Quadratic programming contact formulation for elastic bodies using boundary element method

Abstract

A method for the analysis of contact of deformable bodies based on the boundary element method (BEM) has been presented in this paper. The contact problem is stated in the form of a convex quadratic programming (QP) problem written in terms of the contact tractions on the contact surface. A strategy for the incorporation of the BEM contact analysis into models whose domain may be discretized using the finite element method (FEM) has been investigated. A discussion concerning the merits of the proposed approach is provided and several examples are presented to illustrate the validity of the method. HE frictionless contact of deforming elastic bodies is studied in this paper using the boundary element method (BEM). Because of the underlying complexity of the contact problems, only a limited number of analytical solutions, which are restricted to simple geometries and loading conditions, exist in the literature. An overview of the existing analytical solutions may be found in Ref. 1. A numerical discretization of the governing relationships must be employed for the analysis of complex contact configurations. The numerical method used for such discretization in the past has mainly been the finite element method (FEM), which has been employed for obtaining approximations of the solutions to contact variational inequalities.2 A solution to the variational inequalities may be obtained by a constrained minimization of the governing functionals, such as the total potential energy functional, over an admissible set of functions. 2 The fact that domain-based functionals are easier to model with the FEM and the wide applicability of the method have resulted in significant research on contact problems based on the FEM formulations. The contact constraints in such formulations may be introduced by 1) methods where the kinematic contact constraints are directly imposed in an incremental loading of the structure,37 2) Lagrangian-based methods,810 and 3) penalty methods.2'11 Alternative minimum principles defined on the contact boundary also provide effective methods for contact analysis. These formulations may be efficiently discretized using both the FEM and the BEM; for example, see Refs. 12-14. The use of the BEM is advantageous because of the accuracy of the boundary stresses that it provides and the reduced dimensionality of the resulting discretized BEM models compared with those obtained for the domain-based methods.15 The solution to the contact problems may also be approached directly by satisfying the complementarity conditions on the contact boundary.16 Contact problems can be alternatively stated in the form of mathematical programming (MP) problems. Such formulations are derived either from boundary variational formulations or from the direct discretization of the contact complementarity conditions. Since the first publications1719 on the subject, considerable research has been performed, mainly in FEM framework.12'13'20 Recent studies14' 16<21-22 have illustrated the potential and the advantages of the combined BEM and MP approach for the analysis of contact