Arbitrary Unstructured Meshes Research Articles

Acceleration of Anisotropic Scattering Computations Using Coupled Ordinates Method (COMET)

Traditional sequential solution procedures for solving coupled heat transfer and radiation problems are known to perform very poorly when the optical thickness is high. This poor performance can be traced to the fact that sequential procedures cannot handle strong inter-equation coupling resulting from large absorption and scattering coefficients. This paper extends the use of the coupled ordinates method (COMET) to problems involving anisotropic scattering, where inter-equation coupling results from the in-scattering terms. A finite volume formulation for arbitrary unstructured meshes is used to discretize the radiative transfer equation. An efficient point-solution procedure is devised which is suitable for different scattering phase functions, and eliminates the cubic dependence of the operation count on the angular discretization. This solution scheme is used as a relaxation sweep in a multigrid procedure designed to eliminate long-wavelength errors. The method is applied to scattering problems and shown to substantially accelerate convergence for moderate to large optical thicknesses.

L2-stability of the upwind first order finite volume scheme for the Maxwell equations in two and three dimensions on arbitrary unstructured meshes

We investigate sufficient and possibly necessary conditions for the L 2 stability of the upwind first order finite volume scheme for Maxwell equations, with metallic and absorbing boundary conditions. We yield a very general sufficient condition, valid for any finite volume partition in two and three space dimensions. We show this condition is necessary for a class of regular meshes in two space dimensions. However, numerical tests show it is not necessary in three space dimensions even on regular meshes. Stability limits for time and space schemes with higher orders of accuracy are numerically investigated.

Coupled Ordinates Method for Multigrid Acceleration of Radiation Calculations

The finite volume and discrete ordinates metbods are known to converge increasingly slowly as the optical thickness is increased. This is a result of the sequential nature of the solution procedure; the equations for energy and the directional intensities are solved one by one, assuming prevailing values for other variables. The coupled ordinates method is proposed whereby the discrete energy and intensity equations at each cell are solved simultaneously, assuming spatial neighbors to be known. The point-coupled procedure is used as a relaxation sweep in a multigrid scheme. The formulation of coarse-level discrete equations admits arbitrary nonconvex polyhedra, making the scheme suitable for use with arbitrary unstructured polyhedral meshes. The scheme is shown to substantially accelerate convergence over a range of optical thicknesses

A Taylor–Galerkin Method for Simulating Nonlinear Dispersive Water Waves

A new numerical scheme for computing the evolution of water waves with a moderate curvature of the free surface, modeled by the dispersive shallow water equations, is described. The discretization of this system of equations is faced with two kinds of numerical difficulties: the nonsymmetric character of the (nonlinear) advection–propagation operator and the presence of third order mixed derivatives accounting for the dispersion phenomenon. In this paper it is shown that the Taylor–Galerkin finite element method can be used to discretize the problem, ensuring second order accuracy both in time and space and guaranteeing at the same time unconditional stability. The properties of the scheme are investigated by performing a numerical stability analysis of a linearized model of the scalar 1D regularized long wave equation. The proposed scheme extends straightforwardly to the fully nonlinear 2D system, which is solved here for the first time on arbitrary unstructured meshes. The results of the numerical simulation of a solitary wave overpassing a vertical circular cylinder are presented and discussed in a physical perspective.

Delaunay triangulation and 3D adaptive mesh generation

Delaunay triangulation and its complementary structure the Voronoi polyhedra form two of the most fundamental constructs of computational geometry. Delaunay triangulation offers an efficient method for generating high-quality triangulations. However, the generation of Delaunay triangulations in 3D with Watson's algorithm, leads to the appearance of silver tetrahedra, in a relatively large percentage. A different method for generating high-quality tetrahedralizations, based on Delaunay triangulation and not presenting the problem of sliver tetrahedra, is presented. The method consists in a tetrahedra division procedure and an efficient method for optimizing tetrahedral meshes, based on the application of a set of topological Delaunay transformations for tetrahedra and a technique for node repositioning. The method is robust and can be applied to arbitrary unstructured tetrahedral meshes, having as a result the generation of high-quality adaptive meshes with varying density, totally eliminating the appearance of sliver elements. In this way it offers a convenient and highly flexible algorithm for implementation in a general purpose 3D adaptive finite element analysis system. Applications to various engineering problems are presented

Adaptive methods in computational magnetics

A review of adaptive methods in 2-D and 3-D for the efficient solution of electromagnetic problems with the finite element method, developed by the authors, is presented in the paper. The adaptive methods presented consist mainly of two processes: mesh refinement and error estimation. A highly efficient technique for the refinement of arbitrary unstructured triangular and tetrahedral meshes, based on Delaunay triangulation, directly applicable and suitable in an automatic adaptive procedure, is described. The use of various error estimation criteria for electromagnetic problems is presented. The implementation of an automatic adaptive procedure with the finite element method, incorporating the above techniques, is provided and applied to the efficient solution of various 2-D and 3-D electromagnetic field problems.

An approach to refining three‐dimensional tetrahedral meshes based on Delaunay transformations

AbstractA technique for refining three‐dimensional tetrahedral meshes is proposed in this paper. The proposed technique is capable of treating arbitrary unstructured tetrahedral meshes, convex or non‐convex with multiple regions resulting in high quality constrained Delaunay triangulations. The tetrahedra generated are of high quality (nearly equilateral). Sliver tetrahedra, which present a real problem to many algorithms are not produced with the new method. The key to the generation of high quality tetrahedra is the iterative application of a set of topological transformations based on the Voronoi–Delaunay theory and a reposition of nodes technique. The computational requirements of the proposed technique are in linear relationship with the number of nodes and tetrahedra, making it ideal for direct employment in a fully automatic finite element analysis system for 3‐D adaptive mesh refinement. Application to some test problems is presented to show the effectiveness and applicability of the new method.