Abstract
Some optimal design problems in topology optimization eventually lead to degenerate convex minimization problems \(E(v):=\int _\varOmega W(\nabla v)dx-\int _\varOmega fvdx\) for \(v\in H_{0}^{1}(\varOmega ) \) with possibly multiple minimizers u, but with a unique stress \(\sigma :=DW(\nabla u)\). The discretization of degenerate convex minimization problems experience numerical difficulties with a singular or nearly singular Hessian matrix. This paper studies a modified discretization by adding a stabilization term to the discrete energy. It will be proven that this stabilization technique leads to a posteriori error control on unstructured triangulations, and so enables the use of adaptive algorithms.
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Acknowledgements
This work was partly developed while the first author was visiting the Department of Mathematics at Humboldt University. We would like to thank professor C. Carstensen for his helpful suggestions.
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This author’s research was supported by National Natural Science Foundation of China (Nos. 11571226, 11001168, and 61201113).
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Liu, D.J., Jiang, D.D., Liu, Y. et al. Stabilized FEM for Some Optimal Design Problem. J Sci Comput 73, 228–241 (2017). https://doi.org/10.1007/s10915-017-0409-8
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DOI: https://doi.org/10.1007/s10915-017-0409-8