最小二乘有限元法和有限体积法在CFD中的应用比较

为比较最小二乘有限元法（Least Square Finite Element Method,LSFEM）和有限体积法在CFD应用中的优劣,采用最小二乘法离散不可压N-S方程的有限元模型,得到正定对称线性系统,采用高效的预处理共轭梯度法求解方程组;利用LSFEM和基于有限体积法的FLUENT分别计算Kovasznay流动、定常二维和三维后台阶流动以及非定常圆柱绕流等4个实例并比较计算结果.结果表明,LSFEM比有限体积法的收敛性和精确性更好,在CFD领域的应用价值很高.     

Topology optimization of heat conduction problems using the finite volume method

This note addresses the use of the finite volume method (FVM) for topology optimization of a heat conduction problem. Issues pertaining to the proper choice of cost functions, sensitivity analysis, and example test problems are used to illustrate the effect of applying the FVM as an analysis tool for design optimization. This involves an application of the FVM to problems with nonhomogeneous material distributions, and the arithmetic and harmonic averages have here been used to provide a unique value for the conductivity at element boundaries. It is observed that when using the harmonic average, checkerboards do not form during the topology optimization process. Preliminary results of the work reported here were presented at the WCSMO 6 in Rio de Janeiro 2005, see Gersborg-Hansen et al. (2005b).     

Godunov-type upwind flux schemes of the two-dimensional finite volume discrete Boltzmann method

A simple unified Godunov-type upwind approach that does not need Riemann solvers for the flux calculation is developed for the finite volume discrete Boltzmann method (FVDBM) on an unstructured cell-centered triangular mesh. With piecewise-constant (PC), piecewise-linear (PL) and piecewise-parabolic (PP) reconstructions, three Godunov-type upwind flux schemes with different orders of accuracy are subsequently derived. After developing both a semi-implicit time marching scheme tailored for the developed flux schemes, and a versatile boundary treatment that is compatible with all of the flux schemes presented in this paper, numerical tests are conducted on spatial accuracy for several single-phase flow problems. Four major conclusions can be made. First, the Godunov-type schemes display higher spatial accuracy than the non-Godunov ones as the result of a more advanced treatment of the advection. Second, the PL and PP schemes are much more accurate than the PC scheme for velocity solutions. Third, there exists a threshold spatial resolution below which the PL scheme is better than the PP scheme and above which the PP scheme becomes more accurate. Fourth, besides increasing spatial resolution, increasing temporal resolution can also improve the accuracy in space for the PL and PP schemes.     

对称区域边界处理方法及基于表面粒子提取的表面张力计算

流固边界处理一直是流体模拟的研究重点,边界力法和虚粒子法是研究流固边界 的常用方法。边界力法通过对铺设在边界上的粒子施加排斥力防止粒子穿透,但边界力的计算 限制了模拟速度。虚粒子法在边界处生成虚粒子,随着粒子数的增加所需的虚粒子数也随之增 加,导致计算速度下降,且会出现流体与边界分离的现象。为此,提出一种对称区域边界处理 方法,在保证逼真度的前提下满足实时性要求,随着粒子数的增加,其耗时增长也明显比其他 传统方法慢,更适合对复杂场景的模拟,同时避免了边界处流体与边界分离的现象。CSF 方法 是处理表面张力常用的方法,可将表面张力看作体积力进行计算,大大减弱了表面形状对曲率 计算的影响,而事实上曲率的计算只与表面的形状有关。为此,对CSF 方法进行了改进,提出 了一种基于表面粒子提取的表面张力计算方法,减小了传统CSF 方法计算曲率的误差,提高了 计算速度。模拟仿真的效果验证了该方法的有效性。     

Development of a mixed control volume – Finite element method for the advection–diffusion equation with spectral convergence

In this paper we attack the problem of devising a finite volume method for computational fluid dynamics and related phenomena which can deal with complex geometries while attaining high-orders of accuracy and spectral convergence at a reasonable computational cost. As a first step towards this end, we propose a control volume finite element method for the solution of the advection–diffusion equation. The numerical method and its implementation are carefully tested in the paper where h- and p-convergence are checked by comparing numerical results against analytical solutions in several relevant test-cases. The numerical efficiency of a selected set of operations implemented is estimated by operation counts, ill-conditioning of coefficient matrices is avoided by using an appropriate distribution of interpolation points and control-volume edges.     

Numerical modeling of the InAs quantum dot with application of coordinate transformation and the finite difference method

In order to resolve the three dimensional Schrödinger equation, we report in this paper a method providing sufficient accuracy, stability and flexibility with respect to the size and shape of the quantum dot. This numerical method, already used in the two-dimensional case, is based on a suitable combination of coordinate transformation and the finite difference method. It provides an efficient and simple approach for the energy and wavefunction calculations of quantum nanostructures. The proposed method is used to investigate the electron and hole energy levels as well as their wave functions in InAs/GaAs strained and unstrained quantum dots with the aim to attain the 1.55 μm wavelength with realistic dot size. The optical transition energies and the oscillator strengths are also studied. The obtained results are in agreement with several previous works.