A sixth-order method for large deflection bending analysis of complex plates with multiple holes

Abstract

The challenge of solving nonlinear problems in multi-connected domains with high accuracy has garnered significant interest. In this paper, we propose a unified wavelet solution method for accurately solving nonlinear boundary value problems on a two-dimensional (2D) arbitrary multi-connected domain. We apply this method to solve large deflection bending problems of complex plates with holes. Our solution method simplifies the treatment of the 2D multi-connected domain by utilizing a natural discretization approach that divides it into a series of one-dimensional (1D) intervals. This approach establishes a fundamental relationship between the highest-order derivative in the governing equation of the problem and the remaining lower-order derivatives. By combining a wavelet high accuracy integral approximation format on 1D intervals, where the convergence order remains constant regardless of the number of integration folds, with the collocation method, we obtain a system of algebraic equations that only includes discrete point values of the highest order derivative. In this process, the boundary conditions are automatically replaced using integration constants, eliminating the need for additional processing. Error estimation and numerical results demonstrate that the accuracy of this method is unaffected by the degree of nonlinearity of the equations. When solving the bending problem of multi-perforated complex-shaped plates under consideration, it is evident that directly using higher-order derivatives as unknown functions significantly improves the accuracy of stress calculation, even when the stress exhibits large gradient variations. Moreover, compared to the finite element method, the wavelet method requires significantly fewer nodes to achieve the same level of accuracy. Ultimately, the method achieves a sixth-order accuracy and resembles the treatment of one-dimensional problems during the solution process, effectively avoiding the need for the complex 2D meshing process typically required by conventional methods when solving problems with multi-connected domains.