scholarly journals Numerical Analysis of an Osseointegration Model

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 87 ◽  
Author(s):  
Jacobo Baldonedo ◽  
José R. Fernández ◽  
Abraham Segade
Keyword(s):  
A Priori ◽  
Linear Convergence ◽  
Uniqueness Result ◽  
Regularity Conditions ◽  
The Finite Element Method ◽  
Implicit Euler Scheme ◽  
Strongly Coupled ◽  
Endosseous Implants ◽  
Fully Discrete ◽  
Variational Form

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.

scholarly journals Quasistatic Porous-Thermoelastic Problems: An a Priori Error Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1436
Author(s):  
Jacobo Baldonedo ◽  
José R. Fernández ◽  
José A. López-Campos
Keyword(s):  
A Priori ◽  
Linear Convergence ◽  
Regularity Conditions ◽  
The Finite Element Method ◽  
Discrete Energy ◽  
Implicit Euler Scheme ◽  
Fully Discrete ◽  
Fully Discrete Approximation ◽  
Thermoelastic Problems ◽  
Discrete Stability

In this paper, we deal with the numerical approximation of some porous-thermoelastic problems. Since the inertial effects are assumed to be negligible, the resulting motion equations are quasistatic. Then, by using the finite element method and the implicit Euler scheme, a fully discrete approximation is introduced. We prove a discrete stability property and a main error estimates result, from which we conclude the linear convergence under appropriate regularity conditions on the continuous solution. Finally, several numerical simulations are shown to demonstrate the accuracy of the approximation, the behavior of the solution and the decay of the discrete energy.

Numerical analysis of a piezoelectric bone remodelling problem

2012 ◽  
Vol 23 (5) ◽  
pp. 635-657 ◽  
Author(s):  
J. R. FERNÁNDEZ ◽  
J. M. GARCÍA-AZNAR ◽  
R. MARTÍNEZ
Keyword(s):  
Variational Formulation ◽  
Bone Remodelling ◽  
A Priori ◽  
Linear Convergence ◽  
Coupled System ◽  
Regularity Conditions ◽  
The Finite Element Method ◽  
Discrete Approximations ◽  
Fully Discrete ◽  
Additional Regularity

Although in recent years bone piezoelectricity has been normally neglected, lately a new interest has appeared to show the importance of bone piezoelectricity in wet bone's complex response to loading. Here we numerically study a problem, including a strain-adaptive bone remodelling and the piezoelectricity. Its variational formulation leads to a coupled system composed of two linear variational equations for displacements and electric potential, and a parabolic variational inequality for the apparent density. Fully discrete approximations are now introduced by using the finite element method to approximate spatial variable and the explicit Euler scheme to discretise time derivatives. Some a priori error estimates are proved and the linear convergence of the algorithm is deduced under additional regularity conditions. Finally, some one- and two-dimensional numerical simulations are described to show the accuracy of the proposed algorithm and the behaviour of the solution.

scholarly journals Analysis of a Poro-Thermo-Viscoelastic Model of Type III

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1214 ◽  
Author(s):  
Noelia Bazarra ◽  
José A. López-Campos ◽  
Marcos López ◽  
Abraham Segade ◽  
José R. Fernández
Keyword(s):  
A Priori ◽  
Viscoelastic Model ◽  
Linear Convergence ◽  
Hyperbolic Partial Differential Equations ◽  
Regularity Conditions ◽  
Thermomechanical Model ◽  
Weak Formulation ◽  
Type Iii ◽  
Spatial Approximation ◽  
Constitutive Parameters

In this work, we numerically study a thermo-mechanical problem arising in poro-viscoelasticity with the type III thermal law. The thermomechanical model leads to a linear system of three coupled hyperbolic partial differential equations, and its weak formulation as three coupled parabolic linear variational equations. Then, using the finite element method and the implicit Euler scheme, for the spatial approximation and the discretization of the time derivatives, respectively, a fully discrete algorithm is introduced. A priori error estimates are proved, and the linear convergence is obtained under some suitable regularity conditions. Finally, some numerical results, involving one- and two-dimensional examples, are described, showing the accuracy of the algorithm and the dependence of the solution with respect to some constitutive parameters.

