Equilateral Triangular Mesh Research Articles

ELIMINATION OF NUMERICAL DISPERSION FROM ELECTROMAGNETIC TIME DOMAIN ANALYSIS BY USING RESOURCE EFFICIENT FINITE ELEMENT TECHNIQUE

Time domain analysis of electromagnetic wave propagation is required for design and characterization of many optical and microwave devices. The FDTD method is one of the most widely used time domain methods for analysing electromagnetic scattering and radiation problems. However, due to the use of the Finite Difference grid, this method suffers from higher numerical dispersion and inaccurate discretisation due to staircasing at slanted and curve edges. The Finite Element (FE)-based meshing technique can discretize the computational domain offering a better approximation even when using a small number of elements. Some of the FE-based approaches have considered either an implicit solution, higher order elements, the solution of a large matrix or matrix lumping, all of which require more time and memory to solve the same problem or reduce the accuracy. This paper presents a new FE-based method which uses a perforated mesh system to solve Maxwell’s equations with linear elements. The perforated mesh reduces the requirement on memory and computational time to less than half of that compared to other FE-based methods. This paper also shows a very large improvement in the numerical dispersion over the FDTD method when the proposed method is used with an equilateral triangular mesh.

An accelerated scheme with high quality mesh based on Lloyd iteration

High quality mesh plays an important role for finite element methods in science computation and numerical simulation. Whether the mesh quality is good or not, to some extent, it determines the calculation results of the accuracy and efficiency. Different from classic Lloyd iteration algorithm which is convergent slowly, a novel accelerated scheme was presented, which consists of two core parts: mesh points replacement and local edges Delaunay swapping. By using it, almost all the equilateral triangular meshes can be generated based on centroidal Voronoi tessellation (CVT). Numerical tests show that it is significantly effective with time consuming decreasing by 40%. Compared with other two types of regular mesh generation methods, CVT mesh demonstrates that higher geometric average quality increases over 0.99.

Centroidal Voronoi tessellation‐based finite element superconvergence

AbstractIn this article, a finding on finite element superconvergence is reported. The Laplacian operator with Dirichlet boundary condition is considered. The linear finite element solutions have an O(h2+α)(α≈0.5)‐superconvergence in l2 norm at nodes on an almost equilateral triangular mesh generated based on centroidal Voronoi tessellation, for an arbitrary 2D bounded domain. Extensive numerical examples are presented to demonstrate the superconvergence property. Copyright © 2008 John Wiley & Sons, Ltd.

Viscoelastic flow past a confined cylinder: Instability and velocity inflection

The flow of the Oldroyd-B fluid past a circular cylinder in a channel is simulated by a parallelized finite volume method (FVM) using fine unstructured meshes. Convergent solutions are obtained up to a high Deborah number of De = 1.6 with a high aspect ratio mesh, while it can only be obtained for De = 0.6 with a more or less equi-lateral mesh. The governing equations are re-written using the discrete elastic viscous split stress formulation together with an independent interpolation of the vorticity (DEVSS- ω ) . The numerical procedure is based on the semi-implicit method for pressure-linked equations revised (SIMPLER) algorithm, using a collocated FVM discretization. To enhance the numerical accuracy, mixed meshes (structured and unstructured) are used and adapted to the geometry and the flow physics. The results show that an instability featuring stress oscillations on the top of the cylinder is produced at a critical De number ( De ≈ 0.6 ) . This critical value is in agreement with the experiments in the literature. The oscillation generated in the shear layer on the cylinder surface at high De number is convected downstream into the wake behind the cylinder. This feature of flow oscillation agrees well with experiments reported in the literature. The mechanism of these phenomena can be explained reasonably by a boundary layer analysis. The key finding is that the instability is due to an inflectional velocity profile, near the cylinder, generated by normal stress on the cylinder surface at high De number, which can only be captured with fine meshes. According to a newly proposed theory (so called energy gradient theory), inflectional velocity profile leads to flow instability and consequently allows the convection of the oscillatory flow within the shear layer downstream of the wake and results in the flow pulsation in the spanwise direction. Therefore, the origin of the instability for the flow around a cylinder is in the shear layer on the cylinder and not in the wake itself. In addition, it seems that the increasing rate of numerical perturbation is related to the mesh aspect ratio. Computing using thin-long meshes could get convergence even if the flow is oscillatory, while the computing is prohibited by the inflectional instability when equi-lateral triangular meshes are used.

