Handbook of Computational Solid Mechanics: Survey and Comparison of Contemporary Methods

ISBN-13: 9783642803987 / Angielski / Miękka / 2011 / 763 str.

Michal Kleiber

Opis

The objective of this book is to thoroughly document and discuss the influence of the most important computer-oriented techniques on formulating and solving boundary-value problems typical for contemporary solid and structural mechanics. The book is also intended to serve as an up-to-date introduction into current research on the subject. It will be useful to university researchers and graduate students, as well as to industrial engineers interested in effective solution methods in solid mechanics.

Spis treści

I General Introduction.- 1 On Solving Problems of Mechanics by Computer Methods.- 2 Basic Equations of Nonlinear Solid Mechanics.- 2.1 Introductory Comments.- 2.2 Description of Strain.- 2.3 State of Stress.- 2.4 Equations of Motion.- 2.5 Constitutive Equations.- 2.6 Fundamental System of Equations for Nonlinear Mechanics of Deformable Bodies.- 2.7 Variational Formulation.- 2.8 Heat Conduction.- 3 On Approximate Solving Systems of Differential Equations.- References.- II Finite Element Method.- 1 Introduction.- 2 Selected Topics from the Mathematical Theory of Finite Elements.- 2.1 Variational Formulation.- 2.2 Regularity of the Solution. Sobolev Spaces.- 2.3 Existence, Uniqueness and Regularity Results.- 2.4 Fundamental Notions of the Mathematical Theory of Finite Elements.- 2.5 Convergence Analysis for Conforming Elements. Cea’s Lemma.- 2.6 Convergence in Norm L2. The Aubin-Nitsche Argument.- 2.7 Numerical Integration. Strang’s First Lemma.- 2.8 Nonconforming Elements. Strang’s Second Lemma.- 2.9 Steady State Vibrations as an Example of a Non-coercive Problem.- 2.10 Conclusions.- 2.10.1 Relaxing Regularity Assumptions.- 2.10.2 Summary.- 3 Fundamentals of nonlinear analysis.- 3.1 Solid Mechanics Problems.- 3.2 Heat Conduction Problem.- 4 Problems of Dynamics.- 4.1 Classification of Computer Methods Used for Analysis of Dynamic Problems.- 4.2 Formulation of Equations of Motion Using Rigid Finite Element Method.- 4.3 Eigenvalue Problem.- 4.4 Fourier Transformation.- 4.5 Numerical Integration.- 4.6 The Modal Method.- 4.7 Vibration Analysis of Systems with Changing Configuration.- 4.8 Investigation of Dynamic Stability of a Linear System.- 5 Space-Time Element Method.- 5.1 Introductory Remarks and General Relations.- 5.2 Principle of Virtual Action.- 5.3 Rectangular Space-Time Elements.- 5.4 Non-rectangular Space-Time Elements.- 5.5 Uncoupled Equilibrium Equations for Impulses.- 5.6 Non-stationary Heat Flow.- 6 Plasticity Problems.- 6.1 Forms of Constitutive Equations.- 6.1.1 Introductory Comments.- 6.1.2 Thermo-Elasto-Plasticity with Isotropic Hardening...- 6.1.3 Mixed Isotropic-Kinematic Hardening.- 6.1.4 Creep and Visco-Plasticity.- 6.1.5 Elasto-Plasticity with Damage.- 6.1.6 Large Elastic-Plastic Deformations.- 6.1.7 Rigid-Plasticity and Rigid Visco-Plasticity.- 6.2 Solving Plasticity Problems by FEM.- 6.2.1 Boundary-Value Problem of Plasticity.- 6.2.2 Finite Element Equations.- 6.2.3 Time Integration of the Constitutive Equations.- 6.2.4 The Consistent (Algorithmic) Tangent Matrix.- 6.3 Computational Illustrations.- 7 Stability Problems and Methods of Analysis of FEM Equations.- 7.1 Remarks on Stability Analysis of Mechanical Systems Equilibrium States.- 7.1.1 Loads in Stability Analysis of Mechanical Systems.- 7.1.2 Equilibrium Paths and Buckling of Structures.- 7.1.3 A More General Stability Analysis of Structures.- 7.1.4 Instability under Multiple Parameter Loads.- 7.1.5 Instability of Elastic-Plastic Systems.- 7.1.6 Further Remarks.- 7.2 FEM Equations for Structural Stability.- 7.2.1 Generalisation of FE Incremental Equations.- 7.2.2 Eigenproblems in the Buckling Analysis.- 7.2.3 Extended Sets of Equations.- 7.3 Solution Methods for Nonlinear FEM Equations.