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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
PARTITION OF UNITY METHODS
Master the latest tool in computational mechanics with this brand-new resource from distinguished leaders in the field
While it is the number one tool for computer aided design and engineering, the finite element method (FEM) has difficulties with discontinuities, singularities, and moving boundaries. Partition of unity methods addresses these challenges and is now increasingly implemented in commercially available software. Partition of Unity Methods delivers a detailed overview of its fundamentals, in particular the extended finite element method for applications in solving moving boundary problems. The distinguished academics and authors introduce the XFEM as a natural extension of the traditional finite element method (FEM), through straightforward one-dimensional examples which form the basis for the subsequent introduction of higher dimensional problems. This book allows readers to fully understand and utilize XFEM just as it becomes ever more crucial to industry practice.
Partition of Unity Methods explores all essential topics on this key new technology, including:
- Coverage of the difficulties faced by the finite element method and the impetus behind the development of XFEM
- The basics of the finite element method, with discussions of finite element formulation of linear elasticity and the calculation of the force vector
- An introduction to the fundamentals of enrichment
- A revisitation of the partition of unity enrichment
- A description of the geometry of enrichment features, with discussions of level sets for stationary interfaces
- Application of XFEM to bio-film, gradient theories, and three dimensional crack propagation
Perfect for researchers and postdoctoral candidates working in the field of computational mechanics, Partition of Unity Methods also has a place in the libraries of senior undergraduate and graduate students working in the field. Finite element and CFD analysts and developers in private industry will also greatly benefit from this book.
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Partition of Unity Methods
Stéphane P. A. Bordas University of Luxembourg, Luxembourg, UK
Alexander Menk Robert Bosch GmbH, Germany
Sundararajan Natarajan Indian Institute of Technology Madras, India
This edition first published 2024
© 2024 John Wiley & Sons Ltd
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Set in 9.5/12.5pt STIXTwoText by Integra Software Services Pvt. Ltd, Pondicherry, India
