带阻尼项定常Navier-Stokes方程的并行两水平有限元算法

基于两重网格离散和区域分解技术，提出数值求解带阻尼项定常Navier-Stokes方程的三种并行两水平有限元算法。其基本思想是首先在粗网格上求解完全的非线性问题，以获得粗网格解，然后在重叠的局部细网格子区域上并行求解Stokes、 Oseen和Newton线性化的残差问题，最后在非重叠的局部细网格子区域上校正近似解。数值算例验证了算法的有效性。     

大雷诺数Navier-Stokes方程的两水平亚格子模型稳定化方法

基于两重网格离散方法,提出三种求解大雷诺数定常Navier-Stokes方程的两水平亚格子模型稳定化有限元算法.其基本思想是首先在一粗网格上求解带有亚格子模型稳定项的Navier-Stokes方程,然后在细网格上分别用三种不同的校正格式求解一个亚格子模型稳定化的线性问题,以校正粗网格解.通过适当的稳定化参数和粗细网格尺寸的选取,这些算法能取得最优渐近收敛阶的有限元解.最后,用数值模拟验证三种算法的有效性.     

裂缝性介质多尺度深度学习模型

结合人工神经网络建立裂缝介质多尺度深度学习流动模型.基于一套粗网格和一套细网格,通过在粗网格上训练数据,多尺度神经网络能够以较少的自由度训练出准确的神经网络.并在粗网格上通过求解局部流动问题获得多尺度基函数,结合神经网络进一步得到精细网格的解.基于离散裂缝的流动方程可视为多层网络,网络层数依赖于求解时间步数.阐述裂缝介质多尺度机器学习数值计算格式的建立,介绍如何使用多尺度算法构建离散裂缝模型的多尺度基函数,并采用超样本技术进一步提高计算准确性.数值结果表明,多尺度有限元算法与机器学习结合是一种有效的流体流动模拟算法.     

非线性偏微分方程边值问题的优化算法研究与应用

针对椭圆类非线性偏微分方程边值问题,以差分法和动态设计变量优化算法为基础,以离散网格点未知函数值为设计变量,以离散网格点的差分方程组构建为复杂程式化形式的目标函数.提出一种求解离散网格点处未知函数值的优化算法.编制了求解未知离散点函数值的通用程序.求解了具体算例.通过与解析解对比,表明了本文提出求解算法的有效性和精确性,将为更复杂工程问题分析提供良好的解决方法. 关键词： 非线性偏微分方程 边值问题 动态设计变量优化算法 程序设计     

薄板弯曲大变形高阶非线性偏微分方程推导与优化算法研究

针对薄板弯曲大变形问题, 运用变分原理, 建立了薄板弯曲大变形问题的高阶非线性偏微分方程. 运用有限差分法和动态设计变量优化算法原理, 以离散坐标点的上未知挠度为设计变量, 以离散坐标点的差分方程组构建目标函数, 提出了薄板弯曲大变形挠度求解的动态设计变量优化算法, 编制了相应的优化求解程序. 分析了具有固定边界、均布载荷下的矩形薄板挠度的典型算例. 通过与有限元的结果对比, 表明了本文求解算法的有效性和精确性, 提供了直接求解实际工程问题的基础.     

多网格法求解Euler方程的叶栅绕流计算

一、前言 求解Euler方程是叶栅跨音速流数值模拟的重要方法之一.为了提高收敛速度,目前Euler方程多网格算法有了很大的发展.这种方法利用不同的网格尺寸有利于消除不同波长误差的原理,将数值解放在粗、细几种网格上反复进行,最终保证解具有细网格的离散精度,而求解过程又有粗网格的收敛率.我们的研究表明,正确地设计不同网格     

隐-显积分因子间断Galerkin方法求解二维辐射扩散方程

提出一种求解二维非平衡辐射扩散方程的数值方法.空间离散上采用加权间断Galerkin有限元方法,其中数值流量的构造采用一种新的加权平均;时间离散上采用隐-显积分因子方法,将扩散系数线性化,然后用积分因子方法求解间断Galerkin方法离散后的非线性常微分方程组.数值试验中在非结构网格上求解了多介质的辐射扩散方程.结果表明:对于强非线性和强耦合的非线性扩散方程组,该方法是一种非常有效的数值算法.     

