On the Use of Drilling Degrees of Freedom to Stabilise the Augmented Finite Element Method
Abstract
:1. Introduction
2. Finite Elements with Drilling Degrees of Freedom
2.1. Design and Formulation of Allman’s Elements
2.2. Accuracy of Allman’s Elements
2.2.1. Short Cantilever Beam Test
2.2.2. Convergence Rate in the Energy Norm
3. Augmented Finite Elements with Drilling Degrees of Freedom
3.1. Strong Form
3.2. Discretised Weak Form
3.3. Condensed Discretised Equilibrium Equations and Nested Global/Local Solving Procedure
Algorithm 1 Calculate given , and |
|
3.4. Treatment of Stiffness Matrices Singularities
3.5. Working with Various Element Shapes
4. Numerical Experiments
4.1. Traction-Free Strong Discontinuities
4.1.1. The Stabilising Effects of Drilling Degrees of Freedom
4.1.2. Convergence of the AFEM and the SAFEM in the Energy and L2 Norms
4.2. Cohesive Strong Discontinuities
4.2.1. A Discontinuous Patch Test
4.2.2. Mode I Delamination Test
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Transformation Matrices
Appendix B. Weakly Coupled Bilinear Cohesive Law
Appendix B.1. Traction–Separation Law
Appendix B.2. Damage-Opening Relation
Appendix B.3. Mixed-Mode Failure Criterion
Appendix B.4. Tangent Cohesive Stiffness
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Regular Mesh | Distorted Mesh | Number of DOFs | |
---|---|---|---|
exact solution | 10. | 10. | - |
T3 | 2.55017 | 1.4409 | 20 |
T3A T3A | 7.65647 | 5.60141 | 30 |
T6 | 9.85754 | 9.35355 | 54 |
Q4 | 6.85744 | 4.02667 | 20 |
Q4A | 9.45564 | 9.15315 | 30 |
Q4A | 9.45511 | 9.15279 | 30 |
Q8 | 9.88888 | 9.91559 | 46 |
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Essongue, S.; Couégnat, G.; Martin, E. On the Use of Drilling Degrees of Freedom to Stabilise the Augmented Finite Element Method. Appl. Mech. 2023, 4, 1140-1171. https://doi.org/10.3390/applmech4040059
Essongue S, Couégnat G, Martin E. On the Use of Drilling Degrees of Freedom to Stabilise the Augmented Finite Element Method. Applied Mechanics. 2023; 4(4):1140-1171. https://doi.org/10.3390/applmech4040059
Chicago/Turabian StyleEssongue, Simon, Guillaume Couégnat, and Eric Martin. 2023. "On the Use of Drilling Degrees of Freedom to Stabilise the Augmented Finite Element Method" Applied Mechanics 4, no. 4: 1140-1171. https://doi.org/10.3390/applmech4040059