A scaled boundary finite element based node-to-node scheme for contact problems.

The analysis of contact problems is a major concern in many engineering applications. It is one of the most difficult topics due to unknown contact areas and inequality constraints. For numerical simulations, the dissimilar discretization of contact interfaces is inevitable due to the tangential slippage in large sliding contact problems. Therefore, it is impossible to maintain the node-to-node (NTN) contact. Various treatments have been proposed to enforce the contact constraints on nonmatching contact interfaces. Their implementations, however, either fail the patch test or require sophisticated algorithms and techniques. This thesis presents a novel NTN contact scheme based on the scaled boundary finite element method (SBFEM). Nonmatching meshes can be converted to matching ones through polytope elements constructed in a SBFEM manner, allowing the use of the simplicity and robustness of a purely nodal based contact formulation. For an individual scaled boundary finite element, the number of edges and faces is not limited and new nodes can be inserted on the element boundary arbitrarily. Only its boundary is discretized and it can be easily extended to include higher order approximations. The contact constraints are enforced by means of complementarity formulations, which are solved as a mixed complementarity problem. This mathematical description not only satisfies the non-penetration condition exactly, but also allows an accurate representation of the second-order Coulomb's friction cone in three dimensions without linearization. Outer iterations of the active contact set are not required. The proposed method is also applied to large sliding contact problems with a mesh updating scheme. The inserted nodes are regarded as auxiliary nodes for the current step. They are removed in a future step to avoid overly-refined mesh. The state variables of newly inserted nodes are updated through interpolation of neighboring nodes. The proposed method is verified by contact problems with analytical solutions, such as the patch test and Mindlin-Hertz contact problem. Significant improvements in higher order elements are obtained by the method when compared with the finite element code ABAQUS. Applications of the method are extended to practical contact problems, where complex geometries are modeled through quadtree or octree meshe