Efficient Use of Iterative Solvers in Nested Topology Optimization

Transcription

1 Efficient Use of Iterative Solvers in Nested Topology Optimization Oded Amir, Mathias Stolpe and Ole Sigmund Technical University of Denmark Department of Mathematics Department of Mechanical Engineering June 2, 2009 Efficient Use of Iterative Solvers in Nested Topology Optimization 1/15

2 Introduction - what is this presentation about? Nested approach to structural optimization, with focus on topology optimization. Iterative solution of the nested equation system using Krylov subspace solvers. Case studies are from the field of structural mechanics (linear elasticity), so we focus on the use of Preconditioned Conjugate Gradients as the iterative solver. Main theme Using approximate solutions in the analysis problem can save significant computing time, without affecting the accuracy of the optimization significantly. Efficient Use of Iterative Solvers in Nested Topology Optimization 2/15

3 Introduction - the nested approach The nested approach to topology optimization, demonstrated on a minimum compliance problem: min ρ c(ρ) = f T u (compliance) s.t.: N v e ρ e V e=1 (volume constraint) 0 < ρ min ρ e 1 (element densities) with: K(ρ)u = f (equilibrium) * The main computational bottleneck - solving the (typically large) system of equations Ku = f (and additional adjoint systems in other problems). * Other structural optimization problems may share the same formulation. Efficient Use of Iterative Solvers in Nested Topology Optimization 3/15

4 Introduction - motivation for this study In practice, when solving large problems, most of us turn to the iterative family of Krylov subspace solvers, e.g. the Conjugate Gradients method with effective preconditioning (PCG): Low memory requirements. Suitable for parallel computing. The challenge: Krylov subspace solvers typically require a large number of iterations in order to converge to an accurate solution of the nested problem. Then this should be repeated for every design iteration. Efficient Use of Iterative Solvers in Nested Topology Optimization 4/15

5 Proposed approximation The common convergence criterion for PCG (and similar methods): * A typical value of the f Ku k 2 = r k 2 tolerance ɛ is < ɛ f 2 f 2 * u k is in practice, the accurate solution. The proposed approximation is u m, m < k f Ku m 2 f 2 = r m 2 f 2 >> ɛ * How should the PCG cycle m be chosen? * Is it enough to use a slack ɛ? Efficient Use of Iterative Solvers in Nested Topology Optimization 5/15

6 PCG performance - minimum compliance (1) Relative residual, f T u and u T Ku Remarks * Initial guess = 0. * f T u i = u T i Ku i for all PCG iterations i. * c ρ e i = u T i K ρ e u i. Efficient Use of Iterative Solvers in Nested Topology Optimization 6/15

7 PCG performance - minimum compliance (2) Relative residual, f T u and u T Ku Remarks * Initial guess = u(ρ old ). * In general: f T u i u T i Ku i. Efficient Use of Iterative Solvers in Nested Topology Optimization 7/15

8 Alternative convergence criteria Question: How to choose the approximation u m so that the objective (f T u m ) and sensitivities (-u T m K ρ e u m ) are sufficiently accurate? Initial guess = 0 Measure the relative change in K-norm of the solution: u T mku m u T m 1 Ku m 1 u T m 1 Ku m 1 < ɛ Initial guess = u(ρ old ) Measure the relative difference between compliance and K-norm of the solution f T u m u T mku m u T < ɛ mku m Efficient Use of Iterative Solvers in Nested Topology Optimization 8/15

9 The force inverter problem The nested approach to topology optimization, demonstrated on a force inverter problem: min c(ρ) = l T u (output) ρ s.t.: N v eρ e V e=1 (volume constraint) 0 < ρ min ρ e 1, (element densities) with: K(ρ)u = f (equilibrium) * The aim is to maximize an output displacement, designated by the vector l. * Sensitivity analysis requires the solution of an adjoint system: K(ρ) T λ = l. * c ρ e = λ T K ρ e u. Efficient Use of Iterative Solvers in Nested Topology Optimization 9/15

10 PCG performance - force inverter (1) Relative residual, l T u, f T λ and λ T Ku Remarks * Both r.h.s. solved together by block-pcg. * Initial guess = 0. * l T u i = f T λ i = λ T i Ku i for all PCG iterations i. Efficient Use of Iterative Solvers in Nested Topology Optimization 10/15

11 PCG performance - force inverter (2) Relative residual, l T u, f T λ and λ T Ku Remarks * Initial guess = u(ρ old ), λ(ρ old ). * In general: l T u i λ T i Ku i. * In general: f T λ i λ T i Ku i. Efficient Use of Iterative Solvers in Nested Topology Optimization 11/15

12 Alternative convergence criteria Question: How to choose the approximations u m and λ m so that the objective (l T u m ) and sensitivities (-λ T m K ρ e u m ) are sufficiently accurate? Initial guess = 0 Measure the relative change in the value of the objective: l T u m l T u m 1 l T u m 1 < ɛ Initial guess = u(ρ old ), λ(ρ old ) Measure also the relative difference between the objective and f T λ i : l T u m f T λ m f T λ m < ɛ Efficient Use of Iterative Solvers in Nested Topology Optimization 12/15

13 Example: large scale minimum compliance * 324,000 elements, 1.03E6 DOF. * Preconditioned with IC(0). * 40% reduction in PCG iterations. * After 50 design iterations: 0.02% error in objective value. * After 50 design iterations: Same topology, some differences in boundary regions. Efficient Use of Iterative Solvers in Nested Topology Optimization 13/15

14 Example: force inverter * 7,200 elements, 14,762 DOF. * Preconditioned with IC(0). * 30% reduction in PCG iterations. * After 100 design iterations: 1.5% error in objective value. * After 100 design iterations: Same topology but different shape. Efficient Use of Iterative Solvers in Nested Topology Optimization 14/15

15 Concluding remarks Achieved savings: 40% in 3D minimum compliance, 30% in 2D force inverter. Improving accuracy of force inverter problems and extending to 3D are among future tasks. Further interesting extensions: * Other physical models, objective functions. * Other Krylov solvers besides PCG. The key point: convergence measures should be related to the objective function and to the corresponding design sensitivities. Efficient Use of Iterative Solvers in Nested Topology Optimization 15/15