变分法

An important problem of calculus of variations used in constrained differential systems, i.e., the commutation relation of differential operator and variational oprerator, is investigated by means of Frobenius theorem of integrability.

利用Frobenius可积性定理，研究微分约束系统一个重要的变分法问题：微分运算与变分运算的对易关系。

First, For the coronary artery stent, two kinds of loads, including the pressure named 〓 derivatively from the balloon and the counterforce named 〓 on the wall edge of the stent, have been taken into account in the analysis of the stent.〓 and 〓 were changing as the stent expanding. So, the expansion process of the stent could be considered as the process composing of many extremely short equilibrium processes, where 〓 and 〓 could be regarded as constants and the stresses, the strains and the displacements were obtained using NFEA in every processes. Then, the stresses, the strains and the displacements obtained in every processes or steps were added up to get the final structure of the coronary stent.

首先，解决了支架有限元分析的载荷非线性问题，认为医用心血管支架受到两种载荷的作用：球囊直接传递的压力〓和球囊对不锈钢支架丝壁边缘的反作用力〓，这两种载荷在支架分析中必须都考虑进去；〓为非线性载荷，随着支架扩张不断变化，处理时将支架扩张的整个过程看成由37个很短的平衡过程组成，在每个过程中取〓为常量，采用有限元法获得应力、应变、位移，最后将各过程的应力、应变、位移迭加起来就获得了不锈钢支架总的应力、应变和位移；本研究用变分法和有限元法模拟杆件拉伸证明了上述方法是合理的。

Recently Wang and Xiao discussed the excitation state energy of polaron in polyatomic crystals for the first time.

近年来国内外对多原子极性晶体中磁极化子性质的研究十分活跃，Zorkani等采用变分法计算了束缚磁极化子的基态能量，Kandemir等采用束缚朗道态讨论了二维大磁极化子的基态和第一激发态能量，国内一些学者采用微扰法和新颖算符法讨论了多原子极性晶体中表面和体磁极化子的性质。

We re-examine a problem about action principle which may be easily ignored in the end. After some detailed study, we find that it is correct to do variation of the action with respect to scale factors for Bianchi IX model up to some higher derivative Lagrangian corrections.

我们最后重新检验了在变分法里头容易被忽略的一个问题，最后我们发现，在Lagrangian放入一些高阶修正项的Bianchi II， VIII， IX模型都可以直接对尺度因子变分而得到正确的场方程式。

The abstract result contains several concrete results in the literature and can also be used to deal with some new cases for resonant differential equations.In the introduction, we briefly introduce the development process of the variational methods. In Chapter 2, we list some basic knowledges refering to the variational methods, including the Sobobev space,—△ operator, the weak solution and the minimizing sequence methods and some minimax theorems. In Chapter 3, we introduce the research process of Hamiltonian system of second order and the semilinear elliptic problems, using the methods introduced previously. In Chapter 4, we prove the main theorem of the thesis, and apply it to the problems in the previous Chapter, and can also be applied to some new resonant cases.

在前言中，简要介绍了变分法的产生、发展过程，在第二章中我们介绍了有关变分法的一些基本知识，包括Sobolev空间，—△算子，弱解，极小化序列方法和一些极小极大定理，在第三章中我们介绍了非线性项有界或满足次线性条件，以及它满足推广的Ahmad-Lazer-Paul条件时，二阶Hamiltonian系统和半线性椭圆问题的研究历程，最后在第四章中我们证明了本论文的主要定理，并把它应用到第三章的问题中，使得前面的几种共振的情形都可以统一到这个抽象的结果中。

The method of approaching regular polygon cross section waveguide is proprosed with circle symmetry cross section waveguide.

Marcatili方法、逐点匹配方法、变分法、离散时域差分法，计算量均很大；微扰法通用性强，但规范

Ritz variation method was used to find the numerical relation between the energy near the ground-state of the hydrogen molecular ion H 2 + and the changes of the variation parameter and the bond length,the computation formula of bond length and ground-state energy for H 2 + was also obtained by means of the method of parabolic interpolation .

用Ritz变分法求出了氢分子离子H2 +基态能量附近的能量随变分参数和分子键长变化的数值关系，并用抛物线插值法获得了H2 +键长和基态能量的值及其计算公式，比文献 [1，2 ]更接近于实验值。

It is different from the finite element method which uses the calculus of variation to construct the element-based approximate equilibrium equation. It does consider the equilibriums of forces on the inter-element boundary, which are neglected in the finite element method through the weak formulation.

这种方法跟有限元素法不同，因为有限元素法是采用变分法建立各个元素的近似平衡方程式，这种方法也考虑元素与元素相连接的边界上的力平衡条件，而在有限元素法的推导过程中我们忽略了这种条件。

Relation of hydrogenic impurity binding energy,effective Bohr a _ and virial theorem value and quantum dots size and studied variationally by using variational wave functions of containing two parameters.

选用含有2个变分参数的波函数，利用变分法计算了无限深球形量子点中施主杂质态的束缚能、杂质有效玻尔半径、维里定理值等随量子点尺度的变化关系。

Based on DEM data of different development periods of a indoor watershed model obtained from the integration of artificial simulated rainfall experiment and high accuracy digital photogrammetry technique, the fractal dimensions of topographic feature under different development periods of the indoor watershed model based on variation method were calculated and the relationships between watershed topographic feature fractal dimension and topographic factors such as watershed gully density as well as ground lacerative degree was discussed preliminarily.

以室内流域模型人工模拟降雨试验和高精度摄影测量获得的模型流域不同发育时段的数字高程模型数据为基础，采用变分法计算了流域地貌形态分形维数，并对分形维数与地貌因子沟谷密度和地面割裂度的关系进行初步探讨。

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。