Linked e-resources
Details
Table of Contents
Preface
1. Introduction
1.1 Problem definition
1.2 Overall objectives
1.3 Specific tasks
1.4 The central role of the interpolation functions
1.5 A closer look at the interpolation functions
1.6 Physically interpretable interpolation functions in action
1.7 The overall significance of the physically interpretable notation
1.8 Examples of model refinement and the need for adaptive refinement
1.9 Examples of adaptive refinement and error analysis
1.10 Summary
1.11 References
2. An overview of finite element modeling characteristics
2.1 Introduction
2.2 Characteristics of exact finite element results
2.3 More demanding loading conditions
2.4 Discretization errors in an initial model
2.5 Error reduction and uniform refinement
2.6 Error reduction and adaptive refinement
2.7 The effect of element modeling capability on discretization errors
2.8 Summary and future applications
2.9 References
2A. Elements of two-dimensional modeling
2A1. Introduction
2A2. Submodeling refinement strategy
2A3. Initial model
2A4. Adaptive refinement results
2A5. Summary
2A6. References
2B. Exact solutions for two longitudinal bar problems
2B1. Introduction
2B2. General solution of the governing differential equation
2B3. Application of a free boundary condition
2B4. Second application of separation of variables
2B5. Solution for a constant distributed load
2B6. Solution for a linearly varying distributed load
2B7. Summary
3. Identification of finite element strain modeling capabilities
3.1 Introduction
3.2 Identification of the strain modeling capabilities of a three-node bar element
3.3 An introduction to physically interpretable interpolation polynomials
3.4 Identification of the physically interpretable coefficients
3.5 The decomposition of element displacements into strain components
3.6 A common basis for the finite element and finite difference methods
3.7 Modeling capabilities of the four-node bar element
3.8 Identification and evaluation of element behavior
3.9 Evaluation of a two-dimensional strain model
3.10 Analysis by inspection in two dimensions
3.11 Summary and conclusion
3.12 Reference
4. The source and quantification of discretization errors
4.1 Introduction
4.2 Background concepts, the residual approach to error analysis
4.3 Quantifying the failure to satisfy point-wise equilibrium
4.4 Every finite element solution is an exact solution to some problem
4.5 Summary and conclusion
4.6 Reference
5. Modeling inefficiency in irregular isoparametric elements
5.1 Introduction
5.2 An overview of isoparametric element strain modeling characteristics
5.3 Essential elements of the isoparametric method
5.4 The source of strain modeling errors in isoparametric elements
5.5 Strain modeling characteristics of isoparametric elements
5.6 Modeling errors in irregular isoparametric elements
5.7 Results for a series of uniform refinements
5.8 Summary and conclusion
5.9 References
6. Introduction to adaptive refinement
6.1 Introduction
6.2 Physically interpretable error estimators
6.3 A model refinement strategy
6.4 A demonstration of uniform refinement
6.5 A demonstration of adaptive refinement
6.6 An application of an absolute error estimator
6.7 Summary
6.8 References
7. Strain energy-based error estimators, the Z/Z error estimator
7.1 Introduction
7.2 The basis of the Z/Z error estimator, a smoothed strain representation
7.3 The Z/Z elemental strain energy error estimator
7.4 The Z/Z error estimator
7.5 A modified locally normalized Z/Z error estimator
7.6 A demonstration of the Z/Z error estimator
7.7 A demonstration of adaptive refinement
7.8 Summary and conclusion
7.9 References
7A. Gauss points, super convergent strains, and Chebyshev polynomials
7A1. Introduction
7A2. Modeling behavior of three-node elements
7A3. Gauss points and Chebyshev polynomials
7A4. References
7B. An unsuccessful example of adaptive refinement
7B1. Introduction
7B2. Example 1
7B3. Example 2
7B4. Summary
8. A high resolution point-wise residual error estimator
8.1 Introduction
8.2 An overview of the point-wise residual error estimator
8.3 The theoretical basis for the point-wise residual error estimator
8.4 Computation of the point-wise residual error estimator
8.5 Formulation of the finite difference operators
8.6 The formulation of the point-wise residual error estimator
8.7 A demonstration of the point-wise finite difference error estimator
8.8 A demonstration of adaptive refinement
8.9 A temptation to avoid and a reason for using child meshes
8.10 Summary and conclusion
8.11 Reference
9. Modeling characteristics and efficiencies of higher order elements
