Topology optimization of the two smallest p-Laplacian eigenvalues by a level set method

This paper considers optimization of the two smallest Dirichlet p-Laplacian eigenvalues subject to geometric volume or perimeter constraint. A relaxed topology optimization model with shape sensitivity analysis is presented. A level set method and finite element discretization are employed to numerically solving the model problems. The p-Laplacian eigenvalue is efficiently solved using quasi-Newton method, which achieves accurate results even for extreme values of p. Numerical results are provided to indicate that for the first eigenvalue, the optimal shapes are disks in 2D and balls in 3D under both constraints, independent of p. For the second eigenvalue, optimizers have two identical disks or balls independent of p under volume constraint, while optimized shapes evolve from ellipsoidal to flatter configurations under perimeter constraint as p increases.