Automated adaptive analysis of the biphasic equations for soft tissue mechanics using a posteriori error indicators

Abstract

AbstractHydrated soft tissues of the human musculoskeletal system, such as articular cartilage in diarthrodial joints or the annulus fibrosis of the inter vertebral disk, are accurately represented by a biphasic continuum model consisting of an incompressible solid phase (collagen and proteoglycan) and incompressible, inviscid fluid (interstitial fluid) and derive from the continuum theory of mixtures. These tissues exhibit a viscoelastic‐type response which is caused by the diffusive drag of the fluid phase as it flows past the solid phase. In this study an automated, adaptive, finite element solution of the governing biphasic equations is presented. The finite element formulation is based on a mixed‐penalty approach in which the penalty form of the continuity equation for the mixture is included in the weak form. Pressure, solid and fluid velocities are interpolated independently, and the coefficients of the C−1 pressure field are eliminated at the element level. The resulting matrix form is a system of first order differential equations which is solved via standard finite difference methods. Mesh generation and updating, including both refinement and coarsening, for the two‐dimensional examples examined in this study are performed using Finite Quadtree. The adaptive analysis is based on an error indicator which is the L2 norm of the difference between the finite element solution and a projected finite element solution. Total stress, calculated as the sum of the solid and fluid phase stresses, is used in the error indicator. Rezoning is accomplished by transferring the finite element solution for the primary variables onto the locally updated mesh using a projected field. These projected values allow the finite difference algorithm to proceed in time using the updated mesh. The accuracy and effectiveness of this adaptive finite element analysis is demonstrated using two‐dimensional axisymmetric problems corresponding to the uncomined compression of a cylindrical disk of soft tissue and the indentation of a thin sheet of soft tissue. The method is shown to effectively capture the steep gradients in both problems and to produce solutions in good agreement with independent, converged, numerical solutions.