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16 results on '"Jorge San Martín"'

1. Bloch wave spectral analysis in the class of generalized Hashin–Shtrikman micro-structures

Author

Loredana Bălilescu, Carlos Conca, Tuhin Ghosh, Jorge San Martín, and Muthusamy Vanninathan

Subjects
Applied Mathematics, Modeling and Simulation
Abstract

In this paper, we use spectral methods to introduce Bloch waves for studying the homogenization process in the non-periodic class of generalized Hashin–Shtrikman micro-structures (see Ref. 35), which incorporates both translation and dilation with a family of scales, including one subclass of laminates. We establish the classical homogenization result by providing the spectral representation of the homogenized coefficients. It offers a new lead towards extending the Bloch spectral analysis to general micro-structures, including the class of non-periodic media or the homogenization of Heisenberg operators (see Refs. 8 and 9).

Published
2022
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2. A semilinear system with positivity conditions

Author

Jorge San Martín, Raúl Gormaz, and Carlos Conca

Subjects
Numerical Analysis, Pure mathematics, Matrix (mathematics), Nonlinear system, Control and Optimization, Applied Mathematics, Modeling and Simulation, Linear system, Uniqueness, Lexicographical order, System of linear equations, Mathematics
Abstract

This paper studies a semi-linear system of equations in $$ {\mathbb{R}}^{N} $$ , which comes from a mathematical model for a new tax system proposed in Chile’s 2014 Tax Reform. The system of equations involves a non negative coefficients matrix and simultaneously relates the unknown vector with its positive part, and hence the nonlinear nature of the problem. In addition to find appropriate conditions for the existence and the uniqueness of a solution, in this paper an algorithm is proposed to obtain it by solving at most $$ N $$ linear systems of size $$ N \times N $$ . The proof of the results is based on a monotony method with respect to the usual lexicographic order in $$ {\mathbb{R}}^{N} $$ .

Published
2018
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3. A mathematical basis for the graphene

Author

Carlos Conca, Jorge San Martín, and Viviana Solano

Subjects
Physics, Differential equation, Graphene, Applied Mathematics, Mathematical analysis, law.invention, Computational Mathematics, symbols.namesake, Honeycomb structure, law, Lattice (order), symbols, Hamiltonian (quantum mechanics), Eigenvalues and eigenvectors
Abstract

We present a new basis of representation for the graphene honeycomb structure that facilitates the solution of the eigenvalue problem by reducing it to one dimension. We define spaces in these geometrical basis that allow us to solve the Hamiltonian in the edges of the lattice. We conclude that it is enough to analyze a one-dimensional problem in a set of coupled ordinary second-order differential equations to obtain the behavior of the solutions in the whole graphene structure.

Published
2019
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4. An optimal control approach to ciliary locomotion

Author

Marius Tucsnak, Jorge San Martín, Takéo Takahashi, Departamento de Ingeniería Matemática, Facultad de Ciencias Fisicas y Matemáticas, Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), and ANR-11-BS03-0002,HAMECMOPSYS,Approche Hamiltonienne pour l'analyse et la commande des systèmes multiphysiques à paramètres distribués(2011)

Subjects
0209 industrial biotechnology, Control and Optimization, media_common.quotation_subject, Boundary (topology), 02 engineering and technology, controllability, 01 natural sciences, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], optimal control, symbols.namesake, 020901 industrial engineering & automation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Sensitivity (control systems), 0101 mathematics, Eccentricity (behavior), Astrophysics::Galaxy Astrophysics, Mathematics, media_common, Applied Mathematics, Mathematical analysis, Ode, Scalar (physics), Stokes equations, Reynolds number, Optimal control, 010101 applied mathematics, Controllability, Gegenbauer functions, symbols, ciliates, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

