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.

On the Existence of Solutions to the Generalized Marguerre-von Kármán Equations

2006 ◽  
Vol 11 (1) ◽  
pp. 83-100 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Liliana Gratie
Keyword(s):  
Operator Equation ◽  
Shallow Shell ◽  
Existence Of Solutions ◽  
Vertical Displacement ◽  
Existence Of A Solution ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Nonlinearly Elastic Plate ◽  
Von Kármán ◽  
Nonlinearly Elastic

Using techniques from asymptotic analysis, the second author has recently identified equations that generalize the classical Marguerre-von Kármán equations for a nonlinearly elastic shallow shell by allowing more realistic boundary conditions, which may change their type along the lateral face of the shell. We first reduce these more general equations to a single “cubic” operator equation, whose sole unknown is the vertical displacement of the shell. This equation generalizes a cubic operator equation introduced by M. S. Berger and P. Fife for analyzing the von Kármán equations for a nonlinearly elastic plate. We then establish the existence of a solution to this operator equation by means of a compactness method due to J. L. Lions.

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.

scholarly journals A Review on Analytical Modeling for Collapse Mode Capacitive Micromachined Ultrasonic Transducer of the Collapse Voltage and the Static Membrane Deflections

Micromachines ◽  
2021 ◽  
Vol 12 (6) ◽  
pp. 714
Author(s):  
Jiujiang Wang ◽  
Xin Liu ◽  
Yuanyu Yu ◽  
Yao Li ◽  
Ching-Hsiang Cheng ◽  
...  
Keyword(s):  
Analytical Modeling ◽  
Ultrasonic Transducer ◽  
Governing Equations ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Collapse Mode ◽  
Capacitive Micromachined Ultrasonic Transducer ◽  
Gap Height ◽  
Von Kármán ◽  
Analytical Expressions

Analytical modeling of capacitive micromachined ultrasonic transducer (CMUT) is one of the commonly used modeling methods and has the advantages of intuitive understanding of the physics of CMUTs and convergent when modeling of collapse mode CMUT. This review article summarizes analytical modeling of the collapse voltage and shows that the collapse voltage of a CMUT correlates with the effective gap height and the electrode area. There are analytical expressions for the collapse voltage. Modeling of the membrane deflections are characterized by governing equations from Timoshenko, von Kármán equations and the 2D plate equation, and solved by various methods such as Galerkin’s method and perturbation method. Analytical expressions from Timoshenko’s equation can be used for small deflections, while analytical expression from von Kármán equations can be used for both small and large deflections.

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