G. Montero

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.

Adaptive strategies using standard and mixed finite elements for wind field adjustment

In order to find a map of wind velocities, this study tries to obtain an incompressible wind field that adjusts to an experimental one: also verifying the corresponding boundary conditions of physical interest. This problem has been solved by several authors using finite differences or standard finite element techniques. In this paper, this problem is solved by two different adaptive finite element methods. The first makes use of standard finite element techniques, using linear interpolation of a potential function. In the second, a direct computation of the velocity field is undertaken by means of a mixed finite element method. Several error indicators are proposed for both formulations together with an adaptive strategy. We have applied both methods to several typical test problems, as well as to realistic data corresponding to the Island of Fuerteventura, with satisfactory results from a numerical point of view.

3-D modelling of wind field adjustment using finite differences in a terrain conformal coordinate system

Abstract A mass consistent model is developed in order to obtain a wind field which adjusts to an initial one. For the construction of the initial field, horizontal interpolation is considered at the level of the measure stations over the terrain. From these values, vertical wind profiles are built according to atmospheric stability conditions, terrain roughness, geostrophic wind, atmospheric stratification. The problem is formulated in terms of an incompressible fluid with no-flow-through conditions on the terrain. The adjustment is carried out by minimizing a least-square function. Lagrange multipliers technique leads to an elliptic problem, which is discretized using finite-differences schemes after applying a proposed system of terrain conformal coordinate. This change not only simplifies the mesh generation, but leads to simpler boundary conditions and allows the definition of the vertical spacing with more resolution near the terrain without affecting the efficiency. Finally, the numerical model is applied to realistic data corresponding to the Island of La Palma (Canary Islands).

A finite difference model for air pollution simulation

A 3-D model for atmospheric pollutant transport is proposed considering a set of coupled convection-diffusion-reaction equations. The convective phenomenon is mainly produced by a wind field obtained from a 3-D mass consistent model. In particular, the modelling of oxidation and hydrolysis of sulphur and nitrogen oxides released to the surface layer is carried out by using a linear module of chemical reactions. The dry deposition process, represented by the so-called deposition velocity, is introduced as a boundary condition. Moreover, the wet deposition is included in the source term of the governing equations using the washout coefficient. Before obtaining a numerical solution, the problem is transformed using a terrain conformal coordinate system. This allows to work with a simpler domain in order to build a mesh that provides finite difference schemes with high spatial accuracy. The convection-diffusion-reaction equations are solved with a high order accurate time-stepping discretization scheme which is constructed following the technique of Lax and Wendroff. Finally, the model is tested with a numerical experiment in La Palma Island (Canary Islands).

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.

Approximate inverse computation using Frobenius inner product

Parallel preconditioners are presented for the solution of general linear systems of equations. The computation of these preconditioners is achieved by orthogonal projections related to the Frobenius inner product. So, minM∈∥AM−I∥ F and matrix M0∈ corresponding to this minimum ( being any vectorial subspace of ℳn(ℝ)) are explicitly computed using accumulative formulae in order to reduce computational cost when subspace is extended to another one containing it. Every step, the computation is carried out taking advantage of the previous one, what considerably reduces the amount of work. These general results are illustrated with the subspace of matrices M such that AM is symmetric. The main application is developed for the subspace of matrices with a given sparsity pattern which may be constructed iteratively by augmenting the set of non-zero entries in each column. Finally, the effectiveness of the sparse preconditioners is illustrated with some numerical experiments. Copyright