ISSN:

1937-1632

eISSN:

1937-1179

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## Discrete & Continuous Dynamical Systems - S

December 2013 , Volume 6 , Issue 6

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2013, 6(6): i-i
doi: 10.3934/dcdss.2013.6.6i

*+*[Abstract](1842)*+*[PDF](67.9KB)**Abstract:**

The tenth edition of the French-Romanian Conference on Applied Mathematics took place in Poitiers, France, on august, 2010, and gathered around 130 scientists, mainly from France and Romania, but also from several other countries.

This series of conferences was initiated in 1992 in Iasi. It was then decided to organize it every two years, alternatively in France and in Romania. The eleventh edition was organized in Bucarest, Romania, in august, 2012, and the twelfth one will be organized in Lyons, France, in 2014.

This issue of DCDS S contains carefully refereed contributions from participants of the conference.

2013, 6(6): 1457-1471
doi: 10.3934/dcdss.2013.6.1457

*+*[Abstract](2583)*+*[PDF](377.9KB)**Abstract:**

We introduce four variants of a multigrid method for quasi-variational inequalities composed by a term arising from the minimization of a functional and another one given by an operator. The four variants of the method differ from one to another by the argument of the operator. The method assume that the closed convex set is decomposed as a sum of closed convex level subsets. These methods are first introduced as subspace correction algorithms in a general reflexive Banach space. Under an assumption on the level decomposition of the closed convex set of the problem, we prove that the algorithms are globally convergent if a certain convergence condition is satisfied, and estimate the global convergence rate. These general algorithms become multilevel or multigrid methods if we use finite element spaces associated with the level meshes of the domain and with the domain decompositions on each level. In this case, the methods are multigrid $V$-cycles, but the results hold for other iteration types, the $W$-cycle iterations, for instance. We prove that the assumption we made in the general convergence theory holds for the one-obstacle problems, and write the convergence rate depending on the number of level meshes. The convergence condition in the theorem imposes a upper bound of the number of level meshes we can use in algorithms.

2013, 6(6): 1473-1485
doi: 10.3934/dcdss.2013.6.1473

*+*[Abstract](1964)*+*[PDF](384.5KB)**Abstract:**

In this paper we employ a Cosserat model for rod-like bodies and study the governing equations of thin thermoelastic porous rods. We apply the counterpart of Korn's inequality in the three-dimensional elasticity theory to prove existence and uniqueness results concerning the solutions to boundary value problems for thermoelastic porous rods, both in the dynamical theory and in the equilibrium case.

2013, 6(6): 1487-1506
doi: 10.3934/dcdss.2013.6.1487

*+*[Abstract](2031)*+*[PDF](684.6KB)**Abstract:**

We consider here different models of dissipative Korteweg-de Vries (KdV) equations on the torus. Using a proper wave function $\Gamma$, we compare numerically the long time behavior effects of the damping models and we propose a hierarchy between these models. We also introduce a method based on the solution of an inverse problem to rebuild

*a posteriori*the damping operator using only samples of the solution.

2013, 6(6): 1507-1524
doi: 10.3934/dcdss.2013.6.1507

*+*[Abstract](2400)*+*[PDF](401.1KB)**Abstract:**

This paper deals with the mathematical analysis and the subspace approximation of a system of variational inequalities representing a unified approach to several quasistatic contact problems in elasticity. Using an implicit time discretization scheme and some estimates, convergence properties of the incremental solutions and existence results are presented for a class of abstract implicit evolution variational inequalities involving a nonlinear operator. To solve the corresponding semi-discrete and the fully discrete problems, some general subspace correction algorithms are proposed, for which global convergence is analyzed and error estimates are established.

2013, 6(6): 1525-1537
doi: 10.3934/dcdss.2013.6.1525

*+*[Abstract](2441)*+*[PDF](351.0KB)**Abstract:**

In this paper, we present a geometric representation of the 2D elasticity tensors using the representation theory of linear groups. We use Kelvin's representation in which $\mathbb{O}(2)$ acts on the 2D stress tensors as subgroup of $\mathbb{O}(3) $. We present the method in the simple case of the stress tensors and we recover Mohr's circle construction. Next, we apply it to the elasticity tensors. We explicitly give a linear frame of the elasticity tensor space in which the representation of the rotation group is decomposed into irreducible subspaces. Thanks to five independent invariants choosen among six, an elasticity tensor in 2D can be represented by a compact line or, in degenerated cases, by a circle or a point. The elasticity tensor space, parameterized with these invariants, consists in the union of a manifold of dimension $5$, two volumes and a surface. The complet description requires six polynomial invariants, two linear, two quadratic and two cubic.

