12. Efficient Reduction of PDEs Defined on Domains with Variable Shape

In this work we propose a new, general and computationally cheap way to tackle parametrized PDEs defined on domains with variable shape when relying on the reduced basis method. We easily describe a domain by boundary parametrizations, and generate domain (and mesh) deformations by means of a solid extension, obtained by solving a linear elasticity problem. The proposed procedure is built over a two-stages reduction: (1) first, we construct a reduced basis approximation for the mesh motion problem; (2) then, we generate a reduced basis approximation of the state problem, relying on finite element snapshots evaluated over a set of reduced deformed configurations. A Galerkin-POD method is employed to construct both reduced problems, although this choice is not restrictive. To deal with unavoidable nonaffine parametric dependencies arising in both the mesh motion and the state problem, we apply a matrix version of the discrete empirical interpolation method, allowing to treat geometrical deformations in a non-intrusive, efficient and purely algebraic way. In order to assess the numerical performances of the proposed technique, we address the solution of a parametrized (direct) Helmholtz scattering problem where the parameters describe both the shape of the obstacle and other relevant physical features. Thanks to its easiness and efficiency, the methodology described in this work looks promising also in view of reducing more complex problems.