scholarly journals A Nonconforming Finite Element Approximation for the von Karman equations

2016 ◽  
Vol 50 (2) ◽  
pp. 433-454 ◽  
Author(s):  
Gouranga Mallik ◽  
Neela Nataraj
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Nonconforming Finite Element ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Von Kármán

ON THE GENERALIZED VON KÁRMÁN EQUATIONS AND THEIR APPROXIMATION

2007 ◽  
Vol 17 (04) ◽  
pp. 617-633 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
LILIANA GRATIE ◽  
SRINIVASAN KESAVAN
Keyword(s):  
Original Problem ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Von Karman ◽  
Strict Positivity ◽  
Von Kármán Equations ◽  
Conforming Finite Element ◽  
Nonlinearly Elastic Plate ◽  
Von Kármán ◽  
Nonlinearly Elastic

We consider here the "generalized von Kármán equations", which constitute a mathematical model for a nonlinearly elastic plate subjected to boundary conditions "of von Kármán type" only on a portion of its lateral face, the remaining portion being free. As already shown elsewhere, solving these equations amounts to solving a "cubic" operator equation, which generalizes an equation introduced by Berger and Fife. Two noticeable features of this equation, which are not encountered in the "classical" von Kármán equations are the lack of strict positivity of its cubic part and the lack of an associated functional. We establish here the convergence of a conforming finite element approximation to these equations. The proof relies in particular on a compactness method due to J. L. Lions and on Brouwer's fixed point theorem. This convergence proof provides in addition an existence proof for the original problem.

scholarly journals Morley finite element method for the von Karman obstacle problem

2021 ◽  
Author(s):  
Carsten Carstensen ◽  
Sharat Gaddam ◽  
Neela Nataraj ◽  
Amiya K Pani ◽  
Devika Shylaja
Keyword(s):  
Finite Element ◽  
Obstacle Problem ◽  
A Priori ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Small Data ◽  
Polygonal Domain ◽  
Optimal Convergence ◽  
Von Karman ◽  
Von Kármán

This paper focusses on the  von Karman equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semilinear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation. The first part establishes the well-posedness of the von Karman obstacle problem and also discusses the uniqueness of the solution under an a priori and an a posteriori smallness condition on the data. The second part of the article discusses the regularity result of Frehse from 1971 and combines it with the regularity of the solution on a polygonal domain. The third part of the article shows an a priori error estimate for optimal convergence rates for the Morley finite element approximation to the von Karman obstacle problem for small data. The article concludes with numerical results that illustrates the requirement of smallness assumption on the data for optimal convergence rate.

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