We consider a class of low Reynolds number swimmers, of prolate spheroidal shape, which can be seen as simplified models of ciliated microorganisms. Within this model, the form of the swimmer does not change, the propelling mechanism consisting in tangential displacements of the material points of swimmer's boundary. Using explicit formulas for the solution of the Stokes equations at the exterior of a translating prolate spheroid the governing equations reduce to a system of ODE's with the control acting in some of its coefficients (bilinear control system). The main theoretical result asserts the exact controllability of the prolate spheroidal swimmer. In the same geometrical situation, we consider the optimal control problem of maximizing the efficiency during a stroke and we prove the existence of a maximum. We also provide a method to compute an approximation of the efficiency by using explicit formulas for the Stokes system at the exterior of a prolate spheroid, with some particular tangential velocities at the fluid-solid interface. We analyze the sensitivity of this efficiency with respect to the eccentricity of the considered spheroid and show that for small positive eccentricity, the efficiency of a prolate spheroid is better than the efficiency of a sphere. Finally, we use numerical optimization tools to investigate the dependence of the efficiency on the number of inputs and on the eccentricity of the spheroid. The ``best'' numerical result obtained yields an efficiency of $30.66\%$ with $13$ scalar inputs. In the limiting case of a sphere our best numerically obtained efficiency is of $30.4\%$, whereas the best computed efficiency previously reported in the literature is of $22\%$.

Published
2016
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5. Analysis of a simplified model of rigid structure floating in a viscous fluid

Author

Takéo Takahashi, Jorge San Martín, Marius Tucsnak, Debayan Maity, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Centre de modélisation mathématique (CMM), Universitad de Chile-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-16-CE92-0028,INFIDHEM,Systèmes interconnectés de dimension infinie pour les milieux hétérogènes(2016), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Function space, Differential equation, Applied Mathematics, Mathematical analysis, return to equilibrium, General Engineering, Viscous shallow water equations, fluid-structure interaction, floating structure, Viscous liquid, 01 natural sciences, 010305 fluids & plasmas, 010101 applied mathematics, Viscosity, AMS subject classifications 35Q35 74F10, Flow (mathematics), Linearization, Modeling and Simulation, 0103 physical sciences, Fluid–structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], strong solutions, 0101 mathematics, Shallow water equations, Mathematics
Abstract

International audience; We study the interaction of surface water waves with a floating solid constraint to move only in the vertical direction. The first novelty we bring in is that we propose a new model for this interaction, taking into consideration the viscosity of the fluid. This is done supposing that the flow obeys a shallow water regime (modeled by the viscous Saint-Venant equations in one space dimension) and using a Hamiltonian formalism. Another contribution of this work is establishing the well-posedness of the obtained PDEs/ODEs system in function spaces similar to the standard ones for strong solutions of viscous shallow water equations. Our well-posedness results are local in time for every initial data and global in time if the initial data are close (in appropriate norms) to an equilibrium state. Moreover, we show that the linearization of our system around an equilibrium state can be described, at least for some initial data, by an integro-fractional differential equation related to the classical Cummins equation and which reduces to the Cummins equation when the viscosity vanishes and the fluid is supposed to fill the whole space. Finally, we describe some numerical tests, performed on the original nonlinear system, which illustrate the return to equilibrium and the influence of the viscosity coefficient.

Published
2019
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6. On the reconstruction of obstacles and of rigid bodies immersed in a viscous incompressible fluid

Author

Erica L. Schwindt, Takéo Takahashi, Jorge San Martín, Departamento de Ingeniería Matemática [Santiago] (DIM), Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE), Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), and Université d'Orléans (UO)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Convex hull, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mathematics::Analysis of PDEs, Boundary (topology), Inverse problem, Rigid body, 01 natural sciences, 010101 applied mathematics, Physics::Fluid Dynamics, Nonlinear system, Classical mechanics, complex geometrical solutions, Geometrical inverse problems, fluid-structure interaction, Navier–Stokes system, enclosure method, Obstacle, Fluid–structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Point (geometry), 0101 mathematics, Mathematics
Abstract

We consider the geometrical inverse problem consisting in recovering an unknown obstacle in a viscous incompressible fluid by measurements of the Cauchy force on the exterior boundary. We deal with the case where the fluid equations are the nonstationary Stokes system and using the enclosure method, we can recover the convex hull of the obstacle and the distance from a point to the obstacle. With the same method, we can obtain the same result in the case of a linear fluid-structure system composed by a rigid body and a viscous incompressible fluid. We also tackle the corresponding nonlinear systems: the Navier–Stokes system and a fluid-structure system with free boundary. Using complex spherical waves, we obtain some partial information on the distance from a point to the obstacle.

