Applications of Mathematics and Informatics in Natural Sciences and Engineering

AMINSE 2019

**Mariam Avalishvili**, University of Georgia, Tbilisi.

*Variational Approach for Construction and Investigation of Hierarchical Models of Thermoelastic Piezoelectric Structures*

**Abstract:** In this talk, we present results of investigation of mathematical models of thermoelastic piezoelectric shells and curvilinear bars consisting of inhomogeneous anisotropic material with regard to magnetic field. We consider multilayer shells with variable thickness, which may vanish on a part of the lateral boundary, and multilayer curvilinear bars with variable rectangular cross-section, which may degenerate into segment or point at the butt ends of bar, and obtain variational formulations of the boundary and initial-boundary value problems in curvilinear coordinates corresponding to the static and dynamical linear three-dimensional models of piezoelectric thermoelastic solids with regard to magnetic field, when density of surface force, and normal components of electric displacement, magnetic induction and heat flux vectors are given along certain parts of the boundary and on the remaining parts of the boundary mechanical displacement, electric and magnetic potentials, and temperature vanish. By applying extension and generalization of I. Vekua dimensional reduction method and variational formulations we construct hierarchies of static and dynamical two-dimensional models for shells and one-dimensional models for curvilinear bars. For the boundary and initial-boundary value problems corresponding to the constructed two-dimensional and one-dimensional models we give the results on the existence and uniqueness of solutions in suitable weighted Sobolev spaces or spaces of vector-valued distributions. Moreover, we present the results on the relationship between the constructed two-dimensional and one-dimensional models and the original three-dimensional problems.

This is joint work with Gia Avalishvili (Tbilisi State University) and was supported by Shota Rustaveli National Science Foundation (SRNSF) [Grant number 217596, Construction and investigation of hierarchical models for thermoelastic piezoelectric structures].

**Ayech Benjeddou**, Institut Supérieur de Mécanique de Paris (SUPMECA) & Université de Technologie de Compiègne (UTC)/Centre National de la Recherche Scientifique (CNRS).

*Progress in Mathematical and Numerical Modelling of Piezoelectric Smart Structures*

**Abstract:** Based on the author’s works, this plenary lecture discusses developed mathematical and numerical models forpiezoelectric smart structures analysis. After recalling some early (late 1990s) asymptotic theories-basedmathematical results regarding the mechanical displacements and electric potential and displacements variationsthrough the thickness of thin piezoelectric smart structures, the related (around 2000s) displacement-potential finiteelements (FE) and advanced partial mixed electromechanical variational formulations are briefly presented. Thefocus is then put on the developments regarding the more recent (2010s) Hamiltonian partial mixed FE-state spacesymplectic approach and semi-analytical distributed transfer function and iterative extended Kantorovich methodsfor the piezoelectric smart multilayer cross-ply and angle-ply composite plates and sandwich beams static and free-vibration analyses under theoretical and realistic mechanical and electric boundary conditions.

**Lucian Beznea**, Simion Stoilow Institute of Mathematics of the Academy & University of Bucharest.

*Connections between the Dirichlet and the Neumann problem for integrable boundary data*

**Abstract:**We provide an explicit solution of the generalized solution of the Neumann problem for the Laplace operator, based on a representation of the solution on the unit ball in \mathbb{R}^n, n\ge 1, in terms of the solution of an associated Dirichlet problem, in the case of integrable boundary data.

We also provide a new approach to Brosamler's formula which gives a probabilistic representation of the solution of the Neumann problem for the Laplacian in terms of the reflecting Brownian motion.

The talk is based on joint works with Mihai N. Pascu (Brasov, Romania) and Nicoale R. Pascu (Kennesaw State University, USA).

