Vol. 9, No. 2, 2014

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ISSN: 1559-3959
Solutions of the von Kármán plate equations by a Galerkin method, without inverting the tangent stiffness matrix

Honghua Dai, Xiaokui Yue and Satya N. Atluri

Vol. 9 (2014), No. 2, 195–226
Abstract

Large deflections of a simply supported von Kármán plate with imperfect initial deflections, under a combination of in-plane loads and lateral pressure, are analyzed by a semianalytical global Galerkin method. While many may argue that the dominance of the finite element method in the marketplace may make any other attempts to solve nonlinear plate problems to be redundant and obsolete, semi- and precise analytical methods, when possible, simply serve as benchmark solutions if nothing else. Also, since parametric variations are simpler to access through such analytical methods, they are more useful in studying the physics of the phenomena. In the present method, the Galerkin scheme is first applied to transform the governing nonlinear partial differential equations of the von Kármán plate into a system of general nonlinear algebraic equations (NAEs) in an explicit form. The Jacobian matrix, the tangent stiffness matrix of the system of NAEs, is explicitly derived, which speeds up the Newton–Raphson iterative method if it is used. The present global Galerkin method is compared with the incremental Galerkin method, the perturbation method, the finite element method and the finite difference method in solving the von Kármán plate equations to compare their relative accuracies and efficiencies. Buckling behavior and jump phenomenon of the plate are detected and analyzed. Besides the classical Newton–Raphson method, an entirely novel series of scalar homotopy methods, which do not need to invert the Jacobian matrix (the tangent stiffness matrix), even in an elastostatic problem, and which are insensitive to the guesses of the initial solution, are introduced. Furthermore, we provide a comprehensive review of the newly developed scalar homotopy methods, and incorporate them into a uniform framework, which renders a clear and concise understanding of the scalar homotopy methods. In addition, the performance of various scalar homotopy methods is evaluated through solving the Galerkin-resulting NAEs. The present scalar homotopy methods are advantageous when the system of NAEs is very large in size, when the inversion of the Jacobian may be avoided altogether, when the Jacobian is nearly singular, and the sensitivity to the initially guessed solution as in the Newton–Raphson method needs to be avoided, and when the system of NAEs is either over- or under-determined.

Keywords
von Kármán plate equations, initial imperfection, global Galerkin method, nonlinear algebraic equations, scalar homotopy methods, buckling behavior
Milestones
Received: 16 October 2013
Revised: 8 March 2014
Accepted: 7 April 2014
Published: 30 May 2014
Authors
Honghua Dai
College of Astronautics
Northwestern Polytechnical University
Xi’an 710072
China
Xiaokui Yue
College of Astronautics
Northwestern Polytechnical University
Xi’an 710072
China
Satya N. Atluri
Center for Aerospace Research and Education
University of California, Irvine
Irvine, CA 92697
United States