×

Numerical solution of elliptic shape optimization problems using wavelet-based BEM. (English) Zbl 1061.49028

Summary: In this article we study the numerical solution of elliptic shape optimization problems with additional constraints, given by domain or boundary integral functionals. A special boundary variational approach combined with a boundary integral formulation of the state equation yields shape gradients and functionals which are expressed only in terms of boundary integrals. Hence, the efficiency of (standard) descent optimization algorithms is considerably increased, especially for the line search. We demonstrate our method for a class of problems from planar elasticity, where the stationary domains are given analytically by N. V. Banichuk and B. L. Karihaloo [Int. J. Solids Structures 12, 267–273 (1976; Zbl 0319.73028)]. In particular, the boundary integral equation is solved by a wavelet Galerkin scheme which offers a powerful tool. For optimization we apply gradient and quasi-Newton type methods for the penalty as well as for the augmented Lagrangian functional.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49M15 Newton-type methods
49M30 Other numerical methods in calculus of variations (MSC2010)
65N38 Boundary element methods for boundary value problems involving PDEs
65T60 Numerical methods for wavelets
74P05 Compliance or weight optimization in solid mechanics

Citations:

Zbl 0319.73028
PDFBibTeX XMLCite
Full Text: DOI