Contents
Cover
Title Page
Copyright Page
List of Contributors
Preface
Acknowledgments
1 Introduction
1.1 The Finite Element Method
1.2 Suitability of the Finite Element Method
1.3 Some Limitations of the FEM
1.4 The Idea of Enrichment
1.5 Conclusions
References
2 A Step-by-Step Introduction to Enrichment
2.1 History of Enrichment for Singularities and Localized Gradients
2.1.1 Enrichment by the “Method of Supplementary Singular Functions”
2.1.2 Finite Element with a Singularity
2.1.3 Partition of Unity Enrichment
2.1.4 Mesh Overlay Methods
2.1.5 Enrichment for Strong Discontinuities
2.2 Weak Discontinuities for One-dimensional Problems
2.2.1 Conventional Finite Element Solution
2.2.2 eXtended Finite Element Solution
2.2.3 eXtended Finite Element Solution with Nodal Subtraction/Shifting
2.2.4 Solution
2.3 Strong Discontinuities for One-dimensional Problem
2.4 Conclusions
References
3 Partition of Unity Revisited
3.1 Completeness, Consistency, and Reproducing Conditions
3.2 Partition of Unity
3.3 Enrichment
3.3.1 Description of Geometry of Enrichment Features
3.3.2 Choice of Enrichment Functions
3.3.3 Imposition of boundary conditions
3.3.4 Numerical Integration of theWeak Form
3.4 Numerical Examples
3.4.1 One-Dimensional Multiple Interface
3.4.2 Two-Dimensional Circular Inhomogeneity
3.4.3 Infinite Plate with a Center Crack Under Tension
3.5 Conclusions
References
4 Advanced Topics
4.1 Size of the Enrichment Zone
4.2 Numerical Integration
4.2.1 Polar Integration
4.2.2 Equivalent Polynomial Integration
4.2.3 Conformal Mapping
4.2.4 Strain Smoothing in XFEM
4.3 Blending Elements and Corrections
4.3.1 Blending Between Different Partitions of Unity
4.3.2 Interpolation Error in Blending Elements
4.3.3 Addressing Blending Phenomena
4.4 Preconditioning Techniques
4.4.1 The First Preconditioner Proposed for the XFEM
4.4.2 A domain Decomposition Preconditioner for the XFEM
References
5 Applications
5.1 Linear Elastic Fracture in Two Dimensions with XFEM
5.1.1 Inclined Crack in Tension
5.1.2 Example of a Crack Inclusion Interaction Problem
5.1.3 Effect of the Distance Between the Crack and the Inclusion
5.2 Numerical Enrichment for Anisotropic Linear Elastic Fracture Mechanics
5.3 Creep and Crack Growth in Polycrystals
5.4 Fatigue Crack Growth Simulations
5.5 Rectangular Plate with an Inclined Crack Subjected to Thermo-Mechanical Loading
References
6 Recovery-Based Error Estimation and Bounding in XFEM
6.1 Introduction
6.2 Error Estimation in the Energy Norm. The ZZ Error Estimator
6.2.1 The SPR Technique
6.2.2 The MLS Approach
6.3 Recovery-based Error Estimation in XFEM
6.3.1 The SPR-CX Technique
6.3.2 The XMLS Technique
6.3.3 The MLS-CX Technique
6.3.4 On the Roles of Enhanced Recovery and Admissibility
6.4 Recovery Techniques in Error Bounding. Practical Error Bounds.
6.5 Error Estimation in Quantities of Interest
6.5.1 Recovery-based Estimates for the Error in Quantities of Interest
6.5.2 The Stress Intensity Factor as QoI: Error Estimation
References
7
ϕ
-FEM: An Efficient Simulation Tool Using Simple Meshes for Problems in Structure Mechanics and Heat Transfer
7.1 Introduction
7.2 Linear Elasticity
7.2.1 Dirichlet Conditions
7.2.2 Mixed Boundary Conditions
7.3 Linear Elasticity with Multiple Materials
7.4 Linear Elasticity with Cracks
7.5 Heat Equation
7.6 Conclusions and Perspectives
References
8 eXtended Boundary Element Method (XBEM) for Fracture Mechanics and Wave Problems
8.1 Introduction
8.2 Conventional BEM Formulation
8.2.1 Elasticity
8.2.2 HelmholtzWave Problems
8.3 Shortcomings of the Conventional Formulations
8.4 Partition of Unity BEM Formulation
8.5 XBEM for Accurate Fracture Analysis
8.5.1 Williams Expansions
8.5.2 Local XBEM Enrichment at Crack Tips
8.5.3 Results
8.5.4 Auxiliary Equations and Direct Evaluation of Stress Intensity Factors
8.5.5 Fracture in Anisotropic Materials
8.5.6 Conclusions
8.6 XBEM for ShortWave Simulation
8.6.1 Background to the Development of PlaneWave Enrichment
8.6.2 PlaneWave Enrichment
8.6.3 Evaluation of Boundary Integrals
8.6.4 Collocation Strategy and Solution
8.6.5 Results
8.6.6 Choice of Basis Functions
8.6.7 Scattering from Sharp Corners
8.7 Conditioning and its Control
8.8 Conclusions
References
9 Combined Extended Finite Element and Level Set Method (XFE-LSM) for Free Boundary Problems
9.1 Motivation
9.2 The Level Set Method
9.2.1 The Level Set Representation of the Embedded Interface
9.2.2 The Basic Level Set Evolution Equation
9.2.3 Velocity Extension
9.2.4 Level Set Function Update
9.2.5 Coupling the Level Set Method with the XFEM
9.3 Biofilm Evolution
9.3.1 Biofilms
9.3.2 Biofilm Modeling
9.3.3 Two-Dimensional Model
9.3.4 Solution Strategy
9.3.5 Variational Form
9.3.6 Enrichment Functions
9.3.7 Interface Conditions
9.3.8 Interface Speed Function
9.3.9 Accuracy and Convergence
9.3.10 Numerical Results
9.4 Conclusion
Acknowledgment
References
10 XFEM for 3D Fracture Simulation
10.1 Introduction
10.2 Governing Equations
10.3 XFEM Enrichment Approximation
10.4 Vector Level Set
10.5 Computation of Stress Intensity Factor
10.5.1 Brittle Material
10.5.2 Ductile Material
10.6 Numerical Simulations
10.6.1 Computation of Fracture Parameters
10.6.2 Fatigue Crack Growth in Compact Tension Specimen
10.7 Summary
References
11 XFEM Modeling of Cracked Elastic-Plastic Solids
11.1 Introduction
11.2 Conventional von Mises Plasticity
11.2.1 Constitutive Model
11.2.2 Asymptotic Crack Tip Fields
11.2.3 XFEM Enrichment
11.2.4 Numerical Implementation
11.2.5 Representative Results
11.3 Strain Gradient Plasticity
11.3.1 Constitutive Model
11.3.2 Asymptotic Crack Tip Fields
11.3.3 XFEM Enrichment
11.3.4 Numerical Implementation
11.3.5 Representative Results
11.4 Conclusions
References
12 An Introduction to Multiscale analysis with XFEM
12.1 Introduction
12.1.1 Types of Multiscale Analysis
12.2 Molecular Statics
12.2.1 Atomistic Potentials
12.2.2 A simple 1D Harmonic Potential Example
12.2.3 The Lennard-Jones Potential
12.2.4 The Embedded Atom Method
12.3 Hierarchical Multiscale Models of Elastic Behavior – The Cauchy-Born Rule
12.4 Current Multiscale Analysis – The Bridging Domain Method
12.5 The eXtended Bridging Domain Method
12.5.1 Simulation of a Crack Using XFEM
References
Index
End User License Agreement
List of Tables
CHAPTER 02
Table 2.1 A brief overview...
Table 2.2 Overview of enrichment...
Table 2.3 Bi-material problem...
CHAPTER 03
Table 3.1 Choice of enrichment...
CHAPTER 04
Table 4.1 Standard combination of...
Table 4.2 Material properties for..m
CHAPTER 05
Table 5.1 Normalized SIFs at...
Table 5.2 Normalized SIFs at...
Table 5.3 Elastic constants for..n
Table 5.4 Euler angles describing...
Table 5.5 Square plate with...
CHAPTER 06
Table 6.1 Comparison of features...
CHAPTER 09
Table 9.1 Errors calculated on...
CHAPTER 10
Table 10.1 Orthotropic material properties...
Table 10.2 Bi-material properties...
Table 10.3 Material properties of...
List of Illustrations
CHAPTER 01
Figure 1.1 Sources of error...
Figure 1.2 A piecewise constant...
Figure 1.3 Temperature distribution...
Figure 1.5 Mesh example...
Figure 1.7 The orthogonal (with...
Figure 1.8 The mesh size...
Figure 1.9 L-shaped domain...
Figure 1.10 Function...
Figure 1.12 Approximation along the...
Figure 1.13 Magnitude of gradient...
Figure 1.14 Domains and...
Figure 1.16 with restricted support...
Figure 1.17 A discontinuous function...
Figure 1.18 Several functions that...
Figure 1.19 Recombination of the...
Figure 1.20 Function with a...
CHAPTER 02
Figure 2.1 Two-dimensional quarter...
Figure 2.2 The representation of...
Figure 2.3 In the discontinuous...
Figure 2.4 A bi-material...
Figure 2.5 A bi-material...
Figure 2.6 Bi-material bar...
Figure 2.7 Standard shape function...
Figure 2.8 Enrichment function for...
Figure 2.9 Bi-material enrichment...
CHAPTER 03
Figure 3.1 Set of standard...
Figure 3.2 Signed distance function...
Figure 3.3 Level set description...
Figure 3.4 Solid shape represented...
Figure 3.5 Crack definition by...
Figure 3.6 A typical finite...
Figure 3.7 Weak discontinuity: different...
Figure 3.8 Heaviside function to...
Figure 3.9 Near-tip asymptotic...
Figure 3.10 Near-tip asymptotic...
Figure 3.11 Structural domain with...
Figure 3.12 Sub-triangles used...
Figure 3.13 Physical element (a...
Figure 3.14 One-dimensional bar...
Figure 3.15 Displacement along the...
Figure 3.16 Bi-material boundary...
Figure 3.17 Two-dimensional circular...
Figure 3.18 Bi-material circular...
Figure 3.19 Infinite plate with...
Figure 3.20 Contours and domain...
Figure 3.21 Plate with an...
Figure 3.22 Convergence of the...
CHAPTER 04
Figure 4.1 Schematic represesntation of...
Figure 4.2 Physical element containing...
Figure 4.3 Triangular element cut...
Figure 4.4 A rectangular grid...
Figure 4.5 Integration over an...
Figure 4.6 SFEM: description of...
Figure 4.7 Partially enriched elements...
Figure 4.8 Finite element discretization...
Figure 4.9 Avoiding problem in...
