变分方程

Based on the variational principle of virtual voltage and virtual displacement derived in this thesis and the classical theory of linear laminated plates, and by using the constitutive equations of elastic piezoelectric materials, and the DKQ quadrilateral plate bending element, the finite element modeling for the fabric reinforced laminated plates with nonlinear voltage-electric strain relationship of piezoelectric actuators has been established for the first time in this thesis. And the expressions for the electric loadings with obvious mechanical meaning and convenient for computation are established, which can be used to the deformation analysis of laminated plates with piezoelectric actuators under specified voltages.

以本文直接导出的压电结构的虚电势—虚位移变分原理为基础，依据经典线性层合板理论和压电弹性体的本构方程，采用基于离散法线假设的DKQ四结点任意四边形薄板弯曲单元，首次建立了考虑非线性电压—电应变关系的含压电致动器的纤维增强型层合板的有限元分析模型，并推导建立了力学意义明确、便于计算的电荷载表达式，可用于指定电压条件下纤维增强型叠层合成板的变形分析。

This paper focuses on four important aspects as follows. The first one is the establishment of variational quations which provides the mathematics theories for the finite element method of HMT; the second one is for the study of scalar/node and vector/edge finite element method; the third one is about how to solve faster for the large-sacled matrix system of scalar/node and vector/edge finite element; and the last one is achieving the forward modeling with high precision of HMT method.

主要包括四部分：建立与高频大地电磁法边值问题相等价的变分问题，为有限元数值计算奠定数理基础；标量/矢量有限元方法的研究；加速标量/矢量有限元矩阵系统方程迭代求解技术的研究；实现高频大地电磁参数精确快速正演模拟的研究。

By theory says, the variational principle and the Schrodinger equation are equal.

从原理上讲，变分原理与薛定谔方程等价。

The equation of the cylindrical shell subjected to underwater explosion shock wave is derived from Hamilton variation principle, considering the fluid-structure interaction .

首先，计及流—固耦合作用的影响，文中由Hamilton变分原理推导出了圆柱壳受爆炸载荷作用的控制方程。

A variational method with differential forms gives better precision for numerical solution of transport critical problem based on spherical geometry, and its computation seems simple than other approximate methods.

对于球几何问题，讨论了一种基于微分方程的变分形式求解方法，数值计算显示达到了较满意的精度，而且计算过程比起其他近似方法要简单

Based on instantaneous potential energy principle and spline interpolate function, the dynamic equation of flat spherical shell is derived.

基于瞬时变分原理与样条函数理论，建立扁球壳的动力方程。

The basic equations of structure are derived from Hamilton variational principle and the constitution of material utilizes the increment theory.

基于Hamilton变分原理导出圆柱壳的运动方程，本构关系采用增量理论，借助增量数值算法求解动力方程组。

With the application of the Pro-Hellinger-Reissner variational principle, dual equations in the symplectic space are yielded to present an analytical method for the bending of annular sector plates.

然后应用类 Hellinger-Reissner变分原理，导出了辛几何空间的对偶方程，从而将环扇形薄板弯曲问题导入到辛对偶求解体系。

The reduced Lagrange density of nonlocal nonlinear Schrodinger equation is obtained by expanding the real symmetric response function in Taylor's series in strongly nonlocal Kerr media.

在非局域非线性克尔介质中，通过对介质实对称响应函数的泰勒展开，简化了非局域非线性薛定谔方程所对应的 Lagrange密度，进而利用变分法对光束的传输问题进行了分析。

There arise many techniques in the study of existence of solution for operator equations,such as topological transversality method,upper and lower solutions method,monotone iterative method, and variation method,and bountiful results are obtained.

在算子方程解的存在性研究中产生了许多方法，如拓扑度方法、上下解方法、单调迭代法、变分方法等，获得了丰富的结果。

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。