Non-linear fluid-structure
interaction. In this research topic, on one
hand, we study the convergence of numerical schemes in
fluid-structure interaction theory. We focus on a numerical method for the time disctretization of a boundary value problem that
models the self-propelled motion of a deformable solid in a
viscous incompressible fluid. The
governing equations consist of the Navier-Stokes equations for
the fluid, coupled to Newton's laws for the solid.
The numerical method we propose is based on a global weak
formulation, where the nonlinear term in the Navier-Stokes model
is discretized using the characteristic function. Since the
formulation is global in space, this characteristic function is
extended in an appropriated manner inside of the solid, taking
into account its deformation. We prove the
stability and the convergence of the numerical scheme.
These studies were realized in collaboration with
the Chilean researcher Jorge San Martin from University of Chile and the French
researcher Jean-Francois
Scheid from Institute Elie Cartan, University of
Lorraine. In conclusion, concerning numerical methods in
fluid-structure interaction problems were obtained relevant scientific results which have produced
three
scientific papers, one published to an ISI journal in applied mathematics, the second one will soon appear in an
international Proceedings and the third one will be submitted for publication as soon as possible to important ISI journal:
J. San Martin, J.-F. Scheid,
L. Smaranda, The Lagrange-Galerkin method
in fluid-structure interaction problems, Boundary Value Problems (2013), DOI: 10.1186/10.1186/1687-2770-2013-246.
L. Balilescu, J. San Martin, J.-F. Scheid, Convergence of a discretization
scheme for the motion of a self-propelled deformable
structure in a fluid, to appear in International
Conference of Woman Mathematicians Proceedings (2014).
L. Balilescu, J. San Martin, J.-F. Scheid, Convergence of the Lagrange-Galerkin method for the equations modelling of fish-like swimming, to be submitted (2014).
On the other hand, we also were
interested in the existence and uniqueness of solution
for fluid-structure interaction problems. We are
interested in the case when the boundary condition is
governed by the Coulomb friction law. The
classical results of existence and uniqueness of solution of Navier-Stokes and fluid-structure interaction problems with Dirichlet
or Neumann boundary condition can be found in the
literature in many publications. In one of these
publications, the authors proved that two rigid solids
can not collide if they are surrounded by a viscous
incompressible fluid. In order to get a more realistic
model, we think that the Coulomb friction law is a natural condition, which includes
non-slip boundary condition whereas the tangential stress is bounded, otherwise the sliding motion occurs.
These studies were realized in collaboration with the
Chilean researcher Jorge San Martin from University of
Chile and the French researcher Takeo Takahashi from Institute Elie Cartan,
University of Lorraine. In conclusion, concerning the
existence and uniqueness of solutions in fluid-structure
problems were obtained
relevant scientific results which have produced five scientific papers, one published to an ISI
journal in applied mathematics, the second one will soon
appear in an international Proceedings, the third one
will be submitted for publication till the end of
October 2014 and two others papers will be submitted for
publication as soon as possible to important ISI journals:
L. Balilescu, J. San Martin, T. Takahashi, On the Navier-Stokes equation with Coulomb friction law boundary condition,
to be submitted till the end of October 2014 in Proceedings of The
thirteenth International Conference on Integral Methods in Science and Engineering, Springer (2014).
L. Balilescu, J. San Martin, T. Takahashi,
Fluid-structure interaction system with Coulomb law, to be submitted (2014).
Talks of the project manager
July 22, 2014 -
Invited talk with the title
Burnett coefficients and laminates
at Minisymposium Asymptotic analysis: homogenization and
thin structures at
The thirteenth International Conferenece on Integral
Methods in Science and Engineering,
Karlsruhe Institute of Technology, Karlsruhe, Germania.
August 12, 2014 - Poster with the tiltle
Convergence of a discretization scheme for the
motion of a self-propelled deformable structure in a fluid to
International Congress
of Woman Mathematicians (ICWM),
Seoul, South Korea.
August 16, 2014 - Poster with the tiltle
Numerical analysis for the motion of a selfpropelled
deformable structure in a fluid to
International Congress
of Mathematicians (ICM),
Seoul, South Korea.
August 29, 2014 - Invited talk with the tiltle
Burnett coefficients and laminates to Special Session "Mecanique",
the 12e Colloque Franco-
Roumain de Mathematiques Appliquees, University of Lyon, Lyon, France.
December 12-13, 2014 - Plenary talk with the tiltle
Burnett coefficients and laminates to Conca60, Basque Center
for Applied Mathematics, Bilbao, Spain.
Visiting research positions
June 18th-July 19, 2014 -
Visiting research position in Centro de Modelamiento
Matematico, Universidad de Chile, Santiago, CHILE.
October 15th-December 9th, 2014 -
Visiting research position in Centro de Ciencias Fisicas e Matematicas, Universidade Federal de Santa Catarina,
Florianopolis, BRAZILIA.
