Validation and generation of curved meshes for high-order unstructured methods
Authors:   , Gargallo Peiró Abel 
Published: October 2023

In this thesis, a new framework to validate and generate curved high-order meshes for complex models is proposed. The main application of the proposed framework is to generate curved meshes that are suitable for finite element analysis with unstructured high-order methods. Note that the lack of a robust and automatic curved mesh generator is one of the main issues that has hampered the adoption of high-order methods in industry. Specifically, without curved high-order meshes composed by valid elements and that match the domain boundary, the convergence rates and accuracy of high-order methods cannot be realized. The main motivation of this work is to propose a framework to address this issue.
First, we propose a definition of distortion (quality) measure for curved meshes of any polynomial degree. The presented measures allow validating if a high-order mesh is suitable to perform finite element analysis with an unstructured high-order method. In particular, given a high-order element, the measures assign zero quality if the element is invalid, and one if the element corresponds to the selected ideal configuration (desired shape and nodal distribution). Moreover, we prove that if the quality of an element is not zero, the region where the determinant of the Jacobian is not positive has measure zero. We present several examples to illustrate that the proposed measures can be used to validate high-order isotropic and boundary layer meshes.
Second, we develop a smoothing and untangling procedure to improve the quality for curved high-order meshes. Specifically, we propose a global non-linear least squares minimization of the defined distortion measures. The distortion is regularized to allow untangling invalid meshes, and it ensures that if the initial configuration is valid, it never becomes invalid. Moreover, the optimization procedure preserves, whenever is possible, some geometrical features of the linear mesh such as the shape, stretching, straight-sided edges, and element size. We demonstrate through examples 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.
Third, we extend the definition of distortion and quality measures to curved high-order meshes with the nodes on parameterized surfaces. Using this definition, we also propose a smoothing and untangling procedure for meshes on CAD surfaces. This procedure is posed in terms of the parametric coordinates of the mesh nodes to enforce that the nodes are on the CAD geometry. In addition, we prove that the procedure is independent of the surface parameterization. Thus, it can optimize meshes on CAD surfaces defined by low-quality parameterizations.
Finally, we propose a new mesh generation procedure by means of an a posteriori approach. 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 resulting mesh to produce a valid curved high-order mesh. To conclude, we include several examples to demonstrate that the generated meshes are suitable to perform finite element analysis with unstructured high-order methods.