Variational principles and mathematical methods of solving boundary-value problems for piecewise uniform nonlinear media

1992 ◽  
Vol 62 (1) ◽  
pp. 2557-2560
Author(s):  
S. A. Chernetskii ◽  
N. Yu. Shvaiko
Keyword(s):  
Boundary Value Problems ◽  
Variational Principles ◽  
Boundary Value ◽  
Mathematical Methods ◽  
Nonlinear Media

On dual approximation principles and optimization in continuum mechanics

1969 ◽  
Vol 265 (1162) ◽  
pp. 319-351 ◽  
Keyword(s):  
Mathematical Programming ◽  
Boundary Value Problems ◽  
Continuum Mechanics ◽  
Variational Principles ◽  
Diffusion Theory ◽  
Boundary Value ◽  
Uniqueness Theorems ◽  
Systematic Method ◽  
And Diffusion ◽  
Dual Functions

A unified expression of some of the boundary value problems of continuum mechanics is developed. A central role is given to the notion of a Legendre dual transformation in displaying the simple analytical structure of each problem considered. A systematic method of deriving reciprocal variational principles is described. General boundary value problems governed by inequalities as well as equations are then considered. Convexity of the dual functions related by the Legendre transformation is shown to be the basis of uniqueness theorems and extremum principles. Attention is drawn to the relevance of the literature on mathematical programming theory. M any examples are given, involving new or recent results in elasticity, plasticity, fluid mechanics and diffusion theory.

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