Mesh Quality Metrics
Mesh quality metrics quantify how well-shaped and well-sized elements are, since distorted cells degrade accuracy and solver conditioning.
Why element shape matters
Not all meshes that fill a domain are equally good. Badly shaped elements, thin slivers, highly skewed cells, sharp angles, degrade the accuracy of the discretization and inflate the condition number of the resulting linear system, slowing or destabilizing solvers. Mesh quality metrics are scalar measures that flag such problem elements so a mesh can be assessed and improved before it is used.
Quality is not one number but several, each sensitive to a different kind of distortion. A mesh may score well on size uniformity yet contain a few catastrophically skewed cells that dominate the error.
Common metrics
Aspect ratio measures elongation, the ratio of longest to shortest dimension; large values indicate stretched cells that hurt isotropic problems. Skewness measures deviation from an ideal equilateral or right-angled shape. The scaled Jacobian detects folded or inverted elements: a non-positive value means the element is tangled and the mapping is invalid. Orthogonality measures how perpendicular faces are to the lines connecting cell centers, which matters for finite-volume gradient accuracy.
Consequences and remedies
Skewed and high-aspect-ratio cells raise truncation error and can violate the conditions under which a scheme is stable or a matrix is well conditioned. Smoothing (relocating nodes, as in Laplacian or optimization-based smoothing), topological cleanup (edge and face swaps), and local remeshing repair poor regions. Anisotropic elements are acceptable, even desirable, when aligned with an anisotropic solution feature, so quality must be judged against the physics.
- Aspect ratio flags elongated, stretched elements
- Skewness measures departure from ideal shape
- Scaled Jacobian detects inverted or tangled cells
- Smoothing and swapping repair low-quality regions
Mesh quality checks are a routine part of setting up fusion simulations, where the combination of shaped boundaries and thin layers makes it easy to produce distorted cells that must be caught and repaired.