This article deals with the boundary observability properties of a
space finite-differences semi-discretization of the clamped beam equation. We
make a detailed spectral analysis of the system and, by combining numerical
estimates with asymptotic expansions, we localize all the eigenvalues of the
corresponding discrete operator depending on the mesh size $h$. Then, an
Ingham's type inequality and a discrete multiplier method allow us to deduce
that the uniform (with respect to $h$) observability property holds if and only
if the eigenfrequencies are filtered out in the range ${\cal
O}\left(1/h^4\right)$.
We introduce a direct method allowing to solve numerically inverse
type problems for linear hyperbolic equations posed in
$\Omega\times (0,T)$ - $\Omega$ a bounded subset of
$\mathbb{R}^N$. We consider the simultaneous reconstruction of
both the state and the source term from a partial boundary
observation. We employ a least-squares technique and minimize the
$L^2$-norm of the distance from the observation to any solution.
Taking the hyperbolic equation as the main constraint of the
problem, the optimality conditions are reduced to a mixed
formulation involving both the state to reconstruct and a Lagrange
multiplier. Under usual geometric conditions, we show the
well-posedness of this mixed formulation (in particular the
inf-sup condition) and then introduce a numerical approximation
based on space-time finite elements discretization. We prove the
strong convergence of the approximation and then discuss several
examples in the one and two dimensional case.
We present a sufficient condition under which a weak solution of
the Euler-Lagrange equations in nonlinear elasticity is already a
global minimizer of the corresponding elastic energy functional.
This criterion is applicable to energies $W(F) = \widehat{W}(F^T
F) = \widehat{W}(C)$ which are convex with respect to the right
Cauchy-Green tensor $C = F^T F$, where $F$ denotes the gradient of
deformation. Examples of such energies exhibiting a blow up for
$\det F \to 0$ are given.
In this paper, we study some weak majorization properties with
applications for the trees. A strongly notion of majorization is
introduced and Hardy–Littlewood–Polya’s inequality is generalized.
Based on a new concept of generalized relative convexity, a largee
xtension of Hardy-Littlewood-Pólya theorem of majorization is
obtained. Several applications escaping the classical framework of
convexity are included.
In this paper we propose to study wave propagation, transmission
and reflection in band-gap mechanical metamaterials via the
relaxed micromorphic model. To do so, guided by a suitable
variational procedure, we start deriving the jump duality
conditions to be imposed at surfaces of discontinuity of the
material properties in non-dissipative, linear-elastic, isotropic,
relaxed micromorphic media. Jump conditions to be imposed at
surfaces of discontinuity embedded in Cauchy and Mindlin continua
are also presented as a result of the application of a similar
variational procedure. The introduced theoretical framework
subsequently allows the trans- parent set-up of different types of
micro-macro connections granting the description of both (i)
internal connexions at material discontinuity surfaces embedded in
the considered continua and, as a particular case, (ii) possible
connections between different (Cauchy, Mindlin or relaxed
micromorphic) continua. The established theoretical framework is
general enough to be used for the description of a wealth of
different physical situations and can be used as reference for
further studies involving the need of suitably connecting
different continua in view of (meta-) structural design. In the
second part of the paper, we focus our attention on the case of an
interface between a classical Cauchy continuum on one side and a
relaxed micromorphic one on the other side in order to perform
explicit numerical simulations of wave reflection and
transmission. This particular choice is descriptive of a specific
physical situation in which a classical material is connected to a
phononic crystal. The reflective properties of this particular
interface are numerically investigated for different types of
possible micro-macro connections, so explicitly showing the effect
of different boundary conditions on the phenomena of reflec- tion
and transmission. Finally, the case of the connection between a
Cauchy continuum and a Mindlin one is presented as a numerical
study, so showing that band-gap description is not possible for
such continua, in strong contrast with the relaxed micromorphic
case.
In a series of papers which are either published
[Hadjesfandiari, A., Dargush, G. F., 2011a.
Couple stress theory for solids. Int. J. Solids Struct.
48 (18), 2496–2510; Hadjesfandiari, A., Dargush, G. F.,
2013. Fundamental solutions for isotropic size-dependent
couple stress elasticity. Int. J. Solids Struct. 50 (9),
1253–1265.] or available as preprints [Hadjesfandiari,
A., Dargush, G. F., 2010. Polar continuum mechanics.
