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Abstract: Bone modeling and remodeling have been extensively studied in many fields of research. The process of adaptive bone remodeling can be described mathematically and simulated in a computer model, integrated with the finite element method (FEM). The FEM can determine the long-term behavior of bone and the impact on bone biomechanics produced by prosthetic models. As whole bone remodeling simulations are time consuming, the aim of this paper is to combine FE analyses with numerical extrapolation techniques to accelerate bone remodeling simulations. Two vector extrapolation methods, reduce ranked extrapolation (RRE) and minimal polynomial extrapolation (MPE), are used to reduce the simulation time. These extrapolation techniques are illustrated by numerical examples of 2D and 3D models based on a strain-adaptive bone remodeling theory. [Copyright &y& Elsevier]

Abstract: In this paper we revisit a strain-adaptive bone remodelling problem, assuming that the rate of the apparent density at a particular location is described as a local objective function and depending on a particular stimulus at that location. Normally, continuum mathematical descriptions of adaptive bone remodelling can lead to discontinuous solutions on the global apparent density distribution. To improve this numerical solution, in this work, as the main novelty, we include the diffusion of the bone remodelling into the model. The variational problem is written as a coupled system of a nonlinear variational equation for the displacement field and a parabolic elliptic variational inequality for the apparent density. An existence and uniqueness result is stated. Then, a fully discrete problem is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. A priori error estimates are proved from which, under adequate additional regularity conditions, the linear convergence of the algorithm is deduced. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximation and the behaviour of the solution. [Copyright &y& Elsevier]