As mentioned in the mesh article, a mesh is the discretization of geometry into individual elements. Since the mesh is an approximation of the actual geometry, the mesh density and quality have a significant influence on the accuracy and stability of a simulation. <\/p>\n\n\n\n
The purpose of mesh quality metrics is to assess the suitability of a discrete computational domain (Mesh) for simulations. A mesh is usually considered to be of higher quality compared to another mesh if it improves at least one of the most important simulation properties; Time to convergence, stability or accuracy without affecting the others negatively. <\/p>\n\n\n\n
1. Mesh Quality Metrics <\/h2>\n\n\n\n
We measure the mesh quality, using a set of quality metrics. The quality metrics determine how far we are from an ideal cell shape. Below, you will see some mesh quality metrics used in SimScale.<\/p>\n\n\n\n
1.1 Aspect Ratio<\/h3>\n\n\n\n
Briefly, the aspect ratio is the ratio of a cell’s longest length to the shortest length. The ideal aspect ratio is 1. The smaller it is, the higher the quality of an element is. The calculation method varies by the cell type as follows:<\/p>\n\n\n\n
Hexahedral cell aspect ratio:<\/strong> Ratio of maximum edge length to the minimum edge length. <\/p>\n\n\n\n $$ AspectRatio = \\frac{max(x_1, x_2, …, x_{12})}{min(x_1, x_2, …, x_{12})} $$ <\/p>\n\n\n\n Tetrahedral cell aspect ratio:<\/strong> It’s correlated between the maximum edge length and radius of the cell’s internal sphere.<\/p>\n\n\n\n $$ AspectRatio = \\frac{max(x_1, x_2, …, x_6)}{2 \\cdot \\sqrt 6 \\cdot r} $$ <\/p>\n\n\n Non-orthogonality is the angle between the vector connecting two adjacent cell centers and the normal of the face shared by these cells. The range of non-orthogonality is between 0 (ideal) and 90 (worst). 0 indicates the mesh being orthogonal. Two perfect hexes aligned with each other have non-orthogonality equal to 0. <\/p>\n\n\n A high level of non-orthogonality should be avoided since it causes numerical instability. As a result, the simulations with higher non-orthogonality either take longer to converge, or even diverge. <\/p>\n\n\n\n It is recommended to keep the non-orthogonality below 70. Please consider improving the mesh if the non-orthogonality is above 80. Simulations with the meshes with maximum non-orthogonality above 85 likely diverge.<\/p>\n\n\n\n Volume ratio is the ratio of the volumes of the adjacent cells. Consider two adjacent cells with volumes of \\(V_1\\) and \\(V_2\\). The volume ratio is computed as follows:<\/p>\n\n\n\n $$ Volume\\ ratio = \\frac{max(V_1, V_2)}{min(V_1, V_2)} $$<\/p>\n\n\n\n In reality, a single cell is surrounded by multiple neighbor cells. Since there are multiple volume ratios, we consider the maximum volume ratio as the cell’s volume ratio.<\/p>\n\n\n $$ Volume\\ ratio_i = max( Volume\\ ratio_{j=1}, Volume\\ ratio_{j=2},…, Volume\\ ratio_{j=n}) $$ <\/p>\n\n\n\n The ideal volume ratio is 1. The smaller it is, the higher the quality of an element is. <\/p>\n\n\n\n Tet edge ratio is the ratio between the longest and the shortest edges. This quality metric is only available for the tetrahedral cells. Considering the edge lengths of a cell represented by x, the tet edge ratio is then computed as follows:<\/p>\n\n\n\n $$ Tet edge ratio = \\frac{max(x_1, x_2, … x_6)}{min(x_1, x_2, … x_6) } $$<\/p>\n\n\n\n Skewness is the normalized distance between a line that connects two adjacent cell centroids and the distance from that line to the shared face’s center. The range of skewness is between 0 (ideal) to \\(\\infty\\). Highly skewed cells are not preferred due to the poor accuracy they cause within the interpolated regions. The calculation method is as follows, for any cell type:<\/p>\n\n\n\n $$ Skewness = \\frac{\\mid \\vec{s} \\mid}{\\mid \\vec{d} \\mid}$$<\/p>\n\n\n\n Where \\(\\mid \\vec{s} \\mid\\) is the norm of the skewness vector and \\(\\mid \\vec{d} \\mid\\) is the norm of the vector that connects the adjacent cell centroids, as shown in the Figure below.<\/p>\n\n\n Hexahedral cell quadMaxAngle is the maximum of the angles between neighboring cell edges. The normal range varies between 90 (ideal) to 180 (worst). The acceptable range is 90 to 135.