Investigation of the Thermoelastoplastic State of Locally Heated Shallow Shells of Double Curvature

We develop a numerical-analytic method for calculating the thermoelastoplastic state of shallow shells of arbitrary curvature under conditions of heating by localized heat sources. For solving auxiliary problems that arise in applying the method of additional deformations, we use the Fourier integral transformation and integral representations of a solution. Detailed investigations were performed for shells of cylindrical, spherical, and ellipsoidal shapes. We establish the dependence of the distribution of thermal and residual stresses on the curvatures of a shell and on the mode of heating. We consider the limiting case where the construction has the shape of a plate.     

A representation formula for the voltage perturbations caused by diametrically small conductivity inhomogeneities. Proof of uniform validity

We revisit the asymptotic formulas originally derived in [D.J. Cedio-Fengya, S. Moskow, M.S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems 14 (1998) 553–595; A. Friedman, M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence, Arch. Ration. Mech. Anal. 105 (1989) 299–326]. These formulas concern the perturbation in the voltage potential caused by the presence of diametrically small conductivity inhomogeneities. We significantly extend the validity of the previously derived formulas, by showing that they are asymptotically correct, uniformly with respect to the conductivity of the inhomogeneities. We also extend the earlier formulas by allowing the conductivities of the inhomogeneities to be completely arbitrary L^∞$L^{\infty }$, positive definite, symmetric matrix-valued functions. We briefly discuss the relevance of the uniform asymptotic validity, and the admission of arbitrary anisotropically conducting inhomogeneities, as far as applications of the perturbation formulas to “approximate cloaking” are concerned.     

Nonshallow spherical shells

In this paper, spherical bodies of shell type are discussed, where the displacement vector is independent of the thickness coordinate. Their internal geometry is alterable towards the thickness (nonshallow spherical shells). A planar deformation model for spherical bodies of shell type is obtained. General representations of the system of equilibrium equations are expressed with the help of three holomorphic functions for the spherical shells. The components of the stresses and displacements and the boundary conditions for the components of the stresses and displacements are also expressed with the help of three holomorphic functions. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.     

Electro‐magneto‐encephalography for the three‐shell model : a single dipole in ellipsoidal geometry

Analytic formulae are presented for determining the position and the moment of a single dipole from either Electroencephalography or Magnetoencephalography (MEG) measurements, under the assumption of the three‐shell ellipsoidal model. It is remarkable that for this model all three components of the moment can be determined from the MEG measurements. This means that, in contrast to the spherical model, there exist no silent dipolar sources in MEG for the ellipsoidal model. Copyright © 2012 John Wiley & Sons, Ltd.     

Electroencephalography in ellipsoidal geometry

The human brain is shaped in the form of an ellipsoid with average semiaxes equal to 6, 6.5 and 9 cm. This is a genuine 3-D shape that reflects the anisotropic characteristics of the brain as a conductive body. The direct electroencephalography problem in such anisotropic geometry is studied in the present work. The results, which are obtained through successively solving an interior and an exterior boundary value problem, are expressed in terms of elliptic integrals and ellipsoidal harmonics, both in Jacobian as well as in Cartesian form. Reduction of our results to spheroidal as well as to spherical geometry is included. In contrast to the spherical case where the boundary does not appear in the solution, the boundary of the realistic conductive brain enters explicitly in the relative expressions for the electric field. Moreover, the results in all three geometrical models reveal that to some extend the strength of the electric source is more important than its location.     

An accurate formula for the reconstruction of conductivity inhomogeneities

We carefully derive accurate asymptotic expansions of the steady-state voltage potentials in the presence of a finite number of diametrically small inhomogeneities with conductivities different from the background conductivity. We then apply these accurate asymptotic formulae for the purpose of identifying the location and certain properties of the shape of the conductivity anomaly. Our designed real-time algorithm makes use of constant current sources. It is based on the observation in both the near and far field of the pattern of a simple weighted combination of the input currents and the output voltages. The mathematical analysis provided in this paper indicates that our algorithm is with a very high resolution and accuracy.     

Discontinuous symplectic capacities

We show that the spherical capacity is discontinuous on a smooth family of ellipsoidal shells. Moreover, we prove that the shell capacity is discontinuous on a family of open sets with smooth connected boundaries.     