scholarly journals A type III porous-thermo-elastic problem with quasi-static microvoids

Meccanica ◽  
2021 ◽  
Author(s):  
Noelia Bazarra ◽  
Alberto Castejón ◽  
José R. Fernández ◽  
Ramón Quintanilla
Keyword(s):  
Volume Fraction ◽  
A Priori ◽  
Linear Convergence ◽  
Euler Method ◽  
Point Of View ◽  
Discrete Approximations ◽  
Type Iii ◽  
Fully Discrete ◽  
Backward Euler ◽  
Additional Regularity

AbstractIn this work we study, from the numerical point of view, a one-dimensional thermoelastic problem where the thermal law is of type III. Quasi-static microvoids are also assumed within the model. The variational formulation leads to a coupled linear system made of variational equations and it is written in terms of the velocity, the volume fraction and the temperature. Fully discrete approximations are introduced by using the finite element method and the backward Euler method. A discrete stability property and a priori error estimates are proved, deriving the linear convergence under adequate additional regularity. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution.

scholarly journals Analysis of a contact problem for a viscoelastic Bresse system

2021 ◽  
Author(s):  
Maria Inês Copetti ◽  
Toufic El Arwadi ◽  
Jose Fernández ◽  
Maria Naso ◽  
Wael Youssef
Keyword(s):  
Finite Element ◽  
Contact Problem ◽  
A Priori ◽  
Liapunov Function ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
The Finite Element Method ◽  
Discrete Energy ◽  
Implicit Euler Scheme ◽  
Discrete Stability

In this paper, we consider a contact problem between a viscoelastic Bresse beam and a deformable obstacle. The well-known normal compliance contact condition is used to model the contact. The existence of a unique solution to the continuous problem is proved using the Faedo-Galerkin method. An exponential decay property is also obtained defining an adequate Liapunov function. Then, using the finite element method and the implicit Euler scheme, a finite element approximation is introduced. A discrete stability property and a priori error estimates are proved. Finally, some numerical experiments are performed to demonstrate the decay of the discrete energy and the numerical convergence.

Semi-inverse method for predicting stress–strain relationship from cone indentations

2003 ◽  
Vol 18 (9) ◽  
pp. 2068-2078 ◽  
Author(s):  
A. DiCarlo ◽  
H. T. Y. Yang ◽  
S. Chandrasekar
Keyword(s):  
A Priori ◽  
Model Material ◽  
Material Behavior ◽  
The Finite Element Method ◽  
Stress Strain ◽  
Indentation Tests ◽  
Strain Relationship ◽  
Initial Force ◽  
Relationship Of ◽  
Force Displacement

A method for determining the stress–strain relationship of a material from hardness values H obtained from cone indentation tests with various apical angles is presented. The materials studied were assumed to exhibit power-law hardening. As a result, the properties of importance are the Young's modulus E, yield strength Y, and the work-hardening exponent n. Previous work [W.C. Oliver and G.M. Pharr, J. Mater. Res. 7, 1564 (1992)] showed that E can be determined from initial force–displacement data collected while unloading the indenter from the material. Consequently, the properties that need to be determined are Y and n. Dimensional analysis was used to generalize H/E so that it was a function of Y/E and n [Y-T. Cheng and C-M. Cheng, J. Appl. Phys. 84, 1284 (1999); Philos. Mag. Lett. 77, 39 (1998)]. A parametric study of Y/E and n was conducted using the finite element method to model material behavior. Regression analysis was used to correlate the H/E findings from the simulations to Y/E and n. With the a priori knowledge of E, this correlation was used to estimate Y and n.

