José María Escobar

Simultaneous untangling and smoothing of tetrahedral meshes

Abstract The quality improvement in mesh optimisation techniques that preserve its connectivity are obtained by an iterative process in which each node of the mesh is moved to a new position that minimises a certain objective function. The objective function is derived from some quality measure of the local submesh , that is, the set of tetrahedra connected to the adjustable or free node . Although these objective functions are suitable to improve the quality of a mesh in which there are non- inverted elements, they are not when the mesh is tangled. This is due to the fact that usual objective functions are not defined on all R 3 and they present several discontinuities and local minima that prevent the use of conventional optimisation procedures. Otherwise, when the mesh is tangled, there are local submeshes for which the free node is out of the feasible region , or this does not exist. In this paper we propose the substitution of objective functions having barriers by modified versions that are defined and regular on all R 3 . With these modifications, the optimisation process is also directly applicable to meshes with inverted elements, making a previous untangling procedure unnecessary. This simultaneous procedure allows to reduce the number of iterations for reaching a prescribed quality. To illustrate the effectiveness of our approach, we present several applications where it can be seen that our results clearly improve those obtained by other authors.

Genetic algorithms for an improved parameter estimation with local refinement of tetrahedral meshes in a wind model

The efficiency of a mass-consistent model for wind field adjustment depends on several parameters that arise in various stages of the process. On one hand, those involved in the construction of the initial wind field using horizontal interpolation and vertical extrapolation of the wind measures registered at meteorological stations. On the other hand, the stability parameter which allows from a strictly horizontal wind adjustment to a pure vertical one. In general, the values of all of these parameters are based on empirical laws. The main goal of this work is the estimation of these parameters using genetic algorithms, such that some of the wind velocities observed at the measurement station are regenerated as accurately as possible by the model. In addition, we study the effect of the mesh refinement on the parameter estimation in several numerical experiments.

A 3-D diagnostic model for wind field adjustment

Abstract The main objective of this work is to develop a suitable three-dimensional finite element code for solving the wind field adjustment problem [Solar Energy 54 (1995) 49] in an efficient way. A new triangulation technique is proposed for a 3-D domain, considering the Cartesian coordinate system with both variable horizontal and vertical spacing possibilities in the final mesh. For the initialization process, a general method for horizontal interpolation is devised including the distance and elevation difference between meteorological stations and grid points. A log-linear profile is used for the vertical extrapolation. Special attention is paid to the solver of the linear system of equations (large and sparse) arising from the finite element discretization. An interesting matrix storage scheme is considered and several preconditioners are compared using the BICGSTAB method. Finally, the code is connected to a powerful module of graphic postprocessing which allows better and easier interpretation of the results. The model is checked on a test problem as well as on realistic data corresponding to a zone of the Island of Lanzarote.

Several aspects of three-dimensional Delaunay triangulation

Due to the appearance of slivers, the quality of three-dimensional Delaunay triangulation may be inadequate for the application of the finite-element method (FEM). Otherwise, the round-off errors made by the computer when working with floating point arithmetic may make ineffective the algorithms which construct that triangulation. In this paper a procedure to construct a nearly Delaunay triangulation able to solve these problems is presented. Experimental results and applications are provided.

A New Meccano Technique for Adaptive 3-D Triangulations

This paper introduces a new automatic strategy for adaptive tetrahedral mesh generation. A local refinement/derefinement algorithm for nested triangula-tions and a simultaneous untangling and smoothing procedure are the main involved techniques. The mesh generator is applied to 3-D complex domains whose boundaries are projectable on external faces of a coarse object meccano composed of cuboid pieces. The domain surfaces must be given by a mapping between meccano surfaces and object boundary. This mapping can be defined by analytical or discrete functions. At present we have fixed mappings with orthogonal , cylindrical and radial projections, but any other one-to-one projection may be considered. The mesh generator starts from a coarse tetrahedral mesh which is automatically obtained by the subdivision of each hexahedra, of a meccano hexahedral mesh, into six tetrahedra. The main idea is to construct a sequence of nested meshes by refining only those tetrahedra which have a face on the meccano boundary. The virtual projection of meccano external faces defines a valid triangulation on the domain boundary. Then a 3-D local refinement/derefinement is carried out such that the approximation of domain surfaces verifies a given precision. Once this objective is reached, those nodes placed on the meccano boundary are really projected on their corresponding true boundary, and inner nodes are relocated using a suitable mapping. As the mesh topology is kept during node movement, poor quality or even inverted elements could appear in the resulting mesh. For this reason, we finally apply a mesh optimization procedure. The efficiency of the proposed technique is shown with several applications to complex objects.

