Artículos (Ecuaciones Diferenciales y Análisis Numérico)

URI permanente para esta colecciónhttps://hdl.handle.net/11441/10834

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A 3D isothermal model for nematic liquid crystals 1 with delay terms
(2022-08-01) Caraballo Garrido, Tomás; Cavaterra, Cecilia; Universidad de Sevilla. Departamento de Ecuaciones diferenciales y Análisis numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this paper we consider a model describing the evolution of a nematic liquid crystal flow with delay external forces. We analyze the evolution of the velocity fi eld u which is ruled by the 3D incompressible Navier-Stokes system containing a delay term and with a stress tensor expressing the coupling between the transport and the induced terms. The dynamics of the director eld d is described by a modifi ed Allen-Cahn equation with a suitable penalization of the physical constraint jdj = 1. We prove the existence of global in time weak solutions under appropriate assumptions, which in some cases requires the delay term to be small with respect to the viscosity parameter.
A Bochev-Dohrmann-Gunzburger stabilization method for the primitive equations of the ocean
(Elsevier, 2013-04) Chacón Rebollo, Tomás; Gómez Mármol, María Macarena; Sánchez Muñoz, Isabel María; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Ciencia e Innovación (MICIN). España; European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)
We introduce a low-order stabilized discretization of the Primitive Equations of the Ocean, with a highly reduced computational complexity. We prove stability through a specific inf-sup condition, and weak convergence to a weak solution. We also perform some numerical test for relevant flows.
A characterization result for the existence of a two-phase material minimizing the first eigenvalue
(Elsevier, 2016-09-19) Casado Díaz, Juan; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM309: Control y Homogeneización de Ecuaciones en Derivadas Parciales
Given two isotropic homogeneous materials represented by two constants 0 <α< | |, we consider here the problem consisting in finding a mixture of these materials αχω + β(1 − χω), ω ⊂ RN measurable, with |ω| ≤ κ, such that the first eigenvalue of the operator u ∈ H1 0 ( ) → −divαχω + β(1 − χω) ∇u reaches the minimum value. In a recent paper, [6], we have proved that this problem has not solution in general. On the other hand, it was proved in [1] that it has solution if is a ball. Here, we show the following reciprocate result: If ⊂ RN is smooth, simply connected and has connected boundary, then the problem has a solution if and only if is a ball.
A chemorepulsion model with superlinear production: analysis of the continuous problem and two approximately positive and energy-stable schemes
(Springer, 2021-12-10) Guillén González, Francisco Manuel; Rodríguez Bellido, María Ángeles; Rueda Gómez, Diego Armando; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
We consider the following repulsive-productive chemotaxis model: find 0, the cell density, and 0, the chemical concentration, satisfying 0 in 0 in 0 (1) with 1 2 , a bounded domain ( 1 2 3), endowed with non-flux boundary conditions. By using a regularization technique, we prove the existence of global in time weak solutions of (1) which is regular and unique for 1 2. Moreover, we propose two fully discrete Finite Element (FE) nonlinear schemes, the first one defined in the variables under structured meshes, and the second one by using the auxiliary variable and defined in general meshes. We prove some unconditional properties for both schemes, such as mass-conservation, solvability, energy-stability and approximated positivity. Finally, we compare the behavior of these schemes with respect to the classical FE backward Euler scheme throughout several numerical simulations and give some conclusions.
A common framework for the robust design of tuned mass damper techniques to mitigate pedestrian-induced vibrations in lively footbridges
(Elsevier Ltd, 2021) Jiménez Alonso, Javier Fernando; Soria, José M.; Díaz, Iván M.; Guillén González, Francisco Manuel; Universidad de Sevilla. Departamento de Mecánica de Medios Continuos y Teoría de Estructuras; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)
The dynamic response of modern slender footbridges is usually sensitive to both the pedestrian actions and the uncertainties associated with their inherent structural behavior. Thus, tuned mass dampers have been widely integrated in the design of these structures to guarantee the fulfillment of the vibration serviceability limit state during their overall life cycle. Three different techniques of tuned mass dampers (active, semi-active and passive) are usually considered for this purpose. Although there are algorithms for the robust design of each particular technique, however, this specificity makes difficult the implementation of all these techniques in practical en gineering applications. Herein, the motion-based design method under uncertainty conditions is proposed and further implemented to create a common framework for the robust design of all these techniques when they are employed to mitigate pedestrian-induced vibrations in slender footbridges. According to this method, the design problem may be transformed into the combination of two sequential sub-problems: (i) a reliability multi objective optimization sub-problem; and (ii) a decision-making sub-problem. Subsequently, the performance of this proposal has been validated through a numerical case study in which the dynamic response of a steel footbridge has been controlled by three different tuned mass damper techniques designed according to the proposed common framework.
