INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 40, 1383—1397 (1997) AN ADAPTIVE SUCCESSIVE OVER RELAXATION DOMAIN DECOMPOSITION BY THE BOUNDARY SPECTRAL STRIP METHOD JACOB AVRASHI AND OFER MICHAEL Material Mechanics ¸aboratories, Faculty of Mechanical Engineering, ¹echnion—Israel Institute of ¹echnology, Haifa 32000, Israel GIORA ROSENHOUSE Faculty of Civil Enginering, ¹echnion—Israel Institute of ¹echnology, Haifa 32000, Israel SUMMARY In the present paper a new adaptive successive over relaxation domain decomposition technique is developed for the boundary spectral strip method. The proposed scheme is based on dividing the overall domain of the problem into several subdomains. First each of the subdomains in the BIEM matrices is analysed independently. These matrices together with an arbitrary initial guess of displacements on the interface of each two neighbouring subdomains, enable an iterative and a very efficient solution of the whole problem. An adaptive procedure, based on comparing two norms along the interface of subregions, is carried out to impose successive over relaxation convergence. Numerical results comparing the present scheme with single domain solutions emphasize the capability of the proposed technique regarding accuracy and computational efforts. ( 1997 by John Wiley & Sons, Ltd. KEY WORDS: domain decomposition; BSSM; successive over relaxation 1. INTRODUCTION Treating numerical models in linear or non-linear mechanics of large domains usually reduce to the numerical solution of a large, often poorly conditioned, linear system of algebraic equations. Domain decompositions techniques1 are very attractive nowadays for treating such problems,2 because they can mix direct solvers and iterative solvers and are ideally suited for parallel computers.3,4 These techniques are commonly used in domain discretization methods like the finite element or the finite difference method. Alternative approach to deal with large domain problems is by using the Boundary Element Method (BEM), based on the Boundary Integral Equation Method (BIEM). Applying the BEM means that only the surface of the domain needs to be discretized. However this advantage of reduction in data preparation and number of degrees of freedom is sometimes decreased by the fact that the set of equations generated by the method is fully populated and non-symmetric and often needs special solvers,5—7 or special treatments for matrices arrangement.8 Combining the BIEM approach with the domain decomposition methodology,9 trying to gain the advantages of both methods is the main purpose of the present work. The proposed approach for domain decomposition uses the Boundary Spectral Strip Method (BSSM)10—12 as an approximated solution for the suitable BIEM. Due to the fact that the BSSM saves grid generation along CCC 0029—5981/97/081383—15$17.50 ( 1997 by John Wiley & Sons, Ltd. Received 8 September 1995 Revised 12 February 1996 1384 J. AVRASHI, O. MICHAEL AND G. ROSENHOUSE the boundary, it is very well suited for domain decomposition techniques, since each domain can be treated separately using any desirable independent approximation order. Using a domain type analysis, usually the parent domain is partitioned into many overlapping subdomains. This is not so when the boundary element approach is applied. Based on approximating both primary and secondary variables, the BIEM approach needs only treating with continuity conditions along the interfaces of the subdomains and no overlapping is needed. The present domain decomposition scheme, developed for two-dimensional elastostatics problems, is based on dividing the overall domain of the problem into several subdomains. First, all the matrices of the BSSM formulation are computed independently for each of these subdomains. These matrices together with an arbitrary initial guess of displacements along the interface of each two neighbouring subdomains, enable an iterative and a very efficient solution of the whole problem. An adaptive procedure, based on comparing the non-dimensional energetic and vectorial norms along the interface, is carried out to impose Successive Over Relaxation (SOR) convergence13 of the overall solution. The outline of the paper is as follows: first, a brief review of the BSSM approach and a full description of the iterative domain decomposition procedure is given in Section 2. The convergence criteria are discussed in Section 3 for basic test cases, this is followed by the derivation of an adaptive acceleration procedure in Section 4. Next, numerical results for test cases of 2-D elastostatics problems are presented in Section 5. Finally, we conclude with several aspects and future possibilities of the suggested scheme. 