Smoothing and untangling triangular meshes on surfaces

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

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

Smoothing and untangling of meshes on parameterized surfaces

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 paramètri

Local bisection for conformal refinement of unstructured 4D simplicial meshes

We present a conformal bisection procedure for local refinement of 4D unstructured simplicial meshes with bounded minimum shape quality. Specifically, we propose a recursive refine-to-conformity procedure in two stages, based on marking bisection edges on different priority levels and defining specific refinement templates. Two successive applications of the first stage ensure that any 4D unstructured mesh can be conformingly refined. In the second stage, the successive refinements lead to a cycle in the number of generated similarity classes and thus, we can ensure a bound over the minimum shape quality. In the examples, we check that after successive refinement the mesh quality does not degenerate. Moreover, we refine a 4D unstructured mesh and a space-time mesh (3D + 1D) representation of a moving object

Subdividing triangular and quadrilateral meshes in parallel to approximate curved geometries

A parallel distributed approach to refine a mesh while preserving the curvature of a target geometry is presented. Our approach starts by generating a coarse linear mesh of the computational domain. Second, the former coarse mesh is curved to match the curvature of the target geometry. Then, the curved mesh is partitioned and the subdomain meshes are sent to the slaves. Finally, the curved elements are uniformly subdivided in parallel targeting the geometry approximated by the curved mesh. The result is a distributed finer linear mesh featuring improved geometric accuracy. The key ingredient of our implementation is to approximate the target geometry as a linear mesh equipped with an elemental field corresponding to an element-wise polynomial geometry representation. Thus, the distribution of the curved geometry is equivalent to partitioning the linear mesh and sending the subdomain meshes and the elemental fields to the slaves. The main application of the obtained finer linear mesh is to compute in parallel steady state flow solutions on real topographies. The qualitative results show that for 2D and 3D steady state flow solutions, on real and synthetic topographies, our parallel subdivision approach mitigates the artificial artifacts that might appear with standard straight-sided subdivision methods. We also check the parallel performance of the implementation by performing a weak scalability test in 2D.This work was financially supported by the PRACE project funded in part by the EUs Horizon 2020 research and innovation program (2014-2020) under grant agreement 653838. This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme under grant agreement No 715546. The work of the corresponding author has been partially supported by the Spanish Ministerio de Economía y Competitividad under the personal grant agreement RYC- 2015-01633.Peer ReviewedPostprint (published version

A CFD framework for offshore and onshore wind farm simulation

We present a wind simulation framework for offshore and onshore wind farms. The simulation framework involves an automatic hybrid high-quality mesh generation process, a pre-processing to impose initial and boundary conditions, and a solver for the Reynolds Averaged Navier-Stokes (RANS) equations with two different turbulence models, a modified standard k-epsilon model and a realizable k-epsilon model in which we included the Coriolis effects. Wind turbines are modeled as actuator discs. The wind farm simulation framework has been implemented in Alya, an in-house High Performance Computing (HPC) multi-physics finite element parallel solver. An application example is shown for an onshore wind farm composed of 165 turbines.This work has been supported by the EU H2020 projects New European Wind Atlas ERA-NET PLUS (NEWA), High Performance Computing for Energy (HPC4E, grant agreement 689772), and the Energy oriented Centre of Excellence (EoCoE, grant agreement 676629). We also thank two anonymous reviewers for their constructive comments on the first version of the manuscript.Peer ReviewedPostprint (published version

Representing Urban Geometries for Unstructured Mesh Generation

AbstractWe present a robust and automatic method to generate an idealized surface geometry of a city landscape ready to be meshed for computer simulations. The city geometry is idealized for non viscous flow simulations and targets two main geometrical features: the topography and the city blocks. The procedure is fully automatic and demands no human interaction given the following source data: the city cadastre, a Digital Elevation Model (DEM) of all the target domain, and Light Detection And Ranging (LiDAR) data of the domain region covered by the cadastre. The geometry representation takes three main steps. First, a 2D mesh of the cadastre is generated, where the elements are marked according to street and block regions. Second, using a DEM of the city landscape the topography surface mesh is generated by finding the best surface mesh in the least-squares sense obtained by deforming the previous 2D mesh. Third, we extrude the block facades and we compute a planar ceiling taking into account all the buildings belonging to that city block. We describe the applicability of the geometry representation by presenting the work-flow required to generate an unstructured mesh valid for non-viscous flow or transport simulations. Finally, we illustrate the main application by obtaining a surface and tetrahedral mesh for the city of Barcelona in Spain

Subdividing linear and curved meshes preserving sharp features of a model

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

Adaptive points to estimate the Lebesgue constant on the simplex

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

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