Development Trefftz Method for Problems of Nonhomogeneous Media

Abstract

A new scheme for solving problems in the mechanics of structurally nonhomogeneous media is proposed. This scheme is based on dividing of the initial domain into system of subdomain-blocks similar to finite elements and on approximation of the solution in each block by a systems of functions that exactly satisfy the equation and do not assume unlike the finite element method continuity at the block boundaries. This scheme is based on the Papkovich–Neuber analytical representation through the auxiliary potentials, which makes it possible to construct the complete approximation systems in nonhomogeneous media that analytically satisfy the initial equations and contact conditions on the boundaries of inhomogeneities. Also this scheme is based on the generalization of a direct Trefftz method in the system of subdomain-blocks, which approximates the solution in discontinuous energy space. It is shown that generalized Trefftz method has the ability simultaneously with minimizing of the energy functional to stitch together all the necessary quantities at the block boundaries. They are displacements, surface forces and for gradient elasticity models also derivatives and cohesion moments. This ability is achieved solely due to the analytical representation of the used functions. This analytical representation opens up the possibility of construction of finite element approximations for complex nonhomogeneous media on unstructured meshes and inconsistent shape functions, that analytically accurately reproduce the stress state at the vicinity of inclusions and can be considered as a new technology of finite element approximations.