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Domain Decomposition Methods for Parallel Solution of Shape Sensitivity Analysis Problems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng. 44, 281—303 (1999) DOMAIN DECOMPOSITION METHODS FOR PARALLEL SOLUTION OF SHAPE SENSITIVITY ANALYSIS PROBLEMS MANOLIS PAPADRAKAKIS* AND YIANNIS TSOMPANAKIS Institute of Structural Analysis and Seismic Research, National Technical University of Athens, Athens 15773, Greece ABSTRACT This paper presents the implementation of advanced domain decomposition techniques for parallel solution of large-scale shape sensitivity analysis problems. The methods presented in this study are based on the FETI method proposed by Farhat and Roux which is a dual domain decomposition implementation. Two variants of the basic FETI method have been implemented in this study: (i) FETI-1 where the rigid-body modes of the floating subdomains are computed explicitly. (ii) FETI-2 where the local problem at each subdomain is solved by the PCG method and the rigid-body modes are computed explicitly. A two-level iterative method is proposed particularly tailored to solve re-analysis type of problems, where the dual domain decomposition method is incorporated in the preconditioning step of a subdomain global PCG implementation. The superiority of this two-level iterative solver is demonstrated with a number of numerical tests in serial as well as in parallel computing environments. Copyright ( 1999 John Wiley & Sons, Ltd. KEY WORDS: shape sensitivity analysis; domain decomposition methods; preconditioned conjugate gradient; multiple right-hand sides; reanalysis problems INTRODUCTION Shape optimization aims to improve the shape of a structure, defined with a number of parameters called design variables, by minimizing an objective function subject to certain constraint functions. The shape optimization algorithm proceeds with the following steps: (i) a finite element mesh is generated, (ii) displacements, stresses, frequencies, etc. are evaluated depending on the type of optimization problem, (iii) the gradients, or the sensitivities of the functions are computed by perturbing each design variable by a small amount, (iv) the optimiza- tion problem is solved and the new shape of the structure is defined. These steps are repeated until convergence has occurred. The most time-consuming part of a gradient-based optimization process is devoted to the sensitivity analysis phase. For this reason several techniques have been developed for the efficient calculation of the sensitivities in an optimization problem. The semi-analytical and the finite-difference approaches are the two most widely used types of sensitivity analysis techniques. From the algorithmic point of view the semi-analytical technique results in a typical linear solution problem with multiple right-hand sides in which the stiffness * Correspondence to: Manolis Papadrakakis, Department of Civil Engineering, National Technical University, Zografou Campus, GR 157 73 Athens, Greece. E-mail: [email protected] Contract/grant sponsor: European Union; Contract/grant number: HC&M/9203390 CCC 0029—5981/99/020281—23$17.50 Received 24 June 1996 Copyright ( 1999 John Wiley & Sons, Ltd. Revised 20 February 1998 282 M. PAPADRAKAKIS AND Y. TSOMPANAKIS matrix remains the same, while the finite-difference technique results in a typical re-analysis problem in which the stiffness matrix is modified due to the perturbations of the design variables. Shape optimization problems are usually computationally intensive tasks, where 60—90 per cent of the computations are spent for the solution of equilibrium equations required for the finite element analysis and sensitivity analysis. Although it is widely recognized that hybrid solution methods, based on a combination of direct and iterative solvers for solving linear finite element equations, outperform their direct counterparts, in sequential as well as parallel comput- ing environments, little effort has been devoted until now to their implementation in the field of structural optimization. In a recent paper by Papadrakakis et al., the performance of various iterative solution methods, based on PCG and Lanczos algorithms, in sequential computing environment was demonstrated and compared with the conventional direct skyline solver in a number of topology and shape optimization problems. In the present study two variants of the basic FETI method of Farhat and Roux are implemented for solving sensitivity analysis problems using the semi-analytical approach, while an innovative two-level parallel solution method is proposed for solving sensitivity analysis problems using the global finite-difference approach. In the two variants of the basic FETI method the rigid-body modes of the floating subdomains are computed explicitly, while in the second variant