半解析对偶棱边元及其在波导不连续性问题中的应用

给出电磁波导的对偶变量变分原理,并采用对偶棱边元对波导的横截面进行半解析离散. 将波导中沿纵向均匀的区段视为子结构,运用基于Riccati方程的精细积分算法求出其出口刚度阵,然后与不均匀区段的常规有限元网格拼装即可对波导不连续性问题进行求解. 半解析对偶棱边元的采用可以在最大程度上对有限元网格进行缩减,并且能够在不增加计算量的前提下任意增加子结构的长度,从而可以将截断求解区域的人工边界设置在距离不均匀区段充分远的地方,极大地减少了近似边界条件所带来的误差. 数值算例证明这种方法具有很高的精度与效率. 关键词： 波导的不连续性 半解析辛分析 对偶棱边元 精细积分     

非匹配网格上广义Stokes问题的区域分裂解法及事后误差估算

发展了一种广义Stokes问题的无覆盖区域分裂解法。子域交界面上的约束条件是通过引入一Lagrange乘子而得到弱满足的,在有限元离散子域的交界处网格可以是非匹配的。应用Petrov Galerkin方法解每个子域上的广义Stokes问题,而交界面上的Lagrange乘子则通过共轭梯度法迭代求解,各变量均由线性函数离散。对上述区域分裂解法,还构造了基于求解当地问题的误差事后估算方法。各变量的当地误差估算器定义在二阶非连续鼓包(bump)函数的空间中。最后给出了基于事后误差估算值的自适应网格上的数值结果。     

三维笛卡儿坐标系中Lagrange流体力学的显式相容有限元方法(英文)

将Caramana等人提出的相容算法思想和有限元方法相结合,提出三维笛卡儿坐标系中Lagrange流体力学的显式相容有限元方法.采用三线性六面体单元和交错网格进行空间离散,利用质量集中进行显式求解,无需求解线性代数方程组.时间离散可采用两步显式Runge-Kutta格式.用边人工粘性消除激波振荡,用子网格扰动压力抑制网格的非物理变形.给出若干标准算例.数值算例表明,该方法具有较高的计算精度和计算效率,同时具有很好的对称性和总能量守恒性,总能量计算误差为计算机浮点计算截断误差.     

A Simplified Parallel Two-Level Iterative Method for Simulation of Incompressible Navier-Stokes Equations

Based on two-grid discretization, a simplified parallel iterative finite element method for the simulation of incompressible Navier-Stokes equations is developed and analyzed. The method is based on a fixed point iteration for the equations on a coarse grid, where a Stokes problem is solved at each iteration. Then, on overlapped local fine grids, corrections are calculated in parallel by solving an Oseen problem in which the fixed convection is given by the coarse grid solution. Error bounds of the approximate solution are derived. Numerical results on examples of known analytical solutions, lid-driven cavity flow and backward-facing step flow are also given to demonstrate the effectiveness of the method.     

Two-Level Stabilized Finite Volume Methods for the Stationary Navier-Stokes Equations

In this work, two-level stabilized finite volume formulations for the 2D steady Navier-Stokes equations are considered. These methods are based on the local Gauss integration technique and the lowest equal-order finite element pair. Moreover, the two-level stabilized finite volume methods involve solving one small Navier-Stokes problem on a coarse mesh with mesh size $H$, a large general Stokes problem for the Simple and Oseen two-level stabilized finite volume methods on the fine mesh with mesh size $h$=$\mathcal{O}(H^2)$ or a large general Stokes equations for the Newton two-level stabilized finite volume method on a fine mesh with mesh size $h$=$\mathcal{O}(|\log h|^{1/2}H^3)$. These methods we studied provide an approximate solution $(\widetilde{u}_h^v,\widetilde{p}_h^v)$ with the convergence rate of same order as the standard stabilized finite volume method, which involve solving one large nonlinear problem on a fine mesh with mesh size $h$. Hence, our methods can save a large amount of computational time.     

