Abstract
In this work, we attempt to answer the question posed in Amir O., Sigmund O.: On reducing computational effort in topology optimization: how far can we go? (Struct. Multidiscip. Optim. 44(1):25–29 2011). Namely, we are interested in assessing how inaccurately we can solve the governing equations during the course of a topology optimization process while still obtaining accurate results. We consider this question from a “PDE-based” angle, using a posteriori residual estimates to gain insight into the behavior of the residuals over the course of Krylov solver iterations. Our main observation is that the residual estimates are dominated by discretization error after only a few iterations of an iterative solver. This provides us with a quantitative measure for early termination of iterative solvers. We illustrate this approach using benchmark examples from linear elasticity and demonstrate that the number of Krylov solver iterations can be significantly reduced, even when compared to previous heuristic recommendations, although each Krylov iteration becomes considerably more expensive.
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Notes
This effect is a consequence of the utilization of only first-order information about the optimization problem, which results in computationally inexpensive iterations at the cost of slow rates of convergence. Whereas in some situations, significantly faster converging alternatives exist, see for example (Evgrafov 2015; Rojas-Labanda and Stolpe 2016; Kočvara and Mohammed 2016), we focus on first-order algorithms in this work.
A posteriori FEM error estimates have recently been used in topology optimization in another context (Pimanov and Oseledets 2018).
We denote the space of all bounded linear functionals on V (the dual space of V) by V′.
Strictly speaking, traction forces only enter the model on parts of the boundary, where the displacements are not fixed. We only assume that t is defined on the whole boundary and its components are set to zero on Dirichlet parts of the boundary in order to keep the notation to a bare minimum.
We are not concerned about repeatedly solving the filtering problem, because it is a relatively well-conditioned one, see Lazarov and Sigmund (2011).
Here and elsewhere, we use the notation \(\boldsymbol {r} = [r_{\rho _{h},\boldsymbol {u}_{h}}(\phi _{1}),\dots ,\) \(r_{\rho _{h},\boldsymbol {u}_{h}}(\phi _{N_{h}})] \in \mathbb {R}^{N_{h}}\) for the algebraic representation of the residual \(r_{\rho _{h},\boldsymbol {u}_{h}} \in V_{h}^{\prime }\) with respect to the basis \( \phi _{1},\dots ,\phi _{N_{h}}\) of Vh during PCG iterations.
In the presented case, min{nx,ny} = 60. This quantity is a heuristic number of iterations proposed by Amir et al. in (Amir and Sigmund 2011) based on numerical experiments with early termination of a PCG solver.
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Limkilde, A., Evgrafov, A. & Gravesen, J. On reducing computational effort in topology optimization: we can go at least this far!. Struct Multidisc Optim 58, 2481–2492 (2018). https://doi.org/10.1007/s00158-018-2121-1
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DOI: https://doi.org/10.1007/s00158-018-2121-1