2. BSSM FORMULATION AND ITERATIVE PROCEDURE As stated in the introduction, the Boundary Integral Equations (BIE) can be solved using the Boundary Spectral Strip Method (BSSM).10~12 A brief review on the merits of the BSSM method is introduced in the sequence. The BSSM suggests two approximation ways for BIE evaluation, considering the boundary domain geometry. The first approximation method10 uses a trigonometric approximation which is suitable for periodical geometries. According to this approach each known (boundary conditions) and unknown physical variable along the interface is approximated as follows: N) d+d # + [d cos(mh)#d sin(mh)]; 0)h)2n 0 m1 m2 m/1 (1) where d , d and d are known parameters of the boundary conditions and the unknown 0 m1 m2 degrees of freedom of the unknown variables, and N is the overall number of trigonometric ) harmonies in the BSSM formulation. In the second method a high-order polynomial approximation, which is more adequate for non-periodical geometries is used:11,12 M d+ + d gm; 0)g)1 m m/0 (2) where M is the highest polynomial order. The well-known form HU"GP of the BEM is valid in the BSSM formulation too, but the vectors U and P contain the spectral coefficients of the approximating series of the displacements INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) ( 1997 by John Wiley & Sons, Ltd. AN ADAPTIVE SUCCESSIVE OVER RELAXATION DOMAIN DECOMPOSITION 1385 Figure 1. Description of the domain division—domains, boundaries and imaginary interfaces and tractions along the boundary domain, rather than the prescribed values at the nodes used in the BEM. Consider a given complex domain ) with a non-simple tie boundary !. Let !I denote the k i imaginary boundaries (i"1, . . . , number of imaginary boundaries) which divide the parent domain ) into multi-region (two or more) subdomains ) as shown in Figure 1. By the domain k decomposition approach one splits the problem defined in ) and its boundary !, into two or more subdomains, which are easier for solution (require less memory and less CPU time as compared with the solution for the full domain, and usually have better condition number). We solve each one of those problems independently and afterwards decompose the solutions. Each kth subdomain is denoted by ) and the corresponding boundary will be marked as !. Note that k k + )") and !W !" !I . k k k`1 k i For each subdomain ) the following well-known relation exists: k H U" G P in ) (3) k k k k k This formulation yields the following set of linear equations: A X" F in ) k k k k (4) F" B Q in ) k k k (5) where k ( 1997 by John Wiley & Sons, Ltd. INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) 1386 J. AVRASHI, O. MICHAEL AND G. ROSENHOUSE where A and B are square matrices consisting of influence coefficients, X is the vector of k k k unknowns, Q is the vector of the known boundary conditions, and F is the right-hand side k k vector defined in equation (5). If a multi-region model is considered then on the imaginary boundary interface !I between k i two neighbouring subdomains ) and ) the following continuity conditions of displacements k k`1 u and tractions p must exist G uI" uI k i k`1 i (6) on !I k i pI"! pI k i k`1 i For the BSSM, these continuity conditions (equation (6)) can be easily employed using the simple fact that on each side of an interface the interface local non-dimensional co-ordinate g is measured in an opposite direction, gIP (1!gI ). Thus we have i k~1 i k M n! uI "(!1)m + uI k mi m!(n!m)! k~1 ni n/m (7) M n! pI "(!1)m`1 + pI k mi m!(n!m)! k~1 ni n/m Assuming that the displacements on the interface boundary !I are known for the kth k i domain while for the (k#1)th domain the traction is known, then, for each subdomain, equations (3)—(5) can be converted according to the boundary conditions into a set of linear equations of the form F F G pI " H uI k k k k F F for ) k F F H uI " G pI for ) k`1 k`1 k`1 k`1 k`1 F F (8) Using equations (7) and (8) for the continuity conditions and BSSM formulation, iterative procedure for evaluation of the disagreements in displacements and tractions along the each imaginary boundary !I can be derived, see Algorithm 1. k i Algorithm 1 (See page 1387 top) Note that A~1 · B and A~1 · B are evaluated just once, at the beginning of