A comparison between random and stochastic modeling for a SIR model
(American Institute of Mathematical Sciences, 2017-01) Caraballo Garrido, Tomás; Colucci, Renato; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this article, a random and a stochastic version of a SIR nonautonomous model previously introduced in P. E. Kloeden and V. S. Kozyakin, The dynamics of epidemiological systems with nonautonomous and random coefficients, MESA: Mathematics in Engineering, Science and Aerospace, vol. 2, no. 2 (2011).is considered. In particular, the existence of a random attractor is proved for the random model and the persistence of the disease is analyzed as well. In the stochastic case, we consider some environmental effect on the model, in fact, we assume that one of the coefficients of the system is affected by some stochastic perturbation, and analyze the asymptotic behavior of the solutions. The paper is concluded with a comparison between the two different modeling strategies.
A Comparison Between Two Theories for Multi-Valued Semiflows and Their Asymptotic Behaviour
(2003) Caraballo Garrido, Tomás; Marín Rubio, Pedro; Robinson, James C.; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico
This paper presents a comparison between two abstract frameworks in which one can treat multi-valued semiflows and their asymptotic behaviour. We compare the theory developed by Ball [5] to treat equations whose solutions may not be unique, and that due to Melnik & Valero [25] tailored more for differential inclusions. Although they deal with different problems, the main ideas seem quite similar. We study their relationship in detail and point out some essential technical problems in trying to apply Ball’s theory to differential inclusions.
A comparison of three turbulence models with an application to the West Pacific Warm Pool
(2007) Bennis, Anne-Claire; Gómez Mármol, María Macarena; Lewandowski, Roger; Chacón Rebollo, Tomás; Brossier, Françoise; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Educación y Ciencia (MEC). España
In this work, we compare three turbulence models used to parameterize the oceanic boundary layer. These three models depend on the bulk Richardson number, which is coherent with the studied region, the West Pacific Warm Pool, because of the large mean shear associated with the equatorial undercurrent. One of these models, called R224, is new and the others are Pacanowski and Philander’s model (R213 model) and Gent’s model (R23 model). The numerical implementation is based on a non-conservative numerical scheme. The following (three criteria) are used to compare the models: the surface current intensity, the pycnocline’s form and the mixed layer depth. We initialize the code with realistic velocity and density profiles thanks the TOGA-TAO array (McPhaden, 1995, [21] M. McPhaden, The tropical atmosphere ocean (tao) array is completed, Bull. Am. Meteorol. Soc, 76 (1995), pp. 739–741). In case of static instability zone on the initial density profile, only the R224 model gives realistic results. Afterwards, we study a mixed layer induced by the wind stress. In this case, the R224 results and the Pacanowski and Philander’s results are similar. Furthermore, we simulate a long time case. We obtain a linear solution for all models that is in agreement with Bennis and al [1] A. C. Bennis, T. C. Rebollo, M. G. Marmol, and R. Lewandowski, Stability of some turbulent vertical models for the ocean mixing boundary layer, Applied Mathematical Letters, To Appear (2007).
A corrector for a wave problem with periodic coefficients in a 1D bounded domain
(EDP Sciences, 2015) Casado Díaz, Juan; Couce Calvo, Julio; Maestre Caballero, Faustino; Martín Gómez, José Domingo; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Economía y Competitividad (MINECO). España
We consider a wave problem posed in a bounded open interval of R, where the coefficients, the initial conditions and the right-hand side are highly oscillating, periodic in the space variable and almost periodic in the time one. Our purpose is to find not only the corresponding limit equation but a corrector, i.e. a strong approximation in the H1 topology, which for the wave equation is known to be non-local. In a previous paper we have studied this problem in the whole RN, here we consider the case of a bounded domain in dimension one. Thus the novelty in this paper is the analysis of the boundary conditions.
A corrector for the Sverdrup solution for a domain with islands
(Taylor & Francis, 2004-03) Bresch, Didier; Guillén González, Francisco Manuel; Rodríguez Bellido, María Ángeles; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Comisión Interministerial de Ciencia y Tecnología (CICYT). España
In this paper we look at the influence of the Coriolis force on the quasi-geostrophic equations on a domain with islands. We prove that asymptotically we obtain the solution of the Sverdrup equation with homogeneous Dirichlet conditions on the inward boundary plus a corrector function which takes into account the presence of the islands. This work is motivated by the fact that in oceanography most of the surfaces are not simply connected. This is the case for example for the North Pacific with the Japanese islands. At our knowledge, in all the previous mathematical works, just simply connected domains have been considered. Finally we will give some simple numerical simulations related to the Stommel model to see the importance of the corrector.
