Constrained $L^p$ Approximation of Shape Tensors and its Role for the Determination of Shape Gradients

Abstract: This paper extends our earlier work [arXiv:2309.13595] on the $L^p$ approximation of the shape tensor by Laurain and Sturm. In particular, it is shown that the weighted $L^p$ distance to an affine space of admissible symmetric shape tensors satisfying a divergence constraint provides the shape gradient with respect to the $L^{p^\ast}$-norm (where $1/p + 1/p^\ast = 1$) of the elastic strain associated with the shape deformation. This approach allows the combination of two ingredients which have already been used successfully in numerical shape optimization: (i) departing from the Hilbert space framework towards the Lipschitz topology approximated by $W^{1,p^\ast}$ with $p^\ast > 2$ and (ii) using the symmetric rather than the full gradient to define the norm. Similarly to [arXiv:2309.13595], the $L^p$ distance measures the shape stationarity by means of the dual norm of the shape derivative with respect to the above-mentioned $L^{p^\ast}$-norm of the elastic strain. Moreover, the Lagrange multiplier for the momentum balance constraint constitute the steepest descent deformation with respect to this norm. The finite element realization of this approach is done using the weakly symmetric PEERS element and its three-dimensional counterpart, respectively. The resulting piecewise constant approximation for the Lagrange multiplier is reconstructed to a shape gradient in $W^{1,p^\ast}$ and used in an iterative procedure towards the optimal shape.