Korn's Inequalities for Junctions of Spatial Bodies and Thin Rods

The dependence on the small parameter ε of constants in Korn's inequalities is investigated for domains which are obtained by joining thin rods to an elastic spatial body. The external ends of the rods are clamped. The asymptotic accuracy of the derived inequalities is achieved by certain distribution of weight multipliers and powers of ε in L₂-norms of displacements and their derivatives while the introduced weights differ for longitudinal and transversal components of displacement fields in the rods. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction's body and a great number of ?‐alternating thin rods belonging to two classes. One class consists of rods of finite length, and the second one consists of rods of small length of order . The density of the junction is of order on the rods from the second class and outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ? → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ? → 0, namely the case of ‘light’ concentrated (α ∈ (0,1)), ‘middle’ concentrated (α = 1), and ‘heavy’ concentrated masses (α ∈ (1, + ∞ )) that we divide into ‘slightly heavy’ concentrated (α ∈ (1,2)), ‘intermediate heavy’ concentrated (α = 2), and ‘very heavy’ concentrated masses (α > 2). In the paper, we study in detail the influence of the concentrated masses on the asymptotic behavior if α ∈ (1,2). We construct the leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions and prove the corresponding asymptotic estimates. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotic Analysis and Modeling of the Jointing of a Massive Body with Thin Rods

Asymptotic formulas are obtained for solutions of the anisotropic elasticity problem for a body with cavities into which thin rods are inserted, the outer ends of the rods being rigidly fixed. The surface of the body and the lateral surface of the rods are assumed load-free, but the entire elastic junction is subject to mass forces. The elastic materials are inhomogeneous and the stiffness of the rods may differ greatly from that of the body, their ratio being of the order h with an arbitrary exponent ; for = 0, the junction is homogeneous. Together with the asymptotic formulas, we construct and justify an asymptotic model of the junction. This model is applicable for a wide range of the exponent and preserves the parameter h in the conjugation conditions but is represented by a regularly perturbed problem. Since the leading asymptotic term involves fields with strong singularities, we have to give correct statements of the limit problem for a body with one-dimensional rods. For this purpose, we use the theory of self-adjoint extensions of operators or the technique of weighted spaces with separated asymptotics. The justification of our asymptotic expansions utilizes weighted anisotropic Korn inequalities, which take into account the mutual position of the rods and provide the best possible a priori estimates of the solutions. In contrast to other investigations, we describe, in explicit terms, the dependence of the bounds in the error estimates on the right-hand sides of the original problem. We also discuss the relationship between the asymptotic ansatz formulas and the weighted norms in the asymptotically precise Korn inequality.__________Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 24, pp. 95–214, 2004.     

Homogenization of a parabolic signorini boundary value problem in a thick plane junction

We consider a parabolic Signorini boundary value problem in a thick plane junction Ω _ε which is the union of a domain Ω₀ and a large number of ε−periodically situated thin rods. The Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is done as ε → 0, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove a convergence theorem and show that the Signorini conditions are transformed (as ε → 0) in differential inequalities in the region that is filled up by the thin rods in the limit passage. Bibliography: 31 titles. Illustrations: 1 figure.     

Asymptotic Analysis of an Optimal Control Problem Involving a Thick Two-Level Junction with Alternate Type of Controls

We study the asymptotic behavior (as ε→0) of an optimal control problem in a plane thick two-level junction, which is the union of some domain and a large number 2N of thin rods with variable thickness of order e = O(N^-1).\varepsilon =\mathcal{O}(N^{-1}). The thin rods are divided into two levels depending on the geometrical characteristics and on the controls given on their bases. In addition, the thin rods from each level are ε-periodically alternated and the inhomogeneous perturbed Fourier boundary conditions are given on the lateral sides of the rods. Using the direct method of the calculus of variations and the Buttazzo-Dal Maso abstract scheme for variational convergence of constrained minimization problems, the asymptotic analysis of this problem for different kinds of controls is made as ε→0.     

Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction

A spectral boundary-value problem is considered in a plane thick two-level junction Ω_ε formed as the union of a domain Ω₀ and a large number 2N of thin rods with thickness of order ε = O(N ⁻¹). The thin rods are split into two levels depending on their length. In addition, the thin rods from the indicated levels are ε-periodically alternating. The Fourier conditions are given on the lateral boundaries of the thin rods. The asymptotic behavior of the eigenvalues and eigenfunctions is investigated as ε → 0, i.e., when the number of thin rods infinitely increases and their thickness approaches zero. The Hausdorff convergence of the spectrum is proved as ε → 0, the leading terms of asymptotics are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 195–216, February, 2006.     

Junctions of singularly degenerating domains with different limit dimensions. II

This paper is aimed at finding asymptotic formulas for solutions to the mixed boundary problem for the Poisson equation in a domain obtained by joining singularly degenerating domains. In this paper, which is the second part of the work (the first part was published in No. 18), the main attention is given to three-dimensional problems in which a thin plate or a periodic family of thin rods is joined to a massive body (the distances between the rods are comparable with the diameters of their cross-sections). The distinctive feature of such problems is that an integral equation arises as one of the limit problems. Bibliography: 48 titles. Translated from Trudy Seminara imeni l. G. Petrovskogo, No. 20, pp. 155–195, 1997.     

Asymptotic analysis of elastic curved rods

We consider a sequence of curved rods which consist of isotropic material and which are clamped on the lower base or on both bases. We study the asymptotic behaviour of the stress tensor and displacement under the assumptions of linearized elasticity when the cross‐sectional diameter of the rods tends to zero and the body force is given in the particular form. The analysis covers the case of a non‐smooth limit line of centroids. We show how the body force and the choice of the approximating curved rods can affect the strong convergence and the limit form of the stress tensor for the curved rods clamped on both bases. Copyright © 2006 John Wiley & Sons, Ltd.     

Vibration testing for anomaly detection

In this paper we propose an efficient method to reconstruct a small inclusion buried inside a body using the perturbation of modal parameters measured on the boundary of the body. We design a reconstruction algorithm based on the asymptotic expansions of the eigenvalue perturbations obtained by Ammari and Moskow (Math. Meth. Appl. Sci. 2003; 26 :67–75). We then implement this algorithm and demonstrate its viability and limitations. Copyright © 2008 John Wiley & Sons, Ltd.     

Justification of the asymptotic theory of thin rods. Integral and pointwise estimates

We present a general (without any condition on symmetry) and simplified procedure of obtaining onedimensional equations describing strains of thin rods that can be anisotropic, nonhomogeneous and have periodic structure as well. The presented asymptotics is justified with the help of the weighted Korn inequality, i.e., the difference of the exact solution and an asymptotic solution to the problem of elasticity theory is estimated in the energetic integral metric. Uniform (by the maximum of modulus) estimates for the error of approximation of 3-dimensional displacement fields and stresses are also obtained. As is shown, it is impossible to obtain the pointwise closeness with respect to stresses if the influence of the boundary layer near the end-walls of the rod is not taken into account. Bibliography: 44 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 101–152.     

Derivation of the model of elastic curved rods from three-dimensional micropolar elasticity

In this paper we derive a model of curved elastic rods from the threedimensional linearized micropolar elasticity. Derivation is based on the asymptotic expansion method with respect to the thickness of the rod. The method is used without any a priori assumption on the scaling of the unknowns. The leading term, displacement and microrotation, is identified as the unique solution of a certain one-dimensional problem. Appropriate convergence results are proved.      

Korn's Inequality for an Arbitrary System of Distorted Thin Rods

We derive a weighted and anisotropic Korn inequality for a system of elastic thin rods and verify its asymptotic accuracy. The structure of the norms at the construction elements (rods and knots) is determined from classifying the elements and assigning them to the following categories: movable, hard-movable, and clamped. This classification was never involved while the constraints on the structure of systems of rods in the preceding reserach excluded the presence of movable elements.     

