R. Montenegro

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.

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.

An improved derefinement algorithm of nested meshes

Abstract In this paper we present a new version of the derefinement algorithm developed by Plaza et al . (in Numerical Methods in Engineering , Elsevier Science, Amsterdam, 1992, pp. 225–232; in Algorithms, Software, Architecture , Elsevier Science, Amsterdam, 1992, pp. 409–415; A. Plaza, PhD thesis, University of Las Palmas de Gran Canaria, 1993; Commun. Numer. Meth. Engng. , 1994, 10 , 403–412). 1–4 The purpose is to achieve a better derefinement algorithm with a lesser degree of complexity. We present the theoretical study of this improved derefinement algorithm and of the inverse one for refinement. Firstly, our initial version of the derefinement algorithm is summarized. Then we present the refinement algorithm associated with the improved derefinement one. Finally, automatic control of the sequences of irregular nested triangulations is shown by means of the resolution of an unsteady problem. In this problem the initial mesh has only nine nodes and a combination of refinements and derefinements have been applied to approach both the circular domain and the initial solution.

APPLICATION OF A NONLINEAR EVOLUTION MODEL TO FIRE PROPAGATION

The numerical simulation of fire in forest has been an important objective in recent researches, The rate of spread and shape of a forest fire front is af8ecte.d by many factors. The most important of these are as follows: fuel type and moisture content, wind velocity and variability, forest topography, fire spread mechanism, fuel continuity and the amount of spotting (cf.[ l-21). The development of Geographic Information Systems allows the incorporation of these data to the developed models, The first models took into account constant factors, continuous uniform fuel type, constant wind velocity, moisture and slope. Under these conditions, a fire ignited at a single point reaches a quasi-steady state and progresses toward the down wind direction and expands at a constant rate. These data cannot give precise predictions under variable conditions but are very useful in order to the intuition of the fire controller. Models capable of being incorporated into the computer simulations of fires under variable conditions have been developed, based on cellular automata (cf. [3-7]), and stochastic process [8]. These models can give useful indicators as to fire behavior under such conditions. Combustion phenomena has been extensively studied [9], unsteady flame propagation has been analyzed [lo]. Models based on combustion theories are very difficult to develop because of the diversity of the fuel type and varied chemical composition within a given fuel type. Because of the complexity of the problem, models based rigorously on combustion theory have not been completely developed. In this preliminary work, a first attempt is done to design a computer code for numerical simulation of forest fire spread in landscapes. Basically a convection-diffusion model for temperature and a mass-consistent model for wind field simulation will be assumed. A two-steps chemical mechanism is simplified in order to obtain the heat source. This proposed 2-D model take into account the convection phenomena due to temperature gradients in vertical direction. A numerical solution of the former model is presented using a finite difference method together with the study of stability. This numerical method is contrasted with an adaptive finite element method using reflnementiderefinement techniques (cf. [ 1 1 - 143).

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.