A cure for instabilities due to advection-dominance in POD solution to advection-diffusion-reaction equations
(Elsevier, 2021-01-15) Azaïez, Mejdi; Chacón Rebollo, Tomás; Rubino, Samuele; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM120: Modelado Matemático y Simulación de Sistemas Medioambientales
In this paper, we propose to improve the stabilized POD-ROM introduced in [48] to deal with the numerical simulation of advection-dominated advection-diffusion-reaction equations. In particular, we propose a three-stage stabilizing strategy that will be very useful when considering very low diffusion coefficients, i.e. in the strongly advection-dominated regime. This approach mainly consists in three ingredients: (1) the addition of a “streamline diffusion” stabilization term to the governing projected equations, (2) the modification of the correlation matrix defining the POD modes associated to the advection stabilization term, and (3) an a-posteriori stabilization scheme. Numerical studies are performed to discuss the accuracy and performance of the new method in handling strongly advection-dominated cases.
A decomposition result for the pressure of a fluid in a thin domain and extensions to elasticity problems
(2020-01) Casado Díaz, Juan; Luna Laynez, Manuel; Suárez Grau, Francisco Javier; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Economía y Competitividad (MINECO). España
In order to study the asymptotic behavior of a fluid in a domain of small thickness $\ep$, it is common to use that the norm of the pressure $p_\ep$ in $L^q$, $q>1$, is smaller than $C\|\nabla p_\ep\|_{W^{-1,q}}/\ep$. Our purpose in the present paper is to improve this estimate by showing that in fact $p_\ep$ can be decomposed as the sum of two terms: the first one is of order $1/\ep$ with respect to $\nabla p_\ep$ but it belongs to the Sobolev space $W^{1,q}$ and not only to $L^q$; the second one only belongs to $L^q$ but it is of order one with respect to $\nabla p_\ep$. This result also allows us to improve the classical estimate for Korn's constant in an elastic thin domain providing a decomposition of the deformation which contains terms with a stronger regularity. The advantage of these expansions is that they simplify the study of the asymptotic behavior of continuum mechanics problems in thin domains since they give an additional compactness. As examples we provide two applications in fluid mechanics and linear elasticity.
A density result for the variation of a material with respect to small inclusions
(Elsevier, 2006-03-01) Casado Díaz, Juan; Couce Calvo, Julio; Martín Gómez, José Domingo; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM309: Control y Homogeneización de Ecuaciones en Derivadas Parciales
We consider the family of materials obtained, via homogenization, by replacing a small portion, of size ɛ, of a fixed material by other materials. In a previous paper we have obtained a subset of the set of ‘derivatives’ of this family with respect to ɛ in ɛ . In the present Note we prove that this set is, in fact, dense. This result can be applied, for example, to obtain optimality conditions for composite materials.
A density result for the variation of a material with respect to small inclusions
(Elsevier, 2006-03-01) Casado Díaz, Juan; Couce Calvo, Julio; Martín Gómez, José Domingo; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM309: Control y Homogeneización de Ecuaciones en Derivadas Parciales
We consider the family of materials obtained, via homogenization, by replacing a small portion, of size ɛ, of a fixed material by other materials. In a previous paper we have obtained a subset of the set of ‘derivatives’ of this family with respect to ɛ in ɛ . In the present Note we prove that this set is, in fact, dense. This result can be applied, for example, to obtain optimality conditions for composite materials.
A family of stable numerical solvers for the shallow water equations with source terms
(Elsevier, 2003) Chacón Rebollo, Tomás; Domínguez Delgado, Antonio; Fernández Nieto, Enrique Domingo; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII)
In this work we introduce a multiparametric family of stable and accurate numerical schemes for 1D shallow water equations.These schemes are based upon the splitting of the discretization of the source term into centered and decentered parts.These schemes are specifically designed to fulfill the enhanced consistency condition of Berm udez and V azquez, necessary to obtain accurate solutions when source terms arise.Our general family of schemes contains as particular cases the extensions already known of Roe and Van Leer schemes, and as new contributions, extensions of Steger–Warming, Vijayasundaram, Lax–Friedrichs and Lax–Wendroff schemes with and without flux-limiters.We include some meaningful numerical tests, which show the good stability and consistency properties of several of the new methods proposed.We also include a linear stability analysis that sets natural sufficient conditions of stability for our general methods.