Figure 4.10 Bi-material bar...
Figure 4.11 Blending problem for...
Figure 4.12 Example of a...
Figure 4.13 Structure with slanting...
Figure 4.14 Mesh used to...
Figure 4.15 Mesh in the...
Figure 4.16 Condition number evaluated...
Figure 4.17 Relative error of...
CHAPTER 05
Figure 5.1 Inclined crack in...
Figure 5.2 Variation of stress...
Figure 5.3 A plate with...
Figure 5.4 A plate with...
Figure 5.5 Variation of non...
Figure 5.6 A plate with...
Figure 5.7 Crack tip shielding...
Figure 5.8 Notch formed by...
Figure 5.9 Polycrystalline structure with...
Figure 5.10 von Mises stress...
Figure 5.11 Mesh generation for...
Figure 5.12 Generation of random...
Figure 5.13 Grain structures used...
Figure 5.14 Model of the...
Figure 5.15 Crack development in...
Figure 5.16 Crack lengths calculated...
Figure 5.17 Finite plate with...
Figure 5.18 Crack growth rate...
Figure 5.19 Crack growth rate...
Figure 5.20 Square plate with...
Figure 5.21 Influence of the...
CHAPTER 06
Figure 6.1 Patches intersected by...
Figure 6.2 Westergaard problem. Infinite...
Figure 6.3 Sequence of structured...
Figure 6.4 Sequence of unstructured...
Figure 6.5 Evolution of the...
Figure 6.6 Local effectivity index...
Figure 6.7 Mixed Mode, non...
Figure 6.8 Diffraction criterion to...
Figure 6.9 Evolution of the...
Figure 6.10 Local effectivity index...
Figure 6.15 Local effectivity index...
Figure 6.11 Mixed Mode, non...
Figure 6.12 MLS support with...
Figure 6.13 Satisfaction of boundary...
Figure 6.14 Evolution of the...
Figure 6.16 Mixed Mode, non...
Figure 6.17 Mode I, structured...
Figure 6.18 Mode I, structured...
Figure 6.19 Global effectivity index...
Figure 6.20 SPR-CX technique...
Figure 6.21 SPR-CX technique...
Figure 6.22 Evolution of effectivity...
Figure 6.23 Elastic solid with...
Figure 6.24 Representation of a...
Figure 6.25 Domain of interest...
Figure 6.26 Equivalent forces at...
Figure 6.27 FE (left) and...
Figure 6.28 Evolution of the...
Figure 6.29 Evolution of the...
CHAPTER 07
Figure 7.1 Left: Meshes and...
Figure 7.3 Test case with...
Figure 7.4 Test case with...
Figure 7.5 Test case with...
Figure 7.8 Test case with...
Figure 7.6 Test case with...
Figure 7.7 Test case with...
Figure 7.9 Test case with...
Figure 7.10 Geometry with the...
Figure 7.11 Linear elasticity with...
Figure 7.12 Test case with...
Figure 7.13 Geometry notations to...
Figure 7.14 Test case with...
Figure 7.15 Test case with...
Figure 7.16 Test case for...
Figure 7.17 Test case for...
CHAPTER 08
Figure 8.1 Three-noded, quadratic...
Figure 8.2 (a) Detail of...
Figure 8.3 Definition of coordinate...
Figure 8.4 Example of a...
Figure 8.5 Comparison of unenriched...
Figure 8.6 Comparison of unenriched...
Figure 8.7 Curved crack in...
Figure 8.8 Comparison of solutions...
Figure 8.9 Comparison of solutions...
Figure 8.10 Errors, for the...
Figure 8.11 Convergence of the...
Figure 8.12 Converged solution: contours...
Figure 8.13 Converged solution: real...
Figure 8.14 Comparison of results...
CHAPTER 09
Figure 9.1 Illustration of weighted...
Figure 9.2 Example flow with...
Figure 9.3 Coupling of the...
Figure 9.4 The domain and...
Figure 9.5 Convergence of the...
Figure 9.6 Simulated biofilm growth...
Figure 9.7 Simulated biofilm growth...
Figure 9.8 Simulated biofilm growth...
CHAPTER 10
Figure 10.1 An illustration of...
Figure 10.2 XFEM discretized domain...
Figure 10.3 Definition of level...
Figure 10.4 Arbitrary 3D crack...
Figure 10.5 Virtually extended 3D...
Figure 10.6 A virtual domain...
Figure 10.7 Tetrahedral subdivision of...
Figure 10.8 Tetrahedral subdivision for...
Figure 10.9 A cuboid with...
Figure 10.10 A cuboid with...
Figure 10.11 J-domain position...
Figure 10.12 KI variation along...
Figure 10.13 A cuboid with...
Figure 10.14 KI variation along...
Figure 10.15 A cuboid with...
Figure 10.16 Top view of...
Figure 10.17 KI variation along...
Figure 10.18 KI variation with...
Figure 10.19 Stress contour ( in...
Figure 10.20 KI variation with...
Figure 10.21 A bi-material...
Figure 10.22 KI, KII, and...
Figure 10.23 KI, KII, and...
Figure 10.24 A schematic of...
Figure 10.25 A comparison of...
Figure 10.26 Numerically predicted crack...
CHAPTER 11
Figure 11.1 The von Mises...
Figure 11.2 HHR solution (Hutchinson...
Figure 11.3 Arbitrary contour around...
Figure 11.4 Mode I fracture...
Figure 11.5 Mode I fracture...
Figure 11.6 Mode I fracture...
Figure 11.7 Mixed-mode fracture...
Figure 11.8 Mixed-mode fracture...
Figure 11.9 Mixed-mode fracture...
Figure 11.10 MSG plasticity. Angular...
Figure 11.11 Schematic diagram of...
Figure 11.12 Typical XFEM mesh...
Figure 11.13 Boundary value problem...
Figure 11.14 Mesh sensitivity analysis...
Figure 11.15 Representative finite element...
Figure 11.16 Mesh employed in...
Figure 11.17 XFEM results. Normalized...
Figure 11.18 XFEM results. Normalized...
Figure 11.19 Crack opening displacement...
CHAPTER 12
Figure 12.1 Illustration of a...
Figure 12.2 Illustration of a...
Figure 12.3 (a) Rigid point...
Figure 12.4 Illustration of a...
Figure 12.5 Homogeneous deformation of...
Figure 12.6 Illustration of the...
Figure 12.7 Illustration of a...