- 7.3.1 Incremental Methods.- 7.3.2 Computation of Equilibrium Paths by Incremental Methods.- 7.3.3 Methods of Reduced Basis.- 7.4 Initial Buckling and Evaluation of Solution of Nonlinear and Nonconservative Problems.- 7.5 Numerical Examples.- 7.6 Final Remarks.- References.- III Finite Difference Method.- 1 Introduction.- 1.1 Formulation of Boundary-value Problems for Finite Difference Analysis.- 1.2 FDM Discretization.- 1.3 The Basic FDM Procedure.- 2 The Classical FDM.- 2.1 Domain Discretization.- 2.2 Selection of Stars and Generation of FD Operators.- 2.3 Generation of the FD Equations.- 2.4 Imposition of Boundary Conditions.- 2.5 Solution of Simultaneous FD Equations.- 2.6 Postprocessing — Evaluation of the Required Final Results...- 2.7 Numerical Examples.- 2.8 Advantages and Disadvantages of the Classical FDM.- 3 Curvilinear Finite Difference Method.- 3.1 Introduction.- 3.2 Concept of the CFD.- 3.3 CFD Formulas for Local Partial Derivatives.- 3.4 Transformation of the Local Partial Derivatives to the Global x,y Plane.- 3.5 Remarks.- 4 FD Method Generalized for Arbitrary Irregular Grids.- 4.1 Introduction.- 4.2 Basic GFDM Version.- 4.2.1 Mesh Generation and Modification, Mesh Topology Determination.- 4.2.2 Domain Partition into Nodal Subdomains — Voronoi Tessalation — Delaunay Triangulation and Mesh Topology Determination.- 4.2.3 FD Stars Selection and Classification.- 4.2.4 Local Approximation Technique and FD Stars (Schemes) Generation.- 4.2.5 Generation of the FD Equations.- 4.2.6 Discretization of Boundary Conditions.- 4.2.7 Solution of Simultaneous FD Equations.- 5 Adaptive GFDM Approach.- 5.1 A’posteriori Error Analysis.- 5.2 A’posteriori Solution Smoothing.- 5.3 Adaptive Mesh Generation and Modification.- 5.4 Higher Order Approximation FD Solution Approach.- 5.5 Multigrid Approach.- 5.5.1 Prolongation.- 5.5.2 Restriction.- 5.5.3 Numerical Example.- 5.5.4 Multigrid Adaptive Solution Procedure.- 5.6 Remarks.- 6 On Mathematical Foundations of the FDM for Elliptic Problems.- 6.1 FDM at Regular Meshes.- 6.2 Finite Difference Method at Irregular Meshes.- 7 Final Remarks.- 7.1 Meshless Methods.- References.- IV Boundary Element Method.- 1 Introduction.- 2 Mathematical Foundations.- 2.1 Formal Outline of the Method.- 2.2 Elliptic Equations of the Second Order.- 2.3 Higher-Order Elliptic Equations.- 2.4 Examples of MBIE Application.- 2.4.1 Poisson Equation.- 2.4.2 Biharmonic Equations.- 2.4.3 Equations of Linear Elasticity.- 2.5 BEM as Finite-Dimensional Approximation of MBIE.- 2.6 Supplement.- 3 BEM in Linear Theory of Elasticity.- 3.1 BEM in Statics of Elastic Medium.- 3.1.1 Boundary Integral Equations for Static Elasticity....- 3.1.2 Finite-Dimensional Approximation.- 3.1.3 Particular Form of Volume Forces.- 3.1.4 Bodies of Revolution.- 3.1.5 Anisotropic and Heterogeneous Bodies.- 3.2 BEM in Dynamics of Elastic Medium.- 3.2.1 Boundary Integral Equations for Dynamic Elasticity...- 3.2.2 Time Step Method.- 3.2.3 Integral Transformation Method.- 3.2.4 Steady-State Vibrations in Dynamic Elasticity.- 3.2.5 Alternative BEM Formulation of Dynamic Theory for Elastic Medium.- 3.3 Boundary Element Method in Theory of Plates.- 3.3.1 Statics of Plates.- 3.3.2 Dynamics of Plates.- 3.4 Numerical Examples.- 4 BEM in Nonlinear Problems.- 4.1 Geometrical Nonlinearities.- 4.2 Physical Nonlinearities.- 4.3 Other Nonlinear Problems.- 4.4 Numerical Examples.- 5 BEM in Synthesis Problems.- 5.1 Boundary Element Method in Optimal Control Problems.- 5.1.1 Optimal Control of Constraints.- 5.1.2 Optimal Control of the Boundary.- 5.2 Boundary Element Method in Sensitivity Analysis.- 5.2.1 Boundary Element Method in Sensitivity Analysis of Statically Loaded Bodies.- 5.2.2 Boundary Element Method in Sensitivity Analysis of Dynamically Loaded Bodies.- 5.2.3 Discretization of Sensitivity Boundary Integrals with Boundary Elements.- 5.2.4 Sensitivity Analysis in Boundary Shape Optimization.