Introduction

In order to solve the system of nonlinear partial differential equations governing the nonlinear dynamics of shells and plates, it is necessary to discretize the problem, for example, by using admissible functions. The construction of the basis functions, satisfying the boundary conditions, is a very difficult task in the case of domains with complex shape. In fact, a very small number of studies address the nonlinear dynamics of structural elements of complicated geometry. The R-functions theory, developed by the Ukrainian mathematician V. L. Rvachev, is one of the possible approaches to solving the problem. It allows us to build the basis functions in analytical form for an arbitrary domain and different boundary conditions, including mixed boundary conditions. However, the R-functions are still little known in the Western scientific community.

The R-functions are a powerful tool to obtain meshless discretization of bidimensional and three-dimensional domains of complex shape, and they can be applied to moving boundaries. This approach is particularly convenient to study nonlinear vibration compared with the finite-element method. In fact, it allows us to discretize shells and plates of complex shape with a very small number of degrees of freedom. The method has a great potential to develop commercial computer codes.

In this chapter, the R-functions are introduced and then applied to obtain series expansions of displacements that satisfy exactly the boundary conditions in the case of complex shape of shells and plates.