Numerical method for the bone regeneration model, defined within the evolving 2D axisymmetric physical domain

Abstract

A numerical algorithm, used to obtain a solution for a peri-implant osseointegration model is constructed. The model is formulated in terms of a system of three nonlinear coupled time-dependent advection–diffusion–reaction equations, which are defined within the irregular two dimensional physical domain, which evolves in time. The embedded boundary method and the level set function, which is approximated on the fixed regular rectangular grid, are used to track the changes of the irregular geometry of the physical domain. The method of lines is applied to separate the discretizations in time and in space. The advection, diffusion and reaction terms are discretized separately by means of the cell-centered finite volume method. The exact solution of the Riemann problem for the nonstrictly hyperbolic system without genuine nonlinearity, is obtained. An approach for the determination of the gradients of the unknown variables on the edges of the irregular control volumes is proposed. The explicit second order trapezoidal rule is used for the time integration, since it allows to maintain positivity of the solution, which is critical for the considered problem. Some results of the numerical simulations are presented. Contact and distance osteogenesis are predicted for micro-rough and smooth implants.