隐式格式求解拟压缩性非定常不可压Navier-Stokes方程

采用Rogers发展的双时间步拟压缩方法,数值求解不可压非定常问题.数值通量分别采用三阶精度Roe格式和二阶精度Harten-Yee的TVD格式离散.为了加快收敛,提高求解效率,试验了几种隐式格式(ADI-LU,LGS,LU-SGS).针对经典的低雷诺数(Re=200)圆柱绕流问题,比较了不同隐式方法的计算结果和求解效率,以及两种数值离散格式计算结果的异同.最后采用Roe格式数值求解了两种典型的低速非定常流动问题:绕转动圆柱(ω=1)低雷诺数流动;NACA0015翼型等速拉起数值模拟.     

方柱绕流大涡模拟

采用有限体/有限元混合格式、非结构网格和大涡模拟方法求解可压缩的N-S方程,对Re=22 000的方柱绕流进行数值模拟,并对不同的边界条件进行详细的分析比较.通过对以往研究经验的总结和利用精细的边界条件,使得采用二阶精度的数值格式和较稀疏的网格仍然得到了令人满意的计算结果,甚至优于以往采用密网格的模拟结果.     

定常不可压Navier-Stokes型方程的非线性Galerkin有限元算法的收敛性

给出数值求解二维定常不可压Navier-Stokes型方程的非线性Galerkin有限元算法,并分析了数值解的正则性和收敛性,当粗网格参数H和细网格参数h满足关系式H=O(h^1/2)时,该算法具有和Galerkin有限元算法同阶的收敛精度,然而在计算上比Galerkin有限元算法更为简单,可以节省可观的计算量.最后给出了数值试验,验证了上述结果。     

解对流扩散方程的四阶耗散的单侧逼近指数型格式

本文构造了一种带权的六点格式,讨论了它的稳定性条件,证明了这种格式的解对微分方程的真解具有单侧逼近的性质;当适当选取权数θ=θ₀时,这种格式是一种四阶耗散格式,不仅数值耗散很小,而且满足稳定性条件,不出现非物理振荡;还证明了C.J.Chen的有限分析格式_[1]在一定条件下是这种带权格式的一个特殊情形,因此也具有单侧逼近性质;最后给出了几个算例说明上述性质。     

SAMR网格上扩散方程有限体格式的逼近性与两层网格算法

针对结构自适应加密网格(SAMR)上扩散方程的求解,分析几种有限体格式的逼近性,同时设计和分析一种两层网格算法.首先,讨论一种常见的守恒型有限体格式,并给出网格加密区域和细化/粗化插值算子的条件;接着,通过在粗细界面附近引入辅助三角形单元,消除粗细界面处的非协调单元,设计了一种保对称有限体元(SFVE)格式,分析表明,该格式具有更好的逼近性,且对网格加密区域和插值算子的限制更弱;最后,为SFVE格式构造一种两层网格(TL)算法,理论分析和数值实验表明该算法的一致收敛性.     

粒子输运方程的线性间断有限元方法

将空间线性间断有限元方法应用于动态粒子输运方程的求解.数值算例表明,空间线性间断有限元方法在网格边界的数值精度方面明显高于指数格式和菱形格式,并且通量在时间上的微分曲线相对光滑,避免了指数格式、菱形格式数值解的非物理振荡现象.     

自然对流换热的高精度非结构网格有限体积法

用非结构网格有限体积法求解自然对流换热时,传统的对流项离散格式难以兼顾数值精度与计算效率,我们发展了一种耦合高精度格式的延迟修正方法,用于对流项的离散.高Re数下方腔驱动流数值计算验证了该方法具有较高的计算精度和较好的稳定性.Boussinesq流体的自然对流换热数值模拟,表明该方法能有效克服高Ra数时数值计算发散,可准确捕捉自然对流换热问题中不同偏心率下的等温线和流线分布特征.     

时间分数阶亚扩散方程的高阶差分方法

提出两差分格式求解时间分数阶亚扩散方程.两个格式都是绝对稳定的,收敛阶均为O(τ^q+h²),其中q(q=2-β或2)与方程解的光滑性有关,β(0 < β < 1)是分数阶导数的阶、τ和h分别是时间和空间方向步长.数值实验验证了理论结果的正确性,并与其他方法进行比较,显示了本文方法的有效性和精确性.     

Poisson方程的高精度有限差分格式

本文提出数值求解Poisson方程的含选择因子α的预示校正差分格式,它具有四阶精度。第一种格式处理Drichlet边界条件的Poisson方程,它包括Bramble的差分格式和林群、吕涛提出的差分外推格式。第二种格式处理Neumann边界条件的Poisson方程。对于工程计算常用的粗糙网络,作者建议采用α≲0.5的预示校正差分格式。     

非定常Navier-Stokes方程基于完全重叠型区域分解的有限元并行算法

基于完全重叠型区域分解技巧,提出三种求解非定常Navier-Stokes方程的有限元并行算法.其基本思想是首先对空间施行完全重叠区域分解,然后各个处理器使用向后Euler格式独立并行求解关于时间t的常微分方程;对于非线性的对流项,分别采用半隐格式和全隐格式进行处理.算法中每个处理器所负责的子问题是一个全局问题,它定义在整个求解区域上,但绝大部分自由度来自其所负责的子区域,从而使得算法实现简单,通信需求少.数值算例验证了算法的有效性及其良好的并行性能.     

