Finite & Boundary Element Meth in Eng

Description

The interest in finite element method as a solution technique of the computer age is reflected in the availability of many general and special purpose software based on this technique. This work aims to provide a complete and detailed explanation of the basics of the application areas. Cover Half Title Title Page Copyright Page Dedication Preface Acknowledgement Table of Contents Nomenclature 1. Introduction and Basic Concepts 1.1 Introduction 1.2 Finite Element Method 1.3 Boundary Element Method 1.4 Finite Element Implementation 1.4.1 Force equilibrium approach 1.4.2 Assembly procedure 1.4.3 Formulation using potential energy minimization 1.4.4 Other approaches References 2. Elastic Stress Analysis Using Linear Elements 2.1 Nature of Loading 2.1.1 Concentrated or distributed loads 2.1.2 Body force (gravity etc.) 2.1.3 Loading due to thermal strains etc. 2.1.4 Residual stresses 2.2 Two-Dimensional Analysis 2.2.1 Strain displacement relation 2.2.2 Stress-strain relation 2.2.3 Potential energy 2.3 Three-Dimensional Analysis 2.3.1 Numbering sequence for nodes of elements 2.3.2 Shape function 2.3.3 Strain displacement relation 2.3.4 Stress-strain relation 2.3.5 Solution 2.4 Axi-symmetric Analysis 2.4.1 Shape function 2.4.2 Strain displacement relation 2.4.3 Stress-strain relation 2.4.4 Solution 2.4.5 Nature of expressions 2.4.6 Numerical integration 2.5 Illustrative Examples 2.5.1 Specifying loads and restraints 2.5.2 Stress analysis in crane hook References 3. Some Mathematical Fundamentals and Computer Algorithms 3.1 Introduction 3.2 Scalar, Vector and Tensor 3.2.1 Products of vectors 3.2.2 Summation convention and Kronecker delta 3.2.3 Gradient (or operator V) 3.2.4 Tensors 3.3 Gauss' and Green's Theorems 3.4 Matrices 3.4.1 Transpose of a matrix, square matrix 3.4.2 Matrix multiplication 3.4.3 Inverse of a matrix, solution of simultaneous equations 3.5 Solution of Matrix Equation 3.5.1 Gauss' elimination method 3.5.2 Boundary restraints 3.6 Banded Matrix Solver 3.6.1 Principle of banded solver 3.7 Computer Implementation References 4. Variational Approach and Heat-Flow Analysis (Potential Problem) 4.1 Introduction and Application Examples 4.1.1 General procedure 4.2 Fundamentals of Variational Calculus 4.2.1 Minimization of functional 4.2.2 Euler-Lagrange equation 4.3 Steady-State Analysis 4.3.1 Element characteristics 4.3.2 Solution 4.3.3 Two-dimensional analysis 4.3.4 Axi-symmetric case 4.4 Illustrative Examples 4.4.1 Cutting tool 4.4.2 Continuously cast steel billet 4.4.3 Auto-engine analysis and design 4.5 Heat-transfer coefficient References 5. Weighted Residue Technique and Unsteady-State Heat-Flow Analysis 5.1 Introduction 5.2 Weighted Residue Technique 5.2.1 Form of weighting function 5.3 Application to Steady-State Heat Flow 5.4 Unsteady-State Heat Flow 5.4.1 Shape function in time domain 5.4.2 Matrix relation 5.5 Illustrative Examples 5.5.1 Resistance spot welding 5.5.2 Heat transfer in piston-cylinder assembly References 6. Beams, Plates and Shells 6.1 Introduction 6.2 Bending of Beams 6.2.1 Analysis of beam element 6.2.2 Interelement continuity of displacement and slope—C1 continuity 6.2.3 Displacement function 6.2.4 Strain energy of deformation 6.2.5 Potential energy due to external loads 6.2.6 Stiffness relation 6.2.7 Beam element with general orientation in 3D space 6.3 Bending of Plates 6.3.1 Theory of plate bending 6.4 Finite Element Implementation 6.4.1 External work done 6.5 Other Types of Elements 6.6 Application Example References 7. Non-linear, Curved, Isoparametric Elements and Advanced Plate, Shell Elements 7.1 Introduction 7.2 Basic Requirement of Displacement Function 7.3 Natural Coordinate System 7.3.1 Higher order element shape functions 7.4 Area Coordinates 7.4.1 Higher order elements 7.4.2 Completeness requirement 7.4.3 Continuity requirement 7.5 Curved Elements 7.5.1 An alternative relation 7.5.2 Generalization of alternative relation to curved elements 7.6 Isoparametric Elements 7.6.1 Area coordinates 7.7 Stiffness Matrix 7.8 Numerical Integration 7.8.1 Gauss-Legendre quadrature 7.8.2 Extension to two or three dimensions 7.8.3 Area coordinates 7.8.4 Stiffness matrix in area coordinates 7.9 Area and Volume Integral Using Numerical Integration 7.9.1 Surface integral 7.10 Advanced Plate Elements 7.11 Quadrilateral