Published
2017
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7. On the Navier–Stokes system with the Coulomb friction law boundary condition

Author

Takéo Takahashi, Jorge San Martín, Loredana Bălilescu, Departamento de Matemática = Mathematics Department [Florianópolis] (MTM), Centro de Ciências Físicas e Matemáticas [Florianópolis] (CFM), Universidade Federal de Santa Catarina = Federal University of Santa Catarina [Florianópolis] (UFSC)-Universidade Federal de Santa Catarina = Federal University of Santa Catarina [Florianópolis] (UFSC), Department of Mathematics [Pitesti], Faculty of Mathematics, Universitatea din Pitesti [Roumanie] (UPIT)-Universitatea din Pitesti [Roumanie] (UPIT), Center for Mathematical Modeling (CMM), Universidad de Chile = University of Chile [Santiago] (UCHILE), Departamento de Ingeniería Matemática [Santiago] (DIM), Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015), and Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Cauchy stress tensor, Applied Mathematics, General Mathematics, Weak solution, 010102 general mathematics, Mathematical analysis, General Physics and Astronomy, 010103 numerical & computational mathematics, Mixed boundary condition, 16. Peace & justice, 01 natural sciences, Coulomb's law, Physics::Fluid Dynamics, symbols.namesake, Classical mechanics, Law, symbols, No-slip condition, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Cauchy boundary condition, Uniqueness, Boundary value problem, 0101 mathematics, Mathematics
Abstract

International audience; We propose a new model for the motion of a viscous incompressible fluid. More precisely, we consider the Navier–Stokes system with a boundary condition governed by the Coulomb friction law. With this boundary condition, the fluid can slip on the boundary if the tangential component of the stress tensor is too large. We prove the existence and uniqueness of weak solution in the two–dimensional problem and the existence of at least one solution in the three–dimensional case, together with regularity properties and an energy estimate. We also propose a fully discrete scheme of our problem using the characteristic method and we present numerical simulations in two physical examples.

Published
2017
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8. Fluid-rigid structure interaction system with Coulomb's law

Author

Takéo Takahashi, Loredana Bălilescu, Jorge San Martín, Department of Mathematics [Pitesti], Faculty of Mathematics, Universitatea din Pitesti [Roumanie] (UPIT)-Universitatea din Pitesti [Roumanie] (UPIT), Departamento de Matemática = Mathematics Department [Florianópolis] (MTM), Centro de Ciências Físicas e Matemáticas [Florianópolis] (CFM), Universidade Federal de Santa Catarina = Federal University of Santa Catarina [Florianópolis] (UFSC)-Universidade Federal de Santa Catarina = Federal University of Santa Catarina [Florianópolis] (UFSC), Center for Mathematical Modeling (CMM), Universidad de Chile = University of Chile [Santiago] (UCHILE), Departamento de Ingeniería Matemática [Santiago] (DIM), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015), and Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE)

Subjects
Coulomb's law 2010 Mathematics Subject Classification: 35Q30, 01 natural sciences, Coulomb's law, Physics::Fluid Dynamics, Navier-Stokes system, symbols.namesake, 76D03, Neumann boundary condition, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Fluid-structure system, Boundary value problem, 0101 mathematics, Physics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mixed boundary condition, 16. Peace & justice, Robin boundary condition, 76D27, 010101 applied mathematics, Computational Mathematics, Classical mechanics, Dirichlet boundary condition, No-slip condition, symbols, Cauchy boundary condition, Cauchy theory, Analysis
Abstract

International audience; We propose a new model in a fluid-structure system composed by a rigid body and a viscous incompress-ible fluid using a boundary condition based on Coulomb's law. This boundary condition allows the fluid to slip on the boundary if the tangential component of the stress is too large. In the opposite case, we recover the standard Dirichlet boundary condition. The governing equations are the Navier-Stokes system for the fluid and the Newton laws for the body. The corresponding coupled system can be written as a variational inequality. We prove that there exists a weak solution of this system.