Homogenization of high-contrast two-phase conductivities perturbed by a magnetic field. Comparison between dimension two and dimension three

Homogenized laws for sequences of high-contrast two-phase non-symmetric conductivities perturbed by a parameter $h$ are derived in two and three dimensions. The parameter $h$ characterizes the antisymmetric part of the conductivity for an idealized model of a conductor in the presence of a magnetic field. In dimension two an extension of the Dykhne transformation to non-periodic high conductivities permits to prove that the homogenized conductivity depends on $h$ through some homogenized matrix-valued function obtained in the absence of a magnetic field. This result is improved in the periodic framework thanks to an alternative approach, and illustrated by a cross-like thin structure. Using other tools, a fiber-reinforced medium in dimension three provides a quite different homogenized conductivity.     

Comment on “On the Neumann function and the method of images in spherical and ellipsoidal geometry”

In this note, we point out two errors in the article “On the Neumann function and the method of images in spherical and ellipsoidal geometry” by Dassios and Sten. Two corrections are then proposed.     

Forced Vibration of Three-Layered Spherical and Ellipsoidal Shells under Axisymmetric Loads

A variant of vibration theory for three-layered shells of revolution under axisymmetric loads is elaborated by applying independent kinematic and static hypotheses to each layer, with account of transverse normal and shear strains in the core. Based on the Reissner variational principle for dynamic processes, equations of nonlinear vibrations and natural boundary conditions are obtained. The numerical method proposed for solving initial boundary-value problems is based on the use of integrodifferential approach for constructing finite-difference schemes with respect to spatial and time coordinates. Numerical solutions are obtained for dynamic deformations of open three-layered spherical and ellipsoidal shells, over a wide range of geometric and physical parameters of the core, for different types of boundary conditions. A comparative analysis is given for the results of investigating the dynamic behavior of three-layered shells of revolution by the equations proposed and the shell equations of Timoshenko and Kirhhoff-Love type, with the use of unified hypotheses across the heterogeneous structure of shells.     

Boundary integral formulations for the forward problem in magnetic induction tomography

In this paper we present two models for the forward problem of magnetic induction tomography. In particular, we describe the eddy current model, and a reduced simplified model. The error between the reduced and the full model is analyzed in dependence of parameters such as the frequency and the conductivity. In the case of a piecewise constant conductivity we derive a boundary integral formulation for the reduced model. Finally, we comment on numerical results for the forward problem and give a comparison of both models. Copyrightcopyright 2011 John Wiley & Sons, Ltd.     

Analysis of glass-reinforced plastic bottoms of cylindrical pressure vessels

Solutions are proposed for boundary value problems associated with spherical, ellipsoidal and optimal-design glass-reinforced plastic bottoms filament-wound as an integral part of cylindrical shells.Moscow Region. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 918–926, September–October, 1968.     

Finite element study of nonlinear two-dimensional deoxygenated biomagnetic micropolar flow

In this paper, we consider the two-dimensional fully-developed steady, viscous hydrodynamic flow of a deoxygenated biomagnetic micropolar fluid, in an (X, Y) coordinate system. The momentum conservation equations with zero-pressure gradient are extended to incorporate the X- and Y-components of the biomagnetic body force term with appropriate boundary conditions. The equations are non-dimensionalized using a set of transformations. A finite element solution is obtained to the resulting non-dimensional model and the effects of biomagnetic number (N_H), micropolar microinertia parameter (B) and micropolar viscosity ratio parameter (R) on the X- and Y-direction velocity profiles and micro-rotation (N) is studied in detail. Translational velocities (U, V) are seen to be reduced with an increase in micropolarity (R) and also biomagnetic effects (N_H). Conversely the velocities are increased with a rise in microinertia parameter (B). Several special cases, e.g. Newtonian biomagnetic physiological flow, are also discussed. The model finds applications in blood flow in biomedical device technology (e.g. oxygenators), hemodynamics under strong external magnetic fields, magnetic drug carrier analysis, etc.     

