Green's function in plane anisotropic bimaterials with imperfect interface

^** Email: lsudak{at}ucalgary.ca The fundamental solutions or Green's functions for 2D or 3Danisotropic media with imperfect interface remain a challengingproblem. In this paper, a general method is presented for therigorous solution for the 2D Green's function in an anisotropicelastic bimaterial subject to a line force or a line dislocation.Most significant is the fact that the bonding along the bimaterialinterface is considered to be homogeneous imperfect. Specifically,the tractions are continuous but the displacements are discontinuousand proportional, in terms of interface stiffness parameters,to their respective traction components. Using complex variabletechniques, the basic boundary-value problem for two analyticvector functions is reduced to a coupled linear first-orderdifferential equation for a single analytic vector functiondefined in the lower half space. The coupled linear differentialequation for the single analytic vector function can be subsequentlydecoupled into three independent linear first-order differentialequations for three newly defined analytic functions. Closed-formsolutions for the 2D Green's function are derived in terms ofthe exponential integral. Unlike previous works which involvesome sort of inverse transform method to obtain the physicalquantities from the transform domain, the key feature of thepresent method is that the physical quantities can be readilycalculated without the need to perform any inverse transformoperations.     

基于表面阻抗张量的界面滑移波动态失稳分析

基于Stroh公式和表面阻抗张量理论,提出了研究界面滑移波动态失稳问题的一种新的方法．该方法将表面阻抗张量概念推广到复波速域,并将摩擦接触界面上的边界条件以表面阻抗张量表示．最终将边值问题化归为求解一个复多项式在单位圆内的根．以弹性半空间与刚体平面相对稳态摩擦滑移为例进行了详细的分析,导出了一个4次复特征方程并讨论了方程在单位圆内解的特性,给出了滑移界面波失稳条件的显式解析表达式．     

On the stability of a compressible Rivlin's cube made of transversely isotropic material

^** Email: kostas.soldatos{at}nottingham.ac.uk This paper considers a unit cube made of a compressible, transverselyisotropic elastic material, with the direction of transverseisotropy being aligned normal to one pair of the cube's faces,and investigates the stability of a dilatation equilibrium stateof that cube, with respect to superposed pure homogeneous deformationswith principal directions parallel to the cube edges. This dilatation,intermediate equilibrium state (state I) of the cube is assumedattainable in two different ways. Accordingly, in what is termedas the ‘principal stability problem’ under investigation,state I is considered to be that of uniform dilatation, whichis attained upon loading normally and uniformly the oppositefaces of the unit cube with certain pairs of equal and oppositelydirected forces having appropriately selected magnitudes. Inwhat is termed as the ‘modified stability problem’under investigation, the same compressible, transversely isotropicunit cube is loaded uniformly by three identical pairs of equaland oppositely directed forces acting normally to its faces,and, hence, it attains in state I the shape of a certain rectangularparallelepiped. The necessary and sufficient conditions forstability of state I of the cube deformation are obtained inthe form of three inequalities, which are found to hold regardlessof whether the intermediate equilibrium state I is that of uniformdilatation or that of the aforementioned rectangular parallelepiped(non-uniform dilatation). These, however, lead to quite differentspecific results and conclusions when applied in connectionwith, first, the principal and, then, the modified stabilityproblem of a unit cube made of a particular type of a transverselyisotropic extension of the Blatz–Ko (isotropic) material.     

An explicit expression for the cost of a class of Huffman trees

Let T_n denote a binary tree with n terminal nodes V={υ₁,…,υ_n} and let l_i denote the path length from the root to υ_i. Consider a set of nonnegative numbers W={w₁,…,w_n} and for a permutation π of {1,…,n} to {1,…,n}, associate the weight w_i to the node υ_π(i). The cost of T_n is defined as C(T_n∣W)=Min_π∑ⁿ_i=1w_il_π(i).A Huffman tree H_n is a binary tree which minimizes C(T_n∣W) over all possible T_n. In this note, we give an explicit expression for C(H_n∣W) when W assumes the form: w_i=k for i=1,…,n?m; w_i=x for i=n?m+1,…,n. This simplifies and generalizes earlier results in the literature.     

Unique continuation for the system of elasticity in the plane

We prove the strong unique continuation property for the Lamé system of elastostatics in the plane, , with variable Lamé coefficients , , when is Lipschitz and is measurable.

     

Effect of surface elasticity on the transformation of acoustic disturbances into Tollmien-Schlichting waves in a boundary layer at transonic free-stream velocities

Within the framework of the triple deck theory, the effect of an elastic surface on the characteristics of a wave packet generated by acoustic disturbances in a boundary layer at transonic free-stream velocities is investigated.     

Efficient algorithms for computing the largest eigenvalue of a nonnegative tensor

Consider the problem of computing the largest eigenvalue for nonnegative tensors. In this paper, we establish the Q-linear convergence of a power type algorithm for this problem under a weak irreducibility condition. Moreover, we present a convergent algorithm for calculating the largest eigenvalue for any nonnegative tensors.     

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near cut-off frequencies

A two-dimensional model for extensional motion of a pre-stressedincompressible elastic layer near its cut-off frequencies isderived. Leading-order solutions for displacement and pressureare obtained in terms of the long wave amplitude by direct asymptoticintegration. A governing equation, together with correctionsfor displacement and pressure, is derived from the second-orderproblem. A novel feature of this (two-dimensional) hyperbolicgoverning equation is that, for certain pre-stressed states,time and one of the two (in-plane) spatial variables can changeroles. Although whenever this phenomenon occurs the equationstill remains hyperbolic, it is clearly not wave-like. The second-ordersolution is completed by deriving a refined governing equationfrom the third-order problem. Asymptotic consistency, in thesense that the dispersion relation associated with the two-dimensionalmodel concurs with the appropriate order expansion of the three-dimensionalrelation at each order, is verified. The model has particularapplication to stationary thickness vibration of, or transientresponse to high frequency shock loading in, thin walled bodies.     

An explicit expression for the Fisher information matrix of a multiple time series process

The principal result in this paper is concerned with the derivative of a vector with respect to a block vector or matrix. This is applied to the asymptotic Fisher information matrix (FIM) of a stationary vector autoregressive and moving average time series process (VARMA). Representations which can be used for computing the components of the FIM are then obtained. In a related paper [A. Klein, A generalization of Whittle’s formula for the information matrix of vector mixed time series, Linear Algebra Appl. 321 (2000) 197-208], the derivative is taