In this paper we deal with the numerical analysis of the Lord-Shulman thermoelastic problem with porosity and microtemperatures. The thermomechanical problem leads to a coupled system composed of linear hyperbolic partial differential equations written in terms of transformations of the displacement field and the volume fraction, the temperature and the microtemperatures. An existence and uniqueness result is stated. Then, a fully discrete approximation is introduced using the finite element method and the implicit Euler scheme. A discrete stability property is shown, and an a priori error J. Baldonedo acknowledges the funding by Xunta de Galicia (Spain) under the program Axudasá etapa predoutoral with Ref.

In this paper, we numerically study porosity problems with three different dissipation mechanisms. The root behavior is analyzed for each case. Then, by using the finite element method and the Newmark‐β scheme, fully discrete approximations are introduced and some numerical results are described to show the energy evolution depending on the viscosity coefficient.

There are many works dealing with the dynamics of bone remodeling, proposing increasingly complex and complete models. In the recent years, the efforts started to focus on developing models that not only reproduce the temporal evolution, but also include the spatial aspects of this phenomenon. In this work, we propose the spatial extension of an existing model that includes the dynamics of osteocytes. The spatial dependence is modeled in terms of a linear diffusion, as proposed in previous works dealing with related problems. The resulting model is then written in its variational form, and fully discretized using the well‐known finite element method and a combination of the implicit and explicit Euler schemes. The numerical algorithm is then analyzed, proving some a priori error estimates and its linear convergence. Finally, we extend the examples already published for the temporal model to one and two dimensions, showing the dynamics of the solution in the spatial domain.

In this work, we study a bone remodeling model used to reproduce the phenomenon of osseointegration around endosseous implants. The biological problem is written in terms of the densities of platelets, osteogenic cells, and osteoblasts and the concentrations of two growth factors. Its variational formulation leads to a strongly coupled nonlinear system of parabolic variational equations. An existence and uniqueness result of this variational form is stated. Then, a fully discrete approximation of the problem is introduced by using the finite element method and a semi-implicit Euler scheme. A priori error estimates are obtained, and the linear convergence of the algorithm is derived under some suitable regularity conditions and tested with a numerical example. Finally, oneand two-dimensional numerical results are presented to demonstrate the accuracy of the algorithm and the behavior of the solution.

Bone tissue is a material with a complex structure and mechanical properties. Diseases or even normal repetitive loads may cause microfractures to appear in the bone structure, leading to a deterioration of its properties. A better understanding of this phenomenon will lead to better predictions of bone fracture or bone-implant performance. In this work, the model proposed by Frémond and Nedjar in 1996 (initially for concrete structures) is numerically analyzed and compared against a bone specific mechanical model proposed by García et al. in 2009. The objective is to evaluate both models implemented with a finite element method. This will allow us to determine if the modified Frémond–Nedjar model is adequate for this purpose. We show that, in one dimension, both models show similar results, reproducing the qualitative behaviour of bone subjected to typical engineering tests. In particular, the Frémond–Nedjar model with the introduced modifications shows good agreement with experimental data. Finally, some two-dimensional results are also provided for the Frémond–Nedjar model to show its behaviour in the simulation of a real tensile test.

In this work we study a bone remodeling model for the evolution of the myeloma disease. The biological problem is written as a coupled nonlinear system consisting of parabolic partial differential equations. They are written in terms of the concentrations of osteoblasts and osteoclasts, the density of the relative bone and the concentration of the tumor cells. Then, we deal with the numerical analysis of this variational problem, introducing a numerical approximation by using the finite element method and a hybrid combination of both implicit and explicit Euler schemes. We perform some a priori error estimates and show a few numerical simulations to demonstrate the accuracy of the approximation. Finally, we present the comparison with previous works and the behavior of the solution in two‐dimensional examples.

Passive safety systems of cars include parts on the structure that, in the event of an impact, can absorb a large amount of the kinetic energy by deforming and crushing in a design-controlled way. One such energy absorber part, located in the front structure of a Formula Student car, was measured under impact in a test bench. The test is modeled within the Finite Element (FE) framework including the weld characteristics and weld failure description. The continuous welding feature is almost always disregarded in parts included in impact test models. In this work, the FE model is fully defined to reproduce the observed results. The test is used for the qualitative and quantitative validation of the crushing model. On the one hand, the acceleration against time curve is reproduced, and on the other hand, the plying shapes and welding failure observed in the test are also correctly described. Finally, a model that includes additional elements of the car structure is also simulated to verify that the energy absorption system is adequate according to the safety regulations.

In this work, we consider, from the numerical point of view, a boundary-initial value problem for non-simple porous elastic materials. The mechanical problem is written as a coupled hyperbolic linear system in terms of the displacement and porosity fields. The resulting variational formulation is used to approximate the solution by the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence of the numerical scheme is deduced under adequate regularity conditions. Finally, some numerical simulations are presented to show the accuracy of the finite element scheme studied previously, the evolution of the discrete energy and the behavior of the solution.