Asymptotic Shapes of large Cells in Random Tessellations

We establish a close relationship between isoperimetric inequalities for convex bodies and asymptotic shapes of large random polytopes, which arise as cells in certain random mosaics in d-dimensional Euclidean space. These mosaics are generated by Poisson hyperplane processes satisfying a few natural assumptions (not necessarily stationarity or isotropy). The size of large cells is measured by a class of general functionals. The main result implies that the asymptotic shapes of large cells are completely determined by the extremal bodies of an inequality of isoperimetric type, which connects the size functional and the expected number of hyperplanes of the generating process hitting a given convex body. We obtain exponential estimates for the conditional probability of large deviations of zero cells from asymptotic or limit shapes, under the condition that the cells have large size. This work was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953. Received: May 2005 Accepted: September 2005     

The asymptotic form of the stressed state near a three-dimensional boundary singularity of the “claw” type

Asymptotic formulae are derived for the fields of displacements, strains and stresses near a peak-shaped protrusion in the surface of an anisotropic elactic body (a “claw”-type singularity). The singular constructed are interpreted as forces and torques concentrated at the tip of the peak, while the orders of growth of the displacement depend on the direction of the action of the force (longitudinal or transverse) and of the axis of the torque (twisting or bending) but not on the elastic properties of the material. The asymptotic analysis makes essential use of the observed analogy with one-dimensional models of thin rods of variable cross-section.     

Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions

Expansions in terms of Bessel functions are considered of the Kummer function ₁ F ₁(a; c, z) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the expansions is explained for large values of the parameter a of the Kummer function.     

Displacement of slender axisymmetric bodies: A convenient change of basis

We present a three parameters transformation useful in the study of the finite displacements of thin rods. The changes of orientation of the body are constructed in two invertible independant steps to which simple physical significations can be attributed: deflection and twist. Different expressions are given and a comparison with the Eulerian angles is made.     

Homogenization of a boundary value problem in a thick cascade junction

We consider a homogenization problem in a singularly perturbed two-dimensional domain of a new type that consists of a junction body and many alternating thin rods of two classes. One of the classes consists of rods of finite length, whereas the other contains rods of small length, and inhomogeneous Fourier boundary conditions (the third type boundary conditions) with perturbed coefficients are imposed on boundaries of thin rods. Homogenization theorems are proved. Bibliography: 38 titles. Illustrations: 2 figures. __________ Translated from Problemy Matematicheskogo Analiza, No. 37, 2008, pp. 47–72.     

On quasiphotons of Rayleigh waves (the case of an anisotropic elastic body)

A procedure of constructing analytic formulas for quasiphotons is developed. Quasiphotons are special asymptotic solutions of linear equations, which describe wave processes. These asymptotic solutions correspond to concentrated wave packets propagating along rays on the surface of an elastic body. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 7–18.     

Asymptotic approximation for the solution to a semilinear parabolic problem in a thin star‐shaped junction

A semilinear parabolic problem is considered in a thin 3‐D star‐shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter The purpose is to study the asymptotic behavior of the solution u_ε as ε→0, ie, when the star‐shaped junction is transformed in a graph. In addition, the passage to the limit is accompanied by special intensity factors and in nonlinear perturbed Robin boundary conditions. We establish qualitatively different cases in the asymptotic behavior of the solution depending on the value of the parameters {α_i}and {β_i}. Using the multiscale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε→0. Namely, in each case, we derive the limit problem (ε=0)on the graph with the corresponding Kirchhoff transmission conditions (untypical in some cases) at the vertex, define other terms of the asymptotic approximation and prove appropriate asymptotic estimates that justify these coupling conditions at the vertex, and show the impact of the local geometric heterogeneity of the node and physical processes in the node on some properties of the solution.     

Estimation of parameters in the discrete distributions of order k

This paper considers estimating parameters in the discrete distributions of order k such as the binomial, the geometric, the Poisson and the logarithmic series distributions of order k. It is discussed how to calculate maximum likelihood estimates of parameters of the distributions based on independent observations. Further, asymptotic properties of estimators by the method of moments are investigated. In some cases, it is found that the values of asymptotic efficiency of the moment estimators are surprisingly close to one.