Numerical simulation of thermal bone necrosis during cementation of femoral prostheses (Q809013)

The authors present a mathematical model for describing the thermal and kinetic history of a hip replacement during cementation of the infibulum. The functions to be determined are the temperature u and the degree of polymerization w (0\(\leq w\leq 1)\) on a complex domain D consisting of the prosthesis, a part of the bone and the volume filled by the cement. The mathematical model is as follows: Find u(x,y,z,t), w(x,y,z,t) such that \[ c(\partial u/\partial t)-\nabla \cdot (k \nabla u)=\rho \lambda (dw/dt)X_ c,\quad dw/dt=A(u)\cdot P(u,w), \] where \(X_ c\) stands for the characteristic function of the cement subdomain coupled to initial and boundary conditions \[ u(.,0)=u_ 0(\cdot),\quad k(\partial u/\partial v)+H(u-u_ a)=0\text{ on } \partial D\times (0,T). \] The paper gives a finite element approximation of the problem in cylindrical coordinates. A numerical algorithm is given and various numerical simulations are developed which confirm some critical aspects of this orthopaedic fixing technique. I suggest to the authors to try the Adomian's technique for solving their p.d.e..