A reduced SAND method for optimal design of non-linear structures

Abstract

SUMMARYAn SQP-based reduced Hessian method for simultaneous analysis and design (SAND) of non-linearly behaving structures is presented and compared with conventional nested analysis and design (NAND) methods. It is shown that it is possible to decompose the SAND formulation to take advantage of the particular structure of the problem at hand. The resulting reduced SAND method is of the same size as the conventional NAND method but it is computationally more efficient. The presentation here builds on previous research on SAND methods generalizing the solution approach to cases with both equality and inequality constraints. The new version of the reduced SAND method is tested in the context of weight minimization of 3-D truss structures with geometrically non-linear behaviour.1997 by John Wiley & Sons, Ltd.

Space--Time Least-Squares Petrov--Galerkin Projection for Nonlinear Model Reduction

“…It also renders ROMs ineffective in applications that demand a low temporal dimension for computational tractability. For example, a high temporal dimension can render intrusive uncertainty quantification methods (e.g., stochastic Galerkin [3,25]) and simultaneous analysis and design (SAND) in PDE-constrained optimization [26,36,15,16,37,39,4] computationally intractable, as the dimension of the system of equations arising in such applications scales with the temporal dimension of the problem. Further, rigorous error bounds for these ROMs typically grow exponentially in time [38,28,34,11], which renders certification challenging.…”

A parallel two‐level domain decomposition based one‐shot method for shape optimization problems

“…In addition, the three components have to be solved one after another; such a sequential approach is not desirable on machines with a large number of processors. An alternative approach is the simultaneous analysis and design (SAND), or the so-called one-shot method, which solves the three equations in (4) simultaneously by a nonlinear solver, for example, Newton-Krylov methods [6,9,17,19,[28][29][30]. The main challenges of SAND are that the corresponding Jacobian system in the Newton step is large and very ill-conditioned, but, in general, the method 947 based on Newton-Krylov with a good preconditioner is more robust than the method based on nonlinear Gauss-Seidel [31][32][33].…”

“…To answer the challenges facing in the one-shot methods, one needs to design a preconditioner that can substantially reduce the condition number of the large fully coupled system and, at the same time, provides the scalability for parallel computing. In [19,29,34], the authors introduced a sequential quadratic programming method with some Schur-complement preconditioners for shape optimization problems. In this paper, we introduce a Newton-Krylov type method with multilevel overlapping domain decomposition preconditioners.…”

A study of alternative formulations for optimization of structural and mechanical systems subjected to statics and dynamic loads

“…However, the matrix sparsity in the constraint Jacobian can be exploited and utilized for numerical efficiency Ghattas 1992a, Ringertz 1995). Orozco and Ghattas (1997) also developed a reduced SQP method to optimize geometrically nonlinear truss structures.…”

“…It is wellknown that a crucial step for success of the SAND formulations is the solution of very large scale optimization problems. Therefore considerable focus has been put on the development of new algorithms to solve large-scale optimization problems (Ben-Tal and Bendsøe 1993;Ringertz 1995;Ben-Tal and Roth 1996;Ben-Tal and Nemirovski 1997;Orozco and Ghattas 1997;Jarre et al 1998;Maar and Schulz 2000;Ben-Tal et al 2000;Herskovits et al 2001;Hoppe et al 2002;and others).…”