Finite Element Analysis with Adaptive Mesh Refinement (AMR)

Author: Anurag Bhattacharyya

Dependencies: The dependencies are listed in .github/workflows/Python_env.yml:

• MeshPy
• SciPy

Problem Formulation: The main problem investigated is the 2d plane stress elasticity problem for a L-shaped domain defined mathematically:

$$\begin{split} \Delta^2 u(x, y) & = -F_b \hspace{5 mm} on \hspace{5 mm} \Omega \\\ u & = 0 \hspace{5 mm} on \hspace{5 mm} \Omega_b \end{split}$$

Here $F_b$ is the body force and $\Omega_b$ represents the boundary.

Goals:

In this elasticity problem, there is also a body force acting on the entire structure in the downward direction. From elasticity studies we know that the region connecting the overhanging portion to the top  xed boundary will be highly stressed and as a result the error in these elements will be high as well. I expect that the adaptive mesh re nement process will capture these regions of high internal stresses, consequently more error prone, and the mesh will be re ned in this region to lower the peak in the internal stress values and subsequenly a reduction in the error norm.

Initial stress distribution over the domain:

Implementation:

The adaptive re nement algorithm follows a cycle of solve->estimate->mark->refine. We start with solve. A structural  nite element solver was fully developed in-house coupled with the MeshPy grid generator. Linear triangular elements was used for meshing the design domain having 2 degrees of freedom at each vertex. A body force to simulate loading due to gravity was impelmented. The undeformed mesh (blue) and the deformed mesh (green) is shown in Figure. After calculating the required values on the initial mesh, we estimate the errors in the elements. For this we use a posteriori error estimator.

After the workflow is executed the adaptively refined mesh file is updated in the repo and is visualized below along with the deformed file

Final stress distribution over the refined domain:

Comparing AMR with full domain refinement

Adaptive mesh refinement performance

D.O.Fs	time(s)	$$L_{inf}$$
96	0.02997	0.34533
508	0.15364	0.1145

Whole domain refinement performance

D.O.Fs	time(s)	$$L_{inf}$$
452	0.13805	0.3198
564	0.15363	0.2338
674	0.16926	0.2139
812	0.2161	0.2149

Additional Resources

• Finite Element Analysis: Cook, R.D., 2007. Concepts and applications of finite element analysis. John wiley & sons.
• GitHub Workflows: GitHub Actions for Scientific Data Workflows, Valentina Staneva, eScience Institute, University of Washington
• Meshing: MeshPy