J.R. Fernández

A dynamic viscoelastic contact problem with normal compliance

A dynamic contact problem between a viscoelastic body and a deformable obstacle is numerically considered in this work. The contact is modeled by using the well-known normal compliance contact condition. The variational formulation of this problem is written in terms of the velocity field and it leads to a parabolic nonlinear variational equation. An existence and uniqueness result is stated. Fully discrete approximations are then introduced by using the finite element method to approximate the spatial variable, and a hybrid combination of the implicit and explicit Euler schemes to discretize the time derivatives. An a priori error analysis is recalled. Then, an a posteriori error analysis is provided extending some results already obtained in the study of the heat equation, other parabolic equations and the quasistatic case. Upper and lower bounds are proved. Finally, some two-dimensional numerical simulations are presented to demonstrate the accuracy and the behavior of the error estimators.

Analysis of a problem arising in porous thermoelasticity of type II

ABSTRACT In this article, we consider, from the numerical point of view, a linear thermo-porous-elastic model. The heat conduction is assumed to be of type II. The mechanical problem is written as a coupled system of three hyperbolic partial differential equations for the displacements, the porosity and the thermal displacements. Then, its variational formulation is derived, which is written as a coupled system of three hyperbolic linear variational equations in terms of the velocity, the porous speed and the temperature. An existence and uniqueness result, as well as an energy decay property, is recalled. The fully discrete approximation of the aforementioned problem is introduced by using the finite element method for the spatial approximation and the implicit Euler scheme to discretize the time derivatives. A stability property is proved, from which the energy decay of the discrete energy is deduced. Then, a priori error estimates are obtained, from which, under suitable regularity conditions, the linear convergence of the algorithm is derived. Finally, some numerical simulations are presented to show the accuracy of the approximation and the behavior of the solution.

A porous thermoelastic problem with microtemperatures

ABSTRACT In this article, we consider from the numerical point of view a thermodynamical problem involving a linear elastic material with inner structure, whose particles in addition to the classical displacement possess microtemperatures. The variational formulation of this problem is written as a system of coupled linear parabolic variational equations in terms of velocity, porosity speed, microtemperatures, and temperature. Then, fully discrete approximations are introduced using the finite element method and the backward Euler scheme. A stability property is proved and some a priori error estimates are obtained, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the solution.

An axisymmetric model for the analysis of dynamic surface tension

A quantitative treatment of dynamic surface tension data has been carried out with different mathematical approaches taking into account a diffusion-controlled mechanism. The classical model has been modified in order to achieve a better description of the experimental conditions by considering a finite diffusion domain. The domain has been fixed keeping the restriction that the surfactant concentration in this region should remain constant after the adsorption at the air–water interface, in such a way that the number of surfactant unimers is 30 times the number adsorbed at the interface. The finite diffusion restriction has been used both in 1D and axisymmetric models, the latter one being the most accurate and needing a smaller diffusion domain since it considers surfactant adsorption at a sphere resembling the physical experiments. A distorted sphere geometry taking into account the Laplace–Young equation has also been studied.

Existence and Uniqueness Results for a Kinetic Model in Bulk-Surface Surfactant Dynamics

In this paper we study a new multidimensional mixed-kinetic adsorption model which consists of a nonlinear evolution system of two parabolic partial differential equations: a convective diffusion equation for the bulk surfactant concentration in a bounded domain and a surface diffusion equation for its surface concentration on a compact Lipschitz manifold. The two equations are coupled with a nonlinear relationship, consistent with the Langmuir--Hinshelwood model, which describes the adsorption-desorption transport of surfactant molecules between the bulk phase and one part of its boundary. We provide results on the unique weak solvability of the system and non-negativity of its solution. We use the truncation method combined with the fixed point approach and the theory of time-dependent partial differential equations on manifolds.

On the existence of a solution for an adsorption dynamic model with the Langmuir isotherm

In this paper, we study an adsorption model arising in the dynamics of several surfactants at the air-water interface, where the Langmuir isotherm is employed for modelling the time-dependent surface concentration, providing a nonlinear dynamical boundary condition. Existence of a weak solution is proved by using the Rothe method for a semi-discrete problem in time. After obtaining some a priori estimates and passing to the limit in the time discretization parameter, we conclude that the original Langmuir problem has a bounded solution. An uniqueness result is also given.

Analysis of contact problems of porous thermoelastic solids

ABSTRACT In this article, we study a contact problem between a one-dimensional porous thermoelastic layer and a rigid obstacle. The mechanical problem consists of a coupled system of two hyperbolic partial differential equations and a parabolic one. By defining penalized problems, an energy decay property is obtained. Then, fully discrete algorithms are introduced to approximate both penalized and Signorini problems using the finite element method and the implicit Euler scheme. Stability properties are shown for both problems and a priori error estimates are proved for the penalized problem, from which the linear convergence of the algorithm is derived. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximation and the behavior of the solution.

Numerical analysis of an adsorption dynamic model at the air-water interface

In this paper we deal with the numerical analysis of an adsorption dynamic model arising in a surfactant solution at the air-water interface; the diffusion model is considered together with the so-called Langmuir isotherm. An existence and uniqueness result is stated. Then, fully discrete approximations are introduced by using a finite element method and a hybrid combination of backward and forward Euler schemes. Error estimates are proved from which, under adequate additional regularity conditions, the linear convergence of the algorithm is derived assuming a dependence between both spatial and time discretization parameters. Finally, some numerical simulations are presented in order to demonstrate the accuracy of the algorithm and the behaviour of the solution for two commercially available surfactants. An adsorption dynamic model for surfactants is numerically studied.A priori error estimates are proved by using Gronwalls inequality.Linear convergence is obtained using regularity and an interpolation operator.Numerical results show the accuracy and the peformance of the approximations.