瞬态动力问题的逐单元矩阵分裂和逐步积分方法

本对瞬态动力问题，结合逐步积分方法提出了一类广义的矩阵分裂和逐单元松弛算法，摆脱了有限元法通常需形成总体刚度矩阵，总体质量矩阵和求解大型稀疏方程组的工作，理论分析和计算实例表明，本的广义矩阵分裂是最优的分裂方案，本的算法物理意义明确，但于编写程序推广应用。     

瞬态动力问题的遂单元矩阵分裂和逐步积分方法

本文对瞬态动力问题,结合逐步积分方法提出了一类广义的矩阵分裂和逐单元松弛算法,摆脱了有限元法通常需形成总体刚度矩阵,总体质量矩阵和求解大型稀疏方程组的工作,理论分析和计算实例表明,本文的广义矩阵分裂是最优的分裂方案.本文的算法物理意义明确,便于编写程序推广应用.     

关于Stokes和线性Navier-Stokes方程的广义维数分裂迭代方法

本文研究了鞍点问题的迭代法.在Benzi等人提出的维数分裂(DS)迭代方法的基础上,提出了具有三个参数的广义维数分裂(GDS)迭代法,该方法包含了DS迭代法,理论分析表明该方法是无条件收敛的.通过对有限差分法和有限元法离散的Stokes问题及有限元法离散的Oseen问题的数值结果表明,本文所给方法是有效的.     

多项式分裂可行问题

本文研究多项式分裂可行问题,即由多项式不等式定义的分裂可行问题,包括凸与非凸、可行与不可行的问题;给出多项式分裂可行问题解集的半定松弛表示;研究其半定松弛化问题的性质;并基于这些性质建立求解多项式分裂可行问题的半定松弛算法.本文在较为一般的条件下证明了,如果分裂可行问题有解,则可通过本文建立的算法求得一个解点;如果问题无解,则该算法能够判别问题不可行.最后通过数值实验对算法进行验证.     

椭圆最优控制问题分裂正定混合有限元方法的超收敛性分析

首先利用变分原理和最优化理论得到了原问题的等价最优性条件；其次构造了椭圆最优控制问题分裂正定混合有限元方法的逼近格式；再次通过引入一些重要的中间变量和投影算子，并利用投影算子的相关性质，结合分裂正定混合有限元本身的逼近结果，得到了椭圆最优控制问题分裂正定混合有限元方法的超收敛性；最后数值实验结果验证了所得理论结果的正确性.     

聚合物时温等效模型有限元应用研究

为更好地描述聚合物材料力学性能的温度相关性问题,对目前广泛应用的WLF模型进行改进研究,并引入“零时间”因子提高了粘弹性材料变温松弛模量的获取精度.在此基础上基于ABAQUS用户材料子程序UTRS将时温等效模型应用到数值计算中.根据不同温度水平下的应力松弛实验获得模型参数,并通过等速拉伸实验与数值结果的对比验证了该模型及其有限元方法的可行性及正确性.结果表明:引入“零时间”因子的变温松弛模量精度更高;改进WLF模型对复合推进剂具有更好的适用性和更高的精确度.     

广义异步并行多分裂松弛算法

在本文中，我们设计了求解大型线性代数方程组的适用于ＭＩＭＤ系统的异步并行多分裂松弛算法的一般模型，并在系数矩阵是Ｈ－矩阵的条件下，建立了该一般模型的收敛性理论。     

求解流固耦合问题的一种四步分裂有限元算法

基于arbitrary Lagrangian Eulerian (ALE) 有限元方法,发展了一种求解流固耦合问题的弱耦合算法．将半隐式四步分裂有限元格式推广至求解ALE描述下的Navier-Stokes（N-S）方程,并在动量方程中引入迎风流线（streamline upwind/Petrov-Galerkin, SUPG）稳定项以消除对流引发的速度场数值振荡;采用Newmark-β法对结构方程进行时间离散;运用经典的Galerkin有限元法求解修正的Laplace方程以实现网格更新,每个计算步施加网格总变形量防止结构长时间、大位移运动时的网格质量恶化．运用上述算法对弹性支撑刚性圆柱体的流致振动问题进行了数值模拟,计算结果与已有结果相吻合,初步验证了该算法的正确性和有效性．     

关于子结构法的条件数

子结构法是有限元计算中常用的一种算法,它有很多众所周知的优点,例如:便于内外存交换,因而便于在小计算机上解大问题;形状相同的子结构可以不重复计算刚度矩阵,等等。但人们往往忽略了它的另一优点,即它可以减少舍入误差的积累,提高解的精度,因而用它来解病态问题有一定的效果。 有些人讨论过这一问题.Y.Yamamoto用物理模型模拟有限元刚度矩阵的条件数,得出子结构矩阵的条件数不大于原矩阵条件数的结论。石根华研究了病态问题的解法,指出总体刚度矩阵出现病态的一个很重要的原因是局部有较大的刚性位移,据此他     

对流扩散方程区域分裂特征差分方法

1引言对流扩散方程是许多物理问题的数学模型,研究其稳定的数值解法具有重要的应用价值．而标准的差分法和有限元法通常会失效,出现数值振荡．80年代,Douglas和Russel提出了特征线方法,在一定程度上克服了数值振荡,保证了数值的稳定,尤其对“对流占优”问题,更能突出特征法的优越性,并有了大量的理论成果[1,2,3]．区域分裂是一种解决大规模的科学与工程计算问题的有效方法,Dawson,Du和Dupont对热传导方程给出了非重叠区域分裂格式及分析,由于内边界的显格式,需要一定的稳定性条件Δt≤CH2;而Du等在[5]给出了抛物方程的几种区域分裂格式,对区域分裂法的     

The modulus‐based matrix splitting algorithms for a class of weakly nonlinear complementarity problems

In this paper, we study a class of weakly nonlinear complementarity problems arising from the discretization of free boundary problems. By reformulating the complementarity problems as implicit fixed‐point equations based on splitting of the system matrices, we propose a class of modulus‐based matrix splitting algorithms. We show their convergence by assuming that the system matrix is positive definite. Moreover, we give several kinds of typical practical choices of the modulus‐based matrix splitting iteration methods based on the different splitting of the system matrix. Numerical experiments on two model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of our modulus‐based matrix splitting algorithms. Copyright © 2016 John Wiley & Sons, Ltd.     

