes sense to do the full orthogonalization in the GMRES without restarting, since the system is better conditioned and smaller for BEM compared to FEM. In this method, as with all iterative methods, the dominant part of the computation is the dense matrix-vector multiplication (inner product): p" t4(x«,y)dr(y), (a =!,...,«, /= 1,2,3) (8) which costs O(^) each. Here, v% and q% are the components of the iteration vector corresponding to the unknown components itf and pf, respectively. The summations over j3' and /?" are taken for the elements F^/ and F^// where the components Uk and p& are unknown, respectively. Hence, the amount of computational work is reduced from O(TV^) to O(MN^}, but the memory required is still O(TV^), and it is this memory bottle-neck that hinders the solution of large scale problems using the boundary element method. 4 The panel clustering method The reason why the matrix-vector product of equation (8) is dense is because the observation point x" on the element F^ is related to all the elements F/? on the boundary through the kernels p*^.(x",y) and u*j.(x",y). In this paper, we will apply the panel clustering method [5, 10] in order to reduce the required memory and computational work for computing the above mentioned matrix-vector product. The method makes use of polynomial expansions to approximate the integral kernels for cluster of elements which are sufficiently far from the observation point, thus reducing the amount of computation and required memory, and at the same time achieving the required accuracy which is consistent with the error introduced by discretizing the original boundary integral equation (1) into the boundary element equations (4).

5 Boundary Elements 629 Let 0=1 (9) represent the integral operator or inner product corresponding to the second or third term of the right hand side of equation (8). (u(y) is constant over each element F/? in this case.) Following [5], in the farfield (i.e. when x y is sufficiently large) we will replace fc(x,y) by a piecewise (elementwise) polynomial fc(x,y) with respect to y, which may be chosen as an expansion around a suitable centre y GR3: 6(x,y)= ^KXx;y )$Xy), (10) i lm where Im is an index set of size jj/ < Ciw?. The integer m is the order of the expansion. The functions $,(y) must be independent of x and the centre y. Furthermore, integrals of $,(y) over single boundary elements F/3 must be easily computable. Usually, 4>i(y) is a (piecewise) polynomial over each element F/?. The error of the expansion of equation (10) depends on the order m and on the distance y y from the centre. The exact requirements on k are as follows. Let 770 G (0,1) be given. Then, there are constants d, d such that for all x,y G R^, all 0 < rj < TJQ < 1, and all m G N there are expansions k of the form (10) satisfying fc(x,y)-fc(x,y) < Ci(C^r I *(x,y) for all y-y < 7? x- This inequality provides an estimate of the relative error of k. Next, "clusters" r, which are unions of several boundary elements F/?, are introduced. For each observation point x, the boundary F can be represented by a certain number of elements and clusters: F = FiUr2U...UFpUriUr2U...Ur, (F/j : elements, TJ : clusters). (11) Since the clusters TJ are unions of many elements, the sum p-\- c can be considerably smaller than the number of elements n. The integral of equation (9) can now be expressed as The first term corresponding to the "near field" FI U Y^ U... U Fp will be evaluated directly. The clusters TJ correspond to the "far field", where the

6 630 Boundary Elements expansion (10) around a centre y = z^ of TJ can be exploited, i.e. we can approximate the integral over TJ by replacing k by k to obtain /fc(x,y)«(y)dr(y) = / K,(x;y )$,(y)u(y)dr(y) **' ^1 < /m ' $,(y)u(y)dr(y). (12) Since the quantities y;.(u) = / *i(y)u(y)dr(y), (j = l,...,c) (13)./TJ are independent of x, they will be computed in the first phase for all indices i and clusters T. Then the evaluation of equation (12) can be performed for all element nodes x", 1 < a < n, by which has Jj/m = 0(rn?) terms independent of the size of the cluster TJ, and JJr.(w) can be shared among different x*. By taking a hierarchy of clusters, the number of all possible clusters (consisting of more than one element) for all x", 1 < a < n can be kept under the number of elements n [5]. The error of approximating the integral /r.fc(x,y)u(y)df(y) can be controlled by keeping the size of the clusters sufficiently small compared to the distance from x. A partition of F by equation (11) is called "admissible" with respect to x when the clusters are sufficiently small in this sense. The admissible covering with the smallest number of members is used. Apparently, the initial phase computations of the integrals for J^ (u] in equation (13) have to be done for every iteration of the GMRES method, since the iteration vector v% and % in the matrix-vector product of equation (8) is updated for every iteration. However, this can be avoided. In our case, constant elements are used. Hence, /' («) = / $,(y)«(y)dr(y) J Tj = «/,/ $,(y)dr(y) where ^ " 1 *'

7 Boundary Elements 631 This means that the integrals defining Jf, (1 < j3 < n) need be computed only once, and «/ (u) can be computed for different iteration vectors up for each GMRES iteration by just performing the linear combination of equation (14). The same technique for general high order elements is discussed in [5]. In order to approximate the kernels to an accuracy which is consistent with the error introduced by discretizing the original boundary integral equation by the boundary element method, the order m for the approximating polynomials in (10) should depend on the mesh (element) size A, and hence on n. Since it is sufficient to take m ~ O(logn), the computational work and memory required for one matrix-vector multiplication approximating (9) for all observation points x",(a = l,...,n), is O(nlog^n) and O(nlog^n), respectively [5]. The evaluation of the right hand side of the system of linear equations (7) can also be done using the panel clustering technique to save memory and computational work. This time, the matrix-vector multiplication need be performed only once for the given vectors wf and jof coming from the boundary data. It is important to note that, when treating domains with complex geometry, the distance along the boundary does not necessarily reflect the distance in 3-D Euclidean space. Therefore, it makes more sense to partition the whole domain into a hierarchy of sub-domains and to cluster them, instead of clustering the boundary elements along the boundary. This will also prove useful when evaluating the displacement or stress at an internal point using the panel (domain) clustering method. 5 A modified expression for the elastostatic fundamental solution In order to obtain the polynomial expansion of (10) to the ra-th order for the elastostatic kernel %^ of (2) and p^ of (3), one could expand ^ to the m- th order and r to the (m + l)-th order and combine them by multiplication and linear combination. Alternatively, as in [7], if we recall the derivation of the fundamental solution for the elastostatic problem using Galerkin's vector[l] or note that where d, again, is the dimension of space (d 3 in our case), (2) gives

