Shape and topology optimization involving the eigenvalues of an elastic structure: A multi-phase-field approach

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.

Sequentially coupled shape and topology optimization for 2.5D and 3D beam models

A sequentially coupled shape and topology optimization framework is presented in which the outer geometry and the internal topological layout of beam-type structures are optimized simultaneously. The outer geometry of the beam-type structures is parametrically described by non-uniform rational B-splines (NURBS), which guarantees a highly accurate description of the structural shape and enable an efficient control of the design domain with only a few control points. The computational efficiency of the coupled optimization approach is assured by applying a gradient-based optimization algorithm, for which the sensitivities are derived in closed form. The formulation of the coupled optimization approach is tailored toward 2.5D and full 3D representations of beam structures used in engineering applications. The 2.5D beam model, which has been taken from the literature, uses standard beam elements to simulate the beam response in the longitudinal direction, whereby the cross-sectional properties of the beam elements are calculated from additional 2D finite element method (FEM) analyses. A comparison study of a cantilever beam problem subjected to pure shape optimization and pure topology optimization illustrates that the 2.5D and 3D beam models lead to similar shape and topology designs, but that the 2.5D beam model has a significantly higher computational efficiency. Specifically, the computational times for the 2.5D model are about a factor 70 (shape optimization) and 1.4 (topology optimization) lower than for the 3D model, which indicates that in the coupled optimization approach the optimization of the shape provides the largest contribution to the higher computational efficiency of the 2.5D model. The coupled shape and topology optimization analysis subsequently performed on the 2.5D cantilever beam model demonstrates that the specific order at which the alternating shape and topology optimization increments are performed in the staggered update procedure turns out to have some influence on the computational speed and the value of the minimal compliance computed. Despite these differences, the final beam structures following from the different staggered update procedures illustrate how shape and topology can be efficiently optimized in an integrated, coupled fashion.