Large‐scale topology optimization using preconditioned Krylov subspace methods with recycling

By Shun Wang, Eric de Sturler, Glaucio H. Paulino. 2007

255 citations recorded by the source corpus.

The computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis. We propose fast iterative solvers for large three‐dimensional topology optimization problems to address this problem. Since the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems. In addition, we introduce a MINRES (minimum residual method) version with recycling (and a short‐term recurrence) to make recycling more efficient for symmetric problems. Furthermore, we discuss preconditioning to ensure fast convergence. We show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density. We demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast PC. Copyright © 2006 John Wiley & Sons, Ltd.

Subfields: Civil and Structural Engineering, Computational Theory and Mathematics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/8618dc13b6d6dbd9c44370fa2836b483c136333f

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Citations in the corpus

Polygonal finite elements for topology optimization: A unifying paradigm (2010, background): “[21, 22]) and other solvers, including direct sparse solvers (see, for example, [23]) and iterative solvers tailored for topology optimization problems [24], can also be explored in this work.”
Single-loop system reliability-based topology optimization considering statistical dependence between limit-states (2011, unknown): “For example, researchers make use of powerful computing resources such as parallel computing (Borrvall and Petersson 2001; Evgrafov et al. 2008), approximation procedure (Amir et al. 2009), or fast iterative solvers (Wang et al. 2007; Amir et al. 2010).”
Level‐set topology optimization with many linear buckling constraints using an efficient and robust eigensolver (2016, background): “This leads to large differences in the diagonal entries of the system matrices (such as the stiffness matrix) large condition numbers and ill-conditioning [20].”
Improving multiresolution topology optimization via multiple discretizations (2012, background): “Studies devoted to developing efficient approaches for large-scale topology optimization problems have been focused on the solution of finite element analysis because of its dominance in the computational cost [4, 5].”
Reducing dimensionality in topology optimization using adaptive design variable fields (2010, background): “Most algorithms for reducing computational expense have thus focused on reducing this cost, either by developing advanced solvers [9] or by reducing the number of unknown degrees of freedom via multiple mesh size strategies.”
Improving the efficiency of large scale topology optimization through on‐the‐fly reduced order model construction (2015, methodology): “[13], who used Krylov subspaces, reported savings by a factor of 3.”
Stress-constrained topology optimization: a topological level-set approach (2013, background): “Secondly, 20 the ill-conditioning of the stiffness matrices, once again 21 due to the low-density elements, can lead to high 22 computational costs for iterative solvers [22], [23].”
Efficient use of iterative solvers in nested topology optimization (2010, methodology): “This is opposed to the I-section design which is frequently the optimal shape when the complete wall area is supported (see for example Wang et al. 2007 ; A mir et al. 2009a).”
Benchmarking optimization solvers for structural topology optimization (2015, unknown): “(Wang et al. 2007; Evgrafov et al. 2008; Borrvall and Petersson 2001).”
A survey of structural and multidisciplinary continuum topology optimization: post 2000 (2014, methodology): “…been explored that convert the Hamilton-Jacobi PDE into a simpler set of ordinary differential equations (ODEs) using radial basis functions (RBFs) (Wang and Wang 2006a, b, Wang et al. 2007b) or system of algebraic equations using compactly supported RBFs (CSRBFs) (Luo et al. 2007, 2008c, 2009a).”
A multi-resolution method for 3D multi-material topology optimization (2015, unknown): “…cost of the finite element analysis can be done in a number of ways, by introducing parallel computing [28–31], by employing fast iterative solver [32,33], by performing FE analysis after certain intervals [34,35] or by reducing the number of degrees of freedom in the FE model [36–39].”
Efficient generation of large-scale pareto-optimal topologies (2013, background): “, 110925 DOF; this problem was considered in [11].”