Mixed enrichment for the finite element method in heterogeneous media

Abstract

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractProblems of multiple scales of interest or of locally nonsmooth solutions may often involve heterogeneous media. These problems are usually very demanding in terms of computations with the conventional finite element method. On the other hand different enriched finite element methods such as the partition of unity which proved to be very successful in treating similar problems, are developed and studied for homogeneous media. In this work we present a new idea to extend the partition of unity finite element method to treat heterogeneous materials. The idea is studied in applications to wave scattering and heat transfer problems where significant advantages are noted over the standard finite element method. Although presented within the partition of unity context the same enrichment idea can also be extended to other enriched methods to deal with heterogeneous materials.

Enriched finite elements for initial-value problem of transverse electromagnetic waves in time domain

“…The method consists of enriching the approximation space with oscillatory functions that have better approximation properties compared to the standard low order polynomials usually used in the FEM. The enrichment idea spawned a large body of literature including the work on the Partition of Unity Finite Element Method (PUFEM) [7,18,19,22] and also similar enrichment techniques such as the Generalised Finite Element Method [30,31] the Ultra-Weak Variational Formulation [14,20] and the Discontinuous Enrichment Method [9,16,34]. The enrichment approach is also used in other methods such as the Boundary Element Method [25,26,27].…”

Iterative solvers for generalized finite element solution of boundary‐value problems

“…In this example, we solve a heat equation of the form with discontinuities in the conductivity coefficient D . Similar test examples have also been considered in the work of Diwan et al The circular domain is centered at (1,1) ⊤ with unit radius formed by two materials with different heat conduction properties. Here, the solution u in the boundary‐value problem refers to the temperature media, D refers to the conduction coefficient, f ( t , x ) refers to the thermal source, ( t , x ) refers to the ambient temperature, and u 0 ( x ) refers to the initial temperature.…”

“…As a consequence, often less than 10 enrichment functions were used in the literature; see for instance the works of Mohamed et al 11 and Iqbal et al 27 However, even for a small number of enrichment functions, it can be seen in Figure 2 that, within the range 0 ≤ x ≤ 0.4, all enrichment functions are similar. In this case, if an element falls within this range, the degrees of freedom corresponding to the enrichment functions would lead to an ill-conditioned linear system or even a singular matrix in (10).…”

“…In this section, we describe the changing minimal residual solver based on the Hessenberg reduction algorithm for solving the linear system (10). The CMRH solver was first introduced by Sadok 22 and was further developed and used in the works of Alia et al, 28 Duminil, 29 and Sadok et al 30 among others.…”

“…The CMRH solver was first introduced by Sadok 22 and was further developed and used in the works of Alia et al, 28 Duminil, 29 and Sadok et al 30 among others. The CMRH solver is an alternative method to the classical GMRES algorithm developed in the work of Saad et al 23 In the works of Duminil et al 31 and Heyouni et al, 32 a new implementation of CMRH was developed for solving nonsymmetric linear systems of algebraic equations with dense and large matrices as those resulted from the generalized finite element discretization (10). Comparison studies between the CMRH and GMRES methods in terms of floating point operations and memory requirements for dense matrices can be found in the works of Sadok et al 22,30,32 among others.…”