Time decay for porosity problems

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

Time decay for porosity problems

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGIn 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.Agencia Estatal de Investigación | Ref. PGC2018‐096696‐B‐I00Agencia Estatal de Investigación | Ref. PID2019‐ 105118GB‐I0

On the time decay for the MGT-type porosity problems

In this work we study three different dissipation mechanisms arising in the so-called Moore-Gibson-Thompson porosity. The three cases correspond to the MGT-porous hyperviscosity (fourth-order term), the MGT-porous viscosity (second-order term) and the MGT-porous weak viscosity (zerothorder term). For all the cases, we prove that there exists a unique solution to the problem and we analyze the resulting point spectrum. We also show that there is an exponential energy decay for the first case, meanwhile for the second and third case only a polynomial decay is found. Finally, we present some one-dimensional numerical simulations to illustrate the behaviour of the discrete energy for each caseThis paper is part of the projects PGC2018-096696-B-I00 and PID2019-105118GB-I00, funded by the Spanish Ministry of Science, Innovation and Universities and FEDER "A way to make Europe".Peer ReviewedPostprint (author's final draft

Decay for strain gradient porous elastic waves

We study the one-dimensional problem for the linear strain gradient porous elasticity. Our aim is to analyze the behavior of the solutions with respect to the time variable when a dissipative structural mechanism is introduced in the system. We consider five different scenarios: hyperviscosity and viscosity for the displacement component and hyperviscoporosity, viscoporosity and weak viscoporosity for the porous component. We only apply one of these mechanisms at a time. We obtain the exponential decay of the solutions in the case of viscosity and a similar result for the viscoporosity. Nevertheless, in the hyperviscosity case (respectively hyperviscoporosity) the decay is slow and it can be controlled at least by t-1/2 . Slow decay is also expected for the weak viscoporosity in the generic case, although a particular combination of the constitutive parameters leads to the exponential decay. We want to emphasize the fact that the hyperviscosity (respectively hyperviscoporosity) is a stronger dissipative mechanism than the viscosity (respectively viscoporosity); however, in this situation, the second mechanism seems to be more “efficient” than the first one in order to pull along the solutions rapidly to zero. This is a striking fact that we have not seen previously at any other linear coupling system. Finally, we also present some numerical simulations by using the finite element method and the Newmark-ß scheme to show the behavior of the energy decay of the solutions to the above problems, including a comparison between the hyperviscosity and the viscosity casesPeer ReviewedPostprint (published version

An a priori error analysis of a porous strain gradient model

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.Peer ReviewedPostprint (author's final draft

Decay for strain gradient porous elastic waves

We study the one-dimensional problem for the linear strain gradient porous elasticity. Our aim is to analyze the behavior of the solutions with respect to the time variable when a dissipative structural mechanism is introduced in the system. We consider five different scenarios: hyperviscosity and viscosity for the displacement component and hyperviscoporosity, viscoporosity and weak viscoporosity for the porous component. We only apply one of these mechanisms at a time. We obtain the exponential decay of the solutions in the case of viscosity and a similar result for the viscoporosity. Nevertheless, in the hyperviscosity case (respectively hyperviscoporosity) the decay is slow and it can be controlled at least by t−1/2. Slow decay is also expected for the weak viscoporosity in the generic case, although a particular combination of the constitutive parameters leads to the exponential decay. We want to emphasize the fact that the hyperviscosity (respectively hyperviscoporosity) is a stronger dissipative mechanism than the viscosity (respectively viscoporosity); however, in this situation, the second mechanism seems to be more “efficient” than the first one in order to pull along the solutions rapidly to zero. This is a striking fact that we have not seen previously at any other linear coupling system. Finally, we also present some numerical simulations by using the finite element method and the Newmark-β scheme to show the behavior of the energy decay of the solutions to the above problems, including a comparison between the hyperviscosity and the viscosity cases.Agencia Estatal de Investigación | Ref. PGC2018‐096696‐B‐I00Agencia Estatal de Investigación | Ref. PID2019‐105118GB‐I00Universidade de Vigo/CISU