the procedure, k k k`1 k`1 and they stay without any change during the iteration process. Many parameters may affect the convergence characteristics and accuracy of the above procedure. Those parameters include the initial guess, the degree of approximation, the manner of domain division, acceleration parameters applied on the direct method (see next section) and the problem type. All those parameters are discussed in detail in the next sections. INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) ( 1997 by John Wiley & Sons, Ltd. AN ADAPTIVE SUCCESSIVE OVER RELAXATION DOMAIN DECOMPOSITION 1387 Algorithm 1 3. CONVERGENCE CRITERIA AND BASIC TEST CASES The iterative procedure described in the previous section gives the outline of the present technique. Nevertheless, using this direct procedure does not necessarily mean an unconditionally convergent solution. In order to assure convergence of the solution, an arbitrary acceleration constant u (0(u)1) is introduced to modify the iterative procedure. According to the modified procedure the value of the uI at the j#1 iteration is obtained by a weighted average of the k i ( 1997 by John Wiley & Sons, Ltd. INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) 1388 J. AVRASHI, O. MICHAEL AND G. ROSENHOUSE previous value of uI (at the j iteration) and the value obtained by the direct procedure (Algorithm k i 1) i.e. (9) uI ) #(1!u)( uI ) ( uI ) "u( k ij k`1 i j k i j`1 u"1 yields the original direct procedure (flow chart of algorithm 1) with an accelerated but not necessarily stable procedure. Low values of u may lead into an unconditionally stable (but slower) procedure. In order to look for convergence criteria we first introduce two norms defined along each ith interface. The first norm is a non-dimensional relative vectorial norm defined as E( uI ) !( uI ) E k i j`1 % ElE¢ k i j (10) E( uI ) E k i j`1 % where E E is an Euclidean norm. The second non-dimensional norm is identified with energy and % defined as P P 2 [( uI ) ( pI ) #( uI ) ( pI ) ] d! k ij k ij k i j`1 k i j`1 i (11) EeE¢ !i [( uI ) ( pI ) !( uI ) ( pI ) ] d! i k i j`1 k i j`1 k ij k ij !i Using the BSSM approach for non-periodical geometries (equation (2)), all the integrals of equation (11) are evaluated analytically by P !i ( uI ) ( pI ) jd! "M uI NT [T]M pI N k i j k ij i k mi j k mi j (12) where [T]" T3 0 0 T3 ; ¹I (m, n)"1/(m#n!1) (13) A convergence is achieved when the two norms are smaller than or equal to desirable convergence criteria e . i.e. # max(ElE, EeE)(e (14) # A primary basic test case of a uniformly loading (inplane loading—normal or shear) square plate (Figure 2) was chosen to test the basic characteristics of the present scheme. This test case has an analytical solution for the particular case when the Poisson ratio is taken as l"0, so the error estimation can be conducted accurately. Moreover, taking the zero Poisson ratio we avoid singularities at the corners of the clamped edge. Two kinds of subdivisions were examined for this case: (1) A subdivision to two equal halves along the line ¸ , (2) a subdivision along ¸ type 1 2 lines dividing the square into two asymmetric halves with an interface of the form: y"(1!2a)x#10a; 0)a)1. Results for uniform tension and for uniform shear loads were identical regarding convergence characteristics of the scheme for the basic test case. Six polynomial approximations were employed for each of the two subdivision types. Each of these six approximations is denoted by M , M , where M and M are the degrees of the I O I O polynomial approximation along the interface and the rest of the boundary lines (outer lines), respectively. Adopting the BSSM approach we have only one strip for each line, so there is no INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) ( 1997 by John Wiley & Sons, Ltd. AN ADAPTIVE SUCCESSIVE OVER RELAXATION DOMAIN DECOMPOSITION 1389 Figure 2. The basic test case: a uniformly inplane loaded (normal or shear) square plate—geometry, subdomains, loads and co-ordinates Figure 3. The basic test case: plots of the number of iterations needed to obtain a desirable accuracy, e , of the interface # vectorial norm, ElE, for low values of the u parameter and for any approximation degree (M , M "1, 2, 3, . . . ) I O need for ‘h refinement’ besides the natural ‘p refinement’ of the method. These six approximations combine low-order and high-order polynomials to test the influence of the degree of approximation on the convergence characteristics of the proposed