计算二维静电场的非正交有限差分算法

讨论了计算二维维静电场的非正交有限差分算法，给出了数值计算公式，通过对一些实例的以及与理论解的比较，结果表明非正交有限差分算法具有数值网格的合的特点，只要较少用网格就可以达到较高的精度，是求解复杂边界情况下二维静电问题的一种有效方法。     

求解Euler方程的隐式无网格算法

研究了求解Eluer方程的稳式无网格算法，用点云离散计算区域，代替通常的网格划分；在当地点云上，引入二次平方极小曲面逼近计算空间导数，用Roe的近似Riemann解确定通量；并用LU-SGS算法求解离散得到的Euler方程稳式时间后差联立方程组，数值模拟了二维翼型跨音速绕流，由于无网格算法区域离散只涉及点云，具有灵活性，适合处理复杂的气动外形。     

复杂区域上的一种结构网格生成方法

讨论复杂区域上的一种结构网格生成方法,其主要思想是:以变分形式的Winslow网格生成方法为基础,通过引入网格解扭机制和网格面积均匀化技术,构造出一种新的离散泛函,进而采用一类优化算法求解这一离散泛函的极小化问题,得到所希望的网格.通过分析及大量数值实验表明,这一方法比较健壮,针对二维复杂区域通常能够生成几何品质较优的网格,它在保持Winslow方法优点的同时,克服了它的一些缺点.     

各向异性扩散问题Kershaw格式的守恒保正修复算法

针对各向异性扩散方程Kershaw格式的数值解在正交网格及扭曲网格上计算出负的现象,给出一种守恒的保正修复算法（CENZ）,该算法对简单遇负置零（ENZ）方法进行改进,使修复后的数值解不仅具有非负性,而且保持法向通量的局部守恒性.数值算例表明,该方法不受计算网格类型和扩散系数各向异性比的限制,可用于对任意违背单调性（或保正性）的有限体积格式数值解的修复.     

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。     

有限元与无网格伽辽金耦合法分析二相连续多孔介质

将有限元法和无网格伽辽金法相结合,创造出一种新的数值计算方法,该法既能利用有限元法的优点,又能发挥无网格伽辽金法局部化技术的无单元特性,使其具有计算优势.推导了二相多孔介质的有限元法与无网格伽辽金法耦合的离散方法.给出了两个算例,在计算二维固结方程时,精度较高且处理液体流量准确,是有限元法的有力补充.     

二维非结构网格无粘流场数值计算

1前言非结构化网格流场数值模拟是近十几年来计算流体力学发展的重要标志之一。非结构网格与结构网格相比，在网格划分的灵活性上远远超出，这正好是复杂区域划分所必需的。非结构网格可在某一局部加密，修改而不影响其余部分，用于自适应网格时其方便处也是结构网格所无法比拟的。非结构网格划分的灵活性和加密、修改的方便，使它得以在复杂几何结构流动的计算中得以充分应用，十几年来成为计算流体力学界的潮流，得到了大发展［‘－‘］本文在三角划分的计算域上，进行了二维无粘流场数值计算。采用Roe格式离散二维Euler方程组，三步Run…     

基于Riemann解的二维流体力学Lagrange有限点无网格方法

在高维流体力学计算中,对于多介质大变形等一类问题,采用有网格方法常遇到较大的困难.针对二维问题,研究了一种无网格方法——Lagrange有限点方法:在求解区域上设置适当的离散点集,视其中每一点为流体力学Lagrange点;对于点集的任一点,确定邻点集合,并基于该点同邻点集合的联系,应用Godunov方法将流体力学Lagrange方程进行离散;考虑到算法的稳健性,方法中可设置较多邻点并采用最小二乘法.将该方法应用于典型的数值算例,取得了良好效果.     

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.     

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.     

Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials (FGMs). The three methods are, respectively, the method of fundamental solution (MFS), the boundary knot method (BKM), and the collocation Trefftz method (CTM) in conjunction with Kirchhoff transformation and various variable transformations. In the analysis, Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions. The proposed MFS, BKM and CTM are mathematically simple, easy-to-programming, meshless, highly accurate and integration-free. Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered, and the results are compared with those from meshless local boundary integral equation method (LBIEM) and analytical solutions to demonstrate the efficiency of the present schemes.     

Computational performance of Free Mesh Method applied to continuum mechanics problems

The free mesh method (FMM) is a kind of the meshless methods intended for particle-like finite element analysis of problems that are difficult to handle using global mesh generation, or a node-based finite element method that employs a local mesh generation technique and a node-by-node algorithm. The aim of the present paper is to review some unique numerical solutions of fluid and solid mechanics by employing FMM as well as the Enriched Free Mesh Method (EFMM), which is a new version of FMM, including compressible flow and sounding mechanism in air-reed instruments as applications to fluid mechanics, and automatic remeshing for slow crack growth, dynamic behavior of solid as well as large-scale Eigen-frequency of engine block as applications to solid mechanics.     