Figure 12.8 Illustration of the...
Guide
Cover
Title Page
Copyright Page
Table of Contents
List of Contributors
Preface
Acknowledgments
Begin Reading
Author Index
End User License Agreement
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List of Contributors
Stéphane CotinUniversite de Strasburg, Strasburg France
Ravindra DudduSchool of Engineering, Vanderbilt University Nashville, USA
Michel DuprezUniversite de Strasburg, Strasburg France
Octavio Andrés González-EstradaEscuela de Ingeniería Mecánica Colombia
Juan Jose Ródenas GarcíaUniversidad Politecnica De Valencia Valencia, Spain
Robert GracieUniversity of Waterloo, Ontario, Canada
Vanessa LlerasIMAG, Univ Montpellier, CNRS, Montpellier, France
Alexei LozinskiLaboratoire de Mathématiques, Université de Franche-Comté, Besançon Cedex, France
Emilio Martinez-PañedaImperical College, London, UK
Indra Vir SinghIndian Institute of Technology Roorkee Uttarakhand, India
Jon TrevelyanDurham University, Durham, UK
Killian VuillemotLaboratoire de Mathématiques, Université de Franche-Comté, Besançon Cedex, France
Preface
This book has been a moving target for the past 13 years (2009–2022). We are delighted to see its first edition published. We tell the story of extensions to the finite element method which are now globally accepted, and implemented in industrial simulation software. The book relies on the expertise of the authors and borrows additional know-how and experience from chapters contributed from leading experts in related methods. This makes this book the most complete account of enrichment methods, both in finite elements and boundary element methods. We also discuss the critically important topic of error estimation and adaptivity as well as practical applications.
This book has a long and complex history, typical of academic research. The idea for the book was born around 2007, when Alexander Menk, then funded by Bosch GmbH, was a PhD student with Stéphane Bordas in Glasgow. The original idea was to focus on the extended finite element method, but became a lot more ambitious as the authors investigated other discretization methods, industrialized their work, and collaborated with other research groups.
We start by an introduction on the origin of enriched methods, starting with global enrichment of finite element methods or specialized enrichments (e.g. for fracture mechanics), introduced in the 1970s. We explain how local enrichment methods took precedence, with the introduction of partition of unity methods in the 1990s. We give details on a priori error estimates (Cea’s lemma), to explain how enrichment palliates limitations of polynomial approximations of non-smooth solutions.
By starting with the notion of partition of unity, we motivate how arbitrary functions can be exactly reproduced by the approximation space by multiplying this function with a partition of unity. We compare this approach to other methods, put forward within the context of meshfree methods, such as intrinsic enrichment of moving least squares.
Enrichment techniques require special treatments to support the generality of the enrichment functions used, which are not necessarily continuous, nor even polynomial. We therefore discuss and compare different options for numerical integration, in detail. The locality of enrichment implies the existence of interfaces between enriched and non-enriched regions, within the domain which can lead to decreased optimality in convergence rates, inaccuracies, and spurious oscillations in the solution close to the interface. We describe possibilities to overcome these difficulties.
After tackling these advanced topics, we discuss a wide variety of applications of enrichment schemes, including fracture mechanics, treatment of heterogeneities, and boundary layers.
In order to tackle special topics, we asked experts in free boundary problems, a posteriori error estimation, nonlinear material modeling, fracture simulations, and multiscale methods to provide the reader with up-to-date information on such important areas related to partition of unity enrichment. The book has a chapter on an exciting topic related to the enriched boundary element method for fracture and wave propagation. The book ends with a chapter on an exciting topic related to multiscale modeling of fracture.
This book is dedicated to our teachers and Professors. Stéphane Bordas thinks in particular about Monsieur Martin (math teacher at age 10 and 11), Madame Januel (math teacher at age 16), and Madame Goubet (math teacher in preparatory classes), as well as Monsieur Carsique (math teacher in preparatory classes). But, as when one teaches, two learn, this book is also dedicated to our students, in particular our PhD students, and to our team at large, who all contributed strongly to our education through their own work, research, questionings, philosophical or otherwise. We also thank our mentors Professor Ted Belytschko, Nenad Bićanić, Bhushan Karihaloo, Brian Moran, and Chris Pearce.
Acknowledgments
Writing this book on the partition of unity methods has been an ultra marathon, and as a long-distance runner, I realize the power of taking small steps. I would like to express my heartfelt gratitude to the individuals and organizations who have supported me throughout this 15-year journey.
First and foremost, I extend my appreciation to my mentors and educators who ignited my passion for excellence and instilled in me the importance of understanding the intricate details of any field. I am grateful to Monsieur Martin, Madame Januel, and Madame Goubet for their guidance during my formative years at Lycée Saint Louis in Paris. They showed me the beauty that lies in mastery and the endless quest for knowledge. Brian Moran, my PhD advisor, played a pivotal role by believing in me and introducing me to the field of computational mechanics, particularly solid mechanics. His unwavering support and guidance shaped my career.
The late Ted Belytschko remains a cherished role model whose leadership and influence continue to inspire me. I fondly remember the Friday group meetings at Northwestern University. Bhushan Karihaloo’s trust in my abilities and his decision to entrust me with the leadership of the Institute of Mechanics and Advanced Materials in Cardiff were instrumental in my professional growth. Without his support, our achievements would have been delayed significantly. Thank you for helping me discover my inner potential. I extend my gratitude to the hundreds of co-authors on our research papers who have enriched my understanding of diverse research areas. Special thanks go to my PhD students who placed their trust in me and my research group. They showed me that “when one teaches, two learn.” In particular, I thank Nguyen Vinh Phu, my first Master’s Student, with whom I just co-signed a book on the material point method.
I thank Sundararajan Natarajan who has consistently showed me alternate ways into life and with whom I had long walks, bus rides, and discussions in Glasgow, Cardiff, and Chennai. I also thank Alex Menk, with whom Sundararajan and I co-sign this book. Alex was always an example of pragmatism mixed with mathematical rigor.
My heartfelt thanks go to the funding organizations, including the EU, EPSRC, and FNR (Luxembourg), for providing over €25 million in funding for our research. I am grateful to the universities that trusted me, including the University of Glasgow, EPFL, Cardiff University, and especially the University of Luxembourg, which allowed me to flourish at the heart of Europe.