method. For this primary test the behaviour of the iterative procedure aimed at a very high accuracy of a 10~12 fit (e "10~12) of # both interface norms was studied. Results obtained for subdivision along the ¸ line employing 1 zero initial guess for ( uI "0) show a convergence of the solution to the analytical one along both 1 1 boundaries and interfaces with only one iteration loop for acceleration factor u"1. Results for the second type of subdivision in the basic test case are depicted in Figures 3—6. Results for very low values of the acceleration factor u (0·01)u)0·1), show independence of the convergence characteristics on the approximation degree or the a parameter (identical results ( 1997 by John Wiley & Sons, Ltd. INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) 1390 J. AVRASHI, O. MICHAEL AND G. ROSENHOUSE Figure 4. The geometry effect on domain decomposition of the basic test case—the effect of the a parameter on the needed number of iterations (to obtain a 10~12 fit of the interface vectorial norm), for low order strips (M "M "1) I O Figure 5. Number of iterations needed to obtain a 10~12 fit of the interface energetic norm of the basic test case versus the acceleration factor, for several different BSM approximations (numbers in brackets related to the approximation degree M and M ) I O were obtained for any value of M , M and a). Universal plots of the number of iterations needed I O to obtain a desirable accuracy can be plotted (Figure 3) for all values of the polynomial approximation (M , M "1, 2, 3, . . . ) and geometry. Those plots indicate very slow rate of I O convergence in this range of u (u"0·01, 0·02, . . . , 0·1). The geometrical parameter a has some effect on the number of convergence iterations especially for large values of u as demonstrated in Figure 4 for low-order approximation (M "M "1). Results are presented in Figures 5 and I O 6 for the worst value of the geometric parameter, a, i.e. a"0·4. The dependence of the acceleration factor needed to get a convergent solution on the degree of approximation is demonstrated in Figure 5. In these figures the numbers in brackets related to the approximation degree M and M , respectively. It can be seen that high-order polynomials need lower values I O INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) ( 1997 by John Wiley & Sons, Ltd. AN ADAPTIVE SUCCESSIVE OVER RELAXATION DOMAIN DECOMPOSITION 1391 Figure 6. Number of iterations needed to obtain a 10~12 fit of the interface norms of the basic test case versus the acceleration factor—comparison of vectorial and energetic norms. EeE—energetic norm; ElE—vectorial norm; *N"N !N . (numbers in brackets related to the approximation degree M and M ) %/%3'%5*# 7%#503*!I O of u, and that the order of degree for the interface line (M ) has a major effect on convergence I characteristics, while M plays a minor role. It can also be seen from these graphs that both O energetic and vectorial norms yield basically the same convergence behaviour. In order to get a closer view on the differences between the number of needed iterations of each norm aimed to get a convergent solution, we introduce the *N parameter defined by *N¢N !N (15) %/%3'%5*# 7%#503*!where N and N are the number of iterations, needed for convergence, of the energetic %/%3'%5*# 7%#503*!and vectorial norms, respectively. Figure 6 shows that ‘best’ acceleration factor is obtained for each case when *N"0, which means that best results are obtained whenever the vectorial and the energetic norm have basically the same values. These results are valid for both low-order and high-order approximations as can be seen clearly from Figure 6 (first number indicates M and the second refers to M ). In this I O 'N . figure, negative numbers correspond to the *N only, i.e. N 7%#503*!%/%3'%5*# As mentioned before, all the above results were obtained by employing a zero initial guess in the iterative procedure. Employing different values for the initial guess seems to have only a secondary effect on the iterative procedure. ‘Intelligent’ initial guess may of course decrease the number of iterations, while ‘wild’ guesses may increase this number. Nevertheless, neither ‘intelligent’ nor ‘wild’ guesses effect the convergence characteristics, i.e. whenever convergence is achieved with one initial guess it will be achieved with another guess too. 4. THE ADAPTIVE ACCELERATION PROCEDURE The above results lead to the conclusion that a nearly optimum value of the acceleration factor