Optimization of a regularized distortion measure to generate curved high‐order unstructured tetrahedral meshes

Abstract

SUMMARYWe present a robust method for generating high-order nodal tetrahedral curved meshes. The approach consists of modifying an initial linear mesh by first, introducing high-order nodes, second, displacing the boundary nodes to ensure that they are on the CAD surface, and third, smoothing and untangling the mesh obtained after the displacement of the boundary nodes to produce a valid curved high-order mesh. The smoothing algorithm is based on the optimization of a regularized measure of the mesh distortion relative to the original linear mesh. This means that whenever possible, the resulting mesh preserves the geometrical features of the initial linear mesh such as shape, stretching and size. We present several examples to illustrate the performance of the proposed algorithm. Furthermore, the examples show that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements. We consider cases with polynomial approximations up to degree ten, large deformations of the curved boundaries, concave boundaries, and highly stretched boundary layer elements. The meshes obtained are suitable for high-order finite element analyses.

A distortion measure to validate and generate curved high‐order meshes on CAD surfaces with independence of parameterization

“…where δ is an element-wise constant parameter that is determined automatically according to the size of the corresponding ideal element, see details in [39,69]. Therefore, we consider the regularized distortion measure,…”

“…It is also a crucial step of the standard a posteriori approach used to generate high-order volume meshes suitable for 3D analysis [27,31,32,33,34,35,36,37,38,39]. The goal of this approach is to indirectly generate a high-order volume mesh that matches a curved boundary.…”

“…Specifically, we use this method to provide valid boundary conditions, i.e. curved surface meshes, in our curved high-order tetrahedral mesh generator [39].…”

Distortion and quality measures for validating and generating high-order tetrahedral meshes

“…In [16] a technique is presented to generate meshes composed of tetrahedra of degree two via topology regularization and optimization of two functions (scaled Jacobian measure and a ratio of the length and volume of the elements). Moreover, in [34,12,13,14] we proposed to optimize the high-order nodes location according to an objective function based on the defined distortion measure for high-order meshes. In contrast, in [33,46] it is proposed to optimize a function that is able to avoid Jacobians that are too small or too high.…”

“…In our implementation, we set δ = 0 for valid configurations and δ = 0.01 · σ * for tangled configurations, being σ * the determinant of the corresponding straight-sided ideal element, as proposed in [15,14,11]. Therefore, we consider a modified shape distortion measure,…”