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How to Use Elements Effectively



The main aim of this 'How To' book is to explain the issues involved in designing suitable meshes and selecting appropriate elements for solving such problems. The emphasis is on using the more popular types of element in elastic conditions, although the techniques and mechanics of actual mesh generation software are not covered.

Good use of elements is central to practical finite element application, but coverage of this subject is sparse compared to that of the underlying theories of the method. In fact, it is a considerably more nebulous subject and more complicated to explain. Hence, the present work, based on the author's personal experience, is somewhat subjective and does not profess to establish any definitive format for this subject.

The discussions and presentation of finite element theory involved are aimed at the level of the graduate in engineering or a related discipline, who is one year into a professional engineering career, but with a wide audience of practising and potential finite element users also in mind. Some basic knowledge of the finite element method is assumed.


Contents

1. Introduction 

2. The Main Types of Element 

  • 2.1 Introduction 
  • 2.2 The Main Element Families 
    • 2.2.1 Serendipity Quadrilaterals 
    • 2.2.2 Lagrangian Quadrilaterals 
    • 2.2.3 Triangular Elements
    • 2.2.4 Other Element Families 
  • 2.3 Basic Element Behaviour 
  • 2.4 Numerical Integration 
  • 2.5 Reduced Integration 
  • 2.6 Example of Numerical Integration Aspects 
  • 2.7 Jacobian Transformations 
  • 2.8 Geometric Representation of detJ 
  • 2.9 Summary Comments 

3. Comparison of Main Element Types 

  • 3.1 Introduction 
  • 3.2 Stresses
  • 3.3 Constant Stress versus Linear Stress Elements 
  • 3.4 Triangles and Tetrahedra versus Quadrilaterals and Hexahedra 
  • 3.5 Incompatible Elements
    • 3.5.1 Poor Performance of Linear Displacement Elements 
    • 3.5.2 Improvements Using Incompatible Formulations 
    • 3.5.3 Example of Incompatible Elements 
  • 3.6 Performance Comparisons of Element Types 
    • 3.6.1 The Use of Benchmark Tests 
    • 3.6.2 Examples of Element Convergence 
    • 3.6.3 The Benefits of Benchmark Tests 
  • 3.7 Summary Comments

4. Mesh Design Considerations 

  • 4.1 Introduction 
  • 4.2 Sufficiency of Structure to be Analysed
  • 4.3 Boundary and Load Considerations 
  • 4.4 Mesh Convergence, h and p-Type 
  • 4.5 Pre-processor Mesh Techniques 
  • 4.6 Self-Adaptive Re-Meshing Algorithms 
  • 4.7 Numerical Round-off and Ill-Conditioning 
  • 4.8 Zero Energy Modes
  • 4.9 Basics of Good Mesh Design
    • 4.9.1 Elements to Fit the Geometric Shape 
    • 4.9.2 Not Too Many Elements, Not Too Few
    • 4.9.3 Gradations to Follow Secondary Variable (Stress) Gradients 
    • 4.9.4 Concept of Map Contours to Help Gradation Selection
    • 4.9.5 Practical Examples of the Graded Mesh Developments
  • 4.10 Summary Comments

5. On Element Shape Sensitivity

  • 5.1 Introduction 
  • 5.2 The Price of Meshing the Required Volume 
  • 5.3 Element Distortion Measures 
  • 5.4 Summary of Behaviour in 8-noded Quadrilaterals (QUAD8) 
  • 5.5 Distortion in Other Quadrilaterals
  • 5.6 Distortion in Triangles 
  • 5.7 Distortion in Other Situations
  • 5.8 Distortion in Three Dimensions 
  • 5.9 Extreme Distortions (detJ =0) 
  • 5.10 Examples of How Distortions both Generate and Reduce Solution Errors 
    • 5.10.1 Distortion Comparisons in 2D Quadrilateral Elements
    • 5.10.2 Distortion in a Parabolic Shear Case
    • 5.10.3 Examples of Advantageous Distortions
  • 5.11 Summary of Designing Meshes to Benefit from Distorted Shapes

6. The Value of Single Element and Patch Tests

  • 6.1 Introduction 
  • 6.2 A Brief History of Benchmark Testing
  • 6.3 The Patch Test
  • 6.4 Single Element Test Approaches
  • 6.5 Burrow’s Tests 
  • 6.6 Some Simple Illustrative Tests 
  • 6.7 Summary Comments

7. Mesh Quality Indicators from Results

  • 7.1 Introduction
  • 7.2 Calculation Errors
  • 7.3 Accuracy of Displacements
  • 7.4 Accuracy of Stresses 
    • 7.4.1 Nodal Stresses from Shape Functions
    • 7.4.2 Gauss Point Stresses
    • 7.4.3 Extrapolated Nodal Stress
    • 7.4.4 Superconvergent Patch Recovery
  • 7.5 Using Errors for Mesh Adaption
  • 7.6 Examples of Stress Calculation Errors
  • 7.7 Errors due to Element Distortions
    • 7.7.1 Stress Error Measures
  • 7.8 Examples of Errors due to Element Distortions
  • 7.9 Summary Comments

8. Concluding Remarks 

9. References 





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About

T.K. Hellen

First Published - June 2003

Hardback, 120 Pages