u may be obtained by comparing values of the two different norms along the interface. Those results, especially those presented in Figure 6, show that low values of u yield positive values of *N and poor results regarding the number of iterations, while high values of u yield negative values of *N but still poor results. Therefore, it is desirable to have the minimum value of *N by ( 1997 by John Wiley & Sons, Ltd. INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) 1392 J. AVRASHI, O. MICHAEL AND G. ROSENHOUSE Figure 7. Comparison of the two adaptive algorithms and the direct (non-adaptive) procedure, applied to the basic test case for two values of the interface approximation M —number of iterations needed to obtain a 10~12 fit of the interface I energetic norm versus the initial acceleration factor, u 0 matching the value of u to the sign of EeE!ElE. Several algorithms based on this concept were tested, two of those algorithms were found to be very effective: u "u 1 0 (16) u "u #0·01 sign(EeE!ElE); j'1 j j~1 u "u 1 0 Optimum 2: (17) u "u [1#0·1 sign(EeE!ElE)]; j'1 j j~1 where u is the arbitrary initial value of u, 0(u )1. 0 0 Those adaptive algorithms were first tested or the basic cases, described in the previous sections. Results for both adaptive algorithms are shown in Figure 7, compared to results obtained from the direct, non-adaptive algorithm and for two values of M , M "1, 6. It can I I easily be seen from this figure that both adaptive algorithms enable an almost unconditionally stable procedure for both low order (M "1) and high order (M "6) of the interface approximaI I tion. It can also be seen that the optimum 2 adaptive algorithm has somewhat better results than the optimum 1 algorithm, at least in the basic tests. This conclusion was found to be even more significant later on when more complicated cases were studied (see next section). Optimum 1: 5. APPLICATIONS—NUMERICAL RESULTS The iterative domain decomposition procedure and adaptive algorithms developed in the previous sections for the basic test cases have, of course, to be checked for more complicated problems, including multi-domain problems. Three types of irregular problems (irregular shapes, without a closed-form analytical solution) were chosen to test the performances of the present approach: (a) Problem I—A bi-material flat wing including two immovable circular inclusions at one material (E "7·8; l "0·3) and a circular hole in the other (E "4·8; l "0·2), and 2 2 1 1 subjected to prescribed constant load (Figure 8(a))—this problem cannot be solved by a single domain model. INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) ( 1997 by John Wiley & Sons, Ltd. AN ADAPTIVE SUCCESSIVE OVER RELAXATION DOMAIN DECOMPOSITION 1393 Figure 8. Three irregular problems—geometry, subdomains, loads and coordinates. (a) Problem I—the bi-material flat wing. (b) Problem II—an irregular porous beam. (c) Problem III—an irregular pipe-like plate, including four equal size circular holes, the muolti-domain problem (b) Problem II—An irregular porous beam, clamped at one edge and subjected to a prescribed quadratic load (Figure 8(b)). (c) Problem III—An irregular pipe-like plate, including four circular holes of equal size, clamped at one edge and subjected to a linear load at another end (Figure 8(c)). This plate was modelled by four sub-domains as a multi-domain test case. ( 1997 by John Wiley & Sons, Ltd. INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) 1394 J. AVRASHI, O. MICHAEL AND G. ROSENHOUSE Figure 9. Comparison of the two adaptive algorithms and the direct (non-adaptive) procedure, applied to Problem I—number of iterations needed to obtain 10~6 and 10~12 fit of the interface energetic norm versus the initial acceleration factor, u 0 Numerical results obtained by employing the domain decomposition approach were verified by comparison to results obtained by using a single domain model (Problems II and III) and by checking the convergence of displacements and stresses whenever a single domain solution is not available (Problem I). Hence, from now on, only verified convergent solutions are treated. We note that a requirement for high-accuracy convergence criteria e "10~6, 10~12 is much more # than the needed engineering convergence, and results obtained by using e "10~2, 10~3 yield # very good results, as already shown in Reference 9. Moreover, a requirement of a certain value of e along each imaginary interface, means better results along the other (real) boundaries. # Results for both adaptive algorithms, employed in Problem I, are depicted in Figure 9, compared to results obtained from the non-adaptive algorithm. Those results are given for M "M "6 and two values of e (e "10~6, 10~12) and basically show the same behaviour as I O # # shown before for the basic test case (Figure 7). Moreover, the results for Problem I are even somewhat more stable than those obtained for the basic test case. It is clear now that both adaptive algorithms enable an almost unconditionally stable procedure for any value of u in the 0 range 0·1)u )0·9. 