I would like to express my deep appreciation to my children, Iphigénie, Augustin, Anatole, and Oscar Bordas, who have been constant sources of inspiration. Each of you has a unique quality that reminds me of the beauty and elegance in mathematics, science, and knowledge. Iphigénie, your growth mindset, your grit, competitiveness, and humility in the face of complexity inspire me daily. Augustin, your camaraderie, simplicity, and commitment to authenticity and beauty remind me of what truly matters. Anatole, your good humor, linguistic prowess, your drive to thrive, and your unwavering positive support have been my pillars. Oscar, your perpetual optimism, love for mathematics, chess, and logic along with your infectious positivity brighten every instant of my day. I am also grateful to their mother, Laurelle Demaurex, for her support during the early years of my research group and to help me bring out the best of myself by challenging me in ways I did not anticipate I would be challenged. I also thank her father, Marc-Olivier Demaurex, for showing me the importance of reading between the lines. I also thank Beverly Johnston for daily reminders of the power of logic and sound reasoning over sophism.
Finally, I thank my parents (Pierre Bordas and Christiane Renault) and grandparents (Raymond Renault, Lucette Dusservaix, Suzanne Lebobe, and André Bordas) for their unwavering support and for introducing me to the world of hard work, mathematics, and science. They were the best pacers I could have hoped for in the race of life. I will never forget the advice given by my grandfather Raymond, his generosity, harmony, strength, and positivity which continue to live within us and within my son Oscar, who bears his name. As a conclusion, I leave you with a profound Sufi story that has resonated with me throughout my journey: “You see in people what you yourself are. People are the same everywhere. The real problem is about you. Remember this.”
In our pursuit of knowledge and excellence, we often look outward for answers and inspiration. However, this story reminds us that true understanding begins within ourselves. It is our own perspective, attitudes, and actions that shape our perception of the world and the people in it.
As we reflect on the acknowledgments and the journey chronicled in this book, may we remember that our interactions with others, our mentors, students, colleagues, and loved ones, are a reflection of our own inner world. By continually striving to improve ourselves and fostering a sense of humility, empathy, and gratitude, we can better navigate the complexities of life and contribute to the betterment of the world around us.
Thank you to everyone who has been a part of this remarkable journey. Your support, trust, and shared experiences have enriched my life and this book in immeasurable ways. With heartfelt gratitude and a commitment to ongoing growth, we would like to honor Ivo Babuška, who pioneered the partition of unity methods. His clarity and sagacity have deeply inspired me and motivated me to complete this work.
Stéphane P.A. Bordas
I would like to thank my wife Kathrin Lydia Menk and my children Julius and Magdalena for continuously supporting me with my work and for all the good times we had so far. I also would like to thank my parents and my brother for helping me become the person I am today. Last but not least I would like to thank Stéphane P. A. Bordas for giving me the opportunity to become active in academic research and Sundararajan Natarajan for all the fruitful discussions we had during our time in Glasgow.
Alexander Menk
I would like to thank my parents Geetha Natarajan and Natarajan, my wife Ramya Chandrasekaran, and son Vaibhav for continuously supporting me with my work and standing by me on numerous occasions. I admire Vaibhav’s abundant source of energy, constant thirst to learn and do new things, and especially the attitude of carrying out tasks without the fear of failure and pursuing them until success is attained. I am thankful for my sister Parvathi’s abundant love and affection. My thanks extend to Bhagirath, Nirmal, Mahalakshmi, and Chandrasekaran and Shantha Chandrasekaran for their support, trust, and encouragement. Special thanks to Young philosophers, Pranav, Ayush and Shubha. I would like to thank Stéphane P.A. Bordas for supporting me through different phases as a student, post-doctoral researcher and now as an academic researcher and Alexander Menk for all the good times we had during our time in Glasgow. Finally, I would like to thank the omnipresent for giving me the strength to swim through different hurdles.
Sundararajan Natarajan
We would like to thank our friends Professors Stéphane Cotin, Ravindra Duddu, Michel Duprez, Octavio Andrés González-Estrada, Juan José Ródenas García, Robert Gracie, Vanessa Lleras, Alexei Lozinski, Emilio Martinez-Pañeda, Indra Vir Singh, Jon Trevelyan, and Killian Vuillemot for sharing their expertise by contributing specialized topics on the partition of unity methods. We would like to thank our friends/students Chintan Jansari, Abir Elbeji, Dulce Canha, Saurabh Deshpande, Qiaoling Min, Aravind Kadhambariyil, Zhaoxiang Shen, Meryem Abbad Andaloussi, Geremy Loachamín Suntaxi, Vincent de Wit, Paris Papavasileiou, Sofia Farina, and Ehsan Mikaeili for extensive proofreading and for the numerous suggestions and corrections.
1 Introduction
Stéphane P. A. Bordas1, Alexander Menk2, and Sundararajan Natarajan3
1 University of Luxembourg, Luxembourg, UK2 Robert Bosch GmbH, Germany3 Indian Institute of Technology Madras, India
Physical systems are often modeled using partial differential equations (PDEs). The exact solution or closed form or analytical solutions to these PDEs is only available in special cases for specific geometries. Numerical methods can be used to approximate the exact solution in more general settings. The result of a numerical simulation is rarely exact. Nonetheless, computer-based numerical simulation has revolutionalized industrial product development throughout engineering disciplines. When comparing experiments and simulation with the aim of improving a simulation procedure to give more accurate results, it is necessary to understand the different sources of error. Figure 1.1 shows an overview of errors that occur at different stages of modeling and numerical simulation for a given numerical method.
Figure 1.1 Sources of error in simulation.
One of these numerical methods is called the “finite element method” (FEM). It is most commonly used in structural mechanics, although the field of application is much broader. The historic origins of the FEM cannot be uniquely determined. Mathematicians and engineers seemed to develop similar methods simultaneously which laid the foundations for what is now popularly known as the FEM. In the mathematical community, the developments can be summarized as follows. In 1851, Schellbach obtained an approximate solution to Plateaus problem by using piecewise linear functions on a surface. Variational principles to solve partial differential equations were used by Ritz in 1909. Based on the Ritz method, Courant proposed a triangulation of a two-dimensional (2D) structure to solve the plane torsion problem. The first book providing a solid mathematical basis for the FEM is attributed to Babuška and Aziz (Babuška and Aziz,1972). In the engineering community, the developments of FEM are motivated by physical analogies to describe continuous problems in a discrete fashion. Hrenikoff (Hrennikoff, 1941) combined trusses and beams to model plane elasticity problems. Turner, Clough, Martin, and Topp introduced plane elements in 1956. The term “finite element” was coined by Clough in 1960. The first book about the FEM written from an engineering perspective is attributed to Zienkiewicz (Zienkiewicz, 1971) in 1971. We will assume a certain familiarity of the reader with this method throughout the book, but we want to provide a short introduction to the FEM at this point, in order to introduce the main notations.
