Analysis and optimization in problems of cloaking of material bodies for the Maxwell equations

We consider control problems for the 3D Maxwell equations describing electromagnetic wave scattering in an unbounded inhomogeneous medium that contains a permeable isotropic obstacle with cloaking boundary. Such problems arise when studying cloaking problems by the optimization method. The boundary coefficient occurring in the impedance boundary condition plays the role of a control. We study the solvability of the control problem and derive optimality systems that describe necessary conditions for the extremum. By analyzing the constructed optimality systems, we justify sufficient conditions imposed on the input data providing the uniqueness and stability of optimal solutions.     

Approximate acoustic cloaking in inhomogeneous isotropic space

We consider the approximate acoustic cloaking in an inhomogeneous isotropic background space.By employing transformation media,together with the use of a sound-soft layer lining right outside the cloaked region,we show that one can achieve the near-invisibility by the"blow-up-a-small-region"construction.This is based on novel scattering estimates corresponding to multiple multi-scale obstacles located in an isotropic space.We develop a novel system of integral equations to decouple the nonlinear scattering interaction among the small obstacle components,the regular obstacle components and the inhomogeneous background medium.     

Elastodynamical resonances and cloaking of negative material structures beyond quasistatic approximation

Given the flexibility of choosing negative elastic parameters, we construct material structures that can induce two resonance phenomena, referred to as the elastodynamical resonances. They mimic the emerging plasmon/polariton resonance and anomalous localized resonance in optics for subwavelength particles. However, we study the peculiar resonance phenomena for linear elasticity beyond the subwavelength regime. It is shown that the resonance behaviors possess distinct characters, with some similar to the subwavelength resonances, but some sharply different due to the frequency effect. It is particularly noted that we construct a core–shell material structure that can induce anomalous localized resonance as well as cloaking phenomena beyond the quasistatic limit. The study is boiled down to analyzing the so-called elastic Neumann–Poincaré (N-P) operator in the frequency regime. We provide an in-depth analysis of the spectral properties of the N-P operator on a circular domain beyond the quasistatic approximation, and these results are of independent interest to the spectral theory of layer potential operators.     

Approximation method for the problems of mechanics of inhomogeneous hereditarily elastic bodies

We consider a boundary-value problem of mechanics of inhomogeneous hereditarily elastic bodies formulated as a linear equation with an operator of fractional integration, partial derivatives with respect to time and spatial variables, and polynomial-type coefficients of one of the variables. An approximate solution of this problem is constructed according to Dzyadyk's a-method combined with the use of the Laplace transformation. It is proved that the errors of the approximation of the required function and its derivatives decrease in geometric progression.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1234–1245, September, 1994.     

A method for the approximate solution of quasi-static problems for hardening bodies

A method for the approximate solution of quasi-static problems for hardening elastoplastic bodies is proposed. The constitutive relation of the model is taken in the form of a variational inequality. An approximate solution of the initial problem is constructed in time steps and, by means of the finite element method, is reduced to the solution of a system of two variational inequalities in corresponding finite-dimensional space. It is shown that the solution of this system is equivalent to finding the saddle point of the corresponding quadratic functional. To find the saddle point, Udzawa's algorithm is used, by means of which the process of finding the velocity vector and stress tensor reduces to the successive calculation of these quantities: the velocity vector is determined from the variational inequality corresponding to the equilibrium equation, and the stress tensor is determined from the variational inequality corresponding to the constitutive relation. The latter inequality is reduced to a certain non-linear equation containing the operation of projection onto a closed convex set corresponding to the elastic strains of the medium. In turn, the solution of the non-linear equation is constructed using the method of successive approximations. To illustrate the use of the proposed method, the one-dimensional problem of the quasi-static deformation of a cylindrical tube under a load applied to its internal surface is considered.     

High-frequency projection method for plane problems of wave diffraction on dielectric bodies

A version of the Galerkin incomplete projection method is described for plane problems of wave diffraction on dielectric bodies of arbitrary shape. The proposed method generalizes the Sommerfeld method, which constructs diffraction series rapidly converging at high frequencies for circular and spherical bodies. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 58–67.     

