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On Chebyshev’s method for topology optimization of Stokes flows

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Abstract

We present a locally cubically convergent algorithm for topology optimization of Stokes flows based on a Chebyshev’s iteration globalized with Armijo linesearch. The characteristic features of the method include the low computational complexity of the search direction calculation, evaluation of the objective function and constraints needed in the linesearch procedure as well as their high order derivatives utilized for obtaining higher order rate of convergence. Both finite element and finite volumes discretizations of the algorithm are tested on the standard two-dimensional benchmark problems, in the case of finite elements both on structured and quasi-uniform unstructured meshes of quadrilaterals. The algorithm outperforms Newton’s method in nearly all test cases. Finally, the finite element discretization of the algorithm is tested within a continuation/adaptive mesh refinement framework.

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Notes

  1. In dimension 2, the inclusion in (8) can be further strengthened and thus higher order derivatives may be obtained without assuming any further regularity of u. Alternatively, since the zeroth order term α(ρ)u in (4) is a priori in [L 2(Ω)]d one may appeal to the standard regularity results for the Stokes problem, see for example (Kellogg and Osborn 1976; Amrouche and Girault 1991), to assert that u∈[H s(Ω)]d, 1<s≤2, under the appropriate additional assumptions on f, u 0, and Ω, and in this way further strengthen the inclusion (8) using the Sobolev inequality.

  2. In Evgrafov (2014) we have verified that the state space Newton’s algorithm performs virtually independently of the selected discretization/mesh refinement.

  3. This discretization has been used in the original paper of Borrvall and Petersson (2003). Note that we count the number of elements differently: in (Borrvall and Petersson 2003) the number of u-elements is reported, whereas we list the number of macroelements. As a result the number of degrees of freedom in [Q2]2/Q1 and [Q2-iso-Q1]2/Q1 elements is the same on the same discretization level. In other words our 100×100 discretization [Q2-iso-Q1]2/Q1 corresponds to 200×200 u-elements.

  4. Note the lack of numerical scalability owing to the utilization of the direct linear solver.

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Acknowledgments

The author is grateful to Martin Berggren for pointing out the reference (Carlsson et al. 2009) to us.

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  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, N–7491, Trondheim, Norway

    Anton Evgrafov

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  1. Anton Evgrafov
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Correspondence to Anton Evgrafov.

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Evgrafov, A. On Chebyshev’s method for topology optimization of Stokes flows. Struct Multidisc Optim 51, 801–811 (2015). https://doi.org/10.1007/s00158-014-1176-x

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