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Abstract: In this paper we study from the numerical point of view a strain-adaptive bone remodelling that couples the displacements and the apparent density (the porosity) of the bone. The rate of this density at a particular location is described as an objective function, which depends on a particular stimulus at that location. The variational problem is written as a coupled system of a nonlinear variational equation for the displacement field and a nonlinear parabolic variational inequality for the apparent density. Then, fully discrete approximations are provided by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are proved, from which, under adequate regularity conditions, the linear convergence of the algorithm is deduced. Finally, some numerical simulations involving one- and two-dimensional test examples are presented to demonstrate the accuracy of the approximation and the behaviour of the solution. [Copyright &y& Elsevier]

Abstract: In this paper, we develop a nonconforming finite element methodology using a three-field formulation to analyze a fluid–structure interaction problem. The methodology is used to couple a Lagrangian model describing the structure with the arbitrary Lagrangian–Eulerian strategy used to describe the fluid in order to simulate a full unsteady physical phenomenon. Consistency error estimates are obtained which show that the numerical scheme employed yields a first order approximation for the solution to the fluid–structure interaction problem. Finally, we present a discrete energy estimate to demonstrate the stablity of the proposed method. [Copyright &y& Elsevier]