差分法

In chapter 2,all kinds of numerical methods of computational fracturemechanics are summarized,the development histories and present situation ofboundary collocation method,boundary integral method,finite difference method,boundary element method and finite element method are expounded,and theapplication of finite element method in fracture mechanics are emphaticallydiscussed.

第二章 总结了计算断裂力学中各种数值方法，论述了边界配位法、边界积分法、有限差分法、边界元法及有限元法的发展历史和现状，并着重讨论了有限元法在断裂力学中的应用。

Firstly, the approach formulates a cost functional to turn the inverse problem into a constrained minimization problem according to least squares criterion, then the resulting constrained minimization problem is transformed into an unconstrained minimization problem by using a penalty function technique, and then the closed Fr chet derivatives of the Lagrange function with respect to the properties are derived based on the calculus of variations, finally, one can solve the resulting problem by using any gradient-based algorithm and the finite-difference time-domain method.

该方法首先以最小二乘准则构造目标函数，将逆问题表示为约束最小化问题；接着应用罚函数法转化为无约束最小化问题；然后基于变分计算导出闭式的拉格朗日函数关于特征参数的Fr chet导数；最后借助梯度算法和时域有限差分法迭代反演德拜模型参数。

From the point of vertical interconnect via, the matrix pencil moment method, FDTD, edge-element FEM, microwave network model method and CAD model method of vertical via interconnect were introduced. The analysis of these methods shows their advantages and shortcomings.

从垂直互连通孔入手，介绍了垂直通孔互连的矩阵束矩量法、时域有限差分法、边基有限元法以及微波网络模型法和CAD模型法，分析了这些方法已经取得的结果，并指出了这些方法的优缺点。

If without dissipation, the propagation is essentially a symplectic transformation with one parameter, and, consequently, the numerical calculation methods of the propagation ought to be symplectic, too.

至今 ，已发展起来了许多方法，如计算走时的射线追踪法，计算全波场的有限差分法和有限元法等[1— 5 ]，以及随后发展起来的其他方法，如傅立叶变换法[6 ]、小波变换法[7，8]和各种结合算法[9，10 ]等。

Volume integral equation method has been used to calculate forward and inverse problems of 3D electromagnetic fields in homogeneous half-space.

0引言在电磁法勘探中，利用体积分方程计算三维地电磁场，由于数据点只分布在电阻率异常区域[1~3]而不是整个三维空间，因而与有限元法和有限差分法相比其计算量相对较小，是进行地电参数正反演计算的有效方法。

By using the Eulerian-Lagrangian combined method, the convection-diffusion equation is spitted into the diffusion terms and convection terms. For convection terms, the implicit method of characteristics using time-line interpolation with high accuracy is adopted. While for the diffusion terms the conventional finite difference method is used.

采用欧拉—拉格朗日混合解法，将对流—扩散方程中的扩散项和对流项分裂求解，应用高精度时间序列插值的特征差分隐格式求解对流项，扩散项采用传统的有限差分法。

There are many methods for computation of the slope and deflection of beams such as the integration method, the finite difference method, the Macaulay's method, the Mohr's 'area-moment' method and so on. They possess mostly the characteristics of complex computing or searching schematics.

有多种方法求梁的位移，如积分法、迭加法、初参数方程法、其轭梁法、有限差分法、奇异函数法和莫尔&面积一力矩&法等等，其大都具有或计算复杂或依赖各种不同的图表等特点。

The wave equation of electric in the waveguide can be translated into a coupled difference equation of second order by centered differency approximation.

通过中心差分法可将电场满足的波动方程化为一组耦合二阶差分公式形式。

Firstly, the basic principle of Finite-Difference Time-Domain is introduced and the 2-dimensional FDTD equation is derived from the Maxwells equations, calculated by setting up the appropriate excitation source, cell size, the interval of space and time.

首先，介绍了时域有限差分法的基本原理，从麦克斯韦方程出发，建立二维空间中的时域有限差分方程，通过设置合适的激励源、网格尺寸、空间和时间步长进行计算。

This paper reviews and summarizes the progress in this field as follows: the mathematical model and the difficulty of the analytical solutions is introduced; discusses the latest progress of the numerical solutions is discussed; the finite difference method, the Godunov method, the Boltzmann method, and the KFVS method of one-dimensional dam-break problem are introduced; the TVD method, the discontinuous finite element method, the finite volume method and characteristic method of two-dimensional dam-break problem are discussed; the applicability, the advantage and the disadvantage of each method are analyzed; and the progress in the numerical calculations of flow of fluid with free surface are presented.

回顾和总结了国内外对溃坝水流演进问题的研究进展：介绍了溃坝水流的数学模型及解析解法存在的困难，进而讨论了数值解法的最新进展；论述了求解溃坝水流一维问题的有限差分法、近似黎曼解的Godunov格式法、Boltzmann法、KFVS法和二维问题的TVD格式法、间断有限元法、有限体积法、特征线法，并分析了各种方法的适用范围和优缺点，及讨论了限制函数的使用；介绍了利用自由水面追踪方法计算溃坝水流的研究进展，并根据目前存在的不足和实际工程的需要，提出了进一步研究的方向和发展趋势。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。