Mixed Methods in BEM for Elasto-Plastic Problems

Summary

The BEM is much more suitable than the FEM to solve problems with high field gradients (Brebbia^1,2). However, the FEM assures, contrarily to BEM, symmetrical, positive definite stiffness matrices. This is very advantageous, especially for problems with a high number of degrees of freedom. The question arises now, in which form a coupling of BEM and FEM is possible in order to take profit of these great advantages of BEM against FEM (Zienkiewicz^3,4, Schnack⁵, Wendland⁶). For this purpose, the body is divided in subdomains. One can use, then, BEM elements for domains of high stress concentrations. For these domains, a stress field is defined at first, which satisfies the equilibrium and boundary conditions exactly. The stress field will result in a displacement field, which defines the displacement vector of the boundary for the mentioned subdomains. Independently, for the same boundary, a displacement initial formulation will be formulated, which contains the trial functions of the adjacent finite elements. In the following, an optimal approximation of both of the displacement fields on the boundary will be achieved by a penalty term.

The result is the so-called ‘generalized compatibility condition¹ within mixed methods. Additionally, the special element, formulated by the BEM, must be in equilibrium with the adjacent finite elements. As a consequence, the ‘generalized equilibrium equation¹ will be derived from the principle of virtual work, too.

In case, if no formulation of the stress field and accompanying displacement field is possible in a theoretical way (e. g. Airy’s formulation or Boussinesq-Neuber-Papkovitch formulation), the displacement field can be computed by the collocation method from a formulated traction field by a further integral equation of quasi Fredholm-type.

Results of notch and fracture mechanics show the high rate of convergence by use of this mixed method. Therefore, the method prescribed above is extended to problems with elasto-plastic material law.