SHAPE OPTIMIZATION BY CONSTRAINED FIRST-ORDER SYSTEM LEAST MEAN APPROXIMATION

In this work, the problem of shape optimization, subject to PDE constraints, is reformulated as an Lp best approximation problem under divergence constraints to the shape tensor More precisely, the main result of this paper states that the Lp distance of the above approximation problem is equal to the dual norm of the shape derivative considered as a functional on W1,p\ast (where 1/p + 1/p* = 1). This implies that for any given shape, one can evaluate its distance from being stationary with respect to the shape derivative by simply solving the associated Lp-type least mean approximation problem. Moreover, the Lagrange multiplier for the divergence constraint turns out to be the shape deformation of steepest descent. This provides, as an alternative to the approach by Deckelnick, Herbert, and Hinze, a way to compute shape gradients in W1,p\ast for p* \in (2, \infty ). The discretization of the least mean approximation problem is done with (lowest-order) matrix-valued Raviart--Thomas finite element spaces leading to piecewise constant approximations of the shape deformation acting as a Lagrange multiplier. Admissible deformations in W1,p\ast to be used in a shape gradient iteration are reconstructed locally. Our computational results confirm that the Lp distance of the best approximation does indeed measure the distance of the considered shape to optimality. Also confirmed by our computational tests is the observation of Deckelnick, Herbert, and Hinze [ESAIM Control Optim. Calc. Var., 28 (2022), 2] that choosing p* (much) larger than 2 (which means that p must be close to 1 in our best approximation problem) decreases the chance of encountering mesh degeneracy during the shape gradient iteration.