Level-set, penalization and cartesian meshes: A paradigm for inverse problems and optimal design

The aim of this work is to combine penalization and level-set methods to solve inverse or shape optimization problems on uniform cartesian meshes. Penalization is a method to impose boundary conditions avoiding the use of body-fitted grids, whereas level-sets allow a natural non-parametric description of the geometries to be optimized. In this way, the optimization problem is set in a larger design space compared to classical parametric representation of the geometries, and, moreover, there is no need of remeshing at each optimization step. Special care is devoted to the solution of the governing equations in the vicinity of the penalized regions and a method is introduced to increase the accuracy of the discretization. Another essential feature of the optimization technique proposed is the shape gradient preconditioning. This aspect turns out to be crucial since the problem is infinite dimensional in the limit of grid resolution. Examples pertaining to model inverse problems and to shape design for Stokes flows are discussed, demonstrating the effectiveness of this approach.