A new numerical approach for solving shape optimization fourth-order spectral problems among convex domains

In this work, we deal with a new numerical approach for solving some shape optimization eigenvalue problems governed by the bi-harmonic operator, under volume and convexity constraints. This is based on the new shape derivative formula, recently established in [10,11], which allows to express the shape derivative of optimal shape design problems of minimizing volume cost functionals, in term of support functions. This avoids some computational disadvantages required for the classical shape derivative method involving the vector fields [16, 18], when one use the finite elements discretization for approximating the auxiliary boundary value problems in shape optimization processes. So, we first show the existence of the shape derivative of the eigenvalues for these fourth-order problems with respect to a family of convex domains and express its formula by means of support functions. Thereby we propose a new numerical shape optimization process based on the gradient method performed with the finite element method for approximating the auxiliary eigenvalue boundary value problems. The so obtained numerical results show the efficiency and the ability of the proposed approach in producing good quality solutions for the first ten optimal eigenvalues and their associated optimal shapes.