Published
2017
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9. Bloch wave homogenization of a non-homogeneous Neumann problem

Author

Jaime H. Ortega, Jorge San Martín, and Loredana Smaranda

Subjects
Applied Mathematics, General Mathematics, Mathematical analysis, General Physics and Astronomy, Mixed boundary condition, Robin boundary condition, Poincaré–Steklov operator, symbols.namesake, Dirichlet boundary condition, Neumann boundary condition, symbols, Cauchy boundary condition, Boundary value problem, Mathematics, Bloch wave
Abstract

In this paper, we use the Bloch wave method to study the asymptotic behavior of the solution of the Laplace equation in a periodically perforated domain, under a non-homogeneous Neumann condition on the boundary of the holes, as the size of the holes goes to zero more rapidly than the domain period. This method allows to prove that, when the hole size exceeds a given threshold, the non-homogeneous boundary condition generates an additional term in the homogenized problem, commonly referred to as “the strange term” in the literature.

Published
2007
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10. A New Mathematical Model for Supercooling

Author

Michel Frémond, Jorge San Martín, and Raúl Gormaz

Subjects
Viscous dissipation, Partial differential equation, Applied Mathematics, Solid-state, Thermodynamics, Condensed Matter::Disordered Systems and Neural Networks, Condensed Matter::Soft Condensed Matter, Liquid state, Regularization (physics), Statistical physics, Supercooling, Galerkin method, Analysis, Mathematics
Abstract

In this article we study supercooling from a macroscopic point of view by modeling the evolution of a supercooled body from its liquid state to its solid state. A first model, which would be expected to have discontinuous solutions, is regularized by introducing an intrinsic viscous dissipation. By applying the classical method of Faedo–Galerkin, this regularized model is shown to have a global smooth solution, which describes the state transition of the supercooled body approximately.

Published
2001
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11. A modified Lagrange-Galerkin method for a fluid-rigid system with discontinuous density

Author

Jean-François Scheid, Loredana Smaranda, Jorge San Martín, Centre de modélisation mathématique (CMM), Universitad de Chile-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Departamento de Ingeniería Matemática [Santiago] (DIM), Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE), and Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Characteristic function (probability theory), Discretization, Applied Mathematics, Numerical analysis, 010102 general mathematics, Mathematical analysis, Material derivative, 010103 numerical & computational mathematics, Weak formulation, 16. Peace & justice, Rigid body, 01 natural sciences, Computational Mathematics, Discrete time and continuous time, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Galerkin method, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; In this paper, we propose a new characteristics method for the discretization of the two dimensional fluid-rigid body problem in the case where the densities of the fluid and the solid are different. The method is based on a global weak formulation involving only terms defined on the whole fluid-rigid domain. To take into account the material derivative, we construct a special characteristic function which maps the approximate rigid body at the discrete time level $t_{k+1}$ into the approximate rigid body at time $t_k$. Convergence results are proved for both semi-discrete and fully-discrete schemes.

Published
2012
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12. A time discretization scheme of a characteristics method for a fluid-rigid system with discontinuous density

Author

Jorge San Martín, Loredana Smaranda, Jean-François Scheid, Centre de modélisation mathématique (CMM), Universitad de Chile-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), Robust control of infinite dimensional systems and applications (CORIDA), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematics [Pitesti], Faculty of Mathematics, Universitatea din Pitesti [Roumanie] (UPIT)-Universitatea din Pitesti [Roumanie] (UPIT), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM)

Subjects
Mathematical optimization, Discretization, Characteristic function (probability theory), Rigidity (psychology), 010103 numerical & computational mathematics, General Medicine, Weak formulation, 01 natural sciences, characteristics method, 010101 applied mathematics, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Discrete time and continuous time, Method of characteristics, fluid-structure interactions, Convergence (routing), Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Navier-Stokes equations, Navier–Stokes equations, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; We propose a new characteristics method for the time discretization of a fluid-rigid system in the case when the densities of the fluid and the solid are different. This method is based on a global weak formulation involving only terms defined on the whole fluid-rigid domain. The main idea is to construct a characteristic function which preserves the rigidity of the solid at the discrete time levels. A convergence result for this semi-discrete scheme is then given.

