| Description |
1 online resource (xxi, 459 pages) : illustrations (some color) |
| Series |
Interdisciplinary applied mathematics, 0939-6047 ; 37 |
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Interdisciplinary applied mathematics ; 37.
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| Contents |
Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection with Failure Initiation; Preface; Contents; List of Main Symbols; Chapter 1 Introduction; 1.1 What Is It All About?; 1.2 Principles and Assumptions; 1.3 Layout; 1.4 A Model Problem; 1.4.1 A Path-Independent Integral; 1.4.2 Orthogonality of the ̀̀Primal'' and ̀̀Dual''Eigenfunctions; 1.4.3 Particular Solutions; 1.4.4 Curved Boundaries Intersecting at the Singular Point; 1.5 The Heat Conduction Problem: Notation; 1.6 The Linear Elasticity Problem: Notation |
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Chapter 2 An Introduction to the p- and hp-Versions of the Finite Element Method2.1 The Weak Formulation; 2.2 Discretization; 2.2.1 Blending Functions, the Element Stiffness Matrix and Element Load Vector; 2.2.2 The Finite Element Space; 2.2.3 Mesh Design for an Optimal Convergence Rate; 2.3 Convergence Rates of FEMs and Their Connection to the Regularity of the Exact Solution; 2.3.1 Algebraic and Exponential Rates of Convergence; 2.3.1.1 Numerical Examples; Chapter 3 Eigenpair Computation for Two-Dimensional Heat Conduction Singularities; 3.1 Overview of Methods for Computing Eigenpairs |
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3.2 Formulation of the Modified Steklov Eigenproblem3.2.1 Homogeneous Dirichlet Boundary Conditions; 3.2.2 The Modified Steklov Eigen-problemfor the Laplace Equation with Homogeneous Neumann BCs; 3.3 Numerical Solution of the Modified Steklov Weak Eigenproblem by p-FEMs; 3.4 Examples on the Performance of the ModifiedSteklov Method; 3.4.1 A Detailed Simple Example; 3.4.2 A Crack with Homogeneous Newton BCs(Laplace Equation); 3.4.3 A V-Notch in an Anisotropic Material with Homogeneous Neumann BCs.; 3.4.4 An Internal Singular Point at the Interface of Two Materials |
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3.4.5 An Anisotropic Flux-Free Bimaterial InterfaceChapter 4 GFIFs Computation for Two-Dimensional Heat Conduction Problems; 4.1 Computing GFIFs Using the Dual Singular Function Method; 4.2 Computing GFIFs Using the Complementary Weak Form; 4.2.1 Derivation of the Complementary Weak Form; 4.2.2 Using the Complementary Weak Formulation to Extract GFIFs; 4.2.3 Extracting GFIFs Using the Complementary Weak Formulation and Approximated Eigenpairs; 4.3 Numerical Examples: Extracting GFIFs Using the Complementary Weak Form; 4.3.1 Laplace equation with Newton BCs |
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4.3.2 Laplace Equation with Homogeneous Neumann BCs: Approximate eigenpairs4.3.3 Anisotropic Heat Conduction Equation with Newton BCs; 4.3.4 An Internal point at the Interface of Two Materials; Chapter 5 Eigenpairs for Two-Dimensional Elasticity; 5.1 Asymptotic Solution in the Vicinity of a Reentrant Corner in an Isotropic Material; 5.2 The Particular Case of TF/TF BCs; 5.2.1 A TF/TF Reentrant Corner (V-Notch); 5.2.2 A TF/TF Crack; 5.2.3 A TF/TF Crack at a Bimaterial Interface; 5.3 Power-Logarithmic or Logarithmic Singularities with Homogeneous BCs |
| Summary |
This introductory and self-contained book gathers as much explicit mathematical results on the linear-elastic and heat-conduction ¡solutions in the neighborhood of singular points in two-dimensional domains, and ¡singular edges and vertices in three-dimensional domains. These are presented in an ¡engineering terminology for practical usage. The author treats the mathematical ¡¡formulations from an engineering viewpoint and presents high-order finite-element ¡methods for the computation of singular solutions in isotropic and anisotropic materials, ¡and multi-material interfaces. ¡The proper interpretation of the results in engineering practice ¡is advocated, so that the computed data can be correlated to experimental observations. ¡ The book is divided into fourteen chapters, each containing several sections. Most of it (the first nine Chapters) addresses two-dimensional domains, where only singular points exist. The solution in a vicinity of these points admits an asymptotic expansion composed of eigenpairs and associated¡generalized flux/stress intensity factors (GFIFs/GSIFs), which are being computed analytically when possible or by finite element methods otherwise. Singular points associated with weakly coupled thermoelasticity in the vicinity of singularities are also addressed and thermal GSIFs are computed. The computed data is important in engineering practice for¡predicting failure initiation in brittle¡materials on a daily basis. ¡Several failure laws¡for two-dimensional domains with V-notches are¡presented and their validity is examined by comparison to experimental observations.¡A sufficient simple and reliable condition for predicting failure¡initiation (crack formation) in micron level electronic devices, involving singular¡points, is still a topic of active research and interest, and is addressed herein. ¡ Explicit singular solutions in the vicinity of vertices and edges in three-dimensional domains are provided in the remaining five chapters. New methods for the computation of generalized edge flux/stress intensity functions along¡ singular edges are presented and demonstrated by several example¡ problems from the field of fracture mechanics; including anisotropic domains and bimaterial interfaces. Circular edges are also presented and the author concludes with¡ some remarks on open questions. This well illustrated book will appeal to both applied ¡mathematicians and engineers working in the field of fracture mechanics and¡ singularities |
| Analysis |
Mathematics |
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Engineering mathematics |
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Mechanics, applied |
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Computational Mathematics and Numerical Analysis |
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Theoretical and Applied Mechanics |
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Appl. Mathematics/Computational Methods of Engineering |
| Bibliography |
Includes bibliographical references (pages 447-456) and index |
| Notes |
English |
| Subject |
Singularities (Mathematics)
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Boundary value problems.
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Mathematics.
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Engineering.
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Mechanics.
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Mathematical Concepts
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Mathematics
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Engineering
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Mechanics
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mathematics.
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applied mathematics.
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engineering.
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mechanics (physics)
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MATHEMATICS -- Topology.
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Mechanics
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Mathematics
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Engineering
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Boundary value problems
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Singularities (Mathematics)
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| Form |
Electronic book
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| LC no. |
2011940836 |
| ISBN |
9781461415084 |
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146141508X |
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