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The Essentials of Finite Element Modeling and Adaptive Refinement

Book Description

Finite Element Analysis is a very popular, computer-based tool that uses a complex system of points called nodes to make a grid called a "mesh. " The mesh contains the material and structural properties that define how the structure will react to certain loading conditions, allowing virtual testing and analysis of stresses or changes applied to the material or component design. This groundbreaking text extends the usefulness of finite element analysis by helping both beginners and advanced users alike. It simplifies, improves, and extends both the finite element method while at the same time advancing adaptive refinement procedures. These improvements are made possible due to a change in notation that embeds knowledge of solid continuum mechanics into the equations used to formulate the stiffness matrices; this allows the modeling characteristics of individual elements to be identified by visual inspection. The ability to visually relate the equations involved in element formulation to the physical process they represent is like having an x-ray of the inner workings of the finite element method; it is similar is to the effect that Graphical User Interfaces or GUI's had on computing. As a result, students at any level of finite element study are provided with an understanding of the capabilities and limitations of this powerful analytic tool. The book presents * A more simplified approach to finite element analysis based on computational continuum mechanics * Physically interpretable notation that identifies a common basis for the finite element and the finite difference methods. * New point-wise error estimators that identify errors in terms of quantities of direct interest in solid mechanics

Table of Contents

  1. Cover
  2. Halftitle
  3. The Essentials of Finite Element Modeling and Adaptive Refinement
  4. Copyright
  5. Contents
  6. Preface
  7. 1 INTRODUCTION
    1. 1.1 Problem Definition
    2. 1.2 Overall Objectives
    3. 1.3 Specific Tasks
    4. 1.4 The Central Role of the Interpolation Functions
    5. 1.5 A Closer Look at the Interpolation Functions
    6. 1.6 Physically Interpretable Interpolation Functions in Action
    7. 1.7 The Overall Significance of the Physically Interpretable Notation
    8. 1.8 Examples of Model Refinement and the Need for Adaptive Refinement
    9. 1.9 Examples of Adaptive Refinement and Error Analysis
    10. 1.10 Summary
    11. 1.11 References
  8. 2 AN OVERVIEW OF FINITE ELEMENT MODELING CHARACTERISTICS
    1. 2.1 Introduction
    2. 2.2 Characteristics of Exact Finite Element Results
    3. 2.3 More Demanding Loading Conditions
    4. 2.4 Discretization Errors in an Initial Model
    5. 2.5 Error Reduction and Uniform Refinement
    6. 2.6 Error Reduction and Adaptive Refinement
    7. 2.7 The Effect of Element Modeling Capability on Discretization Errors
    8. 2.8 Summary and Future Applications
    9. 2.9 References
  9. 2A ELEMENTS OF TWO-DIMENSIONAL MODELING
    1. 2A.1 Introduction
    2. 2A.2 SubModeling Refinement Strategy
    3. 2A.3 Initial Model
    4. 2A.4 Adaptive Refinement Results
    5. 2A.5 Summary
    6. 2A.6 References
  10. 2B EXACT SOLUTIONS FOR TWO LONGITUDINAL BAR PROBLEMS
    1. 2B.1 Introduction
    2. 2B.2 General Solution of the Governing Differential Equation
    3. 2B.3 Application of a Free Boundary Condition
    4. 2B.4 Second Application of Separation of Variables
    5. 2B.5 Solution for a Constant Distributed Load
    6. 2B.6 Solution for a Linearly Varying Distributed Load
    7. 2B.7 Summary
  11. 3 IDENTIFICATION OF FINITE ELEMENT STRAIN MODELING CAPABILITIES
    1. 3.1 Introduction
    2. 3.2 Identification of the Strain Modeling Capabilities of a Three-Node Bar Element
    3. 3.3 An Introduction to Physically Interpretable Interpolation Polynomials
    4. 3.4 Identification of the Physically Interpretable Coefficients
    5. 3.5 The Decomposition of Element Displacements into Strain Components
    6. 3.6 A Common Basis for the Finite Element and Finite Difference Methods
    7. 3.7 Modeling Capabilities of the Four-Node Bar Element
    8. 3.8 Identification and Evaluation of Element Behavior
    9. 3.9 Evaluation of a Two-Dimensional Strain Model
    10. 3.10 Analysis by Inspection in Two Dimensions
    11. 3.11 Summary and Conclusion
    12. 3.12 Reference
  12. 4 THE SOURCE AND QUANTIFICATION OF DISCRETIZATION ERRORS
    1. 4.1 Introduction
    2. 4.2 Background Concepts—The Residual Approach to Error Analysis
    3. 4.3 Quantifying the Failure to Satisfy Point-Wise Equilibrium
    4. 4.4 Every Finite Element Solution is an Exact Solution to Some Problem
    5. 4.5 Summary and Conclusion
    6. 4.6 Reference
  13. 5 MODELING INEFFICIENCY IN IRREGULAR ISOPARAMETRIC ELEMENTS
    1. 5.1 Introduction
    2. 5.2 An Overview of Isoparametric Element Strain Modeling Characteristics
