Multiresolution method for bending of plates with complex shapes

A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains. To realize this method, we design a new wavelet basis function, by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains. In the solution of differential equations, various derivatives of the unknown function are denoted as new functions. Then, the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals. Therefore, the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative. During the application of the proposed method, boundary conditions can be automatically included in the integration operations, and relevant matrices can be assured to exhibit perfect sparse patterns. As examples, we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes. By comparing the solutions obtained by the proposed method with the exact solutions, the new multiresolution method is found to have a convergence rate of fifth order. The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method (FEM) with tens of thousands of elements. In addition, because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order, we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.