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We detail how to use Newton’s method for distortion-based curved r-adaption to a discrete high-order metric field. To this end, we consider three existent ingredients. First, a specific-purpose solver for distortion minimization. Second, a log-Euclidean high-order metric interpolation. Third, a point localization procedure for curved high-order meshes. We also extend to discrete metric fields a distortion-based curved r-adaption framework. To extend the framework, we provide, for the log-Euclidean high-order metric interpolation, the first and second derivatives in physical coordinates. These derivatives are required by Newton’s method to solve the distortion minimization. The distortion minimization allows properly matching the anisotropic curved features of a discrete high-order metric. This matching capability might be relevant in global and cavity-based curved (straight-edged) high-order mesh adaption.

We present a novel adaptive sampling method to estimate the Lebesgue constant of nodal sets in n-dimensional simplices. The main application of this estimation is to assess the interpolation capabilities of a nodal distribution. Given such distribution, the Lebesgue constant corresponds to the maximum of the Lebesgue function, which is non-differentiable. Thus, our method estimates the extremum by only evaluating the function values at a set of sample points that are successively adapted to seek the maximum. Remarkably, our adaptive search does not require storing a mesh to query neighbor points. Furthermore, the search automatically stops by considering specific spatial and Lipschitz-based criteria. The examples, up to four dimensions, show that the method is well-suited to estimate the Lebesgue constant of different nodal distributions.

We present a nodal interpolation method to approximate a subdivision model. The main application is to model and represent curved geometry without gaps and preserving the required simulation intent. Accordingly, the technique is devised to maintain the necessary sharp features and smooth the indicated ones. This sharp-to-smooth modeling capability handles unstructured configurations of the simulation points, curves, and surfaces. These surfaces are described by the initial triangulation composed of linear triangles with the same surface identifier, and determine the sharp point and curve features. Automatically, the method suggests a subset of sharp features to smooth which the user modifies to obtain a limit model preserving the initial points. This model reconstructs the curvature by subdividing the initial triangular mesh, with no need of an underlying curved geometry model. Finally, given a polynomial degree and a nodal distribution, the method generates a piece-wise polynomial representation interpolating the limit model. Numerical evidence suggests that this approximation, naturally aligned to the subdivision features, converges to the model geometrically with the polynomial degree for fair nodal distributions.We also apply the method to prescribe the curved boundary of a high-order volume mesh. We conclude that our sharp-to-smooth modeling capability leads to curved geometry representations with enhanced preservation of the simulation intent.

To provide straight-edged and curved piece-wise polynomial meshes that target a unique smooth geometry while preserving the sharp features and smooth regions of the model, we propose a new fast curving method based on hierarchical subdivision and blending. There is no need for underlying target geometry. It is only required a straight-edged mesh with boundary entities marked to characterize the geometry features, and a list of features to recast through a unique sharp-to-smooth modeling capability. The examples show that the method is well-suited to curve large quadratic and quartic meshes in low-memory configurations.

To provide straight-edged and curved piece-wise polynomial meshes that target a unique smooth geometry while preserving the sharp features and smooth regions of the model, we propose a new fast curving method based on hierarchical subdivision and blending. There is no need for underlying target geometry, it is only needed a straight-edged mesh with boundary entities marked to characterize the geometry features, and a list of features to recast. The method features a unique sharp-to-smooth modeling capability not fully available in standard CAD packages. The goal is to obtain a volume mesh that under successive refinement leads to smooth regions bounded by the corresponding sharp features. The examples show that it is possible to refine and obtain smooth curves and surfaces while preserving sharp features determined by vertices and polylines. We conclude that the method is well-suited to curve large quadratic and quartic meshes in low-memory configurations. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 715546. This work has also received funding from the Generalitat de Catalunya under grant number 2017 SGR 1731. The work of the third author has been partially supported by the Spanish Ministerio de Economía y Competitividad under the personal grant agreement RYC-2015-01633.

We present a new hybrid meshing procedure specifically designed to simulate Atmospheric Boundary Layer (ABL) flows featuring Coriolis effects for onshore applications. Two new meshing contributions for ABL flow simulation are proposed. First, a new metric-based adaptive meshing process to discretize the topography is presented. The adaptivity process is specifically written to deal with the discrete nature of topographic data, reparemeterizing smoothly the topography to allow computing the first and second order derivatives required by the desired metrics. Second, we present a new hybrid mesh generation procedure to discretize the ABL. We generate a prismatic boundary layer that captures the gradients of the Surface Boundary Layer, and following, an unstructured mesh is generated to discretize the rest of the domain. We show examples to illustrate the reduction in the element count with respect to standard semi-structured approaches, and we present the simulation result obtained on an onshore topographical scenario.

