The Natural Boundary Conditions as a Variational Basis for Finite Element Methods

Variational formulations that can be employed in the approximation of boundary value problems involving essential and natural boundary conditions are presented in this paper. They are based on trial functions so chosen as to satisfy a priori the governing differential equations of the problem. The essential boundary conditions are used to construct the displacement approximation basis at finite element level. The natural boundary conditions are enforced on average and their integral forms constitute the variational expression of the finite element approach. The shape functions contain both homogeneous and particular terms, which are related through the interpolation technique used. The application in the framework of the finite element method of the approach proposed here is not trouble free, particularly in what concerns the inter-element continuity condition. The Gauss divergence theorem is used to enforce the essential boundary conditions and the continuity conditions at the element boundary. An alternative but equivalent boundary technique developed for the same purpose is presented also. It is shown that the variational statement of the Trefftz approach is recovered when the Trefftz trial functions are so chosen as to satisfy the essential boundary conditions of the problem.