Why does shape of elements matter in finite elements analysis? I have used FEA for a couple of years now, but using it and using it correctly are two different things, safety factor is not the solution to everything. I have the feeling I won't be using it right unless I have a clear answer to that question:
I am aware elements must be close to their ideal shape (based on the Jacobian) in order to get accurate results.. But why? Since I understand it comes from a coordinate transform, unless two vectors of the element become colinear shouldn't the results be accurate no matter its shape?
A step-by-step answer based on n illustrated example (arbitrary stress distribution) would be really appreciated, especially given that it is a relatively common question (but never well answered from what I have seen).
 A: There might be several reasons, some more obvious than others.

The quality of the cell (including its orthogonal quality, aspect ratio, and skewness) also has a significant impact on the accuracy of the numerical solution.

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*Orthogonal quality is computed for cells using the vector from the cell centroid to each of its faces, the corresponding face area vector, and the vector from the cell centroid to the centroids of each of the adjacent cells (see Equation 5–1, Equation 5–2, and Figure 5.22: The Vectors Used to Compute Orthogonal Quality). The worst cells will have an orthogonal quality closer to 0, with the best cells closer to 1. The minimum orthogonal quality for all types of cells should be more than 0.01, with an average value that is significantly higher.


*Aspect ratio is a measure of the stretching of the cell. As discussed in Computational Expense, for highly anisotropic flows, extreme aspect ratios may yield accurate results with fewer cells. Generally, it is best to avoid sudden and large changes in cell aspect ratios in areas where the flow field exhibit large changes or strong gradients.


*Skewness is defined as the difference between the shape of the cell and the shape of an equilateral cell of equivalent volume. Highly skewed cells can decrease accuracy and destabilize the solution. For example, optimal quadrilateral meshes will have vertex angles close to 90 degrees, while triangular meshes should preferably have angles of close to 60 degrees and have all angles less than 90 degrees. A general rule is that the maximum skewness for a triangular/tetrahedral mesh in most flows should be kept below 0.95, with an average value that is significantly lower. A maximum value above 0.95 may lead to convergence difficulties and may require changing the solver controls, such as reducing under-relaxation factors and/or switching to the pressure-based coupled solver.

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