On curving high-order hexahedral meshes

We present a new definition of distortion and quality measures for high-order hexahedral (quadrilateral) elements. This definition leads to two direct applications. First, it can be used to check the validity and quality of a high-order hexahedral (quadrilateral) mesh. Second, it allows the generation of high-order curved meshes composed of valid and high-quality hexahedral (quadrilateral) elements. We describe a method to simultaneously smooth and untangle high-order hexahedral (quadrilateral) meshes by minimizing the proposed mesh distortion. Finally, we analyze the behavior of the proposed distortion measure and we present several results to illustrate the benefits of the mesh generation framework.Peer ReviewedPostprint (author's final draft

Validation and generation of high-order meshes on parameterized surfaces

We present a technique to extend Jacobian-based distortion (quality) measures for planar triangles to high-order isoparametric elements of any interpolation degree on CAD parameterized surfaces. The resulting distortion (quality) measures are expressed in terms of the parametric coordinates of the nodes. These extended distortion (quality) measures can be used to check the quality and validity of a high-order surface mesh. We also apply them to simultaneously smooth and untangle high-order surface meshes by minimizing the extended distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes. Thus, the nodes always lie on the surface. Finally, we include several examples to illustrate the application of the proposed techniquePeer ReviewedPostprint (published version

Defining an2-disparity measure to check and improve the geometric accuracy of noninterpolating curved high-order meshes

We define an2-disparity measure between curved high-order meshes and parameterized manifolds in terms of an2norm. The main application of the proposed definition is to measure and improve the distance between a curved high-order mesh and a target parameterized curve or surface. The approach allows considering meshes with the nodes on top of the curve or surface (interpolative), or floating freely in the physical space (non-interpolative). To compute the disparity measure, the average of the squared point-wise differences is minimized in terms of the nodal coordinates of an auxiliary parametric high-order mesh. To improve the accuracy of approximating the target manifold with a noninterpolating curved high-order mesh, we minimize the square of the disparity measure expressed both in terms of the nodal coordinates of the physical and parametric curved high-order meshes. The proposed objective functions are continuously differentiable and thus, we are able to use minimization algorithms that require the first or the second derivatives of the objective function. Finally, we present several examples that show that the proposed methodology generates high-order approximations of the target manifold with optimal convergence rates for the geometric accuracy even when non-uniform parameterizations of the manifolds are prescribed. Accordingly, we can generate coarse curved high-order meshes significantly more accurate than finer low-order meshes that feature the same resolution.Peer ReviewedPostprint (author's final draft

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

A procedure to quantify the distortion (quality) of a high-order mesh composed of curved tetrahedral elements is presented. The proposed technique has two main applications. First, it can be used to check the validity and quality of a high-order tetrahedral mesh. Second, it allows the generation of curved meshes composed of valid and high-quality high-order tetrahedral elements. To this end, we describe a method to smooth and untangle high-order tetrahedral meshes simultaneously by minimizing the proposed mesh distortion. Moreover, we present a -continuation procedure to improve the initial configuration of a high-order mesh for the optimization process. Finally, we present several results to illustrate the two main applications of the proposed technique.Peer ReviewedPostprint (author’s final draft

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

This is the accepted version of the following article: [Gargallo-Peiró, A., Roca, X., Peraire, J., and Sarrate, J. (2016) A distortion measure to validate and generate curved high-order meshes on CAD surfaces with independence of parameterization. Int. J. Numer. Meth. Engng, 106: 1100–1130. doi: 10.1002/nme.5162], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5162/abstractA framework to validate and generate curved nodal high-order meshes on Computer-Aided Design (CAD) surfaces is presented. The proposed framework is of major interest to generate meshes suitable for thin-shell and 3D finite element analysis with unstructured high-order methods. First, we define a distortion (quality) measure for high-order meshes on parameterized surfaces that we prove to be independent of the surface parameterization. Second, we derive a smoothing and untangling procedure based on the minimization of a regularization of the proposed distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes to enforce that the nodes slide on the surfaces. Moreover, the proposed algorithm repairs invalid curved meshes (untangling), deals with arbitrary polynomial degrees (high-order), and handles with low-quality CAD parameterizations (independence of parameterization). Third, we use the optimization procedure to generate curved nodal high-order surface meshes by means of an a posteriori approach. Given a linear mesh, we increase the polynomial degree of the elements, curve them to match the geometry, and optimize the location of the nodes to ensure mesh validity. Finally, we present several examples to demonstrate the features of the optimization procedure, and to illustrate the surface mesh generation process.Peer ReviewedPostprint (author's final draft

Validation and generation of curved meshes for high-order unstructured methods

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.Postprint (published version