Numerical analysis of a thermoelastic dielectric problem arising in the Moore–Gibson–Thompson theory

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGIn this paper, we numerically study a thermoelastic problem arising in the Moore– Gibson–Thompson theory. Dielectrics effects are also included within the model. The corresponding problem is written in terms of the displacement field, the temperature and the electric potential. A viscous term is added in the heat equation to provide the numerical analysis of the corresponding variational problem. Then, by using the finite element method and the implicit Euler scheme fully discrete approximations are introduced. A discrete stability property and a priori error estimates are obtained. Finally, one- and two-dimensional numerical simulations are shown to demonstrate the accuracy of the approximation and the behavior of the solutionMinisterio de Ciencia; Innovación y Universidades | Ref. PGC2018-096696-B-I00Ministerio de Ciencia; Innovación y Universidades | Ref. PID2019-105118GB-I0

Numerical analysis of a problem of elasticity with several dissipation mechanisms

In this work, we numerically study a problem including several dissipative mechanisms. A particular case involving the symmetry of the coupling matrix and three mechanisms is considered, leading to the exponential decay of the corresponding solutions. Then, a fully discrete approximation of the general case in two dimensions is introduced by using the finite element method and the implicit Euler scheme. A priori error estimates are obtained and the linear convergence is derived under some appropriate regularity conditions on the continuous solution. Finally, some numerical simulations are performed to illustrate the numerical convergence and the behavior of the discrete energy depending on the number of dissipative mechanisms.Universidade de Vigo/CISUGAgencia Estatal de Investigación | Ref. PID2019-105118GB-I0

A MGT thermoelastic problem with two relaxation parameters

In this paper, we consider, from both analytical and numerical viewpoints, a thermoelastic problem. The so-called MGT model, with two different relaxation parameters, is used for both the displacements and the thermal displacement, leading to a linear coupled system made by two third-order in time partial differential equations. Then, using the theory of linear semi-groups the existence and uniqueness to this problem is proved. If we restrict ourselves to the one-dimensional case, the exponential decay of the energy is obtained assuming some conditions on the constitutive parameters. Then, using the classical finite element method and the implicit Euler scheme, we introduce a fully discrete approximation of a variational formulation of the thermomechanical problem. A main a priori error estimates result is shown, from which we conclude the linear convergence under suitable additional regularity conditions. Finally, we present some one-dimensional numerical simulations to demonstrate the convergence of the fully discrete approximation, the behavior of the discrete energy decay and the dependence on a coupling parameter.Agencia Estatal de Investigación | Ref. PID2019-105118GB-I00Universidade de Vigo/CISU

An a priori error analysis of a type III thermoelastic problem with two porosities

In this work, we study, from the numerical point of view, a type III thermoelastic model with double porosity. The thermomechanical problem is written as a linear system composed of hyperbolic partial differential equations for the displacements and the two porosities, and a parabolic partial differential equation for the thermal displacement. An existence and uniqueness result is recalled. Then, we perform its a priori error numerical analysis approximating the resulting variational problem by using the finite element method and the implicit Euler scheme. The linear convergence of the algorithm is derived under suitable additional regularity conditions. Finally, some numerical simulations are shown to demonstrate the accuracy of the approximations and the dependence of the solution on a coupling coefficient.Ministerio de Ciencia, Innovación y Universidades | Ref. PGC2018‐096696‐B‐I00Ministerio de Economía y Competitividad | Ref. MTM2016‐74934‐

Numerical analysis of a swelling poro-thermoelastic problem with second sound

In this paper, we analyze, from the numerical point of view, a swelling porous thermo-elastic problem. The so-called second-sound effect is introduced and modeled by using the simplest Maxwell–Cattaneo law. This problem leads to a coupled system which is written by using the displacements of the fluid and the solid, the temperature and the heat flux. The numerical analysis of this problem is performed applying the classical finite element method with linear elements for the spatial approximation and the backward Euler scheme for the discretization of the time derivatives. Then, we prove the stability of the discrete solutions and we provide an a priori error analysis. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximations, the exponential decay of the discrete energy and the dependence on a coupling parameter

Asymptotic behavior and numerical approximation of a double-suspended bridge system

This paper is devoted to introduce and analyze a new non linear problem describing the vibrations of a double suspended bridge system. The road bed is modeled as a double beam of Woinowsky-Krieger type and the two cables, each connected to a single beam by a distributed system of elastic springs, are modeled as one-sided elastic strings. We achieve the existence and uniqueness of solutions by using the semigroup theory and the exponential decay property is also proved. Then, the model is numerically analyzed, through a variational formulation, by using the finite element method and a first-order time integration scheme. A priori error estimates are obtained and the linear convergence is derived under some suitable additional regularity conditions. Finally, some numerical experiments are performed to verify the behavior of the numerical method.Universidade Vigo/CISU

Analysis of a mathematical model arising in plant disease epidemiology

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGIn this work we study from the mathematical and numerical point of view a problem arising in vector-borne plant diseases. The model is written as a nonlinear system composed of a parabolic partial differential equation for the vector abundance function and a first-order ordinary differential equation for the plant health function. An existence and uniqueness result is proved using backward finite differences, uniform estimates and passing to the limit. The regularity of the solution is also obtained. Then, using the finite element method and the implicit Euler scheme, fully discrete approximations are introduced. A discrete stability property and a main a priori error estimates result are proved using a discrete version of Gronwall’s lemma and some estimates on the different approaches. Finally, some numerical results, in one and two dimensions, are presented to demonstrate the accuracy of the approximation and the behaviour of the solution.Agencia Estatal de Investigación | Ref. PGC2018-096696-B-I0

Numerical approximation of some poro-elastic problems with MGT-type dissipation mechanisms

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGIn this work, we numerically analyze a porous elastic problem including several dissipation mechanisms of MGT type. The resulting variational problem is written in terms of the acceleration and the porosity speed. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the classical 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 approximation is derived. Finally, some numerical simulations are presented to show the accuracy of the approximation, the discrete energy decay and the behavior of the solution.Agencia Estatal de Investigación | Ref. PID2019-105118GB-I00Agencia Estatal de Investigación | Ref. PGC2018-096696-B-I0

Numerical analysis of a dual-phase-lag model with microtemperatures

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGIn the last twenty years, the analysis of problems involving dual-phase-lag models has received an increasing attention. In this work, we consider the coupling between one of these models and the microtemperatures effects. In order to overcome the infinite speed paradox, two relaxation parameters are introduced for each evolution equation related to the temperature and the microtemperatures, leading to a system of linear hyperbolic partial differential equations. Its variational formulation is written in terms of the temperature acceleration and the microtemperatures acceleration. An energy decay property is proved. Next, fully discrete approximations are introduced by using the finite element method and the Euler scheme, proving a stability property and a discrete version of the energy decay, obtaining a priori error estimates and performing one- and two-dimensional numerical simulationsConselho Nacional de Desenvolvimento Científico e Tecnológico, Brasil | Ref. 304709 / 2017-4Ministerio de Ciencia, Innovación y Universidades | Ref. PGC2018-096696-B-I00Ministerio de Ciencia, Innovación y Universidades | Ref. PID2019-105118GB-I0

Analysis of a poro-thermo-viscoelastic model of type III

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.Ministerio de Ciencia, Innovación y Universidades | Ref. PGC2018-096696-B-I00Xunta de Galicia | Ref. ED431C2019/2