Reduced Basis Method for the Stokes Equations in Decomposable Parametrized Domains Using Greedy Optimization

Flow simulations in pipelined channels and several kinds of parametrized configurations have a growing interest in many life sciences and industrial applications. Applications may be found in the analysis of the blood flow in specific compartments of the circulatory system that can be represented as a combination of few deformed vessels from reference ones, e.g. pipes. We propose a solution approach that is particularly suitable for the study of internal flows in hierarchical parametrized geometries. The main motivation is for applications requiring rapid and reliable numerical simulations of problems in domains involving parametrized complex geometries. The classical reduced basis (RB) method is very effective to address viscous flows equations in parametrized geometries (see, e.g., [10]). An interesting alternative foresees a combination of RB with a domain decomposition approach. In this respect, preliminary efforts to reduce the global parametrized problem to local ones have led to the introduction of the so-called reduced basis element method to solve the Stokes problem [6], and more recently to the reduced basis hybrid method [3] and to the static condensation method [7]. In general, we are interested in defining a method able to maintain the flexibility of dealing with arbitrary combinations of subdomains and several geometrical deformations of the latter. A further new contribution to this field is the computation of the reduced basis functions through an optimization greedy algorithm

Didymodon rigidulus subsp. verbanus

Bryophyte

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