An error estimate for finite volume methods for the Stokes equations on equilateral triangular meshes

AbstractWe give here an error estimate for a finite volume discretization of the Stokes equations in two space dimensions on equilateral triangular meshes. This work was initiated by an analogous result presented by Alami‐Idrissi and Atounti for general triangular meshes. However, in this latter article, the result is not actually proven. We state here the restricting assumptions (namely equilateral triangles) under which the error estimate holds, using the tools which were introduced by Eymard, Gallouet and Herbin and used by Alami‐Idrissi and Atounti. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004

Closed-form expressions for the numerical dispersion and reflection in FEM simulations involving biaxial materials

Closed-form expressions for the numerical errors caused by finite-element discretization of problems involving materials of biaxial permittivity and permeability tensors are developed. In particular, we derive expressions for the numerical dispersion and reflection in both first-order node and edge basis function finite-element formulations in an equilateral triangular mesh. Results using these closed-form expressions are compared to practical numerical simulations. The application of these expressions to the analysis of the performance of the perfectly matched layer boundary is suggested.

A triangular mesh random walk for Dirichlet problems

This paper presents a new Monte Carlo technique called the equilateral triangular mesh random walk. A major advantage of the method over the classical Monte Carlo techniques, such as the fixed random walk and the floating random walk, is that it is capable of handling Neumann problems which the classical Monte Carlo methods cannot handle. However, the presentation in this paper is limited to the application of the triangular mesh random walk to Dirichlet problems of Laplace's and Poisson's equations. Numerical experiments involving two-dimensional problems confirm the accuracy of the triangular mesh random walk.

Multigrid solution of the Navier-Stokes equations on triangular meshes

A Navier-Stokes algorithm for use on unstructured triangular meshes is presented. Spatial discretization of the governing equations is achieved using a finite element Galerkin approximation, which can be shown to be equivalent to a finite volume approximation for regular equilateral triangular meshes. Integration steady-state is performed using a multistage time-stepping scheme, and convergence is accelerated by means of implicit residual smoothing and an unstructured multigrid algorithm. Directional scaling of the artificial dissipation and the implicit residual smoothing operator is achieved for unstructured meshes by considering local mesh stretching vectors at each point. The accuracy of the scheme for highly stretched triangular meshes is validated by comparing computed flat-plate laminar boundary layer results with the well known similarity solution, and by comparing laminar airfoil results with those obtained from various well-established structured quadrilateral-mesh codes. The convergence efficiency of the present method is also shown to be competitive with those demonstrated by structured quadrilateral-mesh algorithms.

The Solution of Reactor Diffusion Problems

An iterative method of solving finite difference equations representing multi-group neutron diffusion is described. The method is applicable to many different meshes including rectangular and equilateral triangular meshes and three-dimensional box-like meshes. Normal and adjoint problems with general scatter matrices can be solved. The interaction of the numerical method with the storage scheme and with transfers to and from the backing store is emphasised to show the suitability of the method for solving large problems of practical interest. An iterative method of solving certain matrix eigenvalue problems approximating the multi-group neutron diffusion equations is described. These diffusion equations form a set of simultaneous partial differential equations. There is one equation representing the behaviour of each energy group of neutrons. The equations are coupled together by the fission process and by the scatter process which changes the energy of a neutron causing it to jump from one group to another. It will be convenient to number the fast group g = 1 and to count the groups in order of decreasing velocity so that g = G gives the slow or thermal group. A scatter event which transfers a neutron to a slower group is described as a down-scatter, the opposite being an up-scatter. The techniques to be described were developed primarily for power reactor calculations in which the neutron flux, for a number of groups, at hundreds or possibly thousands of mesh points on a regular lattice may be required. The chosen matrix form of the problem may be so large that it is not possible to hold all the non-zero elements simultaneously in the fast store. Thus the need for efficient algorithms, taking account of the interaction of the numerical method with the characteristics of tape and disc backing stores, provided the motivation for this work. Referring to both instructions and numbers as words, it is assumed throughout that about 32K words of fast store are available in which to run a program. If this is insufficient to hold the whole of a problem then the backing store is brought into use in an efficient, sequential manner that has proved easy to code. The essence of the method is that all fluxes for a line of mesh points are stored together as one block. During each powering iteration the solution of a multi-group inhomogeneous problem is found by successive overrelaxation, the unknowns being relaxed one block at a time. In two-dimensional problems only three blocks of unknowns and coefficients corresponding to one block need be accommodated in the fast store at any stage of the iterations. The multi-group fluxes required in the calculation of a fission or scatter source for a mesh point >