Modulus‐based multigrid methods for linear complementarity problems

By employing modulus‐based matrix splitting iteration methods as smoothers, we establish modulus‐based multigrid methods for solving large sparse linear complementarity problems. The local Fourier analysis is used to quantitatively predict the asymptotic convergence factor of this class of multigrid methods. Numerical results indicate that the modulus‐based multigrid methods of the W‐cycle can achieve optimality in terms of both convergence factor and computing time, and their asymptotic convergence factors can be predicted perfectly by the local Fourier analysis of the corresponding modulus‐based two‐grid methods.     

A wavelet-based stochastic finite element method of thin plate bending

A wavelet-based stochastic finite element method is presented for the bending analysis of thin plates. The wavelet scaling functions of spline wavelets are selected to construct the displacement interpolation functions of a rectangular thin plate element and the displacement shape functions are expressed by the spline wavelets. A new wavelet-based finite element formulation of thin plate bending is developed by using the virtual work principle. A wavelet-based stochastic finite element method that combines the proposed wavelet-based finite element method with Monte Carlo method is further formulated. With the aid of the wavelet-based stochastic finite element method, the present paper can deal with the problem of thin plate response variability resulting from the spatial variability of the material properties when it is subjected to static loads of uncertain nature. Numerical examples of thin plate bending have demonstrated that the proposed wavelet-based stochastic finite element method can achieve a high numerical accuracy and converges fast.     

广义变分原理在有限元半分析法中的应用

本文是在文献[1]中所指出的广义变分原理在合理处理有限元法的边限条件的应用价值还没有受到足够的重视这一思想启发下,应用广义变分原理,选用样条函数与正弦(或余弦)函数乘积型的级数形式再加上多项式,作为板壳的逼近函数,以薄板弯曲问题为例较好地解决了有限元半分析法中出现的耦联问题.由于其未知数个数比有限元法、有限条法均少很多,而精度更高,故为用微机解决一类工程问题,提供了一个有效的方法.     

Unconditionally maximum principle preserving finite element schemes for the surface Allen–Cahn type equations

In this paper, we present two types of unconditionally maximum principle preserving finite element schemes to the standard and conservative surface Allen–Cahn equations. The surface finite element method is applied to the spatial discretization. For the temporal discretization of the standard Allen–Cahn equation, the stabilized semi-implicit and the convex splitting schemes are modified as lumped mass forms which enable schemes to preserve the discrete maximum principle. Based on the above schemes, an operator splitting approach is utilized to solve the conservative Allen–Cahn equation. The proofs of the unconditionally discrete maximum principle preservations of the proposed schemes are provided both for semi- (in time) and fully discrete cases. Numerical examples including simulations of the phase separations and mean curvature flows on various surfaces are presented to illustrate the validity of the proposed schemes.     

Efficient numerical methods for pricing American options under stochastic volatility

Five numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two‐dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M‐matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved using a multigrid method. The projected multigrid method and the componentwise splitting method lead to a sequence of linear complementarity problems with one‐dimensional differential operators that are solved using the Brennan and Schwartz algorithm. The numerical experiments compare the accuracy and speed of the considered methods. The accuracies of all methods appear to be similar. Thus, the additional approximations made in the operator splitting method, in the penalty method, and in the componentwise splitting method do not increase the error essentially. The componentwise splitting method is the fastest one. All multigrid‐based methods have similar rapid grid independent convergence rates. They are about two or three times slower that the componentwise splitting method. On the coarsest grid the speed of the projected SOR is comparable with the multigrid methods while on finer grids it is several times slower. ©John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Modulus‐based matrix splitting iteration methods for a class of implicit complementarity problems

Some modulus‐based matrix splitting iteration methods for a class of implicit complementarity problem are presented, and their convergence analysis is given. Numerical experiments confirm the theoretical analysis and show that the proposed methods are efficient. Copyright © 2016 John Wiley & Sons, Ltd.     

更一般的杂交广义变分原理及有限元模型

本文根据钱伟长教授提出的更一般的广义变分原理,给出了适用于有限元法中的更广义杂交变分原理,并由此建立了新的广义杂交模理. 进一步以变厚度薄板弯曲单元为例,对基于各种不同的广义变分原理建立的各种杂交元做了比较.     

Stability loss in an infinite plate with a circular inclusion under uniaxial tension

Loss of stability under uniaxial tension in an infinite plate with a circular inclusion made of another material is analyzed. The influence exerted by the elastic modulus of the inclusion on the critical load is examined. The minimum eigenvalue corresponding to the first critical load is found by applying the variational principle. The computations are performed in Maple and are compared with results obtained with the finite element method in ANSYS 13.1. The computations show that the instability modes are different when the inclusion is softer than the plate and when the inclusion is stiffer than the plate. As the Young’s modulus of the inclusion approaches that of the plate, the critical load increases substantially. When these moduli coincide, stability loss is not possible.