a PCG iterative solver with a strong preconditioner is also used for the solution of the local problem in each subdomain. These two variants have a beneficial effect on the robustness of basic FETI method for ill-conditioned problems, while the second variant operates under reduced storage requirements. In the present study a two-level iterative method is proposed specially tailored for solving reanalysis type of problems in general, a special case of which is the problem arising when the global finite- difference sensitivity analysis approach is used. For this type of problems the two variants of the basic FETI method are incorporated in the preconditioning step of a global subdomain imple- mentation of the PCG method. This two-level subdomain implementation is applicable in serial and parallel computing environments resulting in a drastic improvement of computing time compared to the conventional one-level methods. SHAPE SENSITIVITY ANALYSIS Sensitivity analysis is the most important and time-consuming part of gradient-based structural optimization. Several techniques have been developed which can be mainly distinguished by their numerical efficiency and their implementation aspects. The primary objective of sensitivity analysis is to compute the derivatives of the displacement field with respect to perturbations of the primary design variables. The methods for sensitivity analysis can be divided into discrete and variational methods. In the variational approach the sensitivity coefficients can be determined by applying basic variational theorems. In this case, the sensitivities are given as boundary and surface integrals which are solved after the structure has been discretized, whereas in the discrete approach they are evaluated using the finite element equations. The implementation for dis- crete methods is simpler than the one for variational techniques. A further classification of the discrete methods is the following: (i) Global finite-difference method where the derivatives needed for the solution of the optimization problem are computed numerically. (ii) Semi-analytical method where the calculation of the sensitivities is performed via analytical and numerical expressions. (iii) Analytical method where the derivatives of the objective and constraint functions are obtained analytically. Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 281—303 (1999) DOMAIN DECOMPOSITION METHODS FOR PARALLEL SOLUTION 283 The decision on which method to implement depends strongly on the type of problem, the organization of the computer program and the access to the source code. The implementation of analytical and semi-analytical methods is more complex and requires access to the source code, whereas when a finite-difference method is applied the formulation is much simpler. Depending on the type of problem and the approach used, the sensitivity analysis together with the finite element analysis can take 60—90 per cent of the total computational effort required to solve the whole optimization problem. An efficient and reliable sensitivity analysis module that can take full advantage of the innovative computer architectures with parallel processing capabilities could therefore result in a considerable reduction of the overall computational effort of the optimization procedure. In the present study the emphasis is given on the application of efficient domain decomposition methods in parallel computing environments for solving the systems of the algebraic equations encountered in the two most commonly used types of sensitivity analysis, namely the global finite difference and the semi-analytical approaches. ¹he Global Finite-Difference (GFD) approach The GFD approach provides a simple way of computing the sensitivity coefficients. This method requires the solution of the linear system of equations Ku"f, where u is the displacement vector, for the original design variables s, and for each perturbed design variable N" #* * sI sI sI, where sI is the magnitude of the perturbation, usually taken in the range of 10\—10\ of the value of the design variable. The design sensitivities for the displacements * * u/ sI are computed using a forward difference scheme: *u u(s #*s )!u(s ) *u *s + " I I I / I * * (1) sI sI #* where u(sI sI) is evaluated by solving the following reanalysis type of problem: #* #* " #* K(sI sI)u(sI sI) f (sI sI) (2) The Semi-Analytical (SA) approach The SA approach is based on the chain rule differentiation of the finite element equations Ku"f : *u *K * f # " K * * u * (3) sI sI sI which when rearranged results in *u K "f * * I (4) sI where *f *K f *" ! u I * * (5) sI sI Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 281—303 (1999) 284 M. PAPADRAKAKIS AND Y. TSOMPANAKIS * * * * fI* represents a pseudo-load vector. The derivatives of
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