A convergent adaptive finite element method for an optimal design problem |
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Authors: | Sören Bartels Carsten Carstensen |
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Institution: | 1. Institute for Numerical Simulation, Rheinische Friedrich-Wilhelms-Universit?t Bonn, Wegelerstra?e 6, 53115, Bonn, Germany 2. Institute of Mathematics, Humboldt-Universit?t zu Berlin, Unter den Linden 6, 10099, Berlin, Germany
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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 degenerate convex
minimisation problem. The numerical analysis is therefore delicate for possibly multiple primal variables u but unique derivatives σ : = DW(D
u). Even fine a posteriori error estimates still suffer from the reliability-efficiency gap. However, it motivates a simple
edge-based 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 AFEM.
Supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin. |
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Keywords: | 65N30 65N15 65N12 |
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