A 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 The first author was sponsored by CUR from DIUE of the Generalitat de Catalunya and the European Social Fund (grants FI-DGR’11 and BE-DGR’12), and by the Ferran Sunyer i Balaguer Foundation (grant FSB’13). The work of the corresponding author was partially supported by the Boeing Co. & US Air Force Office of Scientific Research & European Commission through the Boeing-MIT Alliance & the Computational Math Program & Marie Skłodowska-Curie Actions (HiPerMeGaFlowS individual fellowship grant).

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.

In this work it has been developed the basis of shape quality metrics for triangular meshes on surfaces. It has also been worked out a simultaneous smoothing-untangling procedure based on the ideas introduced by Escobar. This procedure is designed through the minimization of an objective function (inverse of the quality of the mesh). We have increased the computational efficiency of the original method by adding a new algorithm for the search of the optimal projection plane. Moreover we have carried out a more robust discussion of the theoretical construction of the objective function for this new algorithm. Finally, several examples have been presented in order to illustrate the capabilities of the proposed method.

We 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.

We propose an a posteriori approach for generating curved meshes for viscous flow simulations composed by high-order tetrahedra. The proposed approach is performed in the following three steps: (1) generate a linear tetrahedral mesh for inviscid flow; (2) insert a boundary layer mesh, composed by linear tetrahedra, on the viscous part; and (3) convert the linear tetrahedral mesh to a curved and high-order mesh for viscous flow. This approach provides high-order tetrahedral meshes with boundary layer parts that are composed by elements that are: curved, valid, and of any interpolation degree.

A procedure to quantify the distortion (quality) of a high-order mesh composed by 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 by valid and high-quality high-order tetrahedral elements. To this end, we describe a method to simultaneously smooth and untangle high-order tetrahedral meshes by minimizing the proposed mesh distortion. Finally, we present several results to illustrate the two main applications of the proposed technique.

The aim of this work is to develop a simultaneous smoothing-untangling procedure for parametrized surfaces. To this end we will extend the definition of the quality metrics for planar meshes to parametrized surface meshes. Then, we will develop a minimization approach on the parametric space that will allow smoothing and untangling the surface mesh. To this end, first we will increase the robustness of the standard untangling techniques. Then, we will focus on the improvement of the computational efficiency of the developed approach. Finally, several examples will be presented in order to illustrate the capabilities of the proposed method.. L'objectiu d'aquest projecte és la millora de la qualitat de malles d'elements finits generades sobre superficies paramètriques. Aquestes malles podran estar formades per triangles o quadrilàters. Aquesta millora es realitzarà mitjançant la minimització numèrica d'una mesura de la qualitat de la malla definida en l'espai paramètric

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.

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 technique

Barcelona Supercomputing Center In this work, two new methods are presented: first, a given linear tetrahedral volume mesh is refined in such a way that its boundary tends to an almost everywhere C^2-continuous surface, and second, a curved high-order tetrahedral mesh which boundary approximates an almost everywhere C^2 surface is generated from a linear tetrahedral mesh. The aim of these curved meshes is to provide an almost everywhere C^2 smooth parametrization of curved boundaries suitable for finite element analysis with unstructured high-order methods. To illustrate the applicability of the methods, several curved high-order volume meshes have been generated from linear meshes obtained by digitizing real volumes.

Barcelona Supercomputing Center

Barcelona Supercomputing Center In this work, two new methods are presented: first, a given linear tetrahedral volume mesh is refined in such a way that its boundary tends to an almost everywhere C^2-continuous surface, and second, a curved high-order tetrahedral mesh which boundary approximates an almost everywhere C^2 surface is generated from a linear tetrahedral mesh. The aim of these curved meshes is to provide an almost everywhere C^2 smooth parametrization of curved boundaries suitable for finite element analysis with unstructured high-order methods. To illustrate the applicability of the methods, several curved high-order volume meshes have been generated from linear meshes obtained by digitizing real volumes.