2006-01-01Buch DOI: 10.18452/2748
A convergent adaptive finite element method for an optimal design problem
Abstract. The optimal design problem for maximal torsion stiffness of an infinite bar of given geometry and unknown distribution of two materials of prescribed amounts is one model example in topology optimisation. It eventually leads to a degenerated convex minimisation problem. The numerical analysis is therefore delicate for possibly multiple primal variables $u$ but unique derivatives $\sigma := DW(Du)$. Even sharp a posteriori error estimates still suffer from the reliability-efficiency gap. However, it motivates a simple edgebased adaptive mesh-refining algorithm (AFEM) that is not a priori guaranteed to refine everywhere. Its convergence proof is therefore based on energy estimates and some refined convexity control. Numerical experiments illustrate even nearly optimal convergence rates of the proposed adaptive finite element method (AFEM).
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