<\/p>\n\n\n\n Tetrahedral cell triMaxAngle is the maximum of the angles between neighboring cell edges. The normal range varies between 60 (ideal) to 180 (worst). <\/p>\n\n\n\n Tetrahedral cell triMinAngle is the minimum of the angles between neighboring cell edges. The normal range varies between 0 (ideal) to 60 (worst).<\/p>\n\n\n\n There are mainly two applications for this type of quality assessment:<\/strong><\/p>\n\n\n\n The following picture gives a preview of what the result of checking the mesh quality could look like:<\/p>\n\n\n The quality of a mesh can be assessed by different metrics that are usually based upon the geometrical properties of the mesh cells (e.g. Aspect ratio) or upon the relation between neighboring cells (e.g. non-orthogonality). As the suitability of a mesh for a simulation is also strongly influenced by the simulated physics, the simulation solver, and the geometrical domain, the quality metrics can only give an indication if a mesh is a good fit for a simulation setup. As an example, you can find that simple incompressible CFD simulations are usually much less sensitive towards poor quality cells than compressible flow with high-density changes.<\/p>\n\n\n\n Figure 6 shows the meshing logs of a good and bad quality mesh. The marked points indicate:<\/p>\n\n\n\n The following table shows the maximum value for some of the meshing parameters beyond which the simulation is very likely to diverge. Low-quality meshes will most likely cause numerical errors and either fail the simulations or cause unreliable results. Note that some of the parameters are more\/less relevant depending on the type of analysis to be performed (CFD\/FEA).<\/p>\n\n\n\n <\/p>\n\n\n\n![\"high](\"https:\/\/frontend-assets.simscale.com\/media\/2021\/09\/Aspect-ratio-1-1024x603.jpg\")
<\/a>1.2 Non-orthogonality<\/h3>\n\n\n\n
![\"comparison](\"https:\/\/frontend-assets.simscale.com\/media\/2021\/09\/non-orthogonality-1024x290.jpg\")
<\/a>1.3 Volume Ratio<\/h3>\n\n\n\n
![\"small](\"https:\/\/frontend-assets.simscale.com\/media\/2021\/09\/Volume_ratio-2-1024x406.jpg\")
<\/a>1.4 Tet Edge Ratio<\/h3>\n\n\n\n
1.5 Skewness<\/h3>\n\n\n\n
![\"Visual](\"https:\/\/frontend-assets.simscale.com\/media\/2024\/06\/skewness-definition.jpg\")
<\/a>1.6 QuadMaxAngle<\/h3>\n\n\n\n
1.7 TriMaxAngle<\/h3>\n\n\n\n
1.8 TriMinAngle<\/h3>\n\n\n\n
2. Quality Assessment<\/h2>\n\n\n\n
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This is to ensure the mesh is a good fit for the upcoming simulation run. Evaluating at this level is more for CFD-experts, less experienced users can usually rely on the default settings.<\/li>\n\n\n\n
If your solution diverges and the simulation fails, quality assessment gives you powerful tools at hand to identify problematic areas that can be, for example, fixed by adjusting the CAD model or refining certain areas in the mesh.<\/li>\n<\/ol>\n\n\n\n![\"visualisation](\"https:\/\/frontend-assets.simscale.com\/media\/2024\/07\/aspect-ratio-mesh-quality.png\")
<\/a>2.1 Meshing Log<\/h3>\n\n\n\n
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![\"visualization](\"https:\/\/frontend-assets.simscale.com\/media\/2021\/12\/meshing_log-2-1024x536.png\")
<\/a>Best Practices for Mesh Quality Metrics:<\/h4>\n\n\n\n
![\"meshing](\"https:\/\/frontend-assets.simscale.com\/media\/2021\/12\/low_quality-1024x440.png\")
<\/a>![\"meshing](\"https:\/\/frontend-assets.simscale.com\/media\/2021\/12\/high_quality-1024x442.png\")
<\/a>![\"simscale](\"https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/tree.png\")
![\"global](\"https:\/\/frontend-assets.simscale.com\/media\/2024\/07\/mesh-value-whole-domain.png\")
<\/a>![\"demonstration](\"https:\/\/frontend-assets.simscale.com\/media\/2024\/07\/mesh-quality-aspect-ratio-scaled.jpg\")
<\/a>![\"demonstration](\"https:\/\/frontend-assets.simscale.com\/media\/2024\/07\/isolate-element.png\")
<\/a>![\"relaxation](\"https:\/\/frontend-assets.simscale.com\/media\/2021\/12\/image-12.png\")
<\/a>![\"gradient](\"https:\/\/frontend-assets.simscale.com\/media\/2021\/12\/gradient-1.png\")
<\/a>![\"divergence](\"https:\/\/frontend-assets.simscale.com\/media\/2021\/12\/divergence.png\")
<\/a>![\"laplacian](\"https:\/\/frontend-assets.simscale.com\/media\/2021\/12\/laplacian.png\")
<\/a>