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.
Summary.A mesh smoothing method based on Riemannian metric comparison is presented in this paper. This method minimizes a cost function constructed from a measure of metric non-conformity that compares two metrics: the metric that transforms the element into a reference element and a specified Riemannian metric, that contains the target size and shape of the elements. This combination of metrics allows to cast the proposed mesh smoothing method in a very general frame, valid for any dimension and type of element. Numerical examples show that the proposed method generates high quality meshes as measured both in terms of element characteristics and also in terms of orthogonality at the boundary and overall smoothness, when compared to other known methods.
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