A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow.

Abstract We propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to deal with complex parametrized domains in an efficient and straightforward way. The impact of the proposed approach is threefold. First, problems involving parametrizations of complex geometrical shapes and/or large domain deformations can be efficiently solved at full-order by means of the SBM. This unfitted boundary method permits to avoid remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary. Second, the computational effort is reduced by the development of a Reduced Order Model (ROM) technique based on a POD-Galerkin approach. Third, the SBM provides a smooth mapping from the true to the surrogate domain, and for this reason, the stability and performance of the reduced order basis are enhanced. This feature is the net result of the combination of the proposed ROM approach and the SBM. Similarly, the combination of the SBM with a projection-based ROM gives the great advantage of an easy and fast to implement algorithm considering geometrical parametrization with large deformations. The transformation of each geometry to a reference geometry (morphing) is in fact not required. These combined advantages will allow the solution of PDE problems more efficiently. We illustrate the performance of this approach on a number of two-dimensional Stokes flow problems. Highlights • A POD-Galerkin approach to handle problems involving parametrizations of complex geometrical shapes and/or large domain deformations has been developed. • On the contrary to traditional RB-Galerkin approaches, the proposed methodology do not require the transformation of each geometry to a reference one. • The tedious handling of cut cells is avoided at the full order level by the SBM formulation, keeping the good properties that other related methods like CutFEM have demonstrated. • The SBM provides a smooth mapping from the true to the surrogate domain, and for this reason,the stability and performance of the reduced basis are enhanced. • The efficiency and accuracy of the proposed reduced order model have been numerically verified against the full order model. [ABSTRACT FROM AUTHOR]