2013
Papers
Numerical analysis in non-linear fluid-structure
interactions. In this research topic, we study
the convergence of some numerical schemes in
fluid-structure interaction theory. First of all, we generalize our previous
results recently published in Numerische Mathematik (2012) of the equations modelling the motion of a rigid
solid immersed into a viscous incompressible fluid. Precisely, we focus on a numerical method for the time
disctretization of a boundary value problem that
models the self-propelled motion of a deformable solid in a
viscous incompressible fluid. The
governing equations consist of the Navier–Stokes equations for
the fluid, coupled to Newton's laws for the solid.
The numerical method we propose is based on a global weak
formulation, where the nonlinear term in the Navier-Stokes model
is discretized using the characteristic function. Since the
formulation is global in space, this characteristic function is
extended in an appropriated manner inside of the solid, taking
into account its deformation. We prove the
stability and the convergence of the numerical scheme.
These studies were realized in collaboration with
the Chilean researcher Jorge San Martín from University of Chile and the French
researcher Jean-François
Scheid from Institute Elie Cartan, University of
Lorraine.
In conclusion, in this research topic were obtained
relevant scientific results which have produced a scientific paper published in an ISI
journal in applied mathematics and another paper will be submitted as soon as possible to an important ISI journal:
J. San Martín, J.-F. Scheid,
L. Smaranda, The Lagrange-Galerkin method
in fluid-structure interaction problems, Boundary Value Problems 2013, 2013:246, doi:10.1186/1687-2770-2013-246.
J. San Martín, J.-F. Scheid,
L. Smaranda, Convergence of the Lagrange-Galerkin method for the equations modelling of fish-like swimming, to be submitted (2014).
Talks of the project manager
May 10, 2013 -
Invited talk with the title
Numerical analysis in fluid-structure
interaction problems
at
Workshop for Young Researchers in Mathematics,
Ovidius University of Constanta, Constanta, Romania.
June 27, 2013 - Invited talk with the
tiltle
On numerical discretization for the motion of a self-propelled deformable structure in a viscous incompressible fluid to
AMS Special Session on "Mathematical Models in Materials Science and Engineering", the Joint International Meeting of the AMS and the Romanian Mathematical Society,
Alba Iulia, Romania.
August 9, 2013 - Invited talk with the tiltle
Convergence of the Lagrange-Galerkin method for fluid-structure interaction problems to Special Session "PDE
and Incompressible Fluid Flow ",
the Mathematical Congress of the Americas, Guanajuato, Mexic.
August 27, 2013 - Talk with the tiltle
Bounds on dispersion tensor in periodic media to International Conference on Applied Mathematics, Modeling and Computational Science, Wilfrid Laurier University, Waterloo, Ontario, Canada.
August 27, 2013 - Talk with the tiltle
Convergence of the Lagrange-Galerkin method for the equations modelling of fish-like swimming to International Conference on Applied Mathematics, Modeling and Computational Science, Wilfrid Laurier University, Waterloo, Ontario, Canada.
September 27, 2013 - Invited talk with the tiltle
Convergence of the Lagrange-Galerkin method for fluid-structure interaction problems to
Scientific Seminar "Caleta Numerica", Institute of Mathematics, Catholic University of Valparaiso, Chile.
Visiting research positions
August 12-25, 2013 -
Visiting research position in Centro de Modelamiento
Matemático, Universidad de Chile.
September 02-30, 2013 -
Visiting research position in Centro de Modelamiento
Matemático, Universidad de Chile.
November 10, 2013-December 01, 2013 -
Visiting research position in Laboratoire des Sciences des Proceds et Materiaux, Universite Paris 13.
2012
Papers
Numerical analysis in non-linear fluid-structure
interactions. In this research topic, we study
the convergence of some numerical schemes in
fluid-structure interaction theory. Precisely, we
have finished our problem in which we discretize in time
and space the equations modelling the motion of a rigid
solid assumed to be a ball of radius 1 immersed into a
viscous incompressible fluid. Moreover, we continue our research program with some generalizations of the previous problem.
We next focus on a numerical method for the
disctretization in time variable of an initial and boundary value problem that
models the self-propelled motion of one deformable solid in a
bidimensional viscous incompressible fluid. In the model, we
suppose that the solid is subjected to a known deformation field
representing the action of the aquatic organism muscles. The
governing equations consist of the Navier–Stokes equations for
the fluid, coupled to Newton's laws for the solid.
The numerical method we propose is based on a global weak
formulation, where the nonlinear term in the Navier-Stokes model
is discretized using the characteristic function. Since the
formulation is global in space, this characteristic function is
extended in an appropriated manner inside of the creature, taking
into account its deformation. We prove the
stability and the convergence of the numerical scheme. We think
that our numerical method is consistent enough with the motion of
the creature and for this reason, the disctretization in space
variable should be successfully implemented using different
techniques as for instance, finite element method with fixed or
moving mesh or finite volume method.
