On the accuracy of finite element solutions of problems with traction boundary conditions*

In the variational formulation of finite element models for mechanics problems, boundary conditions are classified as either essential or natural. The traction boundary conditions are identified as natural in many commonly-used finite element models. A valid finite element mode must satisfy explicitly the essential boundary conditions, since the latter are satisfied implicitly as a posteriori conditions of the formulation. Though assumed solutions can be arbitrary regarding natural boundary conditions, the arbitrariness can affect the accuracy of solutions, because the weighted error residuals of all the Euler equations of the variational principle should be minimized to obtain solutions. As a possible alternative to refining finite element meshes for improving the accuracy of solutions, an explicit enforcement of the natural boundary conditions can be considered. In this work, the nature of the implicit enforcement of the natural boundary conditions of a problem through a variational formulation is investigated. Systematic methods to enforce natural boundary conditions explicitly to improve the accuracy of solutions are developed for two finite element models. Consider a boundary value problem defined by VZu = f in a domain D and some boundary conditions at boundary B. The problem has a variational principle that the stationary condition of a scalar functional I with respect to an admissible solution u (which satisfies u = h on B, if such a condition exists) yields the interior equation and a natural boundary condition, au/an = g on B where n is the outward normal to B. The functional I is: