Structural Optimization by the Homogenization Method
We discuss a method of structural optimization in the context of linear elasticity. We seek the optimal shape of an elastic body which is both of minimum weight and maximal stiffness under specified loadings. Mathematically, a weighted sum of the elastic compliance and of the weight is minimized among all possible shapes. This problem is known to be “ill-posed”, namely there is generically no optimal shape and the solutions computed by classical numerical algorithms are highly sensitive to the initial guess and mesh-dependent. Our method is based on the homogenization theory which makes this problem well-posed by allowing microperforated composites as admissible designs. A new numerical algorithm is thus obtained which allows to capture an optimal shape on a fixed mesh. Such a procedure is called topology optimization since it places no explicit or implicit restriction on the topology of the optimal shape, i.e. on its number of holes or members.
KeywordsHomogenization Method Fixed Mesh Elastic Compliance Homogenization Theory Structural Optimization Problem
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