Finite element error analysis of a viscoelastic Timoshenko beam with thermodiffusion effects

In this paper, a thermomechanical problem involving a viscoelastic Timoshenko beam is analyzed from a numerical point of view. The so-called thermodiffusion effects are also included in the model. The problem is written as a linear system composed of two second-order-in-time partial differential equations for the transverse displacement and the rotational movement, and two first-order-in-time partial differential equations for the temperature and the chemical potential. The corresponding variational formulation leads to a coupled system of first-order linear variational equations written in terms of the transverse velocity, the rotation speed, the temperature and the chemical potential. The existence and uniqueness of solutions, as well as the energy decay property, are stated. Then, we focus on the numerical approximation of this weak problem by using the implicit Euler scheme to discretize the time derivatives and the classical finite element method to approximate the spatial variable. A discrete stability property and some a priori error estimates are shown, from which we can conclude the linear convergence of the approximations under suitable additional regularity conditions. Finally, some numerical simulations are performed to demonstrate the accuracy of the scheme, the behavior of the discrete energy decay and the dependence of the solution with respect to some parameters

CMMSE: numerical analysis of a chemical targeting model

Treating specific tissues without affecting other regions is a difficult task. It is desirable to target the particular tissue where the chemical has its biological effect. To study this phenomenon computationally, in this work we numerically study a mathematical model which is written as a nonlinear system composed by three parabolic partial differential equations. The variables involved in the model are the concentration of the chemical, the concentration of the binding protein and the concentration of the chemical bound to the protein. Our aim is to propose a fully discrete approximation of this problem, using the Finite Element Method and a semi-implicit Euler scheme, in order to solve it numerically. This discrete problem is analysed, obtaining a discrete stability property and some a priori error estimates that show the algorithm converges linearly if the continuous solution is regular enough. Also, some representative examples are shown, as well as the numerical verification of the convergence.Agencia Estatal de Investigación | Ref. PGC2018-096696-B-I00Universidade de Vigo/CISU

Análise de modelos matemáticos de dano en remodelación ósea: resolución mediante métodos numéricos.

