Volumetric expressions of the shape gradient of the compliance in structural shape optimization
This work provides theoretical derivations for shape sensitivity in structural optimization, which is relevant for engineers and mathematicians working on topology optimization, but the contribution is incremental as it extends existing shape calculus techniques.
The paper derives volumetric expressions for the shape gradient of compliance in structural shape optimization under volume constraints, using both pure displacement and dual mixed formulations of linear elasticity. Numerical simulations with the Boundary Variation Algorithm provide a qualitative comparison of the two gradient expressions.
In this article, we consider the problem of optimal design of a compliant structure under a volume constraint, within the framework of linear elasticity. We introduce the pure displacement and the dual mixed formulations of the linear elasticity problem and we compute the volumetric expressions of the shape gradient of the compliance by means of the velocity method. A preliminary qualitative comparison of the two expressions of the shape gradient is performed through some numerical simulations using the Boundary Variation Algorithm.