9.1 Introduction
9.2 Adaptive refinement examples (4.0% termination criterion)
9.3 Adaptive refinement examples (0.4% termination criterion)
9.4 In-situ identification of the five-node element modeling behavior
9.5 Strain contributions of the basis set components
9.6 Comparative modeling behavior of four-node elements
9.7 Summary, conclusion, and recommendations for future work
10. Formulation of a 10-node quadratic strain element
10.1 Introduction
10.2 Identification of the linearly independent strain gradient quantities
10.3 Identification of the elemental strain modeling characteristics
10.4 Formulation of the strain energy expression
10.5 Identification and evaluation of the required integrals
10.6 Expansion of the strain energy kernel
10.7 Formulation of the stiffness matrix
10.8 Summary and conclusion
10A. A numerical example for a 10-node stiffness matrix
10A1. Introduction
10A2. Element geometry and nodal numbering
10A3. Formulation of the transformation to nodal displacement coordinates
10A4. Formulation and evaluation of the strain energy expression
10A5. Formulation of the stiffness matrix
10A6. Summary and conclusion
10B. Matlab formulation of the 10-node element stiffness matrix
10B1. Introduction
10B2. Driver program for forming the stiffness matrix for a 10-node element
10B3. Form phi and phi inverse for 10-node element
10B4. Form integrals in stiffness matrix using Green's theorem
10B5. Form strain energy kernel for 10-node element
10B6. Plot geometry and nodes for 10-node element
10B7. Function to transform Matlab matrices to form for use in Word
11. Performance-based refinement guides
11.1 Introduction
11.2 Theoretical overview for finite difference smoothing
11.3 Development of the refinement guide
11.4 Problem description
11.5 Examples of adaptive refinement
11.6 An efficient refinement guide based on nodal averaging
11.7 Further comparisons of the refinement guides
11.8 Summary and conclusion
11.9 References
12. Summary and research recommendations
12.1 Introduction
12.2 An overview of advances in adaptive refinement
12.3 Displacement interpolation functions revisited: a reinterpretation
12.4 Advances in the finite element method
12.5 Advances in the finite difference method
12.6 Recommendations for future work and research opportunities
12.7 Reference
Index.
1. Introduction
1.1 Problem definition
1.2 Overall objectives
1.3 Specific tasks
1.4 The central role of the interpolation functions
1.5 A closer look at the interpolation functions
1.6 Physically interpretable interpolation functions in action
1.7 The overall significance of the physically interpretable notation
1.8 Examples of model refinement and the need for adaptive refinement
1.9 Examples of adaptive refinement and error analysis
1.10 Summary
1.11 References
2. An overview of finite element modeling characteristics
2.1 Introduction
2.2 Characteristics of exact finite element results
2.3 More demanding loading conditions
2.4 Discretization errors in an initial model
2.5 Error reduction and uniform refinement
2.6 Error reduction and adaptive refinement
2.7 The effect of element modeling capability on discretization errors
2.8 Summary and future applications
2.9 References
2A. Elements of two-dimensional modeling
2A1. Introduction
2A2. Submodeling refinement strategy
2A3. Initial model
2A4. Adaptive refinement results
2A5. Summary
2A6. References
2B. Exact solutions for two longitudinal bar problems
2B1. Introduction
2B2. General solution of the governing differential equation
2B3. Application of a free boundary condition
2B4. Second application of separation of variables
2B5. Solution for a constant distributed load
2B6. Solution for a linearly varying distributed load
2B7. Summary
3. Identification of finite element strain modeling capabilities
3.1 Introduction
3.2 Identification of the strain modeling capabilities of a three-node bar element
3.3 An introduction to physically interpretable interpolation polynomials
3.4 Identification of the physically interpretable coefficients
3.5 The decomposition of element displacements into strain components
3.6 A common basis for the finite element and finite difference methods
3.7 Modeling capabilities of the four-node bar element
3.8 Identification and evaluation of element behavior
3.9 Evaluation of a two-dimensional strain model
3.10 Analysis by inspection in two dimensions
3.11 Summary and conclusion
3.12 Reference
4. The source and quantification of discretization errors
4.1 Introduction
4.2 Background concepts, the residual approach to error analysis
4.3 Quantifying the failure to satisfy point-wise equilibrium
