Fundamentals of Enriched Finite Element Methods

1st Edition - November 9, 2023
Authors: Alejandro M. Aragón, C. Armando Duarte
Language: English
Paperback ISBN:
9 7 8 - 0 - 3 2 3 - 8 5 5 1 5 - 0
eBook ISBN:
9 7 8 - 0 - 3 2 3 - 8 5 5 1 6 - 7

Fundamentals of Enriched Finite Element Methods provides an overview of the different enriched finite element methods, detailed instruction on their use, and their real-w… Read more

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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.

Cover image
Title page
Table of Contents
Copyright
Dedication
Preface
1: Introduction
Abstract
1.1. Enriched finite element methods
1.2. Origins and milestones of e-FEMs
References
Part One: Fundamentals
2: The finite element method
Abstract
2.1. Linear elastostatics in 1-D
2.2. The elastostatics problem in higher dimensions
2.3. Heat conduction
2.4. Problems
References
3: The p-version of the finite element method
Abstract
3.1. p-FEM in 1-D
3.2. p-FEM in 2-D
3.3. Non-homogeneous essential boundary conditions
3.4. Problems
References
4: The Generalized Finite Element Method
Abstract
4.1. Finite element approximations
4.2. Generalized FEM approximations in 1-D
4.3. Applications of the GFEM
4.4. Shifted and scaled enrichments
4.5. The p-version of the GFEM
4.6. GFEM approximation spaces
4.7. Exercises
References
5: Discontinuity-enriched finite element formulations
Abstract
5.1. A weak discontinuity in 1-D
5.2. A strong discontinuity in 1-D
5.3. Relationship to GFEM
5.4. The discontinuity-enriched FEM in multiple dimensions
5.5. Convergence
5.6. Weak and strong discontinuities
5.7. Recovery of field gradients
References
Part Two: Applications
6: GFEM approximations for fractures
Abstract
6.1. Governing equations: 3-D elasticity
6.2. GFEM approximation for fractures
6.3. Convergence of linear GFEM approximations: 2-D edge crack
6.4. Convergence of linear GFEM approximations: 3-D edge crack
References
7: Generalized enrichment functions for weak discontinuities
Abstract
7.1. Formulation
7.2. Discussion and further reading
References
8: Immerse boundary (fictitious domain) problems
Abstract
8.1. Formulation
References
9: Non-conforming mesh coupling and contact
Abstract
9.1. Formulation
9.2. Examples
9.3. Further reading
References
10: Interface-enriched topology optimization
Abstract
10.1. Formulation
10.2. Compliance minimization
10.3. Fracture resistance
10.4. Discussion and further reading
References
Part Three: Computational aspects
11: Stability of approximations
Abstract
11.1. Conditioning control of GFEM matrices
11.2. Stability of interface-enriched generalized finite element formulations
References
12: Computational aspects
Abstract
12.1. A basic FEM code structure
12.2. e-FEM considerations
References
13: Approximation theory for partition of unity methods
Abstract
13.1. Approximation theory for the GFEM
13.2. A priori error estimates for partition of unity approximations
References
A: Recollections of the origins of the GFEM
A.1. The hp-cloud method
A.2. The name partition of unity method is coined
A.3. A partition of unity method for problems with singularities
A.4. Wrapping up my PhD at UT Austin
A.5. Work on partition of unity methods at COMCO and Altair Engineering
A.6. Early work on partition of unity methods at Texas A&M University
A.7. Early work on partition of unity methods at Northwestern University
References
Index

Alejandro M. Aragón

Alejandro M. Aragón is an Associate Professor in the Department of Precision and Microsystems Engineering at Delft University of Technology in the Netherlands. His research stands at the intersection of engineering and computer science, with a primary focus on pioneering novel enriched finite element methods. This cutting-edge technology, seamlessly integrated in widely applicable software, is leveraged to address complex engineering challenges. Specifically, Dr. Aragón’s innovations have been employed in the analysis and design of a diverse spectrum of (meta)materials and structures, including biomimetic and composite materials, as well as acoustic/elastic metamaterials, photonic/phononic crystals, and even edible fracture metamaterials. Since 2015 he has also been teaching advanced courses on finite element analysis at TU Delft. Dr. Aragón boasts a strong industrial network and holds two patents for the inventive use of acoustic/elastic metamaterials and phononic crystals for noise attenuation.

Affiliations and expertise

Associate Professor, Precision and Microsystems Engineering, Delft University of Technology, Netherlands

C. Armando Duarte

C. Armando Duarte is a Professor in the Department of Civil and Environmental Engineering at the University of Illinois at Urbana-Champaign. Prior to joining the UIUC, he was an assistant professor in the Department of Mechanical Engineering at the University of Alberta, Canada, and a visiting professor in the Department of Structural Engineering at the University of Sao Paulo, Brazil. He has five years of industrial experience and has made fundamental and sustained contributions to the fields of computational mechanics and methods, in particular to development of Meshfree, Partition of Unity, and Generalized/eXtended Finite Element Methods. He proposed the first partition of unity method to solve fracture problems and pioneered the use of asymptotic solutions of elasticity equations of cracks as enrichment functions for this class of methods. His group has developed a 3D GFEM for the simulation of hydraulic fracture propagation, interaction, and coalescence, and he has published more than 95 scientific articles and book chapters, and also coedited 2 books on computational methods. He has papers featured on the ScienceDirect top 25 Hottest Articles of Computer Methods in Applied Mechanics and Engineering and Engineering Fracture Mechanics.

Affiliations and expertise

Professor, Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign Urbana, IL, USA