**Ramaz Botchorishvili**, Tbilisi State University.

*Deep Residual Networks and Numerical Linear Advection*

**Abstract:**Some deep neural networks can be viewed as a numerical approximation of partial differential equations and dynamical systems. Motivated by numerical partial differential equations several new architectures of neural networks are proposed. Some results also exist regarding connections between neural networks and linear advection equation. An award-winning deep residual network for image classification is interpreted as numerical solution of linear transport equation usingmethod of characteristic with specific velocity vector. Some numerical schemes for linear advection equation with constant coeffcient are considered as a deep neural network with specific architecture and the approach is used for accelerating numerical computations. Here we consider different aspects, we study deep convolutional residual network with the help of numerical linear advection. We prove universal approximation property of considered deep residual network. In particular, we prove that deep residual network with specific convolutional kernel, depth and width can approximate continuously differentiable functions at arbitrary precision. We consider several residual networks motivated by finite volume discretization and provide error estimates for them. We formulate classification and regression tasks as parameter identification problem for the linear advection equation. We show how to incorporate stable adjoint numerical solver in backpropagation algorithm for these tasks.

**Maribel Fernandez**, King's College London.

*Strategic Graph Rewriting in an Interactive Modelling Framework*

**Abstract:**In this talk I will describe the use of strategic port graph rewriting as a basis for the implementation of visual modelling tools. The goal is to facilitate the specification and analysis of complex systems, in particular programs. A system is represented by an initial graph and a collection of graph rewrite rules, together with a user-defined strategy to control the application of rules. The traditional notions of strategies for functional languages and term-rewriting languages have been adapted to deal with the more general setting of graph rewriting, and some new constructs have been included in the strategy language to deal with graph traversal and management of rewriting positions in the graph. We give a formal semantics for the language, examples of application, and a brief description of its implementation: the graph transformation and visualisation tool PORGY.

**Alice fialowski**, Eotvos Lorand University.

*Metric Lie Algebras and Applications*

**Abstract:** In my talk I will consider Lie algebras with an invariant symmetric nondegenerate bilinear form. These are important in many areas of mathematical physics. They include beside semisimple Lie algebras other interesting algebras, like the diamond and oscillator algebra. I will chatacterize them in low dimension, and present their metric deformations.

**George Jaiani**, Institute of Applied Mathematics & Department of Mathematics, Tbilisi State University.

*On Hierarchical Models for Piezoelectric Bars*

**Abstract:** The present talk is devoted to construction and investigation of hierarchical models for piezoelectric termo-viscoelastic Kelvin-Voigt bars with rectangular cross-sections. In particular, in (0,0) approximation static and oscillation problems are discussed. A special attention is given to analysis of peculiarities of nonclassical setting boundary conditions (BCs) in the case of cusped bars. Namely, the criteria are established for piezoelectric transversely isotropic cusped bars when on one end or on both ends of the bar no data need to be prescribed. Weighted BCs are set as well. On the face surfaces of the bar under consideration stress vectors and outward normal components of the electric displacement vectors are prescribed, while at the ends of the bar all the admissible (in sense of well-posedness of boundary value problems) BCs, including mixed ones, with respect to weighted (0,0) moments of the components of the mechanical displacement vectors and electric potential, and (0,0) moments of the components of the stress and electric displacement vectors are prescribed.

This work was supported by Shota Rustaveli National Science Foundation (SRNSF) [Grant number 217596, Construction and investigation of hierarchical models for thermoelastic piezoelectric structures].

**Rolf Jeltsch**, ETH Zurich.

*Recent Developments on Numerical Solutions for Hyperbolic Systems of Conservation Laws*

**Abstract:** In 1757 Euler developed the famous Euler equations describing the flow of a compressible gas. This is a system of hyperbolic conservation laws in three space dimensions. However until recently one could not show convergence of numerical schemes to the ’classical’ weak entropy solutions. By adapting the concept of measure-valued and statistical solutions to multidimensional systems Siddhartha Mishra and his coauthors could recently show convergence of numerical schemes. Mishra has presented these results at the ICM 2018 in Rio
de Janeiro. I will present even newer developments which he presented at the ICIAM Congress 2019. After a brief introduction to the field these developments will be described.