4.4 Every finite element solution is an exact solution to some problem
4.5 Summary and conclusion
4.6 Reference
5. Modeling inefficiency in irregular isoparametric elements
5.1 Introduction
5.2 An overview of isoparametric element strain modeling characteristics
5.3 Essential elements of the isoparametric method
5.4 The source of strain modeling errors in isoparametric elements
5.5 Strain modeling characteristics of isoparametric elements
5.6 Modeling errors in irregular isoparametric elements
5.7 Results for a series of uniform refinements
5.8 Summary and conclusion
5.9 References
6. Introduction to adaptive refinement
6.1 Introduction
6.2 Physically interpretable error estimators
6.3 A model refinement strategy
6.4 A demonstration of uniform refinement
6.5 A demonstration of adaptive refinement
6.6 An application of an absolute error estimator
6.7 Summary
6.8 References
7. Strain energy-based error estimators, the Z/Z error estimator
7.1 Introduction
7.2 The basis of the Z/Z error estimator, a smoothed strain representation
7.3 The Z/Z elemental strain energy error estimator
7.4 The Z/Z error estimator
7.5 A modified locally normalized Z/Z error estimator
7.6 A demonstration of the Z/Z error estimator
7.7 A demonstration of adaptive refinement
7.8 Summary and conclusion
7.9 References
7A. Gauss points, super convergent strains, and Chebyshev polynomials
7A1. Introduction
7A2. Modeling behavior of three-node elements
7A3. Gauss points and Chebyshev polynomials
7A4. References
7B. An unsuccessful example of adaptive refinement
7B1. Introduction
7B2. Example 1
7B3. Example 2
7B4. Summary
8. A high resolution point-wise residual error estimator
8.1 Introduction
8.2 An overview of the point-wise residual error estimator
8.3 The theoretical basis for the point-wise residual error estimator
8.4 Computation of the point-wise residual error estimator
8.5 Formulation of the finite difference operators
8.6 The formulation of the point-wise residual error estimator
8.7 A demonstration of the point-wise finite difference error estimator
8.8 A demonstration of adaptive refinement
8.9 A temptation to avoid and a reason for using child meshes
8.10 Summary and conclusion
8.11 Reference
9. Modeling characteristics and efficiencies of higher order elements
9.1 Introduction
9.2 Adaptive refinement examples (4.0% termination criterion)
9.3 Adaptive refinement examples (0.4% termination criterion)
9.4 In-situ identification of the five-node element modeling behavior
9.5 Strain contributions of the basis set components
9.6 Comparative modeling behavior of four-node elements
9.7 Summary, conclusion, and recommendations for future work
10. Formulation of a 10-node quadratic strain element
10.1 Introduction
10.2 Identification of the linearly independent strain gradient quantities
10.3 Identification of the elemental strain modeling characteristics
10.4 Formulation of the strain energy expression
10.5 Identification and evaluation of the required integrals
10.6 Expansion of the strain energy kernel
10.7 Formulation of the stiffness matrix
10.8 Summary and conclusion
10A. A numerical example for a 10-node stiffness matrix
10A1. Introduction
10A2. Element geometry and nodal numbering
10A3. Formulation of the transformation to nodal displacement coordinates
10A4. Formulation and evaluation of the strain energy expression
10A5. Formulation of the stiffness matrix
10A6. Summary and conclusion
10B. Matlab formulation of the 10-node element stiffness matrix
10B1. Introduction
10B2. Driver program for forming the stiffness matrix for a 10-node element
10B3. Form phi and phi inverse for 10-node element
10B4. Form integrals in stiffness matrix using Green's theorem
10B5. Form strain energy kernel for 10-node element
10B6. Plot geometry and nodes for 10-node element
10B7. Function to transform Matlab matrices to form for use in Word
11. Performance-based refinement guides
11.1 Introduction
11.2 Theoretical overview for finite difference smoothing
11.3 Development of the refinement guide
11.4 Problem description
11.5 Examples of adaptive refinement
11.6 An efficient refinement guide based on nodal averaging
11.7 Further comparisons of the refinement guides
11.8 Summary and conclusion
11.9 References
12. Summary and research recommendations
12.1 Introduction
12.2 An overview of advances in adaptive refinement
12.3 Displacement interpolation functions revisited: a reinterpretation
12.4 Advances in the finite element method
12.5 Advances in the finite difference method
12.6 Recommendations for future work and research opportunities
12.7 Reference
Index.