8 632 Boundary Elements Here, we denote rjk = Q Q etc. Since the strain corresponding to u*^ is Hooke's law: 4% = 2^^'^ + %W' where A = ^ ^ is the volumetric component, gives 1 Then, the traction p*^ CT^-HJ corresponding to itj^, where nj is the unit outward normal vector at y P, is given by P* = ^ (l^7 <"* + r,^n, + r _,«A ' J^W,') (^ instead of equation (3). Thus, it suffices to expand r(x,y) = x y around the centre of cluster y, and not ^, since the polynomial expansions for u*k and p*^ can be obtained from the linear combination of the partial derivatives r^ and r^&?, respectively. However, in this formulation, it should be noted [4] that, in order to obtain the ra-th order expansion for u^ and p^, one needs to expand r to the (m + 2)-th and (m + 3)-th order, respectively, instead of the (m + l)-th order when expanding the standard expression in (2) and (3). Hence, a careful comparison must be done between the two strategies for the required computational work and memory to obtain the same consistent accuracy. 6 Evaluation of displacement and stress at an internal point The displacement u/(x) and stress <T^(X) at a point in the domain (internal point) can also be computed using the panel clustering technique using and %'(*) = E^ /?%(x,y)dr(y) - IX (3=1 ^p ff=l where (3=1 07T(1 I/)

9 Boundary Elements 633 and V where we have used -j~ = J^- r,,-. Now the panels (or sub-domains) are clustered according to the distance from the internal point x. 7 Polynomial expansion for the panel clustering method Finally, the (ra-th order) polynomial expansion f(y) of r(y) = y x can be obtained by first taking Taylor expansions of r around the centre of the cluster y = (y, 3/2, 2/3)"^, and then expressing them in terms of polynomials of yi instead of hi = %/, y, i.e., f(y) = f(y + h) m-1 E ^,,,,,3(x;y )2/r^^ + ^m(y,h) (17) =0 where h = (h^h^h^.s = 1/1 + i/i + IA, and «(y,h) = The computation of the coefficients «/i^% j ^3 in (17) can be done recurrently as follows. Let r,, = ^-(y ) etc. and r,- = yf %,-, then, r,ija: := --(r,i r,,;w := --(r,^ etc. Once the polynomial expansions (17) of r are obtained, the expansions for r,/ and then, r,/* -> r^m -> u^ and then, r^j -> r^^nf -> P*k can be easily obtained by differentiating by y^ etc. according to (15) and (16). The polynomial expansion for ^ can be obtained similarly to (17) if the standard formulation of (2) and (3) is preferred.

10 634 Boundary Elements Acknowledgement The first author would like to thank Dr. T. Fukui for enlightening remarks. References [1] Brebbia, C.A. and Dominguez, J., Boundary Elements An Introductory Course, Second Edition, Computational Mechanics Publications and McGraw-Hill Book Company, [2] Greengard, L., The Rapid Evaluation of Potential Fields in Particle Systems, MIT Press, Cambridge, [3] Hackbusch, W. Elliptic Differential Equations, Springer- Verlag, 1992, p.207. [4] Hackbusch, W. (private communication), [5] Hackbusch, W. and Nowak, Z.P. On the fast matrix multiplication in the boundary element method by panel clustering, Numerische Mathematik, Vol. 54, pp , [6] Hay ami, K. and Sauter, S. A formulation of the panel clustering method for the three-dimensional elastostatic problem, Proceedings of the JASon BEM, Tokyo, pp , [7] Pierce, A. P. and Napier, J.A.L. A spectral multipole method for efficient solution of large-scale boundary element models in elastostatics, International Journal for Numerical Methods in Engineering, Vol. 38, pp , [8] Rokhlin, V., Rapid solution of integral equations of classical potential theory, Journal of Computational Physics, Vol. 60, pp , [9] Saad, Y. and Schultz, H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SI AM Journal on Scientific and Statistical Computing, Vol. 7, No. 3, pp , [10] Sauter, S., The panel clustering method in 3-d BEM, Wave Propagation in Complex Media, G. Papanicolau ed., IMA- Volumes in Mathematics and its Applications, Vol. 96, pp , Springer- Verlag, [11] Yam ad a, Y. and Hay ami, K., A multipole boundary element method for two dimensional elastostatics, in W. Hackbusch and G. Wittum eds., Boundary Elements: Implementation and Analysis of Advanced Algorithms, Proceedings of the 12th GAMM-Seminar Kiel, January 19 to 21, 1996, Notes on Numerical Fluid Mechanics, Vol. 54, Vie weg- Verlag, Braunschweig, Wiesbaden, Germany, pp , 1996

Transactions on Modelling and Simulation vol 19, 1998 WIT Press, ISSN X

Transactions on Modelling and Simulation vol 19, 1998 WIT Press, ISSN X Cost estimation of the panel clustering method applied to 3-D elastostatics Ken Hayami* & Stefan A. Sauter^ * Department of Mathematical Engineering and Information Physics, Graduate School of Engineering,