Published
2010
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13. Asymptotics for eigenvalues of the laplacian in higher dimensional periodically perforated domains

Author

Jorge San Martín and Loredana Smaranda

Subjects
Applied Mathematics, General Mathematics, Mathematical analysis, General Physics and Astronomy, Homogenization (chemistry), symbols.namesake, Homogeneous, Dirichlet boundary condition, symbols, Asymptotic expansion, Laplace operator, Eigenvalues and eigenvectors, Mathematics, Bloch wave
Abstract

This paper considers the periodic spectral problem associated with the Laplace operator written in \({\mathbb{R}^N}\) (N = 3, 4, 5) periodically perforated by balls, and with homogeneous Dirichlet condition on the boundary of holes. We give an asymptotic expansion for all simple eigenvalues as the size of holes goes to zero. As an application of this result, we use Bloch waves to find the classical strange term in homogenization theory, as the size of holes goes to zero faster than the microstructure period.

Published
2010

14. Optimal bounds on dispersion coefficient in one-dimensional periodic media

Author

M. Vanninathan, Carlos Conca, Jorge San Martín, and Loredana Smaranda

Subjects
Applied Mathematics, Modeling and Simulation, Mathematical analysis, Volume fraction, Microstructure, Dispersion coefficient, Homogenization (chemistry), Mathematics
Abstract

In this paper, we consider the macroscopic quantity, namely the dispersion tensor associated with a periodic structure in one dimension (see Refs. 5 and 7). We describe the set in which this quantity lies, as the microstructure varies preserving the volume fraction.

Published
2009

15. A Control Theoretic Approach to the Swimming of Microscopic Organisms

Author

Marius Tucsnak, Takéo Takahashi, Jorge San Martín, Takahashi, Takéo, Centre de Modélisation Mathématique / Centro de Modelamiento Matemático (CMM), Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Centre de modélisation mathématique (CMM), and Universitad de Chile-Centre National de la Recherche Scientifique (CNRS)

Subjects
Applied Mathematics, 010102 general mathematics, Relative velocity, Boundary (topology), Reynolds number, Thrust, Fluid mechanics, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Rigid body, 01 natural sciences, 010101 applied mathematics, symbols.namesake, Classical mechanics, Position (vector), symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Vector field, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics
Abstract

Papier accepté pour la publication dans Quarterly of Applied Mathematics; International audience; In this paper, we give a control theoretic approach to the slow self-propelled motion of a rigid body in a viscous fluid. The control of the system is the relative velocity of the fluid with respect to the solid on the boundary of the rigid body (the thrust). Our main results show that, there exists a large class of finite dimensional input spaces for which the system is exactly controllable, i.e., one can find controls steering the rigid body in any final position with a prescribed velocity field. The equations we use are motivated by models of swimming of micro-organisms like cilia. We give a control theoretic interpretation of the swimming mechanism of these organisms, which takes place at very low Reynolds numbers. Our aim is to give a control theoretic interpretation of the swimming mechanism of micro-organisms (like ciliata) which is one of the fascinating problems in fluid mechanics.

Published
2006

16. Convergence of the Lagrange-Galerkin method for the Equations Modelling the Motion of a Fluid-Rigid System

Author

Jean-François Scheid, Takéo Takahashi, Marius Tucsnak, Jorge San Martín, Departamento de Ingeniería Matemática [Santiago] (DIM), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE)

Subjects
finite element method, 35Q30, 76D05, 65M12, 76M10, fluid-structure interaction, 010103 numerical & computational mathematics, 01 natural sciences, Physics::Fluid Dynamics, symbols.namesake, Multigrid method, Simultaneous equations, Collocation method, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, Numerical Analysis, Independent equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Lagrange-Galerkin method, 16. Peace & justice, Euler equations, Computational Mathematics, Nonlinear system, incompressible Navier-Stokes equations, symbols, Differential algebraic equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Numerical partial differential equations
Abstract

In this paper, we consider a Lagrange--Galerkin scheme to approximate a two-dimensional fluid-rigid body problem. The equations of the system are the Navier--Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the rigid body. In this problem, the equations of the fluid are written in a domain whose variation is one of the unknowns. We introduce a numerical method based on the use of characteristics and on finite elements with a fixed mesh. Our main result asserts the convergence of this scheme.

Published
2005
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