How to use Elements Effectively

Article number 10125654

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