Vladimir Salnikov

French National Centre for Scientific Research | CNRS · La Rochelle University

The G/G WZW model results from the WZW-model by a standard procedure of gauging. G/G WZW models are members of Dirac sigma models, which also contain twisted Poisson sigma models as other examples. We show how the general class of Dirac sigma models can be obtained from a gauging procedure adapted to Lie algebroids in the form of an equivariantly c...

In this paper we consider the possibility to use numerical simulations for a computer assisted analysis of integrability of dynamical systems. We formulate a rather general method of recovering the obstruction to integrability for the systems with a small number of degrees of freedom. We generalize this method using the results of KAM theory and st...

We study the dynamics of an ensemble of noninteracting harmonic oscillators in a nonlinear dissipative environment described by the Nosé-Hoover model, and find the histogram for energy regions of phase space against visiting time by employing numerical simulation. The results agree with the analysis of the Nosé-Hoover equations effected with the me...

In this paper we present some relatively unexpected mathematical questions emerging from the idea to approximate dissipative mechanical systems using Lagrangian formalism. We also explain potential consequences of these constructions for definition of representative space-time volume elements in modelling of metamaterials.

Previously, Wilson surface observables were interpreted as a class of Poisson sigma models. We profit from this construction to define and study the super version of Wilson surfaces. We provide some 'proof of concept' examples to illustrate modifications resulting from appearance of odd degrees of freedom in the target.

We recall the question of geometric integrators in the context of Poisson geometry and explain their construction. These Poisson integrators are tested in some mechanical examples. Their properties are illustrated numerically and compared to traditional methods.

We recall the question of geometric integrators in the context of Poisson geometry, and explain their construction. These Poisson integrators are tested in some mechanical examples. Their properties are illustrated numerically and they are compared to traditional methods.

Following recent results of A.K. and V.S. on $\mathbb Z$-graded manifolds, we give several local and global normal forms results for $Q$-structures on those, i.e. for differential graded manifolds. In particular, we explain in which sense their relevant structures are concentrated along the zero-locus of their curvatures, especially when the negati...

In this paper we address several algebraic constructions in the context of groupoids, algebroids and $\mathbb Z$-graded manifolds. We generalize the results of integration of $\mathbb N$-graded Lie algebras to the honest $\mathbb Z$-graded case and provide some examples of application of the technique based on Harish-Chandra pairs. We extend the co...

In this article we study the possibilities of recovering the structure of port-Hamiltonian systems starting from ``unlabelled'' ordinary differential equations describing mechanical systems. The algorithm we suggest solves the problem in two phases. It starts by constructing the connectivity structure of the system using machine learning methods --...

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and recollect the known results. Then, we define the category of differential graded Lie groups and study its properties. We show how to associate...

We discuss the notion of basic cohomology for Dirac structures and, more generally, Lie algebroids. We then use this notion to characterize the obstruction to a variational formulation of Dirac dynamics.

In this paper we discuss the categorical properties of $\mathbb Z$-graded manifolds. We start by describing the local model paying special attention to the differences in comparison to the $\mathbb{N}$-graded case. In particular we explain the origin of formality for the functional space and spell-out the structure of the power series. Then we make...

In this paper we make an overview of results relating the recent "discoveries" in differential geometry, such as higher structures and differential graded manifolds with some natural problems coming from mechanics. We explain that a lot of classical differential geometric constructions in the context can be conveniently described using the language...

Abstract This article reviews some integrators particularly suitable for the numerical resolution of differential equations on a large time interval. Symplectic integrators are presented. Their stability on exponentially large time is shown through numerical examples. Next, Dirac integrators for constrained systems are exposed. An application on ch...

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and present the construction for (non-graded) differential Lie algebras. Then, we define the category of differential graded Lie groups and study...

In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints – we recall the mathematical construction...

This article reviews some integrators particularly suitable for the numerical resolution of differential equations on a large time interval. Symplectic integrators are presented. Their stability on exponentially large time is shown through numerical examples. Next, Dirac integrators for constrained systems are exposed. An application on chaotic dyn...

