Solutions of the von Kármán plate equations by a Galerkin method, without inverting the tangent stiffness matrix

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 NewtonRaphson 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.

Mathematical Models for Suspension Bridges

“…[40,42,43,82,155,156,162,163,264], as well as in the engineering literature [160,Sect. 8.8.1], see also [86] for some numerical approaches. Concerning the evolution (wave-type) von Kármán equations, for the general theory we refer again to [162,163].…”

Nearly exact and highly efficient elastic-plastic homogenization and/or direct numerical simulation of low-mass metallic systems with architected cellular microstructures

“…To solve the incremental tangent stiffness equations, we employ a Newton homotopy method [Liu et al 2009;Dai et al 2014]. One of the most important reasons that we use the newly developed scalar homotopy method is that this approach does not need to invert the Jacobian matrix (the tangent stiffness matrix) to solve NAEs.…”

“…To solve tangent stiffness equations, we use a Newton homotopy method recently developed to solve a system of fully coupled nonlinear algebraic equations (NAEs) with as many unknowns as desired [Liu et al 2009;Dai et al 2014]. By using these methods, displacements of the equilibrium state are iteratively solved without the inversion of the Jacobian (tangent stiffness) matrix.…”

“…It is well known that the simple Newton's method as well as the Newton-Raphson iteration method require the inversion of the Jacobian matrix, which fail to pass the limit load as the Jacobian matrix becomes singular, and require arclength methodology which are commonly used in commercial off-the-shelf software such as ABAQUS. Furthermore, homotopy methods are useful in the following cases: when the system of algebraic equations is very large in size, when the solution is sensitive to the initial guess, and when the system of nonlinear algebraic equations is either over-or under-determined [Liu et al 2009;Dai et al 2014].…”

Analysis of Nonlinear Vibrations and Dynamic Responses in a Trapezoidal Cantilever Plate Using the Rayleigh‐Ritz Approach Combined with the Affine Transformation

“…The vibration response experiment agreed well with the numerical results. Dai et al [21][22][23] proposed a highly efficient global nonlinear Galerkin method for the analysis of the large-deflection problem of plates. Additionally, a time domain collocation method also has been proposed by Dai et al [24,25] to solve nonlinear oscillatory problems, which is promising in the analysis of the von Karman fluttering plate.…”