Coefficient Jump-Independent Approximation of the Conforming and Nonconforming Finite Element Solutions

A counterexample is constructed. It confirms that the error of conforming finite element solution is proportional to the coefficient jump, when solving interface elliptic equations. The Scott-Zhang operator is applied to a nonconforming finite element. It is shown that the nonconforming finite element provides the optimal order approximation in interpolation, in $L^2$-projection, and in solving elliptic differential equation, independent of the coefficient jump in the elliptic differential equation. Numerical tests confirm the theoretical finding.     

Lower Bounds for Eigenvalues of the Stokes Operator

In this paper, we propose a condition that can guarantee the lower bound property of the discrete eigenvalue produced by the finite element method for the Stokes operator. We check and prove this condition for four nonconforming methods and one conforming method. Hence they produce eigenvalues which are smaller than their exact counterparts.     

A Finite Volume Method Based on the Constrained Nonconforming Rotated Q1-Constant Element for the Stokes Problem

We construct a finite volume element method based on the constrained nonconforming rotated Q₁-constant element (CNRQ₁-P₀) for the Stokes problem. Two meshes are needed, which are the primal mesh and the dual mesh. We approximate the velocity by CNRQ₁ elements and the pressure by piecewise constants. The errors for the velocity in the H¹ norm and for the pressure in the L² norm are O(h) and the error for the velocity in the L² norm is O(h²). Numerical experiments are presented to support our theoretical results.     

Convergence Analysis of Carey Nonconforming Finite Element for the Second-Order Elliptic Problem with the Lowest Regularity

Russian Physics Journal - In this paper, the Carey nonconforming finite element method (NFEM) for the second order elliptic problem is discussed. By means of the different techniques from the...     

A nonconforming finite element method for the Cahn–Hilliard equation

This paper reports a fully discretized scheme for the Cahn–Hilliard equation. The method uses a convexity-splitting scheme to discretize in the temporal variable and a nonconforming finite element method to discretize in the spatial variable. And, the scheme can preserve the mass conservation and energy dissipation properties of the original problem. Some typical phase transition phenomena are also observed through the numerical examples.     

非饱和水流问题的迎风有限体积元方法及数值模拟

考察非饱和水流问题的模型方程,利用线性迎风有限体积元方法建立非饱和流动的守恒形式,并获得该方法形式为O(Δt+h)的误差估计,最后给出数值模拟.     

Error Analysis and Adaptive Methods of Least Squares Nonconforming Finite Element for the Transport Equations

In this paper, we consider a least squares nonconforming finite element of low order for solving the transport equations. We give a detailed overview on the stability and the convergence properties of our considered methods in the stability norm. Moreover, we derive residual type a posteriori error estimates for the least squares nonconforming finite element methods under $H^{−1}$-norm, which can be used as the error indicators to guide the mesh refinement procedure in the adaptive finite element method. The theoretical results are supported by a series of numerical experiments.     

Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations

This paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Compared with the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lower $H^{1+s}$ weak regularity under consideration, where 0 ≤$s$≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and Céa Lemma for conforming finite element methods can not be applied immediately when 0≤$s$≤1/2. To overcome this difficulty, our main idea is introducing an auxiliary $C^1$ finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.     

A Two-Level Method for Pressure Projection Stabilized P1 Nonconforming Approximation of the Semi-Linear Elliptic Equations

In this paper, we study a new stabilized method based on the local pressure projection to solve the semi-linear elliptic equation. The proposed scheme combines nonconforming finite element pairs NCP₁−P₁triangle element and two-level method, which has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures, but have more favorable stability and less support sets. Stability analysis and error estimates have been done. Finally, numerical experiments to check estimates are presented.     

Acoustic modelling of exhaust devices with nonconforming finite element meshes and transfer matrices

Transfer matrices are commonly considered in the numerical modelling of the acoustic behaviour associated with exhaust devices in the breathing system of internal combustion engines, such as catalytic converters, particulate filters, perforated mufflers and charge air coolers. In a multidimensional finite element approach, a transfer matrix provides a relationship between the acoustic fields of the nodes located at both sides of a particular region. This approach can be useful, for example, when one-dimensional propagation takes place within the region substituted by the transfer matrix. As shown in recent investigations, the sound attenuation of catalytic converters can be properly predicted if the monolith is replaced by a plane wave four-pole matrix. The finite element discretization is retained for the inlet/outlet and tapered ducts, where multidimensional acoustic fields can exist. In this case, only plane waves are present within the capillary ducts, and three-dimensional propagation is possible in the rest of the catalyst subcomponents. Also, in the acoustic modelling of perforated mufflers using the finite element method, the central passage can be replaced by a transfer matrix relating the pressure difference between both sides of the perforated surface with the acoustic velocity through the perforations. The approaches in the literature that accommodate transfer matrices and finite element models consider conforming meshes at connecting interfaces, therefore leading to a straightforward evaluation of the coupling integrals. With a view to gaining flexibility during the mesh generation process, it is worth developing a more general procedure. This has to be valid for the connection of acoustic subdomains by transfer matrices when the discretizations are nonconforming at the connecting interfaces. In this work, an integration algorithm similar to those considered in the mortar finite element method, is implemented for nonmatching grids in combination with acoustic transfer matrices. A number of numerical test problems related to some relevant exhaust devices are then presented to assess the accuracy and convergence performance of the proposed procedure.