In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints -- we recall the mathematical construction...

In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints-we recall the mathematical construction in...

This paper presents a free vibration analysis of beams made of fibre-metal laminated beans. Due to its attractive properties, this class of composites has gained more and more importance in the aeronautic field. Several higher-order displacements-based theories as well as classical models (Euler-Bernoulli’s and Timoshenko’s ones) are derived, assum...

In this work we continue the investigation of different approaches to conception and modeling of composite materials. The global method we focus on, is called 'stochastic homogenization'. In this approach, the classical deterministic \emph{homogenization} techniques and procedures are used to compute the macroscopic parameters of a composite starti...

In this paper, several rigorous numerical simulations were conducted to examine the relevance of mean-field micromechanical models compared to the Fast Fourier Transform full-field computation by considering spherical or ellipsoidal inclusions. To be more general, the numerical study was extended to a mixture of different kind of microstructures co...

In this paper we describe multigraded generalizations of some constructions useful for mathematical understanding of gauge theories: we perform a near-at-hand generalization of the Aleksandrov--Kontsevich--Schwarz--Zaboronsky procedure, we also extend the formalism of $Q$-bundles introduced first by A. Kotov and T. Strobl. We compare these approach...

Nowadays, the prediction of mechanical behaviour of reinforced composites is a paramount problem in computational materials science. The macroscale effective coefficients are obtained from the microscale information also known as the so-called corrector problem. A numerical strategy based on integral equation known as the periodic Lippmann-Schwinge...

In this paper we study the effective thermal behaviour of 3D representative volume elements (RVEs) of two-phased composite materials constituted by a matrix with cylindrical and spherical inclusions distributed randomly, with periodic boundaries. Variations around the shape of inclusions have been taken into account, by corrugating shapes, excavati...

In this paper we study the thermal effective behaviour for 3D multiphase composite material consisting of three isotropic phases which are the matrix, the inclusions and the coating media. For this purpose we use an accelerated FFT-based scheme initially proposed in Eyre and Milton (1999) to evaluate the thermal conductivity tensor. Matrix and sphe...

In our previous papers we have described efficient and reliable methods of generation of representative volume elements (RVE) perfectly suitable for analysis of composite materials via stochastic homogenization. In this paper we profit from these methods to analyze the influence of the morphology on the effective mechanical properties of the sampl...

In this paper we describe efficient methods of generation of representative volume elements (RVEs) suitable for producing the samples for analysis of effective properties of composite materials via and for stochastic homogenization. We are interested in composites reinforced by a mixture of spherical and cylindrical inclusions. For these geometries...

We show that in the context of two-dimensional sigma models minimal coupling of an ordinary rigid symmetry Lie algebra g leads naturally to the appearance of the “generalized tangent bundle” \( \mathbb{T} \) M≡TM⊕T * M by means of composite fields. Gauge transformations of the composite fields follow the Courant bracket, closing upon the choice of...

We study some graded geometric constructions appearing naturally in the context of gauge theories. Inspired by a known relation of gauging with equivariant cohomology we generalize the latter notion to the case of arbitrary QQ-manifolds introducing thus the concept of equivariant QQ-cohomology. Using this concept we describe a procedure for analysi...

In this paper, we continue the description of the possibilities to use numerical simulations for mathematically rigorous computer-assisted analysis of integrability of dynamical systems. We sketch some of the algebraic methods of studying the integrability and present a constructive algorithm issued from the Ziglin’s approach. We provide some examp...

In this short note we address the problem of integrability of a double pendulum in the constant gravity field. We show its non-integrability using the combination of algebraic and numerical approaches, namely we compute the non-commuting generators of the monodromy group along a particular solution obtained numerically.

We study the dynamics of an ensemble of non-interacting harmonic oscillators in a nonlinear dissipative environment described by the Nos\'e - Hoover model. Using numerical simulation we find the histogram for total energy, which agrees with the analysis of the Nos\'e - Hoover equations effected with the method of averaging. The histogram does not c...