On a reaction–diffusion system modelling infectious diseases without lifetime immunity

2021 ◽  
pp. 1-25
Author(s):  
HONG-MING YIN
Keyword(s):  
Diffusion Coefficients ◽  
A Priori ◽  
Main Tool ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Sobolev Embedding ◽  
System Modelling ◽  
Strongly Coupled ◽  
Elliptic And Parabolic Equations

In this paper, we study a mathematical model for an infectious disease caused by a virus such as Cholera without lifetime immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction–diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behaviour of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving a priori estimates. The analysis developed in this paper can be employed to study other epidemic models in biological, ecological and health sciences.

Comparison of Two Time-Integration Algorithms for an Anisotropic Damage Model Coupled with Plasticity

2015 ◽  
Vol 784 ◽  
pp. 292-299 ◽  
Author(s):  
Stephan Wulfinghoff ◽  
Marek Fassin ◽  
Stefanie Reese
Keyword(s):  
Evolution Equations ◽  
Damage Model ◽  
Time Integration ◽  
Three Dimensional ◽  
Anisotropic Damage ◽  
Euler Scheme ◽  
The Finite Element Method ◽  
Time Step ◽  
Implicit Euler Scheme ◽  
Integration Algorithms

In this work, two time integration algorithms for the anisotropic damage model proposed by Lemaitre et al. (2000) are compared. Specifically, the standard implicit Euler scheme is compared to an algorithm which implicitly solves the elasto-plastic evolution equations and explicitly computes the damage update. To this end, a three dimensional bending example is solved using the finite element method and the results of the two algorithms are compared for different time step sizes.

ERROR ESTIMATES FOR LINEAR PDEs SOLVED BY WAVELET BASED TAYLOR–GALERKIN SCHEMES

2009 ◽  
Vol 07 (02) ◽  
pp. 143-162 ◽  
Author(s):  
MANI MEHRA ◽  
B. V. RATHISH KUMAR
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Equivalent Norm ◽  
Posteriori Error ◽  
Discrete Problem ◽  
Fully Discrete ◽  
Time Stepping ◽  
Priori Estimates ◽  
Space Error

In this paper, we develop a priori and a posteriori error estimates for wavelet-Taylor–Galerkin schemes introduced in Refs. 6 and 7 (particularly wavelet Taylor–Galerkin scheme based on Crank–Nicolson time stepping). We proceed in two steps. In the first step, we construct the priori estimates for the fully discrete problem. In the second step, we construct error indicators for posteriori estimates with respect to both time and space approximations in order to use adaptive time steps and wavelet adaptivity. The space error indicator is computed using the equivalent norm expressed in terms of wavelet coefficients.

Topology Optimization of Compliant Mechanisms Based on Interpolation Meshless Method and Geometric Nonlinearity

2019 ◽  
Author(s):  
Yonghong Zhang ◽  
Zhenfei Zhao ◽  
Yaqing Zhang ◽  
Wenjie Ge
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Topology Optimization ◽  
Galerkin Method ◽  
Meshless Method ◽  
Geometric Nonlinearity ◽  
Compliant Mechanism ◽  
Linear Convergence ◽  
The Finite Element Method ◽  
Element Method

Abstract In order to prevent mesh distortion problem arising in topology optimization of compliant mechanism with massive displacement, a meshless Galerkin method was proposed and studied in this paper. The element-free Galerkin method (EFG) is more accurate than the finite element method, and it does not need grids. However, it is difficult to impose complex boundaries. This paper presents a topology optimization method based on interpolation meshless method, which retains the advantages of the finite element method (FEM) that is easy to impose boundary conditions and high accuracy of the meshless method. At the same time, a method of gradually reducing step is proposed to solve the problem of non-linear convergence caused by low-density points in topology optimization. Numerical example shows that these techniques are valid in topology optimization of compliant mechanism considering the geometric nonlinearity, and simultaneously these techniques can also improve the convergence of nonlinearity.

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