Our book

Enrich your knowledge on finite element analysis

Delve into the world of enriched finite element formulations for solving problems with discontinuities  in solid mechanics. From multiphase materials to fracture, immersed boundary problems to contact, and topology optimization, this book will teach you everything you need to know to implement enriched finite element methods on any FEA package. 

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About our book

Fundamentals of Enriched Finite Element Methods provides an overview of the different enriched finite element methods, detailed instruction on their use, and their real-world applications, recommending in what situations they are best implemented. It starts with a concise background on the theory required to understand the underlying principles behind the methods before outlining detailed instruction on implementation of the techniques in standard displacement-based finite element codes. The strengths and weaknesses of each are discussed, as are computer implementation details, including a standalone generalized finite element package, written in Python. The applications of the methods to a range of scenarios, including multiphase, fracture, multiscale, and immersed boundary (fictitious domain) problems are covered, and readers can find ready-to-use code, simulation videos, and other useful resources on the companion website of the book.

Key features

  • Reviews various enriched finite element methods, providing pros, cons, and scenarios for best use;
  • Provides step-by-step instruction on implementing these methods;
  • Covers the theory of general and enriched finite element methods.
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I have taught courses on advanced finite element analysis. Below you will find the contents of two of my courses.

Courses

Advanced finite element methods (ME46050)

The finite element method (FEM) is undoubtedly the established procedure for solving boundary and initial value problems in solid mechanics. This courses gives an in-depth introduction to the finite element method. For a simple 1D bar in elastostatics equilibrium, we derive the variational formulation and work out the resulting finite element discrete equations. This is then generalized to higher dimensions and other problems. We also study the p-version of the finite element method (p-FEM). For both methodologies, we look at a priori and a posteriori error estimates, which help us understand their convergence properties. The course covers Chapters 2 and 3 and portions of Chapter 12 of our book.

Learning objectives

  • Derive the finite element discrete equations, starting from the equilibrium equation;
  • Identify the limitations of the finite element method and understand the means to overcome them;
  • Evaluate the performance of the methodologies on academic problems;
  • Extend the software package provided to simulate complex problems.

Videos

  1. Introduction 
  2. Strong formulation for elastostatics in 1D
  3. Weak (variational) formulation
  4. Galerkin formulation
  5. Finite element equations
  6. Elastostatics in 1D: Isoparametric mapping
  7. Heat conduction: Strong, weak, and Galerkin formulations
  8. Elastostatics in higher dimensions: Strong and weak formulations
  9. The p-version of FEM in 1D
  10. h– , p– , and hp-FEM on non-smooth problems and a priori error estimates
  11. A posteriori error estimates and high-order patch tests
  12. The p-version of FEM in 2D

Enriched finite element methods (ME46080)

Modeling problems with complex or evolving geometries rapidly exposes the main pitfalls of the standard finite element method: creating geometry-fitted meshes is often a tedious and error-prone process that can take most of the time in a real-world simulation. In this advanced course on finite element analysis, we delve into enriched finite element formulations. Students will be exposed to state-of-the-art methodologies for solving challenging boundary value problems. Particular emphasis is placed in methodologies for solving problems with discontinuities, including material interfaces, cracks, and voids. For these problems, the enriched formulations are used to completely decouple the geometric features of the problem from the finite element discretization. By enriching the finite element space, it is then possible to analyze problems with the same accuracy and rate of convergence as those of standard FEM on geometry-fitted meshes. This is a hands-on course where students develop on the finite element code used in Advanced Finite Element Methods (ME46050). It covers Chapters 4, 5, and portions of Chapters 6, 7 and 12 of our book.

Learning objectives

  • Discuss state-of-the-art enriched formulations and identify problems where they outperform standard FEM;
  • Extend a standard FEM package to enrich the formulation;
  • Evaluate the different enriched methodologies and judge their performance on real problems;
  • Use the software package to simulate complex problems that require an enriched formulation.

Videos

  1. Introduction 
  2. The Generalized Finite Element Method
  3. Polynomial enrichments and Babuška’s algorithm
  4. GFEM for material interfaces
  5. Computational aspects of GFEM
  6. The Interface-enriched Generalized Finite Element Method
  7. Review of IGFEM and advantages with respect to GFEM
  8. GFEM for fracture mechanics
  9. The Discontinuity Enriched Finite Element Method
  10. Introduction to Partition of Unity methods