These studies were realized in collaboration with
the Chilean researcher Jorge San Martín from University of Chile and the French
researcher Jean-François
Scheid from Institute Elie Cartan, University of
Lorraine.
In conclusion, in this research topic were obtained
relevant scientific results which have produced a scientific paper
published in an important ISI
journal in applied mathematics and another paper is in preparation:
J. San Martín, J.-F. Scheid,
L. Smaranda, A modified Lagrange-Galerkin method
for a fluid-rigid system with discontinuous density, Numerische Mathematik 122, No. 2, pp. 341-382, (2012), DOI: 10.1007/s00211-012-0460-1 2012.
J. San Martín, J.-F. Scheid,
L. Smaranda, Convergence of the Lagrange--Galerkin method for the equations modelling of fish-like swimming, in preparation (2012).
Talks of the project manager
July 3-4, 2012 -
Poster with the title
Convergence of a discretization scheme based on characteristics method for a fluid-rigid
system with variable
density
at
6th European Congress of Mathematics,
Jagiellonian University, Krakow, Poland.
August 25, 2012 - Talk with the
tiltle
Convergence of the Lagrange-Galerkin method for the equations modelling of fish-like swimming to
Special Session Modeles mathematiques et numeriques en mecanique des solides, the 11th
French-Romanian Colloquium in Applied Mathematics,
Bucarest, Romania.
November 6, 2012 - Talk with the
tiltle
Convergence of the Lagrange-Galerkin method for fluid-structure interaction problems
to Scientic Seminar in Department of Mathematics, Federal University of Santa Catarina, Florianopolis, Brasil.
December 17, 2012 - Talk with the
tiltle
Convergence of the Lagrange-Galerkin method for the equations modelling of fish-like swimming to International Conference on the Theory, Methods and Applications of Nonlinear Equations, The Texas A&M University-Kingsville, Texas, USA.
Visiting research positions
May 6, 2012 - June 10, 2012 -
Visiting research position in Centro de Modelamiento Matemático, Universidad de Chile.
October 10, 2012 - November 19, 2012 -
Visiting research position in Centro de Modelamiento
Matemático, Universidad de Chile.
October 5, 2012 - October 12,
2012 - Visiting research position in Department
of Mathematics, Federal University of Santa Catarina.
October-December 2011
Papers
Higher order macro coefficients in periodic structures.The main goal in this research topic
is the study of the variation of the macroscopic quantity called dispersion
tensor or Burnett coefficient associated
with a family of periodic heterogeneous medium
consisting of mixture between two homogeneous materials.
In general, the homogenized and the dispersion
coefficients depend on the microstructure
in a complex manner. It is well-known that in the one dimensional domain, the homogenized coefficient is not dependent
on the microstructure. Nevertheless, in higher dimension, it varies with the microstructure as
it is described in Murat and Tartar's theorem in 1983. Moreover, in our recent publication in M3AS, 2009, we have proved that in the one dimensional
domain, the dispersion tensor varies in a bounded domain. In this first period of
the project, we have generalized the previous result and we have proved for the dispersion tensor in
higher dimension, which varies with microstructure, the following conjecture: in the
laminated structures, this tensor varies together with microstructure
also in a bounded domain. This study was realized in collaboration with the Chilean researchers Carlos Conca and Jorge San
Martín from University of Chile and the Indian researcher Muthusamy Vanninathan from
Tata Institute of Fundamental Research. In conclusion, in this research topic were obtained
relevant scientific results which have published one paper:
C. Conca, J. San Martín, L. Smaranda, M. Vanninathan,
Burnett coefficients and laminates, Applicable Analysis, DOI:10.1080/ 00036811.2011.625017, 2011.
Numerical analysis in non-linear fluid-structure interactions. In this research topic, we study
the convergence of some numerical schemes in
fluid-structure interaction theory. Precisely, we
consider the equations modelling the motion of a rigid
solid assumed to be a ball of radius 1 immersed into a
viscous incompressible fluid. The equations that we have
used are the Navier-Stokes system coupled with Newton's
laws. We have proposed 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 k+1
into the approximate rigid body at time k.
We have proved convergence results for both semi-discrete and fully-discrete schemes. This study was realized in collaboration with
the Chilean researcher Jorge San Martín from University of Chile and the French
researcher Jean-François
Scheid from Institute Elie Cartan, University of
Lorraine.
In conclusion, in this research topic were obtained
relevant scientific results which have produced a scientific paper
conditionally accepted for publication in an important ISI journal in applied mathematics:
J. San Martín, J.-F. Scheid, L. Smaranda,
A modified Lagrange-Galerkin method for a
fluid-rigid system with discontinuous density,
conditionally accepted in Numerische Mathematik,
2011.