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临界变分
相关语句
  variational crisis
     The Variational Crisis and Generalized Variational Principles in Elasticity
     论弹性力学广义变分原理的临界变分状态
短句来源
     On variational crisis and generalized variational principle of elasticity
     论弹性力学广义变分原理的临界变分现象
短句来源
     The generalized variational functional of elastic contact problem is deduced by the semi-inverse method that can eliminate the variational crisis. The paper treats the complex problem caused by variational inequalities ingeniously and provides a new method for more complex contact mechanics.
     应用半反推法推导出摩擦约束弹性广义变分二类变量的广义变分不等原理的能量泛函.由于半反推法不用拉氏乘子,可以避免由于拉氏乘子引起的临界变分现象.本文巧妙地处理了由于变分不等式引起的推导困难,为用半反推法导出更为复杂的接触问题的变分不等原理的泛函提供了一条新的思路.
短句来源
     In the image plane, two special functions──pathline length function Y and energy function Ω──have been defined due to the fact that the control equations can be changed into conservative ones by space translation. Interesting variational crisis come across when the Liu' s systematic method was applied to establish the generalized variational principle. The crisis phenomenon comes from the fact that the defined Lagrange mulipliers are relative.
     应用坐标变换,在映象平面上,巧妙地把控制方程转化为守恒形式.在此基础上,作者定义了2个新型函数──迹线长度函数Y和能量函数Ω.在应用刘高联系统方法建立广义变分原理时,发现了非常有趣的临界变分现象.并且分析了产生临界变分的原因,最后导得了一维变截面非定常可压缩均熵流动的广义变分原理.
短句来源
     This method is proved to be very effective and convenient to establish the generalized variational principles with multi-variables not confronting with the variational crisis.
     通过该法可以方便地构造出各种多变量广义变分原理,并且可以消除临界变分现象。
短句来源
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  critical variational
     Note on the Critical Variational State in Elasticity Theory
     对弹性理论中临界变分状态的一个注记
短句来源
     On the basis of Hellinger-Reissner variational principle, the critical variational constrained condition is relieved by use of the identified multiplier method, and a generalized variational principle with three kinds of independent variable in nonlinear theory of elasticity is obtained.
     并在H-R变分原理的基础上,应用已识别拉氏乘子,将其应力应变关系这个临界变分约束条件解除,得到了非线性弹性理论的一个三类独立变量的广义变分原理。
短句来源
  “临界变分”译为未确定词的双语例句
     An overview of variational crises and its recent developments
     流体力学中的临界变分现象及其消除方法
短句来源
     Recently Prof. Chien Wei-zang[1] pointed out that in certain cases, by means of prdinary Lagrange multiplier method, some of undetermined Lagrange multipliers may turn out to be zero during variation.
     (t) 最近钱伟长教授指出,在某些情况下,用普通的拉氏乘子法,其待定的拉氏乘子在变分中恒等于零,这称为临界变分状态,在这种临界状态中,我们无法用待定拉氏乘子法把变分的约束条件吸收入泛函,从而解除这个约束条件。
短句来源
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  variational crisis
Sometimes, however, one may come across variational crisis(some multipliers vanish identically), failing to achieve his aim.
      
The occurrence of three kinds of variational crisis is demonstrated and methods for their removal are suggested.
      
In this section we first illustrate the variational crisis and the ways to overcome them.
      
  critical variational
Note on the critical variational state in elasticity theory
      


It is known[1] that the minimum principles of potential energy and complementary energy are the conditional variation principles under respective conditions of constraints. By means of method of Lagrange multipliers, we are able to reduce the functionals of conditional variation principles into new functionals of non-conditional variation principles. This method can be described as follows: Multiply undetermined Lagrange multipliers to various constraints, and add these products into original functionals. Considering...

It is known[1] that the minimum principles of potential energy and complementary energy are the conditional variation principles under respective conditions of constraints. By means of method of Lagrange multipliers, we are able to reduce the functionals of conditional variation principles into new functionals of non-conditional variation principles. This method can be described as follows: Multiply undetermined Lagrange multipliers to various constraints, and add these products into original functionals. Considering these undetermines Lagrange multipliers and the original variables in these new functionals as independent variables of variation, the stationary conditions of these functionals give these undetermined Lagrange multipliers in terms of original variables. The substitutions of these results for Lagrange multipliers into above functionals lead to the functionals of these non-conditional variation principles. However, in certain cases, some of undetermined Lagrange multipliers may turn out to be zero during variation. This is a critical state of variation. In this critical state, the corresponding variational constraint can not be eliminated by means of simple Lagrange multiplier method. This is indeed the case when one tries to eliminate the constraint condition of stress-strain relation in variational principle of minimum complementary energy by the method of Lagrange multiplier. By means of Lagrange multiplier method, one can only derive, from minimum complementary energy principle,the Hellinger-Reissner principle[2'3], in which only two types of independent variables, stresses and displacements, exist in the new functional. The strain stress relation remains to be a constraint, from which one derives the strain from given stress. Thus the Hellinger-Reissner principle remains to be a conditional variation principle with one constraint uneliminated.

