Projection-based reduced order models for a cut finite element method in parametrized domains

Abstract

This work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modeling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail. Recent improvements go under the names of Ghost-Cell finite difference methods, Cut-Cell finite volume approach, Immersed Interface, Ghost Fluid, Volume Penalty methods, for which we refer to the review paper [1] and references within. In particular, for what concerns incompressible flows in arbitrary smooth domains, the Immersed Boundary Smooth Extension method has shown high-order convergence for the incompressible Navier-Stokes equations [4].More in detail, extended mesh finite element methods using cut elements are examined in [5,6] for stationary Stokes flow systems, as well as for Navier-Stokes. An analysis for high Reynolds numbers, independent of the local Reynolds, has been carried out in [7,8,9]. XFEM approaches in 2D and 3D Navier-Stokes are reported in [10]. Higher Reynolds number aerodynamics problems in unbounded domains, thin vortex ring and Lattice Green function Immersed Boundary methods are studied in [11,12]. Furthermore, embedded and immersed methods have been used in solving fluid structure

A Monolithic and a Partitioned, Reduced Basis Method for Fluid–Structure Interaction Problems

“…If the structure in the system undergoes large deformations, then an ALE formulation may not be the right formalism within which to study the behavior of the coupled system: indeed, it is known that, for large deformation, the ALE formalism may lead to some complications. In this case, we expect a Cut Finite Element approach to be more suited; see for example [68]: the investigation of the performance of a Cut Finite Element based RBM for FSI within a monolithic approach is currently under investigation (we refer to [8,69] for some preliminary results on computational fluid dynamics problems); to the best of our knowledge, there are no results in the literature concerning the application of Cut Finite Element discretization to the RBM within a partitioned approach instead.…”

Embedded domain Reduced Basis Models for the shallow water hyperbolic equations with the Shifted Boundary Method

“…Therefore, we have decided to preprocess each snapshot with an interpolation strategy, in order to avoid discontinuity caused by deactivated elements. Another possible option to avoid this issue would be to compute an harmonic extension from the boundary to the deactivated nodes [14]. The method proposed here has the advantage of ensuring a sufficient level of smoothness without the solve of the additional partial differential equation problem required by the harmonic extension.…”

Wavelet adaptive proper orthogonal decomposition for large-scale flow data

“…This drawback of the POD is known for transport dominated fields with large gradients and is theoretically studied in [26,40] with help of the Kolmogorov n-width. Although methods like [32] are available for parametric moving discontinuities, we obviate from using them here, since it is not within the scope of this work. However, wavelet adaptation reduces the amount of computational resources needed in favor of additional accuracy, like increased number of modes.…”

“…A full POD would be prohibitive because of its tremendous memory demand of approximately 31 TB (N b = 2 3J max N s Blocks with 0.4 MB each). It should be further noted that, to the best of our knowledge, all previous results in the literature including [9,20,24,25,32,51] have been only applied to 1D or 2D cases.…”