Preprint arXiv:1009.3252; Hadjesfandiari, A. R.,
Dargush, G. F., 2011b. Couple stress theory for solids.
Int. J. Solids Struct. 48, 2496–2510; Hadjesfandiari,
A. R., 2013. On the skew-symmetric character of the
couple-stress tensor. Preprint arXiv:1303.3569;
Hadjesfandiari, A. R., Dargush, G. F., 2015a. Evolution
of generalized couple-stress continuum theories: a
critical analysis. Preprint arXiv:1501.03112;
Hadjesfandiari, A. R., Dargush, G. F., 2015b.
Foundations of consistent couple stress theory. Preprint
arXiv:1509.06299] Hadjesfandiari and Dargush have
reconsidered the linear indeterminate couple stress
model. They are postulating a certain physically
plausible split in the virtual work principle. Based on
this postulate they claim that the second-order couple
stress tensor must always be skew-symmetric. Since they
do not consider that the set of boundary conditions
intervening in the virtual work principle is not unique,
their statement is not tenable and leads to some
misunderstandings in the indeterminate couple stress
model. This is shown by specifying their development to
the isotropic case. However, their choice of
constitutive parameters is mathematically possible and
we show that it still yields a well-posed boundary
value problem.
In this paper we consider the additive logarithmic
finite strain plasticity formulation from the view point
of loss of ellipticity in elastic unloading. We prove
that even if an elastic energy
$F \mapsto W ( F ) = \widehat W( \log U )$ defined in
terms of logarithmic strain $\log U$, where
$U = \sqrt{F^T F}$, happens to be everywhere rank-one
convex as a function of $F$, the new function
$F \mapsto \widetilde W( F ) = W^ (\log U − \log U_p)$
need not remain rank-one convex at some given plastic
stretch $U_p$ (viz. $E_p^{\log} := \log U_p$). This is
in complete contrast to multiplicative plasticity (and
infinitesimal plasticity) in which
$F \mapsto W ( F F_p^{−1} )$ remains rank-one convex at
every plastic distortion $F_p$ if $F \mapsto W ( F )$ is
rank-one convex (
$\nabla u \mapsto \| \text{sym } \nabla u − \varepsilon_p\|^2$
remains convex). We show this disturbing feature of the
additive logarithmic plasticity model with the help of a
recently introduced family of exponentiated Hencky
energies.
A compact Riemann surface $X$ of genus $g>1$ which has a
conformal automorphism $\rho$ of prime order $p$ such
that the orbit space $X/\langle \rho \rangle$ is the
Riemann sphere is called cyclic $p$-gonal. Exceptional
points in the moduli space $M_g$ of compact Riemann
surfaces of genus $g$ are unique surface classes whose
full group of conformal automorphisms acts with a
triangular signature. We study symmetries of exceptional
points in the cyclic $p$-gonal locus in $M_g$ for which
$\text{Aut}(X)/\langle \rho \rangle$ is a dihedral group
$D_n$.
The aim of this paper is to prove a uniform observability
inequality for a finite differences semi-discretization of a
clamped beam equation. A discrete multiplier method is employed
in order to obtain the uniform observability of the eigenvectors
of the matrix driving the semi-discrete system, corresponding to
eigenfrequencies smaller than a precise filtering threshold.
This result can be generalized to the uniform observability of
every filtered solution. Numerical simulations, concerning the
dual controllability problem, illustrate the theoretical
results.
This paper surveys the existence of infinitely many solutions of
a nonlinear Neumann problem involving the m-Laplace operator,
where the constant m satisfies certain alternative inequalities,
and some functions $f(x,u)$ and $g(x,u)$ continuous on
$\overline{\Omega}\times\mathbb{R}$ and on $\partial\Omega\times\mathbb{R}$,
respectively, and odd with respect to u. We work on a domain
$\Omega$ bounded in $\mathbb{R}^{N}$ with smooth boundary. More
specifically, we demonstrate the existence of a sequence of
solutions which diverge to infinity provided that the nonlinear
term is locally superlinear and the existence of a sequence of
solutions which converge to zero provided that the nonlinear
term is locally sublinear.