- 5.3 Numerical Examples.- 6 Final Remarks.- References.- V Optimization Methods.- 1 Numerical Approaches to Structural Optimization.- 1.1 Introduction.- 1.2 Optimal Design Problem Formulation.- 1.3 Mathematical Programming Methods.- 1.3.1 Problem Statement.- 1.3.2 Unconstrained Minimization.- 1.3.3 Linearly Constrained Minimization Problem.- 1.3.4 Minimization Methods for General Nonlinear Programming Problem.- 1.4 Optimality Criteria Approach.- 1.5 Concluding Remarks.- 2 Applications of Linear Programming.- 2.1 Main Notions of Mathematical Programming.- 2.2 Matrix Model of Structure.- 2.2.1 Kinematics and Equilibrium.- 2.2.2 Constitutive Relations — Linear Elastic Model.- 2.2.3 Constitutive Relations — Rigid-Plastic Model.- 2.2.4 Elastic Analysis.- 2.2.5 Ultimate Load Analysis.- 2.3 Optimum Design of Elastic Trusses.- 2.3.1 Requirements of Civil Engineering Code.- 2.3.2 General Formulation of Optimum Design Problem.- 2.3.3 Linearized Problem.- 2.3.4 Gradient Matrices.- 2.3.5 Examples.- 2.4 Optimum Plastic Design.- 2.4.1 Ultimate Yield Factor.- 2.4.2 General Formulation of Optimum Design Problem.- 2.4.3 Coefficients of Volume Function.- 2.4.4 Formulation in Redundant Stresses.- 2.4.5 Examples.- 3 Applications of Nonlinear Programming.- 3.1 Introduction.- 3.2 Design of Minimum Weight Structure as a Nonlinear Programming Problem.- 3.2.1 Equality Constraints.- 3.2.2 Inequality Constraints.- 3.3 Optimality Criteria Method....- 3.4 Explicit Formulation of Kuhn-Tucker Necessary Conditions for Minimum Weight Structures.- 3.4.1 Essence of the Method.- 3.4.2 Minimum of Structure Weight with Account for Static, Stability and Free Vibrations Problems.- 3.4.3 Scheme of the Solution Algorithm.- 3.4.4 Numerical Examples.- 3.4.5 Solution Sensitivity to Variations of Design Variables.- 4 Discrete Programming in Structural Optimization.- 4.1 Introductory Remarks.- 4.2 Graph Representing Finite Number of All Possible Structure Volumes.- 4.3 Minimum Weight Structure Made of Catalogued Members: Concept of an Algorithm.- 4.4 Numerical Examples.- References.- VI Methods of Sensitivity Analysis.- 1 Introduction.- 2 Classification of Structural Design Variables and Parameters.- 3 Fundamental Variational Theorems.- 3.1 Virtual Work Equation.- 3.2 Theorems of Potential and Complementary Energies.- 3.3 Mixed (Hybrid) Variational Principles.- 4 Sensitivity Analysis for Varying Material Parameters.- 5 Structural Shape Variation.- 5.1 Shape Description.- 5.2 Variation of Volume and Surface Integrals.- 5.3 Sensitivity Analysis with Shape Transformation.- 6 Sensitivity Analysis for Beam Structures.- 6.1 Fundamental Equations.- 6.2 Sensitivity of Potential and Complementary Energies.- 6.3 Variation of Arbitrary Functional.- 7 Sensitivity Analysis in Static and Dynamic Thermo-elasticity.- 7.1 Static Sensitivity Analysis.- 7.1.1 Material Parameter Sensitivity.- 7.1.2 Sensitivity Analysis for Shape Variation.- 7.2 Transient and Dynamic Sensitivity Analysis.- 7.2.1 Sensitivity Analysis with Respect to Material Parameters.- 7.2.2 Shape Sensitivity Analysis for Transient Thermome-chanicai Problems.- 7.3Examples.- 8 Numerical Aspects of Sensitivity Analysis.- References.

Blurb

The objective of this book is to thoroughly document and discuss the influence of the most important computer-oriented techniques on formulating and solving boundary-value problems typical for contemporary solid and structural mechanics. The book is also intended to serve as an up-to-date introduction into current research on the subject. It will be useful to university researchers and graduate students, as well as to industrial engineers interested in effective solution methods in solid mechanics. Contents: - Introductory Considerations. - The Finite Element Method. - The Finite Difference Method. - The Boundary Element Method. - Numerical Methods of Optimization. - Methods of Sensitivity Analysis.