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This part contains several results of recent research in shape and topology optimization. It consists of the following three independent sections:

Sergio Conti, Benedikt Geihe, Martin Rumpf, and Rüdiger Schultz combine, in Two-stage stochastic optimization meets two-scale simulation, a two-scale model in elastic shape optimization with a stochastic framework. The microstructured material to be optimized is composed of an elastic material with geometrically simple perforations located on a regular periodic lattice, whose parameters depend on the macroscopic position.

Helmut Harbrecht and Johannes Tausch review, in On shape optimization with parabolic state equation, their results on numerical methods for the efficient solution of shape optimization problems with parabolic state equation. For a specific parabolic shape optimization problem, both the shape calculus and the discretization by means of a modern space-time multipole method are demonstrated. For comparison reasons, also the related stationary shape optimization problem is considered.

Luise Blank, M. Hassan Farshbaf-Shaker, Harald Garcke, Christoph Rupprecht, and Vanessa Styles present, in Multi-material phase field approach to structural topology optimization, how to formulate and solve multi-material structural topology and shape optimization problems within a phase field approach. The first-order optimality system is determined and then numerically solved by an H 1-gradient projection method.