On Chebyshev’s method for topology optimization of Stokes flows
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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.
KeywordsTopology optimization Stokes flows Chebyshev’s algorithm
The author is grateful to Martin Berggren for pointing out the reference (Carlsson et al. 2009) to us.
- Bartish MJ (1969) On some iterative methods of solving functional equations. Sibirskij matematiceskij zurnal 10(3):488–493Google Scholar
- Duff IS (2002) MA57 a new code for the solution of sparse symmetric definite and indefinite systems. Technical Report RAL-TR-2002-024. Rutherford Appleton LaboratoryGoogle Scholar
- Griebel M, Dornseifer T, Neunhoeffer T (1997) Numerical simulation in fluid dynamics: a practical introduction. SIAM 3Google Scholar
- Nechepurenko MI (1954) On Chebyshev’s method for functional equations. Uspekhi Mat Nauk 9(2):163–170Google Scholar
- Othmer C (2008) A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows. Internat J Numer Methods Fluids 58(8)Google Scholar
- Seiboldm B (2008) A compact and fast Matlab code solving the incompressible Navier–Stokes equations on rectangular domains. http://math.mit.edu/cse/codes/mit18086_navierstokes.pdf