The Meccano Method for Automatic Tetrahedral Mesh Generation of Complex Genus-Zero Solids

In this paper we introduce an automatic tetrahedral mesh generator for complex genus-zero solids, based on the novel meccano technique. Our method only demands a surface triangulation of the solid, and a coarse approximation of the solid, called meccano, that is just a cube in this case. The procedure builds a 3-D triangulation of the solid as a deformation of an appropriate tetrahedral mesh of the meccano. For this purpose, the method combines several procedures: an automatic mapping from the meccano boundary to the solid surface, a 3-D local refinement algorithm and a simultaneous mesh untangling and smoothing. A volume parametrization of the genus-zero solid to a cube (meccano) is a direct consequence. The efficiency of the proposed technique is shown with several applications.

Simultaneous aligning and smoothing of surface triangulations

In this work we develop a procedure to deform a given surface triangulation to obtain its alignment with interior curves. These curves are defined by splines in a parametric space and, subsequently, mapped to the surface triangulation. We have restricted our study to orthogonal mapping, so we require the curves to be included in a patch of the surface that can be orthogonally projected onto a plane (our parametric space). For example, the curves can represent interfaces between different materials or boundary conditions, internal boundaries or feature lines. Another setting in which this procedure can be used is the adaption of a reference mesh to changing curves in the course of an evolutionary process. Specifically, we propose a new method that moves the nodes of the mesh, maintaining its topology, in order to achieve two objectives simultaneously: the piecewise approximation of the curves by edges of the surface triangulation and the optimization of the resulting mesh. We will designate this procedure as projecting/smoothing method and it is based on the smoothing technique that we have introduced for surface triangulations in previous works. The mesh quality improvement is obtained by an iterative process where each free node is moved to a new position that minimizes a certain objective function. The minimization process is done on the parametric plane attending to the surface piece-wise approximation and to an algebraic quality measure (mean ratio) of the set of triangles that are connected to the free node. So, the 3-D local projecting/smoothing problem is reduced to a 2-D optimization problem. Several applications of this method are presented.

Simultaneous untangling and smoothing of quadrilateral and hexahedral meshes using an object-oriented framework

In this work, we present a simultaneous untangling and smoothing technique for quadrilateral and hexahedral meshes. The algorithm iteratively improves a quadrilateral or hexahedral mesh by minimizing an objective function defined in terms of a regularized algebraic distortion measure of the elements. We propose several techniques to improve the robustness and the computational efficiency of the optimization algorithm. In addition, we have adopted an object-oriented paradigm to create a common framework to smooth meshes composed by any type of elements, and using different minimization techniques. Finally, we present several examples to show that the proposed technique obtains valid meshes composed by high-quality quadrilaterals and hexahedra, even when the initial meshes contain a large number of tangled elements.

The meccano method for isogeometric solid modeling and applications

We present a new method to construct a trivariate T-spline representation of complex solids for the application of isogeometric analysis. We take a genus-zero solid as a basis of our study, but at the end of the work we explain the way to generalize the results to any genus solids. The proposed technique only demands a surface triangulation of the solid as input data. The key of this method lies in obtaining a volumetric parameterization between the solid and the parametric domain, the unitary cube. To do that, an adaptive tetrahedral mesh of the parametric domain is isomorphically transformed onto the solid by applying a mesh untangling and smoothing procedure. The control points of the trivariate T-spline are calculated by imposing the interpolation conditions on points sited both on the inner and on the surface of the solid. The distribution of the interpolating points is adapted to the singularities of the domain to preserve the features of the surface triangulation. We present some results of the application of isogeometric analysis with T-splines to the resolution of Poisson equation in solids parameterized with this technique.

Performance Evaluation of a Parallel Algorithm for Simultaneous Untangling and Smoothing of Tetrahedral Meshes

A new parallel algorithm for simultaneous untangling and smoothing of tetrahedral meshes is proposed in this paper. We provide a detailed analysis of its performance on shared-memory many-core computer architectures. This performance analysis includes the evaluation of execution time, parallel scalability, load balancing, and parallelism bottlenecks. Additionally, we compare the impact of three previously published graph coloring procedures on the performance of our parallel algorithm. We use six benchmark meshes with a wide range of sizes. Using these experimental data sets, we describe the behavior of the parallel algorithm for different data sizes. We demonstrate that this algorithm is highly scalable when it runs on two different high-performance many-core computers with up to 128 processors. However, some parallel deterioration is observed. Here, we analyze the main causes of this parallel deterioration.