A FETI method with a mesh independent condition number for the iteration matrix
(Elsevier, 2008-02-15) Bernardi, Christine; Chacón Rebollo, Tomás; Chacón Vera, Eliseo; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Educación y Ciencia (MEC). España; Universidad de Sevilla. FQM120: Modelado Matemático y Simulación de Sistemas Medioambientales
We introduce a framework for FETI methods using ideas from the decomposition via Lagrange multipliers of H1 0 (Ω) derived by Raviart-Thomas [22] P.-A. Raviart, J.-M. Thomas, Primal Hybrid Finite Element Metho and complemented with the detailed work on polygonal domains developed by Grisvard [17] P. Grisvard, Singularities in Boundary value problems. Recherches en Mathématiques Appliquées, 22. Masson, 1992.. We compute the action of the Lagrange multipliers using the natural H 1/2 00 scalar product, therefore no consistency error appears. As a byproduct, we obtain that the condition number for the iteration matrix is independent of the mesh size and there is no need for preconditioning. This result improves the standard asymptotic bound for this condition number shown by Mandel-Tezaur in [19] J. Mandel, R. Tezaur, Convergence of a substructuring method with Lagrange multipliers. Numer. Math., 73 (1996), 473–487. Numerical results that confirm our theoretical analysis are presented.
A free boundary tumor model with time dependent nutritional supply
(Elsevier, 2020-06) Sun, Wenlong; Caraballo Garrido, Tomás; Han, Xiaoying; Kloeden, Peter E.; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
A non-autonomous free boundary model for tumor growth is studied. The model consists of a nonlinear reaction diffusion equation describing the distribution of vital nutrients in the tumor and a nonlinear integro-differential equation describing the evolution of the tumor size. First the global existence and uniqueness of a transient solution is established under some general conditions. Then with additional regularity assumptions on the consumption and proliferation rates, the existence and uniqueness of steady-state solutions is obtained. Furthermore the convergence of the transient solutions toward the steady-state solution is verified. Finally the long time behavior of the solutions is investigated by transforming the time-dependent domain to a fixed domain.
A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems
(Ele-Math, 2009-08) Ammar-Khodja, Farid; Benabdallah, Assia; Dupaix, Cédric; González Burgos, Manuel; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Agence Nationale de la Recherche. France; Dirección General de Enseñanza Superior. España; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software
In this paper we present a generalization of the Kalman rank condition for linear ordinary differential systems to the case of systems of n coupled parabolic equations (posed in the time interval (0,T) with T > 0) where the coupling matrices A and B depend on the time variable t . To be precise, we will prove that the Kalman rank condition rank [A|B](t0) = n, with t0 ∈ [0,T], is a sufficient condition (but not necessary) for obtaining the exact controllability to the trajectories of the considered parabolic system. In the case of analytic matrices A and B (and, in particular, constant matrices), we will see that the Kalman rank condition characterizes the controllability properties of the system. When the matrices A and B are constant and condition rank [A|B] = n holds, we will be able to state a Carleman inequality for the corresponding adjoint problem.
A generalized Reynolds equation for micropolar flows past a ribbed surface with nonzero boundary conditions
(EDP Sciences, 2022-06) Bonnivard, Matthieu; Pazanin, Igor; Suárez Grau, Francisco Javier; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Agence Nationale de la Recherche. France; Croatian Science Foundation, Croatia; Ministerio de Economía y Competitividad (MINECO). España
Inspired by the lubrication framework, in this paper we consider a micropolar fluid flow through a rough thin domain, whose thickness is considered as the small parameter $\varepsilon$ while the roughness at the bottom is defined by a periodical function with period of order $\varepsilon^\ell$ and amplitude $\varepsilon^\delta$, with $\delta> \ell > 1$. Assuming nonzero boundary conditions on the rough bottom and by means of a version of the unfolding method, we identify a critical case $\delta={3\over 2}\ell-{1\over 2}$ and obtain three macroscopic models coupling the effects of the rough bottom and the nonzero boundary conditions. In every case we provide the corresponding micropolar Reynolds equation. We apply these results to carry out a numerical study of a model of squeeze-film bearing lubricated with a micropolar fluid. Our simulations reveal the impact of the roughness coupled with the nonzero boundary conditions on the performance of the bearing, and suggest that the introduction of a rough geometry may contribute to enhancing the mechanical properties of the device.
A geometric inverse problem for the Boussinesq system
(American Institute of Mathematical Sciences, 2006-11) Doubova Krasotchenko, Anna; Fernández Cara, Enrique; González Burgos, Manuel; Ortega Palma, Jaime Humberto; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec. diferenciales, Simulación Num.y Desarrollo Software
In this work we present some results for the inverse problem of the identification of a single rigid body immersed in a fluid governed by the stationary Boussinesq equations. First, we establish a uniqueness result. Then, we show the way the observation depends on perturbations of the rigid body and we deduce some consequences. Finally, we present a new method for the partial identification of the body assuming that it can be deformed only through fields that, in some sense, are finite dimensional. In the proofs, we use various techniques, related to Carleman estimates, differentiation with respect to domains, data assimilation and controllability of PDEs.

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