Control of boundary impedance in two-dimensional material-body cloaking by the wave flow method

Control problems are considered for a two-dimensional electromagnetic field model describing electromagnetic wave scattering in a unbounded homogeneous medium containing an anisotropic permeable inclusion with a partially covered (cloaked) boundary. The control is a function involved in the impedance boundary condition on the covered part of the boundary. The solvability of the original mixed transmission problem for the two-dimensional Helmholtz equation and of the control problems is proved. Optimality systems describing necessary extremum conditions are derived. The uniqueness and stability of optimal solutions with respect to certain perturbations of the cost functional and the incident wave are established.     

On the integral transform method in nonlinear heat-conduction problems for bodies of complicated shape

Using the construction procedure of numerical-analytic methods and the boundary element technique, we extend the integral transform method to the solution of nonlinear heat-conduction problems for bodies of nonstandard shape. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 42–46.     

Supersymmetry and acoustic problems

The oscillations of a compressible fluid within a region bounded by a flat plate on one side and an acoustically soft boundary on the other are studied. A supersymmetric structure of the resolvent is established. It is shown that, because of the presence of the supersymmetry, the solution has properties in common with the solution to the quantum-mechanical problem of an electron in a magnetic field. The connection between supersymmetry and the symmetry group of the problem is demonstrated.Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 407–411. March, 1992.     

A point source method for inverse acoustic and electromagnetic obstacle scattering problems

We present the mathematical foundation for a point source methodto solve some inverse acoustic and electromagnetic obstaclescattering problems in three dimensions. We investigate theinverse acoustic scattering problem by a sound-soft and a sound-hardscatterer and the inverse electromagnetic scattering problemby a perfect conductor. Two independent approaches to the methodare presented which reflect its strong relation to basic propertiesof obstacle scattering problems.     

A dual-level method of fundamental solutions for three-dimensional exterior high frequency acoustic problems

Physical essence of the fictitious boundary of the method of fundamental solutions has been a mystery for a long time. In this study, we attempt to explain the reason why fictitious boundary has such a dramatic effect on numerical results. The influence law of the fictitious boundary on numerical results is revealed. Based on this understanding, a dual-level method of fundamental solutions with self-adaptive adjustment coefficients is proposed. The competitive attributes of the method are that it inherits the high numerical efficiency of the method of fundamental solutions, and it improves numerical stability significantly. The effect of the fictitious boundary on numerical results is eliminated by introducing the concept of equivalent slope. It should be noted that the dual-level method of fundamental solutions can simulate exterior high frequency acoustic problems under the lowest sampling frequency allowed by the Shannon's sampling theorem. Numerical experiment with up to non-dimensional wavenumber of 600 has successfully been conducted on a single laptop when one uses 100,000 degrees of freedom.     

A method of solving the integral equation of plane contact problems for semi-bounded bodies

A typical integral equation, which arises when solving linear plane contact problems for semi-bounded bodies, is considered. By using a special representation of the kernel of this equation, an approximate method is developed for solving it that is effective over a wide range of variation of the dimensionless geometrical parameter occurring in the kernel. The method is tested on the problem of the symmetrical compression of an elastic strip along its boundaries by two similar punches.     

Optimization for exaust emission problems by the simplex method

Summary The rapidly decreasing mineral resources make the manufacturers of automobiles to construct engines with as low consumption as possible.On the other hand, the growing consciousness of environment led many countries to fix limiting values for pollutants by law.Laying out the engine for minimal consumption only, in many cases, the emission of pollutants is so high that the legal limiting values cannot be complied with. Therefore, the task consists in determining the minimal consumption of fuel in consideration of the limiting values for pollutants. This is reported on in Problem 1.In Problem 2, a different kind of minimization problem for exaust emission is investigated.Both problems are solved by the simplex method.

---

Zusammenfassung Die schnell abnehmenden Rohstoffvorräte zwingen die Automobilhersteller dazu, Motoren mit möglichst geringem Verbrauch zu konstruieren.Das wachsende Umweltbewußtsein führte andererseits in den verschiedenen Ländern dazu, daß durch gesetzliche Fahrzyklen Schadstoffgrenzwerte festgelegt wurden.Bei rein verbrauchsoptimaler Auslegung des Motors sind in vielen Fällen die Schadstoffemissionen so hoch, daß die gesetzlichen Schadstoffgrenzen nicht eingehalten werden können. Deshalb besteht die Aufgabe darin, den minimalen Kraftstoffverbrauch unter Berücksichtigung der Schadstoffobergrenzen zu bestimmen. Hierüber wird in Problem 1 berichtet.Problem 2 behandelt eine andere Minimierungsaufgabe bei Abgasemissionen.Beide Aufgaben werden mit der Simplexmethode gelöst.