Julien Aniort, Laurent Chupin, Nicolae Cindea, Mathematical
model of calcium exchange during hemodialysis using a citrate
containing dialysate, acepted in Mathematical Medicine and Biology a journal of IMA.
In this paper we propose a mathematical model for the calcium
exchange during hemodialysis. This model combines a first part
describing the flows of two fuids, blood and dialysate, in a
dialyser fiber to a second part which tackle the chemical
reactions between several chemical species presented in these
fluids.
The model governing the fluid flows is obtained by asymptotic
analysis and takes into account the anisotropy of fibers of a
dialyzer. Several rheologies for blood are proposed to highlight
the diferences in flow. The fluid velocity field drives the
convective part in the reaction- diffusion system, modelling the
exchange of five chemical species present in blood and
dialysate.
Finally, several numerical experiments illustrate this model
emphasizing the calcium balance for a citrate containing
dialysate.
We show that the reasoning in favor of a symmetric couple
stress tensor in Yang et al.'s introduction of the modified
couple stress theory contains a gap, but we present a
reasonable physical hypothesis, implying that the couple
stress tensor is traceless and may be symmetric anyway. To
this aim, the origin of couple stress is discussed on the
basis of certain properties of the total stress itself. In
contrast to classical continuum mechanics, the balance of
linear momentum and the balance of angular momentum are
formulated at an infinitesimal cube considering the total
stress as linear and quadratic approximation of a spatial
Taylor series expansion.
Articole trimise spre publicare
M. Malin, I. Roventa. Equilibrium problems in the context of
relative convexity.
In this paper we solve a generalized equilibrium problem and a
Nash equilibrium existence theorem is obtained. The key point is
given by an extension of the classical Ky Fan inequality into
the framework of rela- tive convexity. More precisely, we prove
that Ky Fan type mini-max inequal- ity holds even outside the
framework of convexity conditions, namely relative convexity
assumptions. The strategy used appeals to a Knaster-Kuratowsky-
Mazurkievich argument.
S. Micu, I. Roventa, L. E. Temereanca: Approximation of the
controls for the wave equation with a potential.
This article deals with the approximation of the boundary
controls of of a 1-D linear wave equation with a potential by
using a finite difference space semi-discrete scheme. Due to the
high frequency numerical spurious oscillations, the
semi-discrete model is not uniformly controllable with respect
to the mesh-size and the convergence of the approximate controls
corresponding to initial data in the finite energy space cannot
be guaranteed. In this paper we analyze how do the initial data
to be controlled and their discretization affect the result of
the approximation process. We prove that the convergence of the
scheme is ensured if the continuous initial data are
sufficiently regular or if the highest frequencies of their
discretization have been filtered out. In both cases, the
minimal $L^2$-norm discrete controls are shown to be convergent
to the corresponding continuous one when the mesh size tends to
zero.
Articole în lucru
Pierre Lissy, Ionel Roventa: Optimal filtration for the
approximation of boundary controls for a problem involving
fractional Laplacian.
In general, the high frequency numerical spurious oscillations
of the solutions of a system lead to a loss of the uniform (with
respect to the mesh-size) controllability property of the
semi-discrete model. For a very general problem involving
fractional Laplacian, by filtering the high frequencies of the
initial data in an optimal range, we try to restore the uniform
controllability property.
The key point is to use a similar multiplier with the one
introduced in Pierre Lissy, Ionel Roventa, Optimal filtration for the
approximation of boundary controls for the one-dimensional wave
equation, https://hal.archives-ouvertes.fr/hal-01338619/, where we have
considered a finite-diferences semi-discrete scheme for the
approximation of boundary controls in the case of the one-dimensional
wave equation.
We obtain a relation between the range of filtration and the minimal
time of control needed to ensure the uniform controllability, recovering
in many cases the usual minimal time to control the (continuous) wave
equation.
Our aim is to obtain similar results for a more general problem
involving fractional Laplacian, covering the cases of wave, heat and
beam equations.
Maria Malin, Ionel Roventa, Mihai Adrian Tudor, Iterate polygons in global NPC spaces.
Constantin Niculescu, Ionel Roventa, Weak majorization in global NPC spaces.
Ionel Roventa, Laurentiu Temereanca, Schur convexity properties of the even degree complete
homogeneous symmetric polynomials.
