Remarks on Mixed and Singular Finite Elements and on the Field Boundary Element Method

Abstract

In formulating symmetric variational statements for the continuum problem in solid mechanics, we may treat each of the three fields, viz., the stress tensor, the strain tensor, and the displacement vector, as an independent variable. Furthermore, if the solid is incompressible, it is possible to treat the deviatoric as well as the mean parts of the stress tensor as independent variables (see [11]). Likewise, in the analysis of incompressible, viscous, convective flows governed by the Navier-Stokes equations, each of the four fields, viz., the deviatoric stress, the hydrostatic stress, velocity, and velocity strain, may be treated as an independent variable. On the other hand, in formulating Galerkin finite element methods (with similar trial and test function spaces) from these symmetric variational statements in solid mechanics, it is possible to assume trial functions, say for displacements and stresses, in each element of the solid, that do not obey, a priori, the interelement conditions of displacement-continuity and/or traction-reciprocity. In such cases, it becomes necessary to introduce additional independent fields of tractions and/or displacements at the element-boundaries, as Lagrange multipliers to enforce, a posteriori, these interelement constraints. A comprehensive summary of such finite element methods (FEM), with multiple independent fields being assumed inside each element as well as at its boundary, is given in several reports [1, 7, 11, 5, 30].KeywordsFinite Element MethodSolid MechanicTrial FunctionShallow ShellMixed Finite Element MethodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.