An interface‐enriched generalized FEM for problems with discontinuous gradient fields

Abstract

SUMMARYA new generalized FEM is introduced for solving problems with discontinuous gradient fields. The method relies on enrichment functions associated with generalized degrees of freedom at the nodes generated from the intersection of the phase interface with element edges. The proposed approach has several advantages over conventional generalized FEM formulations, such as a lower computational cost, easier implementation, and straightforward handling of Dirichlet boundary conditions. A detailed convergence study of the proposed method and a comparison with the standard FEM are presented for heat transfer problems. The method achieves the optimal rate of convergence using meshes that do not conform to the interfaces present in the domain while achieving a level of accuracy comparable to that of the standard FEM with conforming meshes. Various application problems are presented, including the conjugate heat transfer problem encountered in microvascular materials.

A gradient-based shape optimization scheme using an interface-enriched generalized FEM

“…A recent addition to the family of Generalized/Extended Finite Element Methods (G/XFEM), the IGFEM was introduced by Soghrati et al [32,33] 75 to capture gradient discontinuities present along material interfaces using nonconforming meshes. In the IGFEM, the enrichment functions and the associated generalized dofs are created by inserting nodes at the intersections of the interface with the boundaries of the non-conforming elements [32].…”

“…In the IGFEM, the enrichment functions and the associated generalized dofs are created by inserting nodes at the intersections of the interface with the boundaries of the non-conforming elements [32]. This contrasts with conventional G/XFEM where the generalized dofs are associated with du-80 plicated nodes of the non-conforming elements [39,40].…”

“…The temperature field in each element intersected by an interface takes the form [32,44] Comput Mech (2007) The derivative of u(x) can be defined everywhere except on the interface itself. The interface is assumed to be smooth; strictly speaking, it is a bounded (n − 1)-dimensional smooth manifold embedded in R n .…”

“…These Eulerian methods incorporate mesh-free [20,21], fictitious domain [22,17,23,15,19], material perturbation [24], level-set [25,16,18,26], and the generalized/extended finite element methods (G/XFEM) [27,28,16,29,30,31]. 20 In this paper, we present an Eulerian based shape optimization scheme that incorporates the Interface-enriched Generalized Finite Element Method (IGFEM) [32,33]. Like its G/XFEM counterparts, the IGFEM is formulated on a fixed mesh, thereby eliminating the mesh distortion problems associated with Lagrangian shape optimization schemes.…”

Embedded solids of any dimension in the X-FEM. Part II – Imposing Dirichlet boundary conditions

“…This approach developed in [21,22] for the case of multi-material and known as the I-GFEM (Interface-GFEM), takes its name from the shift of the new DOFs -called generalised DOFs -to the interfaces. The enforcement of a Dirichlet BC can be done in the classical way of the FEM, using collocation technique thanks to the DOFs and/or generalised DOFs on the interface [23].…”