Free Mesh Method: fundamental conception, algorithms and accuracy study

The finite element method (FEM) has been commonly employed in a variety of fields as a computer simulation method to solve such problems as solid, fluid, electro-magnetic phenomena and so on. However, creation of a quality mesh for the problem domain is a prerequisite when using FEM, which becomes a major part of the cost of a simulation. It is natural that the concept of meshless method has evolved. The free mesh method (FMM) is among the typical meshless methods intended for particle-like finite element analysis of problems that are difficult to handle using global mesh generation, especially on parallel processors. FMM is an efficient node-based finite element method that employs a local mesh generation technique and a node-by-node algorithm for the finite element calculations. In this paper, FMM and its variation are reviewed focusing on their fundamental conception, algorithms and accuracy.     

求解辐射传递的非结构混合有限体积／有限元法

本文给了一种适用于任意非结构网格的有限体积/有限元法的混合算法用于求解多维半透明吸收、发射、散射性灰矩形介质内的辐射传递.该方法使用有限元法进行角度离散,有限体积法进行空间离散.与基于辐射传递离散坐标方程的方法不同的是,该方法在迭代求解的过程中,针对每一个空间体元,所有角度方向的辐射强度同时耦合求出.通过两个算例验证了该解法的正确性.     

Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D poisson equation

We develop a sixth order finite difference discretization strategy to solve the two dimensional Poisson equation, which is based on the fourth order compact discretization, multigrid method, Richardson extrapolation technique, and an operator based interpolation scheme. We use multigrid V-Cycle procedure to build our multiscale multigrid algorithm, which is similar to the full multigrid method (FMG). The multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid. A sixth order accurate coarse grid solution is computed by using the Richardson extrapolation technique. Then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid. Numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method, compared to Sun–Zhang’s sixth order Richardson extrapolation compact (REC) discretization strategy using Alternating Direction Implicit (ADI) method and the standard fourth order compact difference (FOC) scheme using a multigrid method.     

Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods

In this paper, we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method. We use two Newton iterations on the fine grid in our methods. Firstly, we solve an original nonlinear problem on the coarse nonlinear grid, then we use Newton iterations on the fine grid twice. The two-grid idea is from Xu$'$s work [SIAM J. Numer. Anal., 33 (1996), pp. 1759-1777] on standard finite method. We also obtain the error estimates for the algorithms of the two-grid method. It is shown that the algorithm achieves asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy $h=\mathcal{O}(H^{(4k+1)/(k+1)})$.     

间断有限元方法求解一维非平衡辐射扩散方程

研究一维非平衡辐射扩散方程的数值方法.通过求解间断系数热传导方程的广义黎曼问题,得到一种带加权数值流量,基于该数值流量构造了一类新型的间断有限元方法.在时间离散上采用向后Euler方法,形成的非线性方程组采用Picard迭代求解.数值试验表明该方法具有捕捉大梯度的能力,而且能适应扩散系数间断的情形.     

Efficient parallel resolution of the simplified transport equations in mixed-dual formulation

A reactivity computation consists of computing the highest eigenvalue of a generalized eigenvalue problem, for which an inverse power algorithm is commonly used. Very fine modelizations are difficult to treat for our sequential solver, based on the simplified transport equations, in terms of memory consumption and computational time.A first implementation of a Lagrangian based domain decomposition method brings to a poor parallel efficiency because of an increase in the power iterations [1]. In order to obtain a high parallel efficiency, we improve the parallelization scheme by changing the location of the loop over the subdomains in the overall algorithm and by benefiting from the characteristics of the Raviart–Thomas finite element. The new parallel algorithm still allows us to locally adapt the numerical scheme (mesh, finite element order). However, it can be significantly optimized for the matching grid case. The good behavior of the new parallelization scheme is demonstrated for the matching grid case on several hundreds of nodes for computations based on a pin-by-pin discretization.     

Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods

In this paper, we study an efficient scheme for nonlinear reaction-diffusion equations discretized by mixed finite element methods. We mainly concern the case when pressure coefficients and source terms are nonlinear. To linearize the nonlinear mixed equations, we use the two-grid algorithm. We first solve the nonlinear equations on the coarse grid, then, on the fine mesh, we solve a linearized problem using Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H =\mathcal{O}(h^{\frac{1}{2}})$. As a result, solving such a large class of nonlinear equations will not be much more difficult than getting solutions of one linearized system.