带源参数的热传导反问题的无网格方法

利用无网格有限点法求带有源参数的一维热传导反问题，推导了相应的离散方程.与其他基于网格的方法相比，有限点法采用移动最小二乘法构造形函数，只需要节点信息，不需要划分网格，用配点法离散求解方程，可以直接施加边界条件，不需要在区域内部求积分，减小了计算量.用有限点法求解热传导反问题具有数值实现简单、计算量小、可以任意布置节点等优点.最后通过算例验证了该方法的有效性.     

动态断裂力学的无网格流形方法

运用无网格流形方法求解动态断裂力学问题.该方法利用单位分解法和有限覆盖技术建立形函数，形函数的建立不受域内不连续的影响，可较好地求解裂纹问题.对于局部化问题，该方法的形函数构造较其他方法更为有效，避免了其他方法在建立试函数时没有考虑不连续尖端的缺点.由于采用有限覆盖技术建立试函数，该方法克服了不连续对试函数的影响，尤其当不连续变得复杂时，更能显示该方法在处理不连续方面的优点.在求解动态断裂力学问题时，弹性动力学积分弱形式的推导采用加权残数法，空间离散采用基于单位分解法的无网格流形方法，时间离散主要采用Newmark法.最后给出两个数值算例，将计算结果与解析解对比，说明该方法的正确性和可行性. 关键词： 有限覆盖 无网格流形方法 动态断裂力学 动态应力强度因子     

The Method of Fundamental Solutions for Solving Convection-Diffusion Equations with Variable Coefficients

A meshless method based on the method of fundamental solutions (MFS) is proposed to solve the time-dependent partial differential equations with variable coefficients. The proposed method combines the time discretization and the one-stage MFS for spatial discretization. In contrast to the traditional two-stage process, the one-stage MFS approach is capable of solving a broad spectrum of partial differential equations. The numerical implementation is simple since both closed-form approximate particular solution and fundamental solution are easier to find than the traditional approach. The numerical results show that the one-stage approach is robust and stable.     

位势问题改进的无网格局部Petrov-Galerkin法

将滑动Kriging插值法与无网格局部Petrov-Galerkin法相结合,采用Heaviside分段函数作为局部弱形式的权函数,提出改进的无网格局部Petrov-Galerkin法,进一步将这种无网格法应用于位势问题,并推导相应的离散方程.因为滑动Kriging插值法构造的形函数满足Kronecker函数性质,所以本文建立的改进的无网格局部Petrov-Galerkin法可以像有限元法一样直接施加边界条件;由于采用Heaviside分段函数作为局部弱形式的权函数,因此在计算刚度矩阵时只涉及边界积分,而没有区域积分.此外,还对本方法中一些重要参数的选取进行了研究.数值算例表明,本文建立的改进的无网格局部Petrov-Galerkin法具有数值实现简单、计算量小以及方便施加边界条件等优点.     

Mathematical and numerical studies on meshless methods for exterior unbounded domain problems

The method of fundamental solutions (MFS) is an efficient meshless method for solving boundary value problems in an exterior unbounded domain. The numerical solution obtained by the MFS is accurate, while the corresponding matrix equation is ill-conditioned. A modified MFS (MMFS) with the proper basis functions is proposed by the introduction of the modified Trefftz method (MTM). The concrete expressions of the corresponding condition numbers are given in mathematical forms and the solvability by these methods is mathematically proven. Thereby, the optimal parameter minimizing the condition number is also mathematically given. Numerical experiments show that the condition numbers of the matrices corresponding to the MTM and the MMFS are reduced and that the numerical solution by the MMFS is more accurate than the one by the conventional method.     

A complex variable meshless local Petrov-Galerkin method for transient heat conduction problems

On the basis of the complex variable moving least-square （CVMLS） approximation, a complex variable meshless local Petrov-Galerkin （CVMLPG） method is presented for transient heat conduction problems. The method is developed based on the CVMLS approximation for constructing shape functions at scattered points, and the Heaviside step function is used as a test function in each sub-domain to avoid the need for a domain integral in symmetric weak form. In the construction of the well-performed shape function, the trial function of a two-dimensional （2D） problem is formed with a one-dimensional （1D） basis function, thus improving computational efficiency. The numerical results are compared with the exact solutions of the problems and the finite element method （FEM）. This comparison illustrates the accuracy as well as the capability of the CVMLPG method.