**Omar Kikvidze**, Akaki Tsereteli State University.

*The Flow of Nonlinear Viscous Strain-Hardening Material in Variable Section Passages*

**Abstract:** The radial flow of material in conic and wedge-shaped passagesis considered, the entranceand exit of which are boundedbysurfaceswith variable curvature. The curvature of the surfaces depends on distribution of flow velocity in the passage entrance. The constitutive equation of material was obtained on the basis of the work-hardeningtheory.

The solution of a two-dimensional problemis reduced to the integration of a system of theordinary nonlinear differential equations with boundary conditions: X'=F(\alpha,X,q); H_0(X(0))=0, H(X(\alpha_1))=0; where: X - the vector of unknowns, \alpha - an angular coordinate, q parameter was entered artificially.

Integration of the nonlinear differential equations with boundary conditions is carried out numerically by method of the movement towards parameter shooting. The use ofcomponents allows for determining flow velocity, components of a tensor of velocities of deformations, thecomponents of strain velocity tensor andcomponents.

**Temur Kutsia**, RISC, Johannes Kepler University.

*Proximity Constraint Solving*

**Abstract:** Proximity relations are binary fuzzy relations that satisfy reflexivity and symmetry properties, but are not transitive. This relation induces the proximity measure between function symbols, which is further extended to terms. The problem that we discuss in this talk is about solving unification problems modulo proximity: Given two terms, find a substitution that brings the terms "sufficiently close" to each other, i.e., the proximity measure between substitution instances of the terms exceeds a predefined threshold. We impose no extra restrictions on proximity relations, allowing a term in unification to be close to two terms that themselves are not close to each other.

The unification problem has finite minimal complete set of unifiers. We designed an algorithm that computes this set. It works in two phases:
first reducing the equation solving problem to neighborhood constraints over sets of function symbols, and then solving the obtained
constraints. We present the algorithm, illustrate it on examples, and discuss its properties and applications.

This is joint work with Cleo Pau.

**Mircea Marin**, Computer Science Department, West University of Timisoara.

*Specification and Analysis of ABAC Policies in a Rule-Based Framework*

**Abstract:** Access control is a fundamental security requirement for computing environments: It controls the ability of a subject to use an object in some specific manner. Attribute-based access control (ABAC) is a logical access control with great flexibility to specify access control policies as rules which get evaluated against the attributes of participating entities (user/subject or subject/object), operations, and the environment relevant to a request.

The access control policies that can be implemented in ABAC are limited only by the computational language and the richness of the available attributes. Considerable work has been done and a number of formal models have been proposed recently for ABAC, with minimal sets of features that are sufficient to implement many desirable capabilities.

In this talk, we discuss the popular access control models ABAC_alpha and ABAC_beta, and propose to study and analyse them in RhoLog, a rule-based framework developed by us. We specify the logical and operational semantics of their policies in our framework, and show how to use the RhoLog system to decide some properties of interest.

**Bernadette Miara**, Université Paris-Est Marne la Vallée.

*Modelling of Elastic Composites with Several Small Parameters*

**Abstract:**In addition to the thinness \varepsilon of a structure, we introduce another positive small parameter \delta in order to take into account some characteristics of the geometry, or of the heterogeneities of an elastic body. When both parameters vanish simultaneously, new models are obtained as limit of an asymptotic approach (in the framework of linearised elasticity). Three examples are discussed.

With Dr Eduard Rohan (Pilsen University, Czech Republic) we considered a composite body made of periodic elastic inclusions of size \varepsilon; and \delta is the ratio between the elastic tensor components of the inclusions and those of the matrix. We proved that for a strongly heterogeneous composite (with \delta=\varepsilon^{2}) the limit homogeneous model, obtained by letting \varepsilon go to 0, presents a negative "mass density" tensor implying the existence of band-gaps in the propagation of elastic waves. The cases of thin plates ruled by Reissner-Mindlin or Kirchhoff-Love equations are compared.

