Trefftz-polynomial reciprocity based FE formulations

Abstract

The paper contains a general procedure for obtaining of Trefftz polynomials of arbitrary order for 2D or 3D problems by numerical or analytical way. Using Trefftz polynomials for displacement and tractions the unknown displacements and tractions are related by non-singular boundary integral equations. For a multidomain (element) formulation we suppose the displacements to be continuous between the sub-domains and the tractions are connected in a weak (integral) sense by a variational formulation of inter-element equilibrium. The stiffness matrix defined in this way is nonsymmetric and positive semi-definite. The finite elements can be combined with other well known elements. The form of the elements can be, however, more general (the multiply connected form of the element is possible, transition elements which can be connected to more elements along one side are available). It is also very easy and simply possible to assess the local errors of the solution from the traction incompatibilities (the inter-element equilibrium, which is satisfied in a weak sense only, is the only incompatibility in the solution of the linear problem). The stress smoothing is a very useful tool in the post-processing stage. It can improve the accuracy of the stress field by even one order or more comparing to the simple averaging, if the stress gradients in the element are large. Also the convergence of the so obtained stress field increases. The examples with high order gradient field and crack modelling document the efficiency of this FEM formulation. The extension to the solution of other field problems is very simple.