A fast adjoint-free shape gradient for p-Laplacian compliance: algorithmic speedup and engineering benchmarks

Abstract

  Shape optimization of nonlinear membranes and non-Newtonian flow domains governed by the $p$-Laplace operator is hampered by the cost of an auxiliary linearised adjoint solve at every iteration. We show that this solve is unnecessary: a self-adjoint identity reduces the adjoint state to a multiple of the primal state, halving the PDE cost per iteration. We derive a closed-form boundary expression of the shape gradient, extend the analysis to the singular regime $1<p<2$ via Muckenhoupt-weighted Sobolev spaces, and prove subsequential convergence of the resulting descent algorithm. Three engineering benchmarks (cantilever, MBB beam, short bridge) for $p\in\{1.5,2,3,4\}$ deliver compliance reductions of up to 51.8%, demonstrate a CPU speedup of up to 25% over the classical full-adjoint method, and confirm identical iterates to floating-point precision. The framework integrates readily into existing finite-element shape-optimization codes.