Geometrical validity of curvilinear finite elements

Abstract

In this paper, we describe a way to compute accurate bounds on Jacobian determinants of curvilinear polynomial finite elements. Our condition enables to guarantee that an element is geometrically valid, i.e., that its Jacobian determinant is strictly positive everywhere in its reference domain. It also provides an efficient way to measure the distortion of curvilinear elements. The key feature of the method is to expand the Jacobian determinant using a polynomial basis, built using Bézier functions, that has both properties of boundedness and positivity. Numerical results show the sharpness of our estimates.

On discretely entropy stable weight-adjusted discontinuous Galerkin methods: curvilinear meshes

“…We note that, for isoparametric curved elements, J k ∈ P N in general (in 2D, J k ∈ P 2N −2 , while in 3D, J k ∈ P 3N −3 [44]). Thus, to ensure local conservation, we will approximate J k using a degree N polynomial (for example, the interpolant or L 2 projection onto P N ).…”

“…where the subscript i denotes the ith component of the vector quantity. From (44), it can be observed that, because the divergence of a curl vanishes, the continuous GCL condition (14) holds. The central idea of [45,2] is to use (44), but to interpolate before applying the curl…”

Modal analysis of the ultrahigh finesse Haroche QED cavity

“…Finally, the computational domain is meshed using the Gmsh mesh engine. Since our objective is to use high-order FE discretizations, curved mesh elements are mandatory [34], not only to achieve a precise representation of the curved mirror, but also to keep the number of mesh elements to an acceptable value.…”

TriWild

“…Optimization methods are usually used as a post-processing step to attempt to untangle the inverted triangles created during the curving process and to improve their quality. Inverted triangles can be identified by extending the notion of area [Knupp 2000] to high-order function geometric maps [Engvall and Evans 2018;Johnen et al 2013;Poya et al 2016;Roca et al 2012]. Various untangling strategies have been proposed, including geometric smoothing and connectivity modifications [Cardoze et al 2004;Dey et al 1999;Gargallo Peiró et al 2013;George and Borouchaki 2012;Lu et al 2013;Luo et al 2002;Peiró et al 2008;Shephard et al 2005].…”