With Pr Georges Griso (Universite Pierre et Marie Curie, Paris, France) we considered a thin beam (\varepsilon is the ratio between its thickness and its length) made of a periodic distribution of small elastic inclusions along its length (\delta is the size of each inclusions). When both the thickness of the beam and the size of the heterogeneities tend simultaneously to zero, we obtain three different one-dimensional models of beam depending upon the limit of the ratio \displaystyle \frac{\varepsilon}{\delta} of these two small parameters.

With Dr Patrick Ballard (Institut Jean le Rond \partial'Alembert, Paris, France) we considered the case of a slender beam (\varepsilon is, as before, the ratio between its thickness and its length) whose cross-section is also slender (\delta measures this slenderness). When both parameters are of the same order of magnitude we recover Vlassov's one-dimensional model for thin-walled beams (for special profiles of the cross-sections and specific loadings) thanks to an appropriate Korn's inequality.

**Wolfgang Mueller**, Technical University of Berlin.

*An Investigation of Electromagnetic Force Models: Electromagnetic Forces and Moments Acting on Spherical Magnets*

**Abstract:** From Maxwell’s equations balance laws for the electromagnetic
linear momentum, angular momentum, and energy can be found after recasting
and using several identities of vector calculus. Therefore, the obtained
equations are not “new results” but rather identities having the form of a balance
law. However, there is some degree of freedom, (a) during construction of
a particular identity and (b) for the choice of the to-be-balanced quantity, the
non-convective flux, and the production term. In short, one is insecure which
of the various forms is correct under which circumstances. This conundrum is
referred to as the Abraham–Minkowski controversy, who first proposed different
expressions for the electromagnetic linear momentum. The proper choice
of electromagnetic force and torque expressions is of particular importance in
matter where the mechanical and electromagnetic fields couple. The question
arises as to whether a comparison between the predicted deformation behavior
and the observed one can help to decide which electromagnetic force model is
suitable for a material of interest. In this talk we shall briefly review the
controversy and suggest new approaches for its solution on the continuum
level. We will learn from the example of total force and moment calculation of two permanent
magnets interacting with each other.

**Elizbar Nadaraia**, Tbilisi State University.

*On the Limit Distribution of the Integral Square Deviation of a Nonparametric Estimator of the Bernoulli Regression Function for One Sample and Two Independent Samples*

**Abstract:** The limiting distribution of the integral square deviation of a kernel-type nonparametric estimator of the Bernoulli regression function is established. The criterion of testing the hypothesis about the Bernoulli regression function is constructed. The question as to its consistency is studied.The asymptotic power of the constructed test is also studied for certain types of close alternatives. The question is investigated for one sample and two independent samples.

**Gerald Trutnau**, Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University.

*Well-posedness for Degenerate Ito-SDEs with Discontinuous Coefficients*

**Abstract:** For second order elliptic PDE operators L in \mathbb{R}^d, d\ge 2, with degenerate diffusion matrix and locally integrable first order term, we construct a pre-invariant density \rho with some nice Sobolev regularity. This then allows at the same time for a functional analytic frame which describes L as infinitesimal generator of a C_0-semigroup in L^p(\rho dx), p\ge 1, and to derive through elliptic regularity theory for PDEs strong Feller properties for the semigroup and resolvent. Subsequently, we show that the continuous version of the semigroup that we obtained through elliptic regularity, is the transition function of a Hunt process that has continuous sample paths on the one point compactification of \mathbb{R}^d with some point at infinity \Delta. This Hunt process then weakly solves a stochastic differential equation (SDE) with degenerate dispersion and locally integrable drift coefficient up to some explosion time. Finally, we present condtions for the explosion time to be infinite, i.e. conditions for non-explosion, and conditions for uniqueness in law of the solution to the SDE. This is joint work with Haesung Lee (Seoul National University).

**Vaja Tarieladze, George Giorgobiani, Vakhtang Kvaratskhelia**, Muskhelishvili Institute of Computational Mathematics of the Georgian Technical
University.