The mechanical properties of vertebrate bone are complex, and they can be studied at two different scales: macroscopic and microscopic. The macroscopic properties are modelled phenomenologically while the microscopic dynamics are determined by cell populations. This thesis comprises a series of researches where we study mathematical models related to damage and remodelling in bone tissue. We explore both the biomechanical aspects of damage and remodelling from the macroscopic point of view and the microscopic models for cell populations. In the macroscopic part, we develop a novel model that couples damage and bone remodelling effects. Bone remodelling is crucial in order to maintain the integrity of the skeleton through our lives, since it is necessary to repair the damage that appears on our day-to-day activities. However, their coupling had not yet been deeply studied. In this thesis, the first thing we study is the damage model proposed by Frémond and Nedjar in 1996 so as to show its capabilities of modelling bone damage. Then, we present its coupling with the well-known remodelling model from Weinans Huiskes and Grootenboer (1992). In both cases, we present the variational formulation of the models and a discrete approximation of the solution using the finite element method. We also perform numerical analysis to obtain the convergence of the discrete solution. In the second part of this thesis, we study microscopic models that reproduce the remodelling effect at the cell level. Although most of these models only account for temporal effects, we studied a spatio-temporal model proposed by Ayati and col. and provided a discrete approximation of the solution. We also propose a novel spatial extension of the existing temporal model which was developed by Graham and col. Lastly, we study an osseointegration model from Moreo, García-Aznar and Doblaré, relevant to understand the behaviour of implants. All of our studies include a numerical analysis of the convergence of the solutions.Las propiedades mecánicas de los huesos son complejas y se pueden estudiar desde dos puntos de vista: el macroscópico y el microscópico. Las propiedades macroscópicas se modelan fenomenológicamente, mientras que las dinámicas microscópicas vienen dadas por las relaciones entre conjuntos de células. Esta tesis comprende una serie de investigaciones donde estudiamos modelos matmeáticos relacionados con el daño y con la remodelación ósea. Exploramos tanto los aspectos biomecánicos del daño y de la remodelación ósea desde el punto de vista macroscópico como los modelos de poblaciones de células. En la parte microscópica, desarrollamos un nuevo modelo que acopla los efectos de daño y remodelación ósea. La remodelación ósea es crucial para mantener la integridad del esqueleto a lo largo de nuestras vidas, ya que es necesaria para reparar el daño que aparece en las actividades cotidianas. Sin embargo, este acoplamiento entre ambos efectos no ha sido estudiado en profundidad. En esta tesis, lo primero que estudiamos es el modelo de daño propuesto por Frémond y Nedjar en 1996, para mostrar sus posibilidades en el modelado de tejido óseo. Después, presentamos su acoplamiento con el reconocido modelo de Weinans Huiskes y Grootemboer (1992). En ambos casos presentamos la formulación variacional de los modelos y una aproximación discreta de las soluciones por el método de elementos finitos. También llevamos a cabo análisis numéricos para obtener la convergencia de la solución discreta. En la segunda parte de la tesis estudiamos modelos microscópicos que reproducen el fenómeno de remodelación a nivel celular. Aunque la mayoría de estos modelos solamente tienen en cuenta efectos temporales, estudiamos el modelo espacio-temporal propuesto por Ayati y col. y mostramos una aproximación discreta de la solución. También proponemos una nueva extensión espacial del modelo temporal desarrollado por Graham y col. Finalmente, estudiamos un modelo de oseointegración de Moreo, García-Aznar y Doblaré, relevante para el comportamiento de implantes. Todos estos estudios incluyen un análisis numérico de la convergencia de las soluciones.As propiedades mecánicas dos huesos son complexas e pódense estudar dende dous puntos de vista: o macroscópico e o microscópico. As propiedades macroscópicas modélanse fenomenolóxicamente, mentres que as dinámicas microscópicas veñen dadas polas relacións entre conxuntos de células. Esta tese comprende unha serie de investigacións donde estudamos modelos matemáticos relacionados co dano e coa remodelación do tecido óseo. Exploramos tanto os aspectos biomecánicos do dano e da remodelación ósea dende o punto de vista macroscópico como modelos de poblacións de células. Na parte macroscópica, desenvolvemos un novo modelo que acopla os efectos do dano e da remodelación ósea. A remodelación ósea é crucial para manter a integridade do esqueleto ao longo das nosas vidas, xa que é necesaria para reparar o dano que aparece coas actividades cotidiás. Sen embargo, o seu acoplamento aínda non se estudou en profundidade. Nesta tese, o primeiro que estudamos é o modelo de dano proposto por Frémond e Nedjar en 1996, para mostrar as súas posibilidades para modelar dano no tecido óseo. Despois, presentamos o seu acoplamento co recoñecido modelo de Weinans Huiskes e Grootenboer (1992). En ambos casos presentamos a formulación variacional dos modelos e unha aproximación discreta das solucións polo método de elementos finitos. Tamén levamos a cabo unha análise numérica para obter a converxencia da solución discreta. Na segunda parte da tese estudamos modelos microscópicos que reproducen o fenómeno de remodelación a nivel celular. Aínda que a maioría destes modelos só teñen en conta efectos temporais, estudiamos o modelo espacio-temporal proposto por Ayati e col. e mostramos unha aproximación discreta da solución. Tamén propoñemos unha nova extensión espacial do modelo temporal desenvolto por Graham e col. Finalmente, estudiamos un modelo de oseointegración de Moreo, García-Aznar e Doblaré, relevante para o comprendemento do comportamento de implantes. Todos estes estudos inclúen unha análise numérica da converxencia das solucións

Numerical analysis of an osseointegration model

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, one- and two-dimensional numerical results are presented to demonstrate the accuracy of the algorithm and the behavior of the solution.Ministerio de Ciencia, Innovación y Universidades | Ref. PGC2018-096696-B-I00Xunta de Galicia | Ref. ED481A-2019/23