1.1 The Finite Element Method
Assume to be a domain of . Let us take a look at the following boundary value problem:
where is an unknown scalar field and .
Equation (1.1a) is known as Poisson’s equation when and as Laplace equation when . The open domain could be a region in , bounded by a dimensional surface whose outward normal is . We are looking for a scalar function that fulfills Poisson’s equation everywhere in . On the domain boundary , the function should be zero (c.f. Equation (1.1b)). In physics, this equation can be used to model a variety of phenomena, for example, an elasto-static rod under a torsional load, Newtonian gravity, electrostatics, diffusion, the motion of inviscid fluid, Schrödinger’s equation in Quantum mechanics, the motion of biological organisms in a solution and can also be used in surface reconstruction.
Here, let us assume that the scalar function could for instance be a temperature distribution, , that has come to an equilibrium. Then describes the heat supply inside the domain. It is easy to interpret the equation physically under these assumptions. The heat flux is proportional to , the gradient of . Because the system is in equilibrium, the sum of the heat flowing in and out of an infinitesimal subregion should be the same as the heat supplied by the heat sources inside that region. In other words, the divergence of the heat flux should be equal to at any point, which is equivalent to . Assuming a constant temperature distribution at the domain boundary could be reasonable if a material with a high thermal conductivity is attached to the region of interest. To simplify things, we postulate in Equation (1.1b) that this constant temperature is zero. Please note that once this problem is solved, one can add an arbitrary constant to and Equation (1.1a) is still fulfilled. If the problem is posed this way, then any solution must be differentiable twice in . This is a stronger condition on the choice of and moreover in many situations the solution is not differentiable twice, although the underlying physics is the same.
Let us consider an example. Assume that the temperature does not vary in the -direction. In that case, the problem can be posed in a 2D setting. Let the domain be the open unit square . A scalar function defined on the unit square is shown in Figure 1.2. The function is piecewise constant. We take this function to be the heat supply . A discontinuous heat supply is a realistic assumption in several situations. One could imagine an electric current flowing through a metal. Then the heat is generated at every point inside the metal, but not outside. Assuming that the heat generation at some point is proportional to the electrical current, a function containing jumps really is physically meaningful. Experimentally, one would measure a temperature distribution similar to the one shown in Figure 1.3.
Figure 1.2 A piecewise constant function on the unit square as an example for .
Figure 1.3 Temperature distribution.
The -component of the gradient of this temperature distribution is shown in Figure 1.4.
Figure 1.4 Gradient of the temperature distribution along the and the direction.
It is easily observed that the gradient is not differentiable at certain points. Therefore, the temperature distribution in Figure 1.3 is not a solution of Equation (1.1a), although it is the correct solution from a physical point of view.
This motivates the search for another problem description. Equation (1.1a) will subsequently be referred to as the classical formulation of the problem and a solution is called a classical solution. To obtain a new formulation, we multiply equation Equation (1.1a) by a scalar function defined on the domain and integrate over the whole domain to get:
The function is not completely arbitrary, but for now it suffices to assume certain nice properties such that we can perform the necessary integrations and differentiations in the following discussion. Applying partial integration to the left-hand side of Equation (1.2), we obtain:
It is obvious that Equation (1.3) is fulfilled for any function if is a classical solution. Let us assume that there is a function space which contains all the functions that are physically reasonable. By that we mean especially that the classical solution is in if it exists and that all the functions in vanish at the boundary1. Once such a space is known, the problem could be stated in the following abstract form, known as the weak form:
Weak form: Find , such that for all
where and are the symmetric bilinear and linear forms, respectively.
⨀ Example 1. Find the weak form for the following strong form and identify the linear and bilinear form:
where are constants independent of and subject to the following Dirichlet boundary conditions: .
⨀ Example 2. Repeat the above example, subjected to the following Dirichlet and Neumann boundary conditions: .
Using the weak form gives the function in Figure 1.3 a chance of being a solution to the problem because a solution of Equation (1.1a) needs only one time continuously differentiable. It remains to be checked under which circumstances a solution exists, and if this is a unique solution. Therefore, the term “physically meaningful,” used when initially describing , needs to be defined more precisely. To do this, we need to address the integration and the differentiation in Equation (1.3). To motivate why the classical integration and differentiation operations are not useful for our purpose we take a first step toward the numerical solution of the problem. The function space will generally have infinitely many dimensions. To compute an approximation to the exact solution one could try to solve the weak formulation in a finite dimensional subspace of , which we will denote by . The solution associated with this restricted formulation is denoted by . Without further analysis, it is not clear how is related to the solution of Equation (1.3). But one can show that minimizes the error over in a particular norm called the “energy norm.” The problem then becomes:
There exists a basis for , that is, we can write the elements of as:
Then, certainly can also be written in this form:
To determine the unknown coefficients of , we insert Equation (1.7) in Equation (1.5) and postulate that Equation (1.5) holds for all basis functions:
Because of the linearity of the integration operation, these equations are equivalent to postulating that Equation (1.5) holds for all . Due to the linearity of the gradient operator and the integration operation, we can rewrite Equation (1.8) as:
Therefore, to determine the unknown coefficients one has to solve a linear system of equations:
where
Different choices for are possible. One could choose to be the set of all global polynomials defined on up to a certain order. In that case, all points in are related to each other, and hence most of the entries of will be non-zero. It is disadvantageous if the equation systems become large, because memory storage grows with the square of the system size and solving large systems of equations using direct solver is computationally expensive .
In practice, large equation systems are solved using iterative solvers. Iterative solvers multiply a vector with at each iteration. This multiplication takes a large amount of time if is fully populated. It is much easier to deal with sparse equation systems, in which most of the entries are zero. The zero entries do not have to be stored and can be neglected when evaluating the matrix-vector product. Now the question one would try to answer is:
Can we find function spacesthat result in sparse equation systems?
Assume that is covered by a triangular mesh such as the one shown in Figure 1.5.
Figure 1.5 Mesh example.