*Induced Operators by Subgaussian Random Elements*

**Abstract:** To a (real-valued) random variable \xi given on a probability space (\Omega, \mathcal A, \mathbb{P}) let us associate a quantity \tau(\xi)\in [0, +\infty] defined by the equality: \tau(\xi)=\inf\{a\ge 0: {\mathbb E}\,e^{t\xi}\le \,e^{\frac{1}{2}t^2a^2}\quad {\textit{for every}} \quad t\in\mathbb R\,\}. A random variable \xi is called

* Subgaussian if \tau(\xi)<+\infty,

* strictly Subgaussian if it is Subgaussian and \tau^2(\xi)={\mathbb E}\xi^2.

Let SG(\Omega) be the set of all Subgaussian random variables \xi:\Omega\to \mathbb R. It is known that SG(\Omega) is a vector space with respect to the natural point-wise operations, the functional \tau(\cdot) is a norm on SG(\Omega) (provided the random variables which coincide a.s. are identified) and, moreover, (SG(\Omega), \tau(\cdot)) is a Banach space.

Let X be a real Banach space and X^* be its dual. We say that a random element \eta:\Omega\to X

* is weakly Subgaussian, if for every x^* \in X^* the random variable x^*(\eta) is Subgaussian;

* is strictly Subgaussian, if for every x^* \in X^* the random variable x^*(\eta) is strictly Subgaussian;

* is Subgaussian in Fukuda's sense, or F-Subgaussian, if for every x^* \in X^* the random variable x^*(\eta) is Subgaussian and there is a finite constant C\ge 0 such that \tau(x^*(\eta))\le C \left({\mathbb E}|x^*(\eta)|^2\right)^{\frac{1}{2}} for every x^* \in X^*;

* T-Subgaussian, if there exists a Gaussian random element \gamma:\Omega\to X such that {\mathbb E}\,e^{x^*(\eta)}\le {\mathbb E}\,e^{x^*(\gamma)} for all x^* \in X^*.

For every weakly Subgaussian random element \eta:\Omega\to X let us define the induced operator T_\eta:X^*\rightarrow SG(\Omega) by the equality T_\eta x^*=x^* (\eta) for all x^* \in X^*.

We will discus a relationship between these notions together with a proof of the following statement: For any F-Subgaussian random element \eta in a Banach space X the induced operator T_\eta:X^*\rightarrow SG(\Omega) is a 1-summing operator.

**Tamaz Vashakmadze**, Tbilisi State University.

*To the Theory and Practice of Thinwalled Structures*

**Abstract:**The classical theory for elastic thinwalled beams, plates and shells (tpthwalstr) may give rough results different from models according to [1]. These models, in particular, contain the refined theories but not only in elastic case. On other hand we constructed in [2] uniform 3D mathematical models which as particular case contain the systems of Navier-Stokes, Euler, nonlinear PDEs of Solid Mechanics, Maxwell’s dynamical system, principles of the mass and energy conservations, Saint-Venant-Beltrami continuity conditions, Hook’s and Newton’s relations. Differing from the Truesdell theory [3], we [1, 2] present both the balance equations of continuum mechanics and experimental laws uniformly by means of a parameter. Thus we formulate the following principle: any phenomenon discovered for the separate matter has the universal nature for any form of continuum mechanics.

We consider three examples:

1. The anisotropic inhomogeneous elastic structures. For this case the classical topic “tpthinwalstr” presents as fight of opposites, while really there is the presence of the classical category of unity of opposites. In particular, this way gives the possibility to construct easily the mixed theory for the inhomogeneous case.

2. Piezoelectric materials and electrical conductivity. [1, 2] give for basic relations the possibility of constructing models describing also the shock, 2D soliton and Rayleigh-Lamb’s waves.

3. Poroelasticity. Application of our methodology for this case refined the basic equations while Biot’s and Reissner’s models for porosity part used the Pascal-Darcy law having for symbolic determinant the parabolic degeneracy.