Amaury Johnen

Geometrical validity of curvilinear finite elements

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 Bezier functions, that has both properties of boundedness and positivity. Numerical results show the sharpness of our estimates.

The Generation of Valid Curvilinear Meshes

It is now well-known that a curvilinear discretization of the geometry is most often required to benefit from the computational efficiency of high-order numerical schemes in simulations. In this article, we explain how appropriate curvilinear meshes can be generated. We pay particular attention to the problem of invalid (tangled) mesh parts created by curving the domain boundaries. An efficient technique that computes provable bounds on the element Jacobian determinant is used to characterize the mesh validity, and we describe fast and robust techniques to regularize the mesh. The methods presented in this article are thoroughly discussed in Ref. [1, 2], and implemented in the free mesh generation software Gmsh [4, 12].

Geometrical validity of high-order triangular finite elements

This paper presents a method to compute accurate bounds on Jacobian determinants of high-order (curvilinear) triangular finite elements. This method can be used to guarantee that a curvilinear triangle is geometrically valid, i.e., its Jacobian determinant is strictly positive everywhere in its reference domain. It also provides an efficient way to measure the quality of triangles. 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.

Geometrical validity of curvilinear pyramidal finite elements

A method to efficiently determine the geometrical validity of curvilinear finite elements of any order was recently proposed in [1]. The method is based on the adaptive expansion of the Jacobian determinant in a polynomial basis built using Bezier functions, that has both properties of boundedness and positivity. While this technique can be applied to all usual finite elements (triangles, quadrangles, tetrahedra, hexahedra and prisms), it cannot readily be applied to pyramids, due to non-polynomial nature of pyramidal finite element spaces. In this short paper, we extend the results from [1] to pyramidal elements, by making use of the high-order nodal pyramidal finite element proposed by Bergot et al. [10], which exhibits optimal convergence properties inH1-norm. The paper is organized as follows. We begin by briefly recalling the pyramidal finite element space in Section 2, before constructing the function space of the Jacobian determinant in Section 3. Section 4 then introduces a generalized Bezier function basis, which can be used to obtain adaptive bounds on the pyramidal Jacobian determinant. Numerical results showing the sharpness of the estimates are given in Section 5.

Efficient computation of the minimum of shape quality measures on curvilinear finite elements

Abstract We present a method for computing robust shape quality measures defined for finite elements of any order and any type, including curved pyramids. The measures are heuristically defined as the minimum of the pointwise quality of curved elements. Three pointwise qualities are considered: the ICN that is related to the conditioning of the stiffness matrix for straight-sided simplicial elements, the scaled Jacobian that is defined for quadrangles and hexahedra, and a new shape quality that is defined for triangles and tetrahedra. The computation of the minimum of the pointwise qualities is based on previous work presented by Johnen et al. (2013) and Johnen and Geuzaine (2015) and is very efficient. The key feature is to expand polynomial quantities into Bezier bases which allow to compute sharp bounds on the minimum of the pointwise quality measures.

Sequential decision-making approach for quadrangular mesh generation

A new indirect quadrangular mesh generation algorithm which relies on sequential decision-making techniques to search for optimal triangle recombinations is presented. In contrast to the state-of-art Blossom-quad algorithm, this new algorithm is a good candidate for addressing the 3D problem of recombining tetrahedra into hexahedra.

Identifying combinations of tetrahedra into hexahedra: A vertex based strategy

Indirect hex-dominant meshing methods rely on the detection of adjacent tetrahedra an algorithm that performs this identification and builds the set of all possible combinations of tetrahedral elements of an input mesh T into hexahedra, prisms, or pyramids. All identified cells are valid for engineering analysis. First, all combinations of eight/six/five vertices whose connectivity in T matches the connectivity of a hexahedron/prism/pyramid are computed. The subset of tetrahedra of T triangulating each potential cell is then determined. Quality checks allow to early discard poor quality cells and to dramatically improve the efficiency of the method. Each potential hexahedron/prism/pyramid is computed only once. Around 3 millions potential hexahedra are computed in 10 seconds on a laptop. We finally demonstrate that the set of potential hexes built by our algorithm is significantly larger than those built using predefined patterns of subdivision of a hexahedron in tetrahedral elements.