Fundamentals of Enriched Finite Element Methods

  Expected Feb 2022.

We recently signed a contract with Elsevier for the publishing of a book devoted to enriched finite element methods and their applications for the solution of boundary value problems in solid mechanics.

Audience

This book could serve as a reference book for a graduate course in finite element analysis (in any engineering curriculum). This book could also serve as a reference book for researchers working on numerical procedures in solid mechanics, particularly using the finite element method.

Background and purpose

In the early 90s a new family of numerical techniques emerged for the solution of partial differential equations in continuum mechanics. The key idea was to use any a priori knowledge of the solution to a problem to improve (or enrich) the approximation finite element space in a continuous Galerkin framework. The Generalized Finite Element Method became the most known member of this family, a method that has shown incredible flexibility in the construction of finite element approximations. The application of the method to the analysis of problems with discontinuities (e.g., fracture mechanics) was later known as the eXtended Finite Element Method (XFEM), thus providing a means to decouple the problem’s discontinuities from the finite element discretization. This book aims at filling a gap in the existing literature regarding GFEM and other enriched finite element methods. These methods have achieved the required maturity and thus have been implemented in several mainstream FEM software like Abaqus. Consequently, these advanced discretization techniques are used not only from an academic perspective but also by practitioners from several industries, including, among others, automotive, aerospace, and medical. We provide a survey on the different methods under the umbrella of enriched FEM. We discuss the theory thoroughly, and also provide details on the implementation of these techniques in standard displacement-based finite element codes. Finally, we discuss the application of the several techniques to a range of applications, including multi-phase, fracture, multiscale, and immerse boundary (fictitious domain) problems.