Finite & Boundary Element Meth in Eng

✍ Scribed by O.P. Gupta

• Publisher: CRC Press
• Year: 1999
• Tongue: English
• Leaves: 496
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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