问题求解

The forth algorithm is the same as the third one with an addition of finite termination criteria. Although the latter two algorithms have only linear rate of convergence, they are especially suitable for large-scale and sparse problems, with features of simple formula, small storage, sparsity preservation and easy implementation.

前者借助NCP函数把互补问题转化为等价的非光滑方程组，再用带参数的光滑方程组近似这些非光滑方程组，最后用牛顿型方法求解所得到的光滑方程组，希望通过光滑参数趋于零得到原来互补问题的解；后者基于等价不动点格式，构造了一个光滑迭代算法和一个具有有限终止性质的算法，虽然这种迭代算法仅有线性收敛速度，但由于其格式简单、存储量小、保稀疏性、非常易于计算机实现等特点，故较适用于求解大规模稀疏问题。

In chapter 3 constructs firstlya new differential merit function by a perturbation structure of VIP and study theproperties of this merit function.Based on this merit function,a continuation-typeNewton method is proposed,which use the technique of inexact linear search forassuring its global convergence,and improves essentially the method of Taji,Fukushima and Ibaraki which can only solve the strong monotone VIP,and has locallyquadratic convergent rate under some condition.

第三章利用变分不等式问题的一种扰动结构构造了新的可微效用函数，并研究了效用函数的性质；在此基础上，给出了一类求解一般单调变分不等式问题的连续型Newton方法CN，方法采用了不精确的线性搜索技术以确保整体收敛性，从本质上改进了Taji、Fukushima和Ibaraki方法只能求解强单调变分不等式问题的局限性，同时又保持了局部二次收敛率。

In this algorithm, the solving space of logical variables is reduced by logical constraints, and the computational complexity is not raised exponentially when the logical variable number is raised. The scale of the NLP sub-problems is also reduced. The global optimal solution will be obtained, and the local optimal problem of nonlinear system is avoided.

算法的特点为：利用逻辑变量的物理意义来缩小求解空间，避免了计算量随逻辑变量数目呈指数型上升的问题；NLP子问题的规模大大缩小，加快了求解速度；可以得到严格的全局最优解，避免了非线性模型可能带来的局部极值问题。

Firstly, the three-dimension boundary question and its corresponding variation question concerning the point-source electrical field on a 3-D structure are presented. Secondly, the finite element method is used to solve the variational equation. The study area is divided into many tetrahedral elements. Then, we interpolate with a tri-linear function in each element. So, the variational equation is converted into a linear equation system. Finally, taking into account saving of time, we make use of symmetric successive overrelaxation preconditioned conjugate gradient algorithm to solve large linear system, and obtain the potential value of each node, resulting in the calculation of the apparent resistivity values on the ground surface.

首先给出了三维构造中点源电场的边值问题以及相应的变分问题；然后利用有限单元法求解变分问题，采用四面体单元对研究区域进行剖分，在单元中进行三线性函数插值，将变分方程化为线性代数方程组；最后，考虑到节约计算时间，利用对称超松弛预条件共轭梯度迭代算法求解大型线性方程组，得到了各节点的电位值，进而计算出地表的视电阻率。

At present, the basic idea of exact approaches to solve the fuzzy linear programming problems is: firstly, they are transformed into parametric programming problems, then an unique exact optimal solution with highest membership degree can be achieved via the simplex method for the crisp linear programming.

目前已有的一些求解模糊线性规划问题一般为精确解法，它的基本思想是：将模糊线性规划问题通过对隶属函数的转换化为精确的参数规划问题，然后利用单纯形方法求解出唯一的极大化隶属函数的最优解。

The paper proposed a new method for the linear programming problem with interval numbers in the objective function and constraint conditions simultaneously.By p roposing the concept of constraint satisfactory degree,the problem is transform e d into parametric linear programming problems with real coefficients.

文摘：针对目标函数和约束条件均为区间数的线性规划问题，提出了一种基于模糊约束满意度的求解方法，把区间数线性规划问题转化为确定型的一般参数规划问题来求解。

So it can be avoided that the computation accuracy is lowered and the calculation result is hard to be convergent for the odd state of transformation matrix.The computation accuracy of wavelet Ritz method and wavelet Galerkin method has been improved so the special'microscopic'property could be fully expressed. An effective method is offered also to solve the problem of high stress gradient and some other odd state problems in engineering.

从而避免了由于转换矩阵奇异而造成精度下降且计算结果不易收敛的问题，提升了小波Ritz法和小波Galerkin法的求解精度，使小波分析的"显微"特性得以充分发挥，并为应力大梯度问题和工程奇异问题的有效求解提供了强有力的计算手段。

To simplify and reduce the effort required for solving control equations, separation of time-space variables is invoked.

在此基础上，在整个时间空间上构造了迭代求解方法，将问题化为局部状态变量修正和全局控制变量求解问题；从而形成了有限变形问题过程最优控制计算方法的理论基础。

Furthermore,the stability behavior and accuracy characteristics of the algorithmare proven by a spectral decomposition method.

在前人工作的基础上，提出了一种新的用于解决结构动力分析问题的时间积分方法——Taylor级数方法；建立了求解线性问题和非线性项可以表示为多元多项式形式的非线性问题的Taylor级数方法的理论，并给出递归求解通式；阐述了该方法的程序实施过程，给出了计算流程图，并在非线性有限元分析平台NFAP中嵌入了Taylor级数方法的计算模块；利用谱分解的方法分析了该方法的稳定性和精度特性，以封闭的解析形式给出了描述积分法特性的周期延长率和振幅衰减率的表达式；通过对该方法的理论分析和特性研究，阐述了该方法的可行性和高效性。

In order to test spectral element method and Newmark time integral, the codes of program and the feasibility of spectral element method for aeroacoustics, One question which has analytical solution and two typical questions of computational aerocaustics are simulated numerically by the program.

为了验证谱元方程和Newmark时间积分方法的正确性、自己所编程序代码的有效性以及谱元方法求解气动声学问题的可行性，本文使用程序分别求解了具有解析解的波动方程和计算气动声学问题中的两个典型问题。

In the negative and interrogative forms, of course, this is identical to the non-emphatic forms.

。但是，在否定句或疑问句里，这种带有"do"的方法表达的效果却没有什么强调的意思。

Go down on one's knees;kneel down

屈膝跪下。。。下跪祈祷