作者曾指出,弹性理论的最小位能原理和最小余能原理都是有约束条件限制下的变分原理采用拉格朗日乘子法,我们可以把这些约束条件乘上待定的拉氏乘子,计入有关变分原理的泛函内,从而将这些有约束条件的极值变分原理,化为无条件的驻值变分原理.如果把这些待定拉氏乘子和原来的变量都看作是独立变量而进行变分,则从有关泛函的驻值条件就可以求得这些拉氏乘子用原有物理变量表示的表达式.把这些表达式代入待定的拉氏乘子中,即可求所谓广义变分原理的驻值变分泛函. 但是某些情况下,待定的拉氏乘子在变分中证明恒等于零.这是一种临界的变分状态.在这种临界状态中,我们无法用待定拉氏乘子法把变分约束条件吸收入泛函,从而解除这个约束条件.从最小余能原理出发,利用待定拉氏乘子法,企图把应力应变关系这个约束条件吸收入有关泛函时,就发生这种临界状态,用拉氏乘子法,从余能原理只能导出Hellinger-Reissner变分原理,这个原理中只有应力和位移两类独立变量,而应力应变关系则仍是变分约束条件,人们利用这个条件,从变分求得的应力中求应变.所以Hellinger-Reissner变分原理仍是一种有条件的变分原理. 普通的拉氏乘子法,只考虑变分条件的线性项。当...

作者曾指出,弹性理论的最小位能原理和最小余能原理都是有约束条件限制下的变分原理采用拉格朗日乘子法,我们可以把这些约束条件乘上待定的拉氏乘子,计入有关变分原理的泛函内,从而将这些有约束条件的极值变分原理,化为无条件的驻值变分原理.如果把这些待定拉氏乘子和原来的变量都看作是独立变量而进行变分,则从有关泛函的驻值条件就可以求得这些拉氏乘子用原有物理变量表示的表达式.把这些表达式代入待定的拉氏乘子中,即可求所谓广义变分原理的驻值变分泛函. 但是某些情况下,待定的拉氏乘子在变分中证明恒等于零.这是一种临界的变分状态.在这种临界状态中,我们无法用待定拉氏乘子法把变分约束条件吸收入泛函,从而解除这个约束条件.从最小余能原理出发,利用待定拉氏乘子法,企图把应力应变关系这个约束条件吸收入有关泛函时,就发生这种临界状态,用拉氏乘子法,从余能原理只能导出Hellinger-Reissner变分原理,这个原理中只有应力和位移两类独立变量,而应力应变关系则仍是变分约束条件,人们利用这个条件,从变分求得的应力中求应变.所以Hellinger-Reissner变分原理仍是一种有条件的变分原理. 普通的拉氏乘子法,只考虑变分条件的线性项。当这个线性项的拉氏系数等于零时,这个条件就没法吸收入泛函之中。为此,?

Recently Prof. Chien Wei-zang[1] pointed out that in certain cases, by means of prdinary Lagrange multiplier method, some of undetermined Lagrange multipliers may turn out to be zero during variation. This is a critical state of variation. In this critical state, the corresponding variational constraint can not be eliminated by means of simple Lagrange multiplier method. This is indeed the case when one tries to eliminate the constraint condition of stress-strain relation in variational principle of minimum...

Recently Prof. Chien Wei-zang[1] pointed out that in certain cases, by means of prdinary Lagrange multiplier method, some of undetermined Lagrange multipliers may turn out to be zero during variation. This is a critical state of variation. In this critical state, the corresponding variational constraint can not be eliminated by means of simple Lagrange multiplier method. This is indeed the case when one tries to eliminate the constraint condition of stress-strain relation in variational principle of minimum complementary energy by the method of Lagrange multiplier. By means of Lagrange multiplier method, one can only derive, from minimum complementary energy principle, the Hellinger-Reissner principle[2,3], in which only two types of independent variables, stresses and displacements, exist in the new functional. Hence Prof. Chien Wei-zang introduced the high-order Lagrange multiplier method by adding the quadratic terms

(t) 最近钱伟长教授指出,在某些情况下,用普通的拉氏乘子法,其待定的拉氏乘子在变分中恒等于零,这称为临界变分状态,在这种临界状态中,我们无法用待定拉氏乘子法把变分的约束条件吸收入泛函,从而解除这个约束条件。例如用拉氏乘子法,从最小余能原理只能导出Hellinger-Reissner变分原理,这个原理中只有应力和位移两类独立变量,而应力应变关系仍然是变分的约束条件。为了消除这个约束条件,钱伟长教授提出了高次拉氏乘子法,即在泛函中引入二次项来消除应力应变这个约束条件。 本文目的是要证明,如果在泛函中引入如下二次项我们也可以用高次拉氏乘子法解除应力应变这个变分约束条件。用这种方法,我们不仅可以从Hel-linger-Reissner原理的基础上,找到更一般的广义变分原理。在特殊情况下,这个更一般的广义变分原理,可以还原为各种已知的弹性理论变分原理。同样,我们也可以从Hu-Washizu(胡海昌-鹫津久-郎)变分原理,用高次拉氏乘子法,求得比该原理更一般的广义变分原理。

Based on a preliminary study on the identified multiplier method, it is pointed out in this paper that this method is a unite method of functional transformation in variational principle. On the basis of Hellinger-Reissner variational principle, the critical variational constrained condition is relieved by use of the identified multiplier method, and a generalized variational principle with three kinds of independent variable in nonlinear theory of elasticity is obtained.

本文通过对已识别拉氏乘子法的初步探讨,指出了已识别拉氏乘子法是变分原理中泛函变换的统一方法。并在H-R变分原理的基础上,应用已识别拉氏乘子,将其应力应变关系这个临界变分约束条件解除,得到了非线性弹性理论的一个三类独立变量的广义变分原理。

 
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