The global nonlinear galerkin method for the analysis of elastic large deflections of plates under combined loads: A scalar homotopy method for the direct solution of nonlinear algebraic equations

In this paper, the global nonlinear Galerkin method is used to perform an accurate and efficient analysis of the large deflection behavior of a simplysupported rectangular plate under combined loads. Through applying the Galerkin method to the governing nonlinear partial differential equations (PDEs) of the plate, we derive a system of coupled third order nonlinear algebraic equations (NAEs). However, the resultant system of NAEs is thought to be hard to tackle because one has to find the one physical solution from among the possible multiple solutions. Therefore, a suitable initial guess is required to lead to the real solution for given load conditions. The feature of this paper is that we apply the global nonlinear Galerkin method to the governing PDEs and solve the resultant NAEs directly in each load step. To keep track of the physical solution, the initial guess for the current load step is provided by taking the solution of the NAEs for the last step as the initial guess. Besides, the size of the NAEs grows dramatically larger, with the increase of the number of terms of the trial functions, which will cost much more computational efforts. An exponentially convergent scalar homotopy algorithm (ECSHA) is introduced to solve the large set of NAEs. The approach in the present paper is more direct and simpler than either the incremental global Galerkin method, or the incremental local Galerkin method (finite element method) based on a symmetric incremental weak-form; both of which methods lead to the inversion of tangent stiffness matrices and Newton-Raphson iterations in each load step. The present method of exponentially convergent scalar homotopy of directly solving the NAEs is much better than the quadratically convergent Newton-Raphson method. Several numerical examples are provided to validate the feasibility and efficiency of the proposed scheme.