RT Dissertation/Thesis
T1 Numerical analysis and simulations in bone remodeling models
A1 Martínez Fernández, Rebeca
K1 Matemáticas
AB ConclusionsIn 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 boneis considered as an elastic material. A variational formulation was provided, obtainingan elliptic variational equation for the displacement ¯eld and an ordinary di®erentialequation 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 timederivatives, respectively, we obtained a fully discrete problem and we proved an errorestimates result. Moreover, under additional regularity assumptions, we derived thelinear 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 nowthat the bone may come into contact with a rigid or a deformable obstacle. In order tomodel these two contact conditions, we used the classical Signorini condition and thenormal compliance contact law, respectively. The variational formulation was obtainedfor both problems and the convergence of the solution to the contact problem with adeformable obstacle, when the deformability coeficient tends to zero, to the solutionof the Signorini's problem was established. We introduced fully discrete aproximationsand we proved an error estimates result for both problems. Finally, under additionalregularity assumptions, we obtained the linear convergence of the algorithm and somesimulations were also presented.The third chapter dealt with the numerical analysis, including numerical simulationsin one and two dimensions, of a bone remodeling model introduced byWeinans, Huiskesand Grootenboer in [66]. A numerical algorithm for the variational problem, based onthe ¯nite element method to approximate the spatial variable and an Euler scheme todiscretize the time derivatives, was proposed, an error estimate on its solutions wasobtained and its linear convergence was established under suitable regularity assump-tions. The numerical simulations demonstrated the accuracy of the approximationsand some properties related to the behavior of the solution.Finally, in the last chapter, we proposed a new bone remodeling model in which weconsidered the bone as an piezoelectric material. This property of the bone tissue wassuggested in 1957. However, it was not normally used to understand bone remodelingand 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 modelto characterize the evolution of the bone density and the mechanical properties of thebone. 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 wasproposed, coupling the electric and displacement ¯elds. Finally, error estimates wereproved and the linear convergence was established under adequate regularity condi-tions. Again, the numerical results shown the accuracy of the approximations as wellas 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 MATLABcode 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 algorithmsand 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 variationalinequalities (see [44]). However, we remark that the existence and uniqueness resultsof weak solutions for the continuous variational formulations are open problems. In theCowin 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 andKuttler dealt with the model proposed byWeinans, Huiskes and Grootenboer obtainingan existence and uniqueness result for a regularized problem.
PB Universidade de Santiago de Compostela. Servizo de Publicacións e Intercambio Científico
SN 978-84-9887-632-1
YR 2011
FD 2011-06-24
LK http://hdl.handle.net/10347/3107
UL http://hdl.handle.net/10347/3107
LA eng
NO MARTÍNEZ FERNÁNDEZ, Rebeca: «Numerical analysis and simulations in bone remodeling models». Santiago de Compostela: Universidade. Servizo de Publicacións e Intercambio Científico, 2011. ISBN 978-84-9887-632-1
DS Minerva
RD 29 abr 2026