Abstract
Abstract A cost function involving the eigenvalues of an elastic structure is optimized using a phase-field approach, which allows for topology changes and multiple materials.We show continuity and differentiability of simple eigenvalues in the phase-field context. Existence of global minimizers can be shown, for which first order necessary optimality conditions can be obtained in generic situations. Furthermore, a combined eigenvalue and compliance optimization is discussed.
Highlights
The main goal in structural topology optimization is to find the optimal distribution of materials in a so called design domain
In contrast to shape optimization, topological changes including the design and distribution of holes in the structure are allowed in topology optimization
Several mathematical techniques to deal with shape or topology optimization problems can be found in the literature
Summary
The main goal in structural topology optimization is to find the optimal distribution of materials in a so called design domain. In [25, 26] the authors investigate models similar to the one we intend to study In these papers, the density distribution ρ is assumed to depend only on the spatial variable x ∈ Ω meaning that the dependence on the structure (represented by the phase-field φ) is neglected. We prove the existence of eigenvalues and eigenfunctions for the problem (1.1) and we establish essential properties needed for the theory of calculus of variations such as suitable continuity statements This allows us to investigate an optimal control problem where an objective functional. As the energy EGε L is an approximation of the perimeter of the material boundaries, minimizing (1.2) can be related to a shape and topology optimization problem with a perimeter penalization (see, e.g., [32]). In the last part we want to combine the eigenvalue problem with compliance minimization problems
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