A mixed-penalty contact finite element formulation for biphasic soft tissues

As a key step in developing a complete computational model of diarthrodial joint mechanics, this thesis presents a finite element formulation and implementation for contact between hydrated soft tissues. The biphasic theory of Mow and co-workers is chosen as the continuum model which governs cartilage mechanics. Boundary conditions for contact surfaces have been developed by Hou et al. by considering a surface of material discontinuity within the continuum. These differential equations and boundary conditions, together with inequality constraints to define the contact surface, form the strong form of the problem. Currently, a fully linear form of the biphasic theory is used and frictionless contact is assumed.

A weighted residual approach is used to develop a form of the equations suitable for numerical approximation. A penalty form of the mixture continuity relation, following Spilker and Maxian, and the momentum equations for each phase are introduced into the weighted residual. Lagrange multipliers are defined on the contact surface and reflect the appropriate continuity of solid and fluid normal traction. Four equations for the multipliers and two expressing kinematic continuity are added to the weighted residual. Standard procedures result in a weak form of the weighted residual, which is then approximated by the finite element method. Quadratic interpolations are selected for velocity and displacement within an element, as well as for the multipliers on the contact surface. A linear interpolation for pressure, independent from element to element, is used.

Part of the solution procedure is to determine the correct contact surface; two tools are required to accomplish this in the finite element setting. First is an algorithm which discretizes the contact surface and provides a coordinate system in which to perform the required surface integrals. Second is an iterative scheme which recognizes regions of the contact surface exhibiting non-physical behaviour, and modifies the surface definition. These algorithms have been evaluated only with the present two-dimensional element, but are readily extensible to three dimensions. Two types of example problems are presented: those that validate the formulation of the element and those that demonstrate its applicability to clinically relevant geometries.