Axisymmetric poroelastic boundary element methods for biphasic mechanics of articular cartilage

In this study, an axisymmetric Laplace domain boundary element method for modeling linear biphasic articular cartilage mechanics was developed. A boundary integral formulation was derived by writing the associated integral equations in terms of axisymmetric poroelastic fundamental solutions. Formulas for these fundamental solutions were derived from their three-dimensional Cartesian counterparts via transformations from Cartesian to cylindrical polar coordinates. The fundamental solutions of the poroelastic partial differential equations represent the effects at a particular boundary point of placing vector or scalar sources at all points on the boundary. In the axisymmetric formulation, these sources on the axisymmetric boundary are rotated about the z-axis, creating a ring of sources.;Axisymmetric boundary element methods were developed for solving the resulting boundary integral equations in the Laplace transform domain. The axisymmetric boundary was discretized by placing nodal points along a one-dimensional curve using three-node isoparametric quadratic boundary elements. Gaussian quadrature was employed to evaluate integrals over the boundary elements, which give rise to double integrals over strip regions on the axisymmetric surface. Weakly- and strongly-singular integrals were evaluated, separately, via specialized methods. In the case of weakly-singular integrals, transformation to local polar coordinates at the element nodes regularized the integrals. Strongly-singular integrals were evaluated using three known analytical solutions that enabled determination of unknown strongly-singular entries in terms of previously computed matrix entries.;Accuracy of the boundary element methods was demonstrated for configurations of biphasic compressive stress, pure radial stretching, and uniaxial confined compression, where analytical solutions are known. Potential use of the axisymmetric boundary element method and a Laplace inversion technique were illustrated via simulation of confined compression stress relaxation of a biphasic cartilage cell in a cylindrical sample of extracellular matrix.