An Explicit Finite Element Primer

Jacob, P, and Goulding, L

First Published - July 2002
Hardback, 128 Pages

This Primer is one of many documents produced to help new and experienced analysts solve a range of problem types. It complements the books entitled A Finite Element Primer and NAFEMS Introduction to Non-linear Finite Element Analysis and is the first NAFEMS’ booklet on explicit analysis.

Explicit finite element codes have become increasingly more accessible to analysts and engineers for the solution of dynamic problems that are both non-linear and transient in nature. The functionality of these codes has also seen rapid development, as new applications are defined and older ones better understood. This primer document attempts to outline the theory behind some of the more common functionality, and, where possible to present objective methods on how best to apply it in analyses.

The document is basically divided into three sections, Chapters 1 and 2 provide an introductory overview of the evolution and philosophy on how an explicit finite element code is arranged. Chapters 3 to 6 describe the basic functionality used in the generation of a model, namely material properties, loading and boundary conditions, contact and results selection. Contact is included as a separate chapter because it was (and still is) one of the main drivers for the development of explicit codes. Chapters 7 and 8 offer some guidance on methods of optimising the computational performance of a model and on alternative solution sequences that can be used to overcome problems of extreme mesh deformation.

The primer is targeted at readers with one to two years of experience using general finite element packages. It is impossible in a short document such as this to cover all aspects of the functionality and subtleties of each and every commercial code. Consequently, we have deliberately kept the subjects of each chapter general in nature. Additionally, we have deliberately refrained from exhaustive theoretical discourse on material models, preferring to indicate texts that treat this subject in a more rigorous manner in a bibliography section. We have tried to bring out the salient points of the topics of each chapter and where possible present a description of what can go wrong with an analysis, how to identify the problems and most importantly how to correct or minimise them. Finally, the primer is not intended as a substitute for the theory and user manuals of commercial programs and we would always recommend that these be consulted as the first point of reference before commencing the solution to a problem.

Contents

1.0 Introduction 1

• 1.1 Setting the Scene 1
• 1.2 Explicit Versus Implicit Time Integration 2
• 1.2.1 Implicit Integration 2
• 1.2.2 Explicit Integration 3
• 1.3 Problems Suited to Explicit Analysis 4
• 1.4 Historical Development of Explicit Finite Element Methods 8
• 1.5 Stress Wave Propagation Example 9
• 1.6 Consistent Units of Analysis 13
• 1.7 Code Precision 15

2.0 Theoretical Overview 17

• 2.1 Background 17
• 2.2 Element Formulations 17
  • 2.2.1 Common Element Classes 17
  • 2.2.2 Reduced Element Integration 18
  • 2.2.3 Hourglass Control 20
• 2.3 Time Integration Loop 23
• 2.4 Time Integration Stability 24
• 2.5 Elemental Time Step Calculation 27
• 2.6 Damping Control 31
  • 2.6.1 Viscous Damping 31
  • 2.6.2 Artificial Bulk Viscosity 33

3.0 Material Modelling 35

• 3.1 Elastoplastic Material Response 36
  • 3.1.1 Post Yield Stress Variation with Strain 38
  • 3.1.2 Material Yield Criteria and Plastic Straining 41
  • 3.1.3 Dynamic Material Response 46
  • 3.1.4 Material Failure 50
• 3.2 Hyperelastic Material Response 52
• 3.3 Viscous Material Response 54
• 3.4 Rigid Body Response 55

4.0 Loading and Boundary Conditions 59

• 4.1 Nodal and Element Variables 59
  • 4.1.1 Follower Forces 59
  • 4.1.2 Stability Effects of Nodal Forces and Boundary Conditions 60
• 4.2 Initial Conditions 62
  • 4.2.1 Nodal Constraints with Failure Criteria 63
  • 4.2.2 Body Forces 65
• 4.3 Stress Initialisation 65
• 4.4 Transmitting Boundaries 69

5.0 Contact Modelling 71

• 5.1 Contact Definitions 72
• 5.2 Sorting Algorithms 75
• 5.3 Penetration and Contact Force Evaluation 76
  • 5.3.1 Friction Force Calculation 77
• 5.4 Special Cases of Contact 77
  • 5.4.1 Plane Rigid Surface Contact 77
  • 5.4.2 General Rigid Body Contact 78
  • 5.4.3 Tied Contact 79
  • 5.4.4 Contact Combined with Material Erosion 80
• 5.5 Potential Problems in Contact Modelling 81
  • 5.5.1 Penalty Method Contact 81
  • 5.5.2 Contact Stability 83
  • 5.5.3 Penetration Tracking 84

6.0 Selection of Analysis Results 87

• 6.1 Introduction 87
• 6.2 Data Sampling Rates 89
• 6.3 Energy Results 91
• 6.4 Nodal Results 92
• 6.5 Element Results 92

7.0 Model Optimisation 95

• 7.1 Maximising an Analysis Time Step 96
  • 7.1.1 Time Step Scaling 96
  • 7.1.2 Mass Scaling 96
• 7.2 Minimising Computational Time per Time Step 97
  • 7.2.1 Mixed Time Integration 98
  • 7.2.2 Activation and Deactivation of Analysis Functionality 101
  • 7.2.3 Rigid and Deformable Material Switching 101
  • 7.2.4 Parallel Processing 102
• 7.3 Restart File Usage 103
• 7.4 Coupling to External Analysis Codes 104

8.0 Mesh Adaptivity, Rezoning & Euler Solutions Seqs. 105

• 8.1 Mesh Adaptivity 105
• 8.2 Rezoning 107
• 8.3 ALE (Arbitrary Lagrangian Eulerian) Mesh Updating 108
• 8.4 Euler Solution Sequence 109

9.0 Concluding Remarks 111

10.0 Bibliography 113

11.0 References 117