Conclusions
In the course of this Ph.D. thesis we studied several bone remodeling models, trying to develop a complete study from the mathematical and physical points of view.
In Chapter 2, the Cowin and Hegedus model was introduced. In this model, the bone
is considered as an elastic material. A variational formulation was provided, obtaining
an elliptic variational equation for the displacement ¯eld and an ordinary di®erential
equation which describes the evolution of the bone density. Applying the ¯nite ele-
ment method and an Euler scheme to approximate the spatial variable and the time
derivatives, respectively, we obtained a fully discrete problem and we proved an error
estimates result. Moreover, under additional regularity assumptions, we derived the
linear convergence of the algorithm. Numerical simulations in one, two and three dimensions were presented to show the accuracy and the behavior of the approximations.
In the second part of this chapter, we considered a similar problem assuming now
that the bone may come into contact with a rigid or a deformable obstacle. In order to
model these two contact conditions, we used the classical Signorini condition and the
normal compliance contact law, respectively. The variational formulation was obtained
for both problems and the convergence of the solution to the contact problem with a
deformable obstacle, when the deformability coeficient tends to zero, to the solution
of the Signorini's problem was established. We introduced fully discrete aproximations
and we proved an error estimates result for both problems. Finally, under additional
regularity assumptions, we obtained the linear convergence of the algorithm and some
simulations were also presented.
The third chapter dealt with the numerical analysis, including numerical simulations
in one and two dimensions, of a bone remodeling model introduced byWeinans, Huiskes
and Grootenboer in [66]. A numerical algorithm for the variational problem, based on
the ¯nite element method to approximate the spatial variable and an Euler scheme to
discretize the time derivatives, was proposed, an error estimate on its solutions was
obtained and its linear convergence was established under suitable regularity assump-
tions. The numerical simulations demonstrated the accuracy of the approximations
and some properties related to the behavior of the solution.
Finally, in the last chapter, we proposed a new bone remodeling model in which we
considered the bone as an piezoelectric material. This property of the bone tissue was
suggested in 1957. However, it was not normally used to understand bone remodeling
and there are not many models that justify bone remodeling based on bone piezoelec-
tricity. We continued the work developed in the previous chapter, using this model
to characterize the evolution of the bone density and the mechanical properties of the
bone. Then, we extended the classical electro-mechanical dependence adding a func-
tion ®(½) = ½°, which regulates the coupling between the mechanical and electric ¯elds.
This function guarantees that the electric ¯eld increases with the density of the bone.
The variational formulation for this model was derived and a numerical algorithm was
proposed, coupling the electric and displacement ¯elds. Finally, error estimates were
proved and the linear convergence was established under adequate regularity condi-
tions. Again, the numerical results shown the accuracy of the approximations as well
as the behavior of the solution, giving also a numerical justi¯cation of the electro-
mechanical bone remodeling model.
All the algorithms proposed in this Ph.D. thesis were implemented using MATLAB
code and a good number of examples were computed. First, the one-dimensional exam-
ples were chosen in such a way as to show the numerical convergence of the algorithms
and also their linear convergence. Then, two-or three-dimensional examples were per-
formed in order to show the behavior of the models.
The existence and uniqueness of weak solutions for the discrete problems were ob-
tained applying classical results on linear variational equations or nonlinear variational
inequalities (see [44]). However, we remark that the existence and uniqueness results
of weak solutions for the continuous variational formulations are open problems. In the
Cowin and Hegedus model, this result was obtained for a similar variational formula-
tion in which stronger assumptions were made over the data. Recently, Fern¶andez and
Kuttler dealt with the model proposed byWeinans, Huiskes and Grootenboer obtaining
an existence and uniqueness result for a regularized problem.