Measuring the conformity of non-simplicial elements to an anisotropic metric field

Abstract

SUMMARYThis paper extends an approach for measuring the element conformity of simplices to non-simplicial elements of any type, in spaces of arbitrary dimension. Element non-conformity is defined as the difference between a given size specification map, in the form of a Riemannian metric tensor, and the actual metric tensor of the element. An approach to the measurement of non-conformity coefficients of non-simplicial elements based on sub-simplex division is proposed. An analysis of the measure's behaviour presented for quadrilaterals, hexahedra, prisms and pyramids shows that the measure is sensitive to size, stretching and orientation variations, as well as to other types of element shape degeneration. Finally, numerical applications show that the metric conformity measure can be used as a quality measure to quantify the discrepancy between a whole non-simplicial mesh and a complex anisotropic size specification map.

Hybrid mesh adaptation applied to industrial numerical combustion

“…In fact, the Jacobian matrix of any transformation is not constant on a non‐simplex, and proposed that the averaged tensor should be the result of the integration of on the element. However, it has been shown in that integration of the actual metric on non‐simplices can lead to undetected element degeneracies. An alternative way was proposed through the calculation of the current metric of all corner sub‐simplices.…”

“…It was discussed in section 4 that the remeshing toolkit used () lacks the possibility to refine and coarsen meshes in structured zones of the mesh. It is known, see , that the non‐conformity coefficient measures both the distortion of an element in size and shape. But when looking at a value of the non‐conformity coefficient, it is impossible to discern if the element generating that value is non‐conform in terms of size, shape, or a combination of both.…”

“…The method and the related algorithms have been verified by Labbé et al on unstructured meshes , demonstrating that the approach can effectively reduce interpolation error. The norm, called the non‐conformity coefficient , allows the use of the same criterion on many types of elements, resulting in a converging adaptation algorithm on hybrid meshes. To the authors’ knowledge, this study constitutes the first application of anisotropic mesh adaptation on hybrid meshes to compute a complete combustion problem in an industrially relevant configuration.…”

Mesh Smoothing Based on Riemannian Metric Non-Conformity Minimization

“…The quality of a mesh can be measured using the non-conformity measure presented for simplices in [20,21] and extended to non-simplices in [22]. The central idea is that the actual metric M K of an element K, the metric that defines the transformation between the element in its present state and the ideal element described before, must be equal to the specified metric:…”

“…This is caused by the fact that every time a position is tested, the non-conformity measure must be evaluated on all the neighboring elements. In order to compute these non-conformity measures, an average specified metric M s must be evaluated on each element using numerical integration as discussed in [22]. This operation is very costly because each integration point must be localized and interpolated on the background mesh, which is usually the mesh before smoothing, onto which the specified Riemannian metric field M s as been defined.…”