The triangles will be called elements and the intersections of the element edges are called nodes. We could choose to be the space of all functions that are continuous in , linear inside each element, and vanish at the boundary. A basis function is defined as follows:
is 1 at the interior node, i.e., ;
is 0 at all other nodes; , ;
for each element the function values can be obtained by a linear interpolation between the function values above.
Together, if chosen as described above, the basis functions form a basis for the space . Figure 1.6 shows the basis function for a node inside a domain covered by a triangular mesh. One could also cover the domain with elements with more than three sides. This has led to the “polygonal finite element methods.” The shape functions over arbitrary elements are collectively known as “barycentric coordinates.”
Figure 1.6 A typical basis function with its nodal support.
☞ It is interesting to note that there is no unique way to represent the shape functions over polytopes. Any set of functions that satisfy the aforementioned properties is a candidature for basis functions.
With a choice of such basis functions, we would have problems evaluating the elements of matrix , since the functions are not differentiable at the element boundaries. We could neglect this and evaluate the integration and the differentiation only for points inside the elements. The resulting matrix would then be sparse. Since the basis functions are zero in most elements, the same is true for their gradient. Therefore, most of the matrix entries in Equation (1.9) are also zero. From a computational point of view this is exactly what is done in the FEM. Please note that due to the linearity of the basis functions inside each element, the evaluation of the matrix entries can be done in a few computational steps. Since the gradient of the basis functions is constant inside each element, the integrals can be computed by evaluating the function only at the midpoint (Gauss quadrature of order one).
We now return to the discussion about the differentiation and the integration of terms in Equation (1.9). We need modified operations which, in some way, neglect subsets of with measure zero, such as lines and points. With those operations we will be able to formulate the problem in a rigorous manner such that the basis functions we just discussed are part of space . For integration, this can be achieved by using the Lebesgue integral.
Lebesgue Integral
The Lebesgue integral has the same value for functions that differ only on a point set of measure zero. The Lebesgue integral can also be used with functions that are undefined on such sets and still deliver meaningful values. We will denote the space of all infinitely differentiable functions that vanish on the boundary of by . Then, the weak derivative of a scalar function is found if it fulfills for all :
The derivative is defined in the classical sense. Similarly, other partial derivatives of higher order can be defined. The weak derivative, if it exists, is unique. If the classical derivative of a function exists, the weak derivative is the same (neglecting subsets of measure zero).
With our piecewise polynomial approximation functions, the weak derivative is not defined at the element boundaries, while inside the elements it coincides with the classical derivative. Together, the Lebesgue integral and the weak derivative make it possible to evaluate Equation (1.9) for our approximation functions. Moreover, we can now define the space . This space will be denoted by and a function is an element of this space if:
can be evaluated in the Lebesgue sense and is smaller than infinity;
the weak derivative of exists;
and can be evaluated in the Lebesgue sense and are smaller than infinity;
vanishes along the boundary.
☞ The space of square-integrable functions is our starting point. Postulating that the weak partial derivatives exist and that they are square-integrable is necessary, otherwise one is not able to evaluate the weak form. Postulating that the functions vanish at the boundary is necessary to make sure that the essential boundary conditions are fulfilled.
We motivated the definition of from a physical and from a computational point of view. But the definition has much more fundamental consequences from a mathematical point of view. One can define an inner product, such that this space becomes a Hilbert space. Hilbert spaces are inner product spaces in which geometrical operations like projections can be generalized. This makes a geometric interpretation of the FEM-procedure possible. The exact solution is a vector in this space. In this Hilbert space, the existence and the uniqueness of the exact solution can be proven. But more advanced tools from functional analysis are needed for this. Space forms a hyperplane, and the numerical solution is the projection of the exact solution onto that plane. Therefore, in some sense the numerical solution is the best approximation one can get. To increase the accuracy of the numerical solution, one may, for example, increase the dimension of the hyperplane which will decrease the error. In the FEM, this is done by refining the mesh. However, since is infinite dimensional not every mesh refinement strategy results in convergence.
☉ Example 3. Using triangular elements, write a program for an FEM-solver for the Poisson problem on a rectangular square and a right-hand side given by the function in Figure 1.2. Check how the solution behaves for different element sizes.
☉ Example 4. Matlabs minres()-function is an iterative solver. Measure the time it takes to solve the FEM-equations from Exercise 1.1 directly. Compare this with the time minres() needs to solve the equation system. How do the computation times change if a sparse matrix is passed to minres()? How does this difference behave if the mesh is refined?
1.2 Suitability of the Finite Element Method
In this section, we discuss the convergence of the FEM. At the end of this section, you should be able:
to assess what the convergence rate of the FEM depends on;
to estimate the error;
to comment on how it can possibly be improved.
To do this, we refer back to the weak form in the finite dimensional space (Equation (1.5)). We are interested in the relation of to the exact solution . Defining the error due to the finite element approximation . This means that for all functions in , , or, equivalently, that the approximate solution is the orthogonal projection of the exact solution on . This is depicted geometrically in Figure 1.7, wherein, one can think of as a vectorial identity. The space where the exact solution, , lives is represented artificially as a 3D space (in reality it is a space of infinite dimensions). The finite element subspace is a plane (dimension 2). The error, , is a member of , and the finite element solution a member of . Considering the right triangle in the figure, it can be immediately seen that the length of (squared) equals the length of (squared) plus the length of , squared, i.e., 2. This is an important property, if we add a function, which is not in the span of to the basis of and thus increase its dimension, the error will always decrease.
Figure 1.7 The orthogonal (with respect to bilinear form ) projection of the exact solution on the finite element space is the finite element solution . In this figure, you can think of as a vectorial identity. In this figure, the space where the exact solution, , lives is represented artificially as a three-dimensional space (in reality it is a space of infinite dimensions). The finite element subspace is a plane (dimension 2). The error, , is a member of , and the finite element solution a member of . Considering the right triangle in the figure, it can be immediately seen that the length of (squared) equals the length of (squared) plus the length of , squared, i.e., . This identity is similar to the Pythagorean theorem in Euclidean spaces. Interpretation of the error as a projection.
Galerkin Orthogonality is the Galerkin orthogonality relation for the error. From the Galerkin orthogonality, follows that the Galerkin approximation is the “best approximant” with respect to the associated bilinear form . This best approximation property means that the finite element solution is a least square fit of the exact solution in the sense of the bilinear form .
