4.0 Adaptive Mesh Refinement (AMR)
What is it?
Please see example 21 on https://mfem.org/examples/ – this report section is in relation to example 21.
AMR applications in real world problems
MFEM’s Adaptive mesh refinement capabilities https://mfem.org/features/
What are conforming and non-conforming meshes? Why is this good?
Conforming mesh
A serial, unstructured, conforming mesh in MFEM consists of two parts: topological data and geometric (coordinates) data. The primary topology data consists of: a list of vertices, elements, and boundary elements. Each element has a type, such as, triangle, quad, tetrahedron, etc.), an attribute (an integer used to identify subdomains and physical boundaries), and a tuple of vertex indices.
Boundary elements are described in the same way, with the assumption that they define elements with dimension one less than the dimension of the regular elements. Any additional topological data – such as edges, faces, and their connections to the elements, boundary elements and vertices – is derived internally from the primary data.
Nonconforming mesh
Non-conforming meshes, otherwise referred to as meshes with hanging nodes, can be described as conforming meshes with a set of constraints imposed on some of their vertices. Assuming a linear mesh, the requirement is that each constrained vertex has to be the convex combination of a set of parent vertices. Note that, in general, the parent vertices of a constrained vertex can be constrained themselves. However, it is usually required that all the dependencies can be uniquely resolved and all constrained vertices can be expressed as linear combinations of non-constrained ones.
Although conforming meshes are preferable, non-conforming meshes are regularly encountered for a variety of reasons including:
1. Different engineers created the mesh on the different components.
2. The meshing algorithm used on the model did not honor boundary meshing constraints [1, 2].
3. Difficulties in generating the mesh required a different mesh topology on the interfacing surfaces for the two components. For example, utilizing hexahedral sweeping [3, 4], one interface surface may be required to be a linkingsurface requiring a mapped mesh, while the other interface surface may be a source surface allowing a paved mesh. 4. The desired density of elements is different in the two components.
ZZ (Zienkiewicz & Zhu) Error Estimator for linear and nonlinear elasticity
What is it? Detailed description of how it works in relation to linear and nonlinear elasticity equations. Used by example 21 https://mfem.org/examples/
How does it work for mesh methods? With equations and diagrams, this could be for the linear elasticity of a beam problem similar to the one in my attached draft report. There is a source document attached with information on the zz error estimation technique
5.0 Nonlinear elasticity and Hyperelasticity
General applications of nonlinear elasticity used in FEA
How are nonlinear elasticity equations solved using a finite element library?
What is hyper elasticity and how is it related to nonlinear elasticity?
Based on MFEM’s example 10 and 19 on https://mfem.org/examples/
