FEM: Mesh quality and body simplification (Part 4) - Mesh Quality

Have ever you asked: How to check the mesh quality in FEM? What are the criteria to consider to evaluate the mesh quality? In this topic, let's brifly to explain some concepts to have a good mesh quality and how to control it.

-          Elements distribution

How much more homogeneous is the mesh, better. It means: similar type of elements and non-distortion of the elements shape, regular distribution, elements with similar size and smooth transition of sizes, continuity of nodes between parts transitions.

Figure 1 - Element distribution comparison. At left a mesh more homogeneous and below, quantitative element quality analysis. Most of them have more than 88% of quality criteria.

-          Elements Transition

There are some regions in the body which a smaller element is required while for others, not so small elements are enough. Due to this sizing difference, there are transitions regions from smaller to bigger elements. The softer is the transition, better will be the mesh quality.  

Figure 2 - Comparison of element transition. At left a fast transition and at right a smooth transition.

-          Node anchorage

In most of the meshing strategies, nodes are anchored on edges and on vertexes. If there are more than two vertexes in an edge, the nodes will necessarily consider them and a distortion on the mesh can appear due that the node must be at this point. So, to avoid extra elements and distortion of mesh, a continuous edge is recommended in the CAD model.

Figure 3 - Example of nodes anchorage on an edge and how extra vertexes disturbs the mesh.

-          Connection between bodies

Some problems made by several bodies need to be interpreted as an only one big body (rigid, without relative movements). In this case, the mesh must be continuous in all system and share the same nodes (shared topology) on the edges where there are the connections between each body.

Figure 4 - A problem when the bodies need to seem as just only one.

Figure 5 - At left, a system without shared topology, the mesh is generated independently for each body. And at right with shared topology, the mesh is continuous and considers all system.

-          Mesh density in proper regions and geometry simplification

There are some regions that cause an increment of the mesh density. These are shaped transition regions, like fillets, chamfers, small details, markings, holes, recesses etc. As is known, how much small is the element size, bigger is the results accuracy, but by another hand, more computing processing is required. The point is, that there are some regions where we will have maximum tensions and is only in these regions that is interesting we have accuracy and a proper mesh density (local mesh refinement), while that a big density in other regions are not necessary.

Sometimes is not possible to know where are these regions, so one first interaction is required to identify where is necessary to the mesh have its density incremented or not.

Figure 6 - After the first interaction, is identified the region of maximum tension, wherein the mesh density needs to increment.

Figure 7 - The maximum tension in just a node is a signal that the mesh density must increase.

Figure 8 - After increased of the mesh density, the maximum tensile is distributed by several nodes so as provide more reliability of the result.

                For other regions with low tensions, where do not need an increment of the mesh density, bigger elements can be used without losing the resulting quality and even increase the speed of solution with fewer elements and nodes. It is achieved with geometry simplification, removing regions that naturally creates small elements (fillets, chamfers, small details, markings, holes, recesses etc), but do never remove fillets and chamfers in regions of maximum stress as will be explained at section “Number of elements and convergence”.

Figure 9 – The original body created a mesh with 24813 nodes and 14525 elements.

Figure 10 - The simplified body created a mesh with 4007 nodes and 2137 elements, resulting in an 85% reducing.

-          Geometry repairs

Is important we always check the CAD quality before do the meshing. It´s necessary because some inconsistencies or redundancy could be in the geometry. Some examples of problems in the geometry: gaps between surfaces or missing faces, extras vertex on edges, extra or redundancy of edges or surfaces, some residue due CAD operations etc. There are FEM software that identifies these problems and solve automatically. For other situations, some manually geometry manipulation is necessary.

Figure 11 - Examples of geometry failures. From left to right: Surface gaps, face missing and extra edges.   

Figure 12 - A very small unnecessary surface that generated small extra elements, disturbing the mesh. Probably due to residue of CAD operations.

-          Number of elements through the thickness

In a body subjected to bending, there is a recommended rule that a minimum of 4 linear elements or 2 superior order elements should have through the thickness of the body. But if a membrane stress can be observed, with a slight change in stress (tensile and compressive) magnitude through the thickness, only one element through the thickness is enough. If the body has a thin wall, subject to tensile and compressive stress on opposite sides, is recommended to reduce the element size only in critical regions, to do not create excessive unnecessary elements [Autodesk].

Figure 13 - For a body subjected to bending, a minimum of 4 elements through the thickness. But witch a slight change in stress (tensile and compressive) like right one part, only one element through the thickness is enough. FONT: Autodesk.