Polarization tensors of planar domains as functions of the admittivity contrast

(Electric) polarization tensors describe part of the leading order term of asymptotic voltage perturbations caused by low volume fraction inhomogeneities of the electrical properties of a medium. They depend on the geometry of the support of the inhomogeneities and on their admittivity contrast. Corresponding asymptotic formulas are of particular interest in the design of reconstruction algorithms for determining the locations and the material properties of inhomogeneities inside a body from measurements of current flows and associated voltage potentials on the body’s surface. In this work, we consider the two-dimensional case only and provide an analytic representation of the polarization tensor in terms of spectral properties of the double layer integral operator associated with the support of simply connected conductivity inhomogeneities. Furthermore, we establish that an (infinitesimal) simply connected inhomogeneity has the shape of an ellipse, if and only if the polarization tensor is a rational function of the admittivity contrast with at most two poles whose residues satisfy a certain algebraic constraint. We also use the analytic representation to provide a proof of the so-called Hashin–Shtrikman bounds for polarization tensors; a similar approach has been taken previously by Golden and Papanicolaou and Kohn and Milton in the context of anisotropic composite materials.     

弹性动力学的等效内含物法和两椭球异质体的散射场

用Gurtin变分原理推导了一般线弹性动力学的多异质体问题的等效性方程.用“等效内含物法”求得两椭球异质体对入射弹性波的散射场和散射横截面的近似表达式,并给出了计算实例.     

双层Kidder自相似解及其Rayleigh-Taylor不稳定性研究

将单层Kidder自相似解推广到双层,使得两层壳体的交界面两侧存在密度跳跃,使得轻流体向重流体加速产生Rayleigh-Taylor不稳定性;通过采用Lagrange坐标下的Godunov方法进行一维直接数值模拟,将模拟解与双层Kidder自相似基本解进行比较,验证了双层Kidder自相似解的可靠性;最后,通过编制球形内爆的三维扰动的线性稳定性分析程序,对双层Kidder自相似解的Rayleigh-Taylor不稳定性进行了分析计算.计算结果表明:初始扰动越集中于交界面,会造成后期扰动增长得越快,越不稳定;扰动波数越大,扰动增长得越快,越不稳定;从扰动在空间上的发展来看,可压缩性研究表明内外壳体的可压缩性对扰动增长起着相反的作用,外层壳体的可压缩性对Rayleigh-Taylor不稳定起失稳作用,而内层壳体的可压缩性对Rayleigh-Taylor不稳定起致稳作用.     

Ellipsoidal classification via semidefinite programming

We propose a classification approach exploiting relationships between ellipsoidal separation and Support-vector Machine (SVM) with quadratic kernel. By adding a (Semidefinite Programming) SDP constraint to SVM model we ensure that the chosen hyperplane in feature space represents a non-degenerate ellipsoid in input space. This allows us to exploit SDP techniques within Support-vector Regression (SVR) approaches, yielding better results in case ellipsoid-shaped separators are appropriate for classification tasks. We compare our approach with spherical separation and SVM on some classification problems.     

Asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities

We consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution can be arbitrary large. In this paper we derive an asymptotic formula of the solution, which characterizes the gradient blow-up of the solution in terms of conductivities of inclusions as well as the distance between inclusions. The asymptotic formula is expressed in bipolar coordinates in terms of the Lerch transcendent function, and it is valid for inclusions with arbitrary constant conductivities. We illustrate our results with numerical calculations.     

球壳的环向剪切屈曲

通过球壳微元初始屈曲的微分几何分析,推导出一组新的精确的屈曲分支方程,并且应用Galerkin变分法研究铰支球壳承受环向剪切力时的整体稳定性,构造了接近分支点变形状态的屈曲模式,首次求得了从扁球壳到半球壳大范围内的扭转屈曲临界特征值,临界荷载强度和临界应力.     

A model for the corona of a spherical star cluster

A general model for the outer layers of steady spherical star systems is considered. The corona is assumed to be formed by stars flying out of the system along ballistic orbits. Expressions for density runs and velocity dispersions are found. Models of three kinds are studied, namely, systems with isotropic velocity distribution, spheres with purely radial orbits, and models with ellipsoidal velocity distribution. The following isotropic distribution functions were considered as examples: the truncated Maxwellian distribution, King’s model, and the spherical analogue of Perek’s model. The truncated Maxwellian distribution of radial velocities is an example of a radial orbit system. The truncated Schwarzschildian velocity distribution and generalized polytropes introduced by Kuzmin and Veltmann are examples of models with ellipsoidal velocity distribution. Density asymptotics are found for all models provided that the self-gravitation of coronas is neglected.