     

Transient elastodynamic three-dimensional problems in cracked bodies

The general formulation of the transient elastodynamic second boundary value problem in an isotropic linear elastic body with a crack of arbitrary shape by combining the boundary integral equation method and the Laplace transform with respect to time is presented in this paper. Both finite and infinite elastic bodies are considered. A numerical solution of the transformed boundary integral equations is proposed.     

A comparison between the material point method and the finite element method in view of extrusion problems

An appropriate method for the simulation of continuous forming processes is the material point method (MPM) [1],[2] which combines the viewpoints of fluid dynamics and solid mechanics. The MPM and related methods [3] are derived from the particle‐in‐cell methods [4]. Bodies are discretised by Lagragian particles with pointwise mass distributions. The differential equations in their weak form are solved on temporary meshes built of standard finite elements. At the end of each time step the particle positions are updated and the mesh is replaced by a new mesh with a regular shape. The state variables at the nodes of the new mesh are extracted from the state variables at the particles by a transfer algorithm. When particles pass element boundaries, numerical difficulties might be observed. These are eliminated by a smooth approximation of nodal data from material point data. The modified MPM has been implemented together with the FEM in one programme because the similarities of the methods outbalance the differences. On the basis of numerical examples the results of both methods are compared. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variant of the coboundary element method for studying heat-conduction problems in bodies of complicated shape

We propose a scheme for constructing the solution of boundary-value problems for bodies of complicated shape by applying coboundary or boundary elements and integral transforms. We illustrate the method of applying the finite Fourier cosine transform in solving a mixed boundary-value problem for a body having the shape of a trapezoid. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 97–103.     

Propagation of acoustic waves in a fibrous fluid-filled material

The homogenization method is used for the analysis of acoustic wave propagation in a unidirectional fibrous material for the acoustic control of the fiber arrangement in manufacturing composites. Acoustic equations for a rigid periodic structure filled by a nonviscous fluid are obtained by two-dimensional asymptotic expansions. A regular square and triangular arrangement of the fibers with a round cross-section are considered. The analysis reveals that the velocity of acoustic waves is significantly affected by both the volume content and the fiber arrangement.Moscow State Academy of Chemical Engineering, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 5, pp. 651–655 September–October, 1997.     

Application of the method of generalized coupling problems in the thermoelasticity of piecewise-homogeneous bodies under nonideal contact

We analyze the specifics of applying the method of generalized coupling problems to determine and study the temperature fields and the stresses they cause in piecewise-homogeneous bodies under nonideal thermomechanical contact at the interfaces. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 108–116.     

The method of potential functions in problems of the theory of elasticity for bodies with a defect

The method of potential functions using a Fourier transformation in the class of slowly increasing distributions, corresponding to the classical method of complex potentials, is proposed for solving well-known problems of the theory of elasticity for bodies with a defect. It is shown that when a Fourier transformation with respect to all the spatial variables is used, the solution of the dynamic problem of the theory of elasticity can also be represented in terms of a jump in the stresses and displacements at the defect. The correctness of the transformed problem is considered (in terms of an analogue of the Lopatinskii condition). The solution of the system of Helmholtz equations, to which the system of Lamé equations is reduced in the case of the two-dimensional dynamic problem, is expressed in terms of the jump in the stresses and displacements at the defect as a result of solving the corresponding singular integral equations.     

Some optimization problems for elastic bodies

Two optimization problems for finite deformations of an elastic body are formulated. In the first problem, the motion that gives minimum value (maximum value) to a certain functional is chosen from a single family of controllable motions. The necessary conditions for optimality are stated for one specific problem of this type. In the second problem, the minimizing (maximizing) motion is chosen from the class of all possible motions connecting two configurations of the body. The necessary conditions for the optimality of singular solutions are obtained for one specific example of this type.This work was supported by the Pokrajinska Zajednica za Naucni Rad, Novi Sad, Yugoslavia.