How to Use Elements Effectively

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Details:

T.K. Hellen

First Published - June 2003
Hardback, 120 Pages

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 1

2. The Main Types of Element 5

• 2.1 Introduction 5
• 2.2 The Main Element Families 7
  • 2.2.1 Serendipity Quadrilaterals 7
  • 2.2.2 Lagrangian Quadrilaterals 8
  • 2.2.3 Triangular Elements 8
  • 2.2.4 Other Element Families 9
• 2.3 Basic Element Behaviour 10
• 2.4 Numerical Integration 15
• 2.5 Reduced Integration 18
• 2.6 Example of Numerical Integration Aspects 20
• 2.7 Jacobian Transformations 22
• 2.8 Geometric Representation of detJ 24
• 2.9 Summary Comments 25

3. Comparison of Main Element Types 27

• 3.1 Introduction 27
• 3.2 Stresses 27
• 3.3 Constant Stress versus Linear Stress Elements 30
• 3.4 Triangles and Tetrahedra versus Quadrilaterals and Hexahedra 32
• 3.5 Incompatible Elements 34
  • 3.5.1 Poor Performance of Linear Displacement Elements 34
  • 3.5.2 Improvements Using Incompatible Formulations 35
  • 3.5.3 Example of Incompatible Elements 35
• 3.6 Performance Comparisons of Element Types 37
  • 3.6.1 The Use of Benchmark Tests 37
  • 3.6.2 Examples of Element Convergence 37
  • 3.6.3 The Benefits of Benchmark Tests 40
• 3.7 Summary Comments 41

4. Mesh Design Considerations 43

• 4.1 Introduction 43
• 4.2 Sufficiency of Structure to be Analysed 43
• 4.3 Boundary and Load Considerations 45
• 4.4 Mesh Convergence, h and p-Type 46
• 4.5 Pre-processor Mesh Techniques 49
• 4.6 Self-Adaptive Re-Meshing Algorithms 50
• 4.7 Numerical Round-off and Ill-Conditioning 50
• 4.8 Zero Energy Modes 51
• 4.9 Basics of Good Mesh Design 54
  • 4.9.1 Elements to Fit the Geometric Shape 54
  • 4.9.2 Not Too Many Elements, Not Too Few 55
  • 4.9.3 Gradations to Follow Secondary Variable (Stress) Gradients 56
  • 4.9.4 Concept of Map Contours to Help Gradation Selection 59
  • 4.9.5 Practical Examples of the Graded Mesh Developments 61
• 4.10 Summary Comments 65

5. On Element Shape Sensitivity 67

• 5.1 Introduction 67
• 5.2 The Price of Meshing the Required Volume 67
• 5.3 Element Distortion Measures 68
• 5.4 Summary of Behaviour in 8-noded Quadrilaterals (QUAD8) 73
• 5.5 Distortion in Other Quadrilaterals 75
• 5.6 Distortion in Triangles 76
• 5.7 Distortion in Other Situations 76
• 5.8 Distortion in Three Dimensions 76
• 5.9 Extreme Distortions (detJ =0) 78
• 5.10 Examples of How Distortions both Generate and Reduce Solution Errors 79
  • 5.10.1 Distortion Comparisons in 2D Quadrilateral Elements 79
  • 5.10.2 Distortion in a Parabolic Shear Case 82
  • 5.10.3 Examples of Advantageous Distortions 84
• 5.11 Summary of Designing Meshes to Benefit from Distorted Shapes 85

6. The Value of Single Element and Patch Tests 87

• 6.1 Introduction 87
• 6.2 A Brief History of Benchmark Testing 87
• 6.3 The Patch Test 88
• 6.4 Single Element Test Approaches 90
• 6.5 Burrow’s Tests 91
• 6.6 Some Simple Illustrative Tests 93
• 6.7 Summary Comments 96

7. Mesh Quality Indicators from Results 97

• 7.1 Introduction 97
• 7.2 Calculation Errors 97
• 7.3 Accuracy of Displacements 98
• 7.4 Accuracy of Stresses 99
  • 7.4.1 Nodal Stresses from Shape Functions 99
  • 7.4.2 Gauss Point Stresses 100
  • 7.4.3 Extrapolated Nodal Stress 100
  • 7.4.4 Superconvergent Patch Recovery 101
• 7.5 Using Errors for Mesh Adaption 101
• 7.6 Examples of Stress Calculation Errors 102
• 7.7 Errors due to Element Distortions 104
  • 7.7.1 Stress Error Measures 104
• 7.8 Examples of Errors due to Element Distortions 106
• 7.9 Summary Comments 108

8. Concluding Remarks 109

9. References 113