2013, 6(6): 1539-1550
doi: 10.3934/dcdss.2013.6.1539

*+*[Abstract](1687)*+*[PDF](296.5KB)**Abstract:**

In this paper we study the deformation of right porous cylinders subjected to a prescribed thermal field. We assume that the cylinder is filled by an inhomogeneous anisotropic porous material. In the first part of the paper we study the problem of extension-bending-torsion, when the thermal field is independent of the axial coordinate and then we study the problem of extension-bending-torsion-flexure when the thermal field is considered linear in the axial coordinate. The considered problems are reduced to some generalized plane strain problems in the cross-section of the cylinder. Our analysis shows how the considered thermal fields influence the deformation of the porous cylinders.

2013, 6(6): 1551-1567
doi: 10.3934/dcdss.2013.6.1551

*+*[Abstract](1704)*+*[PDF](393.7KB)**Abstract:**

Control synthesis for electrohydraulic servoactuators (EHSA) is achieved using elements of geometric control theory. Based on a Malkin type theorem for switched systems of ordinary differential equations, the existence of stabilizing feedback controllers is prove to hold in the specific case of EHSAs when the relative degree of the nonlinear control system is one unit less than the order of the system. The proof relies on coordinate transformations that bring the system to some canonical form.

2013, 6(6): 1569-1586
doi: 10.3934/dcdss.2013.6.1569

*+*[Abstract](1842)*+*[PDF](409.3KB)**Abstract:**

In this paper, some results obtained on the asymptotic behavior of hard, thin curvilinear interfaces i.e., in cases where the interphase and adherents have comparable rigidities, are presented. The case of hard interfaces is investigated in terms of cylindrical coordinates and some analytical examples are presented.

2013, 6(6): 1587-1598
doi: 10.3934/dcdss.2013.6.1587

*+*[Abstract](1816)*+*[PDF](349.8KB)**Abstract:**

We consider a mathematical model which describes the contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic, the contact is frictionless and is modelled with unilateral constraint. We derive a variational formulation of the model which leads to a history-dependent quasivariational inequality for stress field, associated to a time-dependent convex. Then we prove the unique weak solvability of the model. The proof is based on an abstract existence and uniqueness result obtained in [11].

2013, 6(6): 1599-1608
doi: 10.3934/dcdss.2013.6.1599

*+*[Abstract](2141)*+*[PDF](159.6KB)**Abstract:**

This short contribution aims at introducing the notion of dynamic materials (as initiated by Blekhman and Lurie) and the corresponding allied techniques of homogenization and asymptotic analysis. Main role is played by the canonical conservation laws of energy and wave momentum - the latter most often ignored in the field of continuum mechanics - as follows from an application of the celebrated theorem of E. Noether.

2013, 6(6): 1609-1619
doi: 10.3934/dcdss.2013.6.1609

*+*[Abstract](1934)*+*[PDF](363.2KB)**Abstract:**

We consider a system of rigid bodies subjected to some non penetration conditions characterized by the inequalities $f_{\alpha} (q) \ge 0$, $\alpha \in \{1, \dots, \nu\}$, $\nu \ge 1$, for the configuration $q \in \mathbb{R}^d$. We assume that there is no adhesion and no friction during contact and we model the behaviour of the system at impact by a Newton's law. Starting from the mechanical description of the problem, we derive two mathematical formulations, using either the configuration or the generalized velocity as unknown. Then a velocity-based time-stepping scheme, inspired by the catching-up algorithms, is presented and its convergence in the multi-constraint case (i.e $\nu \ge1$) is stated.

2013, 6(6): 1621-1639
doi: 10.3934/dcdss.2013.6.1621

*+*[Abstract](1683)*+*[PDF](748.1KB)**Abstract:**

We propose a non-local model with dislocation densities and non-Schmid effect in the finite elasto-plasticity framework, which accounts for the dissipation postulate formulated through a principle of the free energy imbalance. Our goal is to characterize the scalar plastic velocities and activation condition in order to be compatible with the principle of the imbalanced free energy. The activation condition is expressed in terms of the generalized resolved stress, which is dependent not only on the Mandel stress measure, but also on the gradient of the scalar dislocation density. We analyze numerically how the model behaves for a simple shear problem into a layer when only one slip system is activated.

2013, 6(6): 1641-1649
doi: 10.3934/dcdss.2013.6.1641

*+*[Abstract](2058)*+*[PDF](278.9KB)**Abstract:**

We analyze the relation between Géry de Saxcé's bipotentials representing non-associated constitutive laws and Fitzpatrick's functions representing maximal monotone multifunctions. Revisiting the model of elastic materials initiated by Robert Hooke, we illustrate that Fitzpatrick's representation of monotone operators coming from convex analysis provides a constructive method to discover the best bipotential for modelling an Implicit Standard Material.

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