Uniqueness for a high order ill posed problem

In this work, we study a high order derivative in time problem. First, we show that there exists a sequence of elements of the spectrum which tends to infinity and therefore, it is ill posed. Then, we prove the uniqueness of solutions for this problem by adapting the logarithmic arguments to this situation. Finally, the results are applied to the backward in time problem for the generalized linear Burgers’ fluid, a couple of heat conduction problems and a viscoelastic model.Agencia Estatal de Investigación | Ref. PGC2018-096696-B-I00Agencia Estatal de Investigación | Ref. PID2019-105118GB-I0

Fast spatial behavior in higher order in time equations and systems

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGIn this work, we consider the spatial decay for high-order parabolic (and combined with a hyperbolic) equation in a semi-infinite cylinder. We prove a Phragmén-Lindelöf alternative function and, by means of some appropriate inequalities, we show that the decay is of the type of the square of the distance to the bounded end face of the cylinder. The thermoelastic case is also considered when the heat conduction is modeled using a high-order parabolic equation. Though the arguments are similar to others usually applied, we obtain new relevant results by selecting appropriate functions never considered before.Agencia Estatal de Investigación | Ref. PGC2018-096696-B-I00Agencia Estatal de Investigación | Ref. PID2019-105118GB-I0

On the time decay for an elastic problem with three porous structures

In this paper, we study the three-dimensional porous elastic problem in the case that three dissipative mechanisms act on the three porosity structures (one in each component). It is important to remark that we consider the case when the material is not centrosymmetric, and therefore, some coupling, not previously considered in the literature concerning the time decay of solutions in porous elasticity, can appear in the system of field equations. The new couplings provided in this situation show a strong relationship between the elastic and the porous components of the material. In this situation, we obtain an existence and uniqueness result for the solutions to the problem using the Lumer-Phillips corollary to the Hille-Yosida theorem. Later, assuming a certain condition determining a “very strong” coupling between the material components, we can use the well-known arguments for dissipative semigroups to prove the exponential stability of the solutions to the problem. It is worth emphasizing that the proposed condition allows bringing the decay of the dissipative porous structure of the problem to the macroscopic elastic structurePeer ReviewedPostprint (published version

On the hyperbolic thermoelasticity with several dissipation mechanisms

In this work, we study a two-dimensional problem involving a thermoelastic body with four dissipative mechanisms. The well-known theory proposed by Lord and Shulman is used. The existence and uniqueness of solution is proved by using theory of linear semigroups. Then, introducing some assumptions of the coupling coefficients, we prove that the energy decay is exponential. An extension to the theory provided by Green and Lindsay is briefly presented and to the three-dimensional case is also commentedPeer ReviewedPostprint (published version

On the instability for an incremental problem in elastodynamics

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGIn this short note, we consider some issues regarding the instability of some elastodynamical problems when the elasticity tensor is not positive definite. By using the so-called logarithmic convexity argument, we prove the instability of solutions when the time derivative of the elasticity tensor is semi-definite negative or it satisfies another restriction on the coefficients. The uniqueness of the solution is also concluded. Finally, a simple one-dimensional example is provided to demonstrate the numerical behaviour of the instability.Ministerio 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

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

Método de exploración secuencial cuasi equilibrado en muestreo en poblaciones finitas

En este trabajo se propone un procedimiento de muestreo de tipo secuencial que proporciona probabilidades de inclusión de primer orden proporcionales al tamaño, sobre bloques de la población, siendo las de segundo orden nulas para pares de elementos similares, lo que se traduce en una disminución del error de muestreo. Para la estimación de la varianza se usa un esquema de replicación que puede ser aplicado en paralelo, bastando una única exploración de la población. Se realiza un estudio computacional comparativo con procedimientos existentes de tipo IIPS, tanto de naturaieza secuencial como no secuencial, mediante estimaciones repetidas sobre poblaciones construidas por simulación.In this paper we propose a sampling sequential procedure with first-order inclusion probabilities wich are proportiona! to size over subpopulations from the entire popuiation, and second-order inclusion probabilities equal to zero for pairs of similar units. We show that this method reduces the sampling error with respect to similar methodolagies. A replication method is used to estimate the variance. Its main advantage is that all the sampling replicatíons are obtained with only one sequential exploration of the population. The relative performance of this method, in relation with another unequal probability sampling schemes, either sequential-type or not, is studied. For that, using simulated populations, the relative errors are calculated under different sampting schemes

n2 of dissipative couplings are sufficient to guarantee the exponential decay in elasticity

In this paper, we prove that the solutions to the problem determined by an elastic material with n2 coupling dissipative mechanisms decay in an exponential way for every (bounded) geometry of the body, where n is the dimension of the domain, and whenever the coupling coefficients satisfy a suitable condition. We also give several examples where the solutions do not decay when the rank of the matrix of the coupling mechanisms is less than n2 (2 in dimension 2 and 6 in dimension 3)Peer ReviewedPostprint (published version