Plate Bending Element 7.11.1 Continuity and completeness requirements 7.11.2 Elemental stiffness matrix 7.11.3 In-plane loading and shell element 7.11.4 Global stiffness matrix 7.12 9DOF Triangular Plate Bending Element 7.12.1 Displacement formulation 7.12.2 Slope formulation 7.13 Other Shell Elements 7.14 Application Examples References 8. Fluid Flow 8.1 Introduction 8.2 Governing Equations in Fluid Mechanics 8.2.1 Continuity condition 8.2.2 Momentum conservation or force equilibrium 8.2.3 Energy equation 8.2.4 Irrotationally condition 8.2.5 Constitutive equations 8.2.6 Summary 8.3 Special Forms of Governing Equation 8.3.1 Viscous flow: Navier-Stokes equation 8.3.2 Creeping viscous flow: Stokes flow 8.4 General Approach to Solution 8.5 Inviscid, Incompressible, Steady Flow 8.6 Inviscid, Incompressible, Irrotational Steady Flow 8.6.1 Two-dimensional flow: stream function 8.6.2 Finite element implementation 8.6.3 General remarks 8.7 Compressible Flow 8.8 Viscous Flow 8.8.1 Stokes flow and penalty function 8.8.2 Finite element formulation 8.9 Illustrative Examples 8.9.1 Cooling water flow in engine cylinder 8.9.2 Molten metal flow in tundish during steel melting References 9. Material Non-linearity Including Plasticity 9.1 Introduction 9.2 Reversible Non-linearity 9.2.1 Direct iteration 9.2.2 Improving convergence through use of relaxation factor 9.2.3 Newton-Raphson method 9.2.4 Newton-Raphson method for multivariable case 9.2.5 Modified Newton-Raphson method 9.2.6 Tangent matrix for heat conduction problem 9.3 Irreversible Non-Linearity (Plasticity) 9.3.1 General elastoplastic behaviour 9.3.2 Three-dimensional plasticity 9.3.3 Post-yield behaviour 9.3.4 Approach to finite element analysis 9.3.5 Another method of presenting experimental stress-plastic strain relation 9.3.6 Incremental elastoplastic stress-strain analysis 9.3.7 Iterative elastoplastic analysis and initial stress method 9.3.8 Radial return method 9.3.9 Mixed incremental and iterative approach 9.3.10 Conclusion: Elastoplastic analysis 9.4 Illustrative Examples References 10. Creeping Viscous Flow and Metal Forming 10.1 Introduction 10.2 Boundary Conditions 10.2.1 Forging 10.3 Constitutive Equations 10.3.1 Stress-strain rate relationship 10.4 Finite Element Formulation 10.4.1 Alternative formulation 10.4.2 Special boundary conditions 10.4.3 Global to local transformation 10.5 Iterative Solution and Special Procedures 10.5.1 Rigid regions 10.6 Illustrative Examples References 11. Boundary Element Method: Potential Problems 11.1 Introduction 11.2 Boundary Element Approach 11.2.1 Fundamental solution 11.2.2 Another form of boundary integral equation 11.2.3 Volume integral of w at source point 11.3 Numerical Implementation 11.3.1 Determination of Ci 11.3.2 Final Relation 11.3.3 Consideration of internal heat generation (body force term) 11.3.4 Three-dimensional analysis 11.3.5 Tackling kernel singularity 11.3.6 Axi-symmetric kernel 11.3.7 Mixed boundary condition 11.4 Analysing Time Domain (Transient Case) 11.4.1 Three-dimensional formulation 11.4.2 Numerical implementation 11.5 Illustrative Examples 11.5.1 Temperature distribution in cutting tool 11.5.2 Thermal design of blast furnace bottom 11.5.3 Laser heating and hardening References 12. Boundary Element Formulation for Elastostatic Problems 12.1 Introduction 12.2 Basic Relation 12.2.1 Boundary condition 12.2.2 Other relations 12.3 Boundary Integral Relation 12.4 Fundamental Solution 12.5 Discretization and Matrix Formulation 12.5.1 Determination of term C (P)m 12.6 Determination of Stresses 12.7 Other Cases 12.8 Illustrative Examples 12.8.1 Loose-fit, loaded pin in hole 12.8.2 Cam-tappet contact problem References 13. Adaptive Mesh Refinement and Large Problem Solvers 13.1 Introduction 13.2 Automatic Mesh Generation 13.2.1 Isoparametric coordinate mapping 13.2.2 Automatic triangulation 13.2.3 Octree-based approach 13.2.4 Element type conversion 13.3 Adaptive Mesh Refinement 13.3.1 Error norm 13.3.2 Estimating error norm in FE analysis 13.3.3 Energy error norm used for adaptive mesh refinement 13.4 Frontal Solver References Appendices 1. Area of Triangle and Volume of Tetrahedron 2. Augmented Matrix and Its Use 3. Vector Representation for Bending Moment and Rotation 4. Penalty Function and Its Application 5. Stokes Flow: Elemental Stiffness Matrix 6. Higher Order Shape Functions Index