Generation of curved high-order meshes with optimal quality and geometric accuracy

We present a novel methodology to generate curved high-order meshes featuring optimal mesh quality and geometric accuracy. The proposed technique combines a distortion measure and a geometric L2-disparity measure into a single objective function. While the element distortion term takes into account the mesh quality, the L2-disparity term takes into account the geometric error introduced by the mesh approximation to the target geometry. The proposed technique has several advantages. First, we are not restricted to interpolative meshes and therefore, the resulting mesh approximates the target domain in a non-interpolative way, further increasing the geometric accuracy. Second, we are able to generate a series of meshes that converge to the actual geometry with expected rate while obtaining high-quality elements. Third, we show that the proposed technique is robust enough to handle real-case geometries that contain gaps between adjacent entities.Peer ReviewedPostprint (published version

Checking and improving the geometric accuracy of non-interpolating curved high-order meshes

Peer ReviewedPostprint (author's final draft

Unstructured and semi-structured hexahedral mesh generation methods

Discretization techniques such as the finite element method, the finite volume method or the discontinuous Galerkin method are the most used simulation techniques in ap- plied sciences and technology. These methods rely on a spatial discretization adapted to the geometry and to the prescribed distribution of element size. Several fast and robust algorithms have been developed to generate triangular and tetrahedral meshes. In these methods local connectivity modifications are a crucial step. Nevertheless, in hexahedral meshes the connectivity modifications propagate through the mesh. In this sense, hexahedral meshes are more constrained and therefore, more difficult to gener- ate. However, in many applications such as boundary layers in computational fluid dy- namics or composite material in structural analysis hexahedral meshes are preferred. In this work we present a survey of developed methods for generating structured and unstructured hexahedral meshes.Peer ReviewedPostprint (published version

Size preserving mesh generation in adaptivity processes

It is well known that the variations of the element size have to be controlled in order to generate a high-quality mesh. Hence, several techniques have been developed to limit the gradient of the element size. Although these methods allow generating high-quality meshes, the obtained discretizations do not always reproduce the prescribed size function. Specifically, small elements may not be generated in a region where small element size is prescribed. This is critical for many practical simulations, where small elements are needed to reduce the error of the numerical simulation. To solve this issue, we present the novel size-preserving technique to control the mesh size function prescribed at the vertices of a background mesh. The result is a new size function that ensures a high-quality mesh with all the elements smaller or equal to the prescribed element size. That is, we ensure that the new mesh handles at least one element of the correct size at each local minima of the size function. In addition, the gradient of the size function is limited to obtain a high-quality mesh. Two direct applications are presented. First, we show that we can reduce the number of iterations to converge an adaptive process, since we do not need additional iterations to generate a valid mesh. Second, the size-preserving approach allows to generate quadri- lateral meshes that correctly preserves the prescribed element size.Peer 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

Management, design and developement of a mesh generation environment using open source software

In this paper we present an object oriented implementation of a general-purpose mesh generation environment for geometry-based simulations. The aim of this application is to unify available legacy code and new research algorithms in only one mesh generation suite. We focus in two aspects that can be of the general interest for managers, designers and developers of similar projects. On the one hand, we analyze the software engineering practices that we have followed in the management and development process. In addition, we detail and discuss the Open Source tools and libraries that we have used. On the other hand, we discuss the design and the data structure of the environment. In particular, we first summarize the topological and geometrical representation. Second, we detail our implementation of the hierarchical mesh generation structure. Third we present our design to mediate collaboration between classes. Finally, we present some of the mesh generation features to show the capabilities of the environment

Bisecting with optimal similarity bound on 3D unstructured conformal meshes

We propose a new method to mark for bisection the edges of an arbitrary 3D unstructured conformal mesh. For these meshes, the approach conformingly marks all the tetrahedra with coplanar edge marks. To this end, the method needs three key ingredients. First, we propose a specific edge ordering. Second, marking with this ordering, we guarantee that the mesh becomes conformingly marked. Third, we also ensure that all the marks are coplanar in each tetrahedron. To demonstrate the marking method, we implement an existent marked bisection approach. Using this implementation, we mark and then locally refine 3D unstructured conformal meshes. We conclude that the resulting marked bisection features an optimal bound of similarity classes per tetrahedron.This project has received funding from the European Research Council (ERC) under the European Union’sHorizon 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 X. Roca 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 new procedure to compute imprints in multi-sweeping algorithms

One of the most widely used algorithms to generate hexahedral meshes in extrusion volumes with several source and target surfaces is the multi-sweeping method. However, the multi-sweeping method is highly dependent on the final location of the nodes created during the decomposition process. Moreover, inaccurate location of inner nodes may generate erroneous imprints of the geometry surfaces such that a final mesh could not be generated. In this work, we present a new procedure to decompose the geometry in many-to-one sweepable volumes. The decomposition is based on a least-squares approximation of affine mappings defined between the loops of nodes that bound the sweep levels. In addition, we introduce the concept of computational domain, in which every sweep level is planar. We use this planar representation for two purposes. On the one hand, we use it to perform all the imprints between surfaces. Since the computational domain is planar, the robustness of the imprinting process is increased. On the other hand, the computational domain is also used to compute the projection onto source surfaces. Finally, the location of the inner nodes created during the decomposition process is computed by averaging the locations computed projecting from target and source surfaces.Postprint (published version

Temperature dependence of dynamic, tunnelling and kinetic isotope effects in formate dehydrogenase

The origin of the catalytic power of enzymes has been a question of debate for a long time. In this regard, the possible contribution of protein dynamics in enzymatic catalysis has become one of the most controversial topics. In the present work, the hydride transfer step in the formate dehydrogenase (FDH EC 1.2.1.2) enzyme is studied by means of molecular dynamic (MD) simulations with quantum mechanics/molecular mechanics (QM/MM) potentials in order to explore any correlation between dynamics, tunnelling effects and the rate constant. The temperature dependence of the kinetic isotope effects (KIEs), which is one of the few tests that can be studied by experiments and simulations to shed light on this debate, has been computed and the results have been compared with previous experimental data. The classical mechanical free energy barrier and the number of recrossing trajectories seem to be temperature-independent while the quantum vibrational corrections and the tunnelling effects are slightly temperature-dependent over the interval of 5–45 °C. The computed primary KIEs are in very good agreement with previous experimental data, being almost temperature-independent within the standard deviations. The modest dependence on the temperature is due to just the quantum vibrational correction contribution. These results, together with the analysis of the evolution of the collective variables such as the electrostatic potential or the electric field created by the protein on the key atoms involved in the reaction, confirm that while the protein is well preorganised, some changes take place along the reaction that favour the hydride transfer and the product release. Coordinates defining these movements are, in fact, part of the real reaction coordinate

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