0 Problem II (together with the multi domain problem—Problem III) was used to compare the numerical results evaluated by the domain decomposition scheme with the respective results obtained from a single domain model. This comparison was conducted regarding the accuracy of displacements and tractions and the consumed computational efforts. Results for the displacements u , u along the lower free boundary, and the tractions p , p along the upper clamped edge x y x y are plotted at Figures 10(a) and 10(b), respectively. Those results were obtained from relatively coarse approximations of M "3, N "3 (equation (1)) for both domain decomposition and O ) single domain models, and M "4 for the domain decomposition model. Both figures show the I percentage difference between the models, and demonstrate the good agreement between the two models. Similar results are obtained along the other boundaries and along the boundaries of the holes, too. Moreover, higher approximations order yield even better agreement of the numerical results. INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) ( 1997 by John Wiley & Sons, Ltd. AN ADAPTIVE SUCCESSIVE OVER RELAXATION DOMAIN DECOMPOSITION 1395 Figure 10. Percentage difference between results obtained from the domain decomposition model and the ‘reference’ single domain solution. (a) Displacements along the lower free boundary of Problem II. (b) Tractions along the upper clamped boundary of Problem II Table I. Domain decomposition versus single domain—computational efforts and consumed CPU time Number of degrees of freedom single domain SD multi domain MD DOF ratio MD/SD Matrix ratio MD/SD CPU time ratio MD/SD Two sub-domains 126 150 186 210 64#74"138 80#90"170 96#110"206 112#126"238 1·15625 1·13333 1·10753 1·13333 0·60292 0·64444 0·61614 0·64444 0·67607 0·74021 0·67376 0·72505 Four sub-domains 120 132 140 164 196 4]42"168 4]42"168 4]50"200 4]50"200 4]58"232 1·40000 1·27273 1·42857 1·19048 1·18367 0·49000 0·40496 0·51020 0·37180 0·35027 0·49947 0·47014 0·48508 0·45345 0·37942 ( 1997 by John Wiley & Sons, Ltd. INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) 1396 J. AVRASHI, O. MICHAEL AND G. ROSENHOUSE Figure 11. Convergence rate of the domain decomposition method employed on Probem III—comparison of the direct, non-adaptive scheme and the two adaptive algorithms Comparison of the consumed computational efforts for multi- and single-domain models is shown in Table I for different approximations order in Problem II and III. Both multi- and single-domain models are modelled by the BSSM having the same polynomial approximation on the outer boundaries. The advantages of the domain decomposition approach over a single domain models can be easily seen from this table, especially for multi-domain problems. These advantages of the present scheme show that they constitute an efficient tool for modeling of large problems. The numerical results of the multi-domain problem (Problem III) show the same behaviour of convergence parameters as already been indicated in the two-domain models (basic test case and Problems I and II). A demonstration of the iterative procedures is shown in Figure 11 for Problem III employing the following parameters: M "M "4, N "2. Results in this figure are I O ) reported for three values of u(u"0·2, 0·3, 0·4) used by the direct, non-adaptive procedure, compared to the two adaptive procedure (with u "0·4). Those results emphasize the remarkable 0 advantage of both adaptive algorithms over the direct method. It is also clear, as in the two-domain models, that the optimum 2 scheme has somewhat better performances than optimum 1. 