In the context of structural optimization via a level-set method we propose a framework to handle geometric constraints related to a notion of local thickness. The local thickness is calculated using the signed distance function to the shape. We formulate global constraints using integral functionals and compute their shape derivatives. We discuss different strategies and possible approximations to handle the geometric constraints. We implement our approach in two and three space dimensions for a model of linearized elasticity. As can be expected, the resulting optimized shapes are strongly dependent on the initial guesses and on the specific treatment of the constraints since, in particular, some topological changes may be prevented by those constraints.

/pdf/thickness-control-in-structural-optimization-via-a-level-set-1kvsvbxawi.pdf

Thickness control in structural optimization via a level set method

Shape gradients of PDE constrained shape functionals can be stated in two equivalent ways. Both rely on the solutions of two boundary value problems (BVPs), but one involves integrating their traces on the boundary of the domain, while the other evaluates integrals in the volume. Usually, the two BVPs can only be solved approximately, for instance, by finite element methods. However, when used with finite element solutions, the equivalence of the two formulas breaks down. By means of a comprehensive convergence analysis, we establish that the volume based expression for the shape gradient generally offers better accuracy in a finite element setting. The results are confirmed by several numerical experiments.

/pdf/comparison-of-approximate-shape-gradients-4jx5b0pdca.pdf

Comparison of approximate shape gradients

https://orbit.dtu.dk/files/101181550/1_s2.0_S0097849314001095_main.pdf

Combined shape and topology optimization of 3D structures

We introduce a model and several constraints for shape and topology optimization of structures, built by additive manufacturing techniques. The goal of these constraints is to take into account the...

/pdf/taking-into-account-thermal-residual-stresses-in-topology-4888fpgmgs.pdf

Taking into account thermal residual stresses in topology optimization of structures built by additive manufacturing

https://hal.archives-ouvertes.fr/hal-01398877/document

Shape optimization of a layer by layer mechanical constraint for additive manufacturing

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Johanna Weiss

Papers

Optimal Shape Design For Elliptic Systems