A level set method for structural topology optimization

A simple evolutionary procedure for structural optimization

In the context of shape optimization, we seek minimizers of the sum of the elastic compliance and of the weight of a solid structure under specified loading. This problem is known not to be well-posed, and a relaxed formulation is introduced. Its effect is to allow for microperforated composites as admissible designs. In a two-dimensional setting the relaxed formulation was obtained in [6] with the help of the theory of homogenization and optimal bounds for composite materials. We generalize the result to the three dimensional case. Our contribution is twofold; first, we prove a relaxation theorem, valid in any dimensions; secondly, we introduce a new numerical algorithm for computing optimal designs, complemented with a penalization technique which permits to remove composite designs in the final shape. Since it places no assumption on the number of holes cut within the domain, it can be seen as a topology optimization algorithm. Numerical results are presented for various two and three dimensional problems.

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Shape optimization by the homogenization method

The role of structural hierarchy in determining bulk material properties is examined. Dense hierarchical materials are discussed, including composites and polycrystals, polymers, and biological materials. Hierarchical cellular materials are considered, including cellular solids and the prediction of strength and stiffness in hierarchical cellular materials.

Materials with structural hierarchy

Two types of solutions may be considered in generalized shape optimization. Absolute minimum weight solutions, which are rather unpractical, consist of solid, empty and porous regions. In more practical solutions of shape optimization, porous regions are suppressed and only solid and empty regions remain. This note discusses this second class of problems and shows that a solid, isotropic microstructure with an adjustable penalty for intermediate densities is efficient in generating optimal topologies.

Generalized shape optimization without homogenization

https://deepblue.lib.umich.edu/bitstream/handle/2027.42/29003/0000032.pdf;sequence=1

A homogenization method for shape and topology optimization

We shall describe a brief review of the structural optimization of a linearly elastic structure, and we shall present a new method to solve the sizing, shape, and layout (topology) problems based on the theory of homogenization Many numerical examples of the optimal design are also presented as well as a mathematical formulation of a relaxed design problem

/pdf/structural-optimization-of-a-linearly-elastic-structure-3d5xhg6v5w.pdf

Structural Optimization of a Linearly Elastic Structure using the Homogenization Method

We shall review how finite element methods have been developed and what subjects researchers in the United States are studying at present in the high-speed computing environment

Katsuyuki Suzuki

Papers

A homogenization method for shape and topology optimization

Structural Optimization of a Linearly Elastic Structure using the Homogenization Method

Recent Development of Finite Element Methods in the High-Speed Computing Environment