The success of the FEM is probably due to its rich mathematical analysis and its elegant framework. This facilitates accurate a priori and a posteriori estimates of the discretization error. Here, we briefly outline the a priori error estimates. Assume that is a norm on the function space . And furthermore assume that the problem has a unique solution (i.e., appropriate essential boundary conditions are prescribed). Then, let be the order of the norm in which the error is measured, a measure of the element size3, is the polynomial order (chosen as 1 in the example above), the regularity of the exact solution , Cea’s Lemma states that the error of the finite element approximation in norm satisfies the following inequality:
Figure 1.8 The mesh size, , is the diameter of the smallest circle enclosing the largest element in the mesh.
In other words, up to a constant which is independent of the mesh size, the solution is the best possible solution in the chosen function space. If no statement can be made about the continuity of the exact solution, mesh refinement guarantees at least a linear convergence, provided the following is true:
The above condition can be rewritten as:
or, equivalently,
This means that that in order for the FEM to converge optimally in norm , we must both:
select the polynomial order larger than .
make sure that the exact solution is more regular than the order of the norm .
If the order of the continuity of the exact solution is high, increasing the polynomial order might be a better strategy than refining the mesh. However, if the regularity of the solution is low, there is no use increasing the polynomial order , because if is small, then will remain the determining term in . This is further discussed in detail in Section 1.3.
☞ A priori estimation discussed above does not account for:
numerical integration error to compute the bilinear and linear form;
interpolating Dirichlet boundary conditions;
approximating the boundary
by piecewise polynomial functions
.
1.3 Some Limitations of the FEM
We have seen earlier that the FEM possesses the best approximation property. The FEM is optimal for problems that are self-adjoint, viz., elliptic or parabolic PDEs. This is because, in such cases, it is possible to show that there exists a quadratic functional, the minimum of which corresponds to the solutions of the PDEs governing the problem. However, in certain situations (even with self-adjoint equations) calculating a numerical approximation using the FEM can become a tedious task, especially when the problem involves features that lead to singular or discontinuous solutions. In this section, we discuss this in detail through (low value of regularity in Equation (1.11)) two examples.
Problem with rough solutions Consider the domain shown in Figure 1.9. The domain has a re-entrant corner. Again, we consider a temperature distribution on this domain. The task is to find the equilibrium temperature distribution in , given the temperature distribution on the boundary . The governing differential equation is given by Equation (1.1a) with and the scalar function is assumed to be the temperature distribution. The problem can be transformed into polar coordinates with a radial component and an angular component . The mapping between the Cartesian and the polar coordinates is given by:
Figure 1.9 L-shaped domain.
It is easily checked that using polar coordinates, Equation (1.1a) becomes:
Let us consider the following function:
The function is visualized in Figure 1.10. In , this function is a solution to the Laplace equation (see Equation (1.16)). Therefore, if we use in Equation (1.16), , then becomes the (unique) solution to the Laplace equation. A numerical solution can be obtained in a similar way as previously described, although slight changes have to be made because the boundary conditions are not homogeneous. This has to be taken into account when the problem is formulated in its weak form using integration by parts.
Figure 1.10 Function .
To see why the numerical solution of this problem can become difficult, we consider along a line segment in given by and as shown in Figure 1.9. Since we are approximating the exact solution with piecewise linear functions in , the numerical solution along the line is also a piecewise linear function.
We choose a mesh as shown in Figure 1.11 to discretize the problem. In the preceding section, we already mentioned that the FEM solution can be interpreted as a projection onto a subspace of the function space in which the problem is formulated. We now try to motivate why, in this particular case, any function of the subspace is a bad approximation of the exact solution close to the re-entrant corner. To do this, we try to adjust the numerical solution to obtain a good approximation along the line . One possibility is to choose the nodal values of to be the values of the exact solution at these locations.
Figure 1.11 Mesh for the L-shaped domain.
Evaluation of along the line in that case is shown in Figure 1.12 together with the exact solution and the error. Obviously, the error is large inside the elements close to the re-entrant corner, further away, the linear approximation resolves the exact solution much better. One could try to adjust the approximation in the vicinity of the re-entrant corner to reduce the error, but obviously there is a limit to this process. The problem is related to the gradient of the solution. In the vicinity of the re-entrant corner, the first derivative is unbounded. The -component of the gradient of is shown in Figure 1.13.
Figure 1.12 Approximation along the line (see Figure 1.9).
Figure 1.13 Magnitude of gradient of in the -direction.
Therefore, in this area, deviates a lot from being a linear function and therefore it cannot be approximated by a straight line very well. One way to address this problem is to refine the mesh and use a larger number of elements. This would increase the ability to approximate the exact solution. On a smaller scale, however, similar problems would occur close to the re-entrant corner. In the presence of (weak) singularities, the approximation properties of piecewise linear functions (or more generally piecewise polynomial functions) are not sufficient. This certainly is a limitation of finite element approaches.
Conforming discretization Another limitation of the FEM is related to the mesh generation. To ensure a certain quality of the mesh, elements whose edges form sharp or obtuse angles should be avoided. Imagining a sharp triangle and the corresponding shape functions we may say that the exact solution cannot be resolved very well along the long edges. Therefore, it would be more efficient to generate a mesh with the same number of elements and equally sized element edges. When considering the deformation of a structure, an even more important reason for avoiding sharp element with large edge size ratios is a phenomena called locking. When bending or shearing occurs, sharp linear elements behave too stiff and the numerical solution is not acceptable. This is a geometric restriction on the mesh choice. Looking again at the mesh in Figure 1.11, we can see that, for this particular domain, generating a mesh with well-shaped elements is an easy task. The mesh can be generated by a very simple algorithm. But this is only possible if the geometry of the domain is simple as well and no other restrictions on the mesh choice are given.
Assume that we want to generate a high-quality mesh for a domain . Consider the geometry shown in Figure 1.14 with open domains and . In the following situations, problems may occur:
Figure 1.14 Domains and .
Case1: The domain is given by ;
Case 2: The domain is given by and the derivative of the exact solution is discontinuous along the interface;
Case 3: The domain is given by and the exact solution or its derivatives are discontinuous along the interface between and .
In all of the cases mentioned above, there would be the need to align the element edges with the interface between and . In the first case, this is necessary in order to discretize the domain appropriately. If the derivative of the exact solution is discontinuous along the interface, the numerical solution should be able to represent such a weak discontinuity. In the framework discussed so far, the derivative of the numerical solution is constant inside each element and weak discontinuities can only form at the element edges. Thus, to represent a weak discontinuity numerically the element edges should align with the interface. If the exact solution is discontinuous along the interface, the substructures and
