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Relating phase field and sharp interface approaches to structural topology optimization

Luise Blank, Harald Garcke, M. Hassan Farshbaf-Shaker, Vanessa Styles (2014)

ESAIM: Control, Optimisation and Calculus of Variations

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for...

Relaxation of vectorial variational problems

Tomáš Roubíček (1995)

Mathematica Bohemica

Multidimensional vectorial non-quasiconvex variational problems are relaxed by means of a generalized-Young-functional technique. Selective first-order optimality conditions, having the form of an Euler-Weiestrass condition involving minors, are formulated in a special, rather a model case when the potential has a polyconvex quasiconvexification.

Reliable solution of an elasto-plastic Reissner-Mindlin beam for Hencky's model with uncertain yield function

Ivan Hlaváček (1998)

Applications of Mathematics

We apply the method of reliable solutions to the bending problem for an elasto-plastic beam, considering the yield function of the von Mises type with uncertain coefficients. The compatibility method is used to find the moments and shear forces. Then we solve a maximization problem for these quantities with respect to the uncertain input data.

Removing holes in topological shape optimization

Maatoug Hassine, Philippe Guillaume (2008)

ESAIM: Control, Optimisation and Calculus of Variations

The gradient based topological optimization tools introduced during the last ten years tend naturally to modify the topology of a domain by creating small holes inside the domain. Once these holes have been created, they usually remain unchanged, at least during the topological phase of the optimization algorithm. In this paper, a new asymptotic expansion is introduced which allows to decide whether an existing hole must be removed or not for improving the cost function. Then, two numerical examples...

Removing holes in topological shape optimization

Philippe Guillaume, Maatoug Hassine (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The gradient based topological optimization tools introduced during the last ten years tend naturally to modify the topology of a domain by creating small holes inside the domain. Once these holes have been created, they usually remain unchanged, at least during the topological phase of the optimization algorithm. In this paper, a new asymptotic expansion is introduced which allows to decide whether an existing hole must be removed or not for improving the cost function. Then, two numerical...

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