    3. 5.3 Essential Elements of the Isoparametric Method
    4. 5.4 The Source of Strain Modeling Errors in Isoparametric Elements
    5. 5.5 Strain Modeling Characteristics of Isoparametric Elements
    6. 5.6 Modeling Errors in Irregular Isoparametric Elements
    7. 5.7 Results for a Series of Uniform Refinements
    8. 5.8 Summary and Conclusion
    9. 5.9 References
  14. 6 INTRODUCTION TO ADAPTIVE REFINEMENT
    1. 6.1 Introduction
    2. 6.2 Physically Interpretable Error Estimators
    3. 6.3 A Model Refinement Strategy
    4. 6.4 A Demonstration of Uniform Refinement
    5. 6.5 A Demonstration of Adaptive Refinement
    6. 6.6 An Application of an Absolute Error Estimator
    7. 6.7 Summary
    8. 6.8 References
  15. 7 STRAIN ENERGY-BASED ERROR ESTIMATORS—THE Z/Z ERROR ESTIMATOR
    1. 7.1 Introduction
    2. 7.2 The Basis of the Z/Z Error Estimator—A Smoothed Strain Representation
    3. 7.3 The Z/Z Elemental Strain Energy Error Estimator
    4. 7.4 The Z/Z Error Estimator
    5. 7.5 A Modified Locally Normalized Z/Z Error Estimator
    6. 7.6 A Demonstration of the Z/Z Error Estimator
    7. 7.7 A Demonstration of Adaptive Refinement
    8. 7.8 Summary and Conclusion
    9. 7.9 References
  16. 7A Gauss Points, Super Convergent Strains, and Chebyshev Polynomials
    1. 7A.1 Introduction
    2. 7A.2 Modeling Behavior of Three-Node Elements
    3. 7A.3 Gauss Points and Chebyshev Polynomials
    4. 7A.4 References
  17. 7B An Unsuccessful Example of Adaptive Refinement
    1. 7B.1 Introduction
    2. 7B.2 Example 1
    3. 7B.3 Example 2
    4. 7B.4 Summary
  18. 8 A HIGH RESOLUTION POINT-WISE RESIDUAL ERROR ESTIMATOR
    1. 8.1 Introduction
    2. 8.2 An Overview of the Point-Wise Residual Error Estimator
    3. 8.3 The Theoretical Basis for the Point-Wise Residual Error Estimator
    4. 8.4 Computation of the Point-Wise Residual Error Estimator
    5. 8.5 Formulation of the Finite Difference Operators
    6. 8.6 The Formulation of the Point-Wise Residual Error Estimator
    7. 8.7 A Demonstration of the Point-Wise Finite Difference Error Estimator
    8. 8.8 A Demonstration of Adaptive Refinement
    9. 8.9 A Temptation to Avoid and a Reason for using Child Meshes
    10. 8.10 Summary and Conclusion
    11. 8.11 Reference
  19. 9 MODELING CHARACTERISTICS AND EFFICIENCIES OF HIGHER ORDER ELEMENTS
    1. 9.1 Introduction
    2. 9.2 Adaptive Refinement Examples (4.0% Termination Criterion) 161
    3. 9.3 Adaptive Refinement Examples (0.4% Termination Criterion) 165
    4. 9.4 In-Situ Identification of the Five-Node Element Modeling Behavior
    5. 9.5 Strain Contributions of the Basis Set Components
    6. 9.6 Comparative Modeling Behavior of Four-Node Elements
    7. 9.7 Summary, Conclusion, and Recommendations for Future Work
  20. 10 FORMULATION OF A 10-NODE QUADRATIC STRAIN ELEMENT
    1. 10.1 Introduction
    2. 10.2 Identification of the Linearly Independent Strain Gradient Quantities
    3. 10.3 Identification of the Elemental Strain Modeling Characteristics
    4. 10.4 Formulation of the Strain Energy Expression
    5. 10.5 Identification and Evaluation of the Required Integrals
    6. 10.6 Expansion of the Strain Energy Kernel
    7. 10.7 Formulation of the Stiffness Matrix
    8. 10.8 Summary and Conclusion
  21. 10A A NUMERICAL EXAMPLE FOR A 10-NODE STIFFNESS MATRIX
    1. 10A.1 Introduction
    2. 10A.2 Element Geometry and Nodal Numbering
    3. 10A.3 Formulation of the Transformation to Nodal Displacement Coordinates
    4. 10A.4 Formulation and Evaluation of the Strain Energy Expression
    5. 10A.5 Formulation of the Stiffness Matrix
    6. 10A.6 Summary and Conclusion
  22. 10B MATLAB FORMULATION OF THE 10-NODE ELEMENT STIFFNESS MATRIX
    1. 10B.1 Introduction
    2. 10B.2 Driver Program for Forming the Stiffness Matrix for a 10-Node Element
    3. 10B.3 Form Phi and Phi Inverse for 10-Node Element
    4. 10B.4 Form Integrals in Stiffness Matrix Using Green’s Theorem
    5. 10B.5 Form Strain Energy Kernel for 10-Node Element
    6. 10B.6 Plot Geometry and Nodes for 10-Node Element
    7. 10B.7 Function to Transform Matlab Matrices to Form for Use in Word
  23. 11 PERFORMANCE-BASED REFINEMENT GUIDES
    1. 11.1 Introduction
    2. 11.2 Theoretical Overview for Finite Difference Smoothing
    3. 11.3 Development of the Refinement Guide
    4. 11.4 Problem Description
    5. 11.5 Examples of Adaptive Refinement
    6. 11.6 An Efficient Refinement Guide Based on Nodal Averaging
    7. 11.7 Further Comparisons of the Refinement Guides
    8. 11.8 Summary and Conclusion
    9. 11.9 References
  24. 12 SUMMARY AND RESEARCH RECOMMENDATIONS
    1. 12.1 Introduction
    2. 12.2 An Overview of Advances in Adaptive Refinement
    3. 12.3 Displacement Interpolation Functions Revisited: A Reinterpretation
    4. 12.4 Advances in the Finite Element Method
    5. 12.5 Advances in the Finite Difference Method
    6. 12.6 Recommendations for Future Work and Research Opportunities
    7. 12.7 Reference
  25. Index
  26. Ad Page
  27. Back Cover