6. CLOSURE The proposed adaptive SOR domain decomposition method divides the full domain problem into several sub-domains that can be solved separately and independently of each other in a very efficient manner. This method uses the BSSM (boundary spectral strip method) along the boundaries and chosen interfaces yielding very good results which save a great part of the grid generation. The investigation of convergence of the present two adaptive algorithms indicates an almost unconditionally stable procedure for both schemes with some advantages, referring the rate of convergence to the so-called optimum 2 algorithm. The proposed algorithms have several advantages that make the suggested method very attractive, especially for large models divided into multi-domain models. Those advantages INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997) ( 1997 by John Wiley & Sons, Ltd. AN ADAPTIVE SUCCESSIVE OVER RELAXATION DOMAIN DECOMPOSITION 1397 include: (1) Significant reduction in consumed computational efforts, memory and CPU time, (2) an unconditionally stable convergence, (3) the convergence characteristics do not depend on the involved parameters (geometry, loads and boundary conditions), (4) results for multi-domain and two-domain problems are similar, (5) there is no need for an ‘intelligent’ initial guess, although such a guess may reduce the number of needed iterations. The present domain decomposition scheme is based on the BSSM formulation,10~12 that for the time being is formulated for smoothed regular approximations along each boundary (real or artificial). In order to solve problems with singularities, such as sharp notches or cracks, one has to employ special singular functions or edge function techniques.14 Treatment of such problems is the issue of a forthcoming work. In the present study, using only regular functions, one has to be careful in choosing the imaginary boundaries. Those artificial boundaries should not be introduced on regions where displacements and stresses have high gradients. The proposed domain decomposition approach, used in the present work for 2-D elastostatics can be used for other problems such as 3-D elastostatics, Laplace equation, 2-D and 3-D elastodynamics, etc. This approach seems to be useful for multi-phase problems too, where a single domain model cannot be used. REFERENCES 1. H. A. Schwartz, ‘Uber einige Abbildungensaufgaben’, J. Reine Angew. Math., 105—120 (1869). 2. P. Le Tallec, ‘Domain decomposition methods in computational mechanics’, Comp. Mech. Adv., 1(2), 121—220 (1994). 3. D. Keyes, ‘Domain decomposition: A bridge between nature and parallel computers’, Symp. on Adaptive, Multilevel and Hierarchial Computational Strategies, ASME ¼inter Annual Meeting, Anaheim, CA, 8—13 November, 1992. 4. J. Bramble, J. Pasciak and J. Xu, ‘Parallel multilevel preconditioners’, Math. Comp., 55, 1—22 (1990). 5. W. J. Mansur, F. C. Araujo and J. E. B. Malaghini, ‘Solution of BEM system of equations via iterative techniques’, Int. j. numer. methods eng., 33, 1823—1841 (1992). 6. J. M. Crotty, ‘A block equation solver for large unsymmetric matrices arising in the boundary integral equation method’, Int. j. numer. methods eng., 18, 997—1017 (1982). 7. R. H. Rigby and M. H. Aliabadi, ‘Out-of-core solver for large, multi-zone boundary element matrices’, Int. j. numer. methods eng., 38, 1507—1533 (1995). 8. J. H. Kane, B. L. Kashava Kumar and S. Saigal, ‘An arbitrary condensing solution strategy for large scale, multi-zone boundary element analysis’, Comp. Methods Appl. Mech. Eng., 79, 219—244 (1990). 9. J. Avrashi, O. Michael and G. Rosenhouse, ‘Domain decomposition technique using the boundary spectral strip method’, Proc. of the 10th Int. Conf. on Boundary Element ¹echnology, BE¹ECH 95, Liege, Belgium, 13—15 September, 1995, pp. 203—210. 10. O. Michael, J. Avrashi and G. Rosenhouse, ‘The boundary strip method in elastostatics and potential equations’, Int. j. numer. methods eng., 39, 527—544 (1996). 11. O. Michael, J. Avrashi and G. Rosenhouse, ‘Boundary strip method (BSM) applied to non-periodic geometrical entities’, Proc. 17th Boundary Element Int. Conf. (BEM17), Madison, WI, 17—19 July 1995, pp. 75—82. 12. O. Michael, J. Avrashi and G. Rosenhouse, ‘A new boundary spectral strip method for non-periodical geometrical entities based on analytical integrations’, Comp. Methods Appl. Mech. Eng., 135, 327—342 (1996). 13. C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, Addison-¼esley Reading, MA, 45th edn, 1989. 14. P. M. Quinlan, ‘The edge-function method in elastostatics’, in B. K. P. Scaife (ed.), Studies in Numerical Analysis, ¹he Royal Irish Academy and Academic Press, 1974, pp. 265—293. . ( 1997 by John Wiley & Sons, Ltd. INT. J. NUMER. METHODS ENG., VOL. 40: 1383—1397 (1997)

1/--страниц