Extended group finite element method

Abstract

In this paper, we develop a nonlinear reduction framework based on our recently introduced extended group finite element method. By interpolating nonlinearities onto approximation spaces defined with the help of finite elements, the extended group finite element formulation achieves a noticeable reduction in the computational overhead associated with nonlinear finite element problems. However, the problem's size still leads to long solution times in most applications. Aiming to make real-time and/or many-query applications viable, we apply model order reduction and complexity reduction techniques in order to reduce the problem size and efficiently handle the reduced nonlinear terms, respectively. For this work, we focus on the proper orthogonal decomposition and discrete empirical interpolation methods. While similar approaches based on the group finite element method only focus on semilinear problems, our proposed framework is also compatible with quasilinear problems. Compared to existing methods, our reduced models prove to be superior in many different aspects as demonstrated in three numerical benchmark problems.

Extended Group Finite Element Method for a port‐Hamiltonian Formulation of the Non‐Isothermal Euler Equations

“…This is computationally demanding, as, e.g., in the context of model reduction, products with projection matrices cannot be precomputed and the reduced quantities still depend on the full order states. To counteract this we use EGFEM, introduced in [4]. EGFEM allows to separate the state-dependent matrices into products of state-independent tensors and nonlinearity vectors, such that the nonlinearities can be efficiently handled by complexity reduction, e.g., with DEIM.…”

“…To set up the corresponding tensors we introduce the basis {η κ } 3n κ=1 . These basis functions are made up of the quadrature weights w κ and Dirac-delta functions related to the quadrature points x κ , i.e., η κ (x) = w κ δ xκ (x), see [4]. The tensor system then has the following form,…”

Numerical appraisal of the unsteady Casson fluid flow through Finite Element Method (FEM)

“…Zhang and Cu [32] explored the condensed generalized nite element method in detail. Tolle and Marheineke [33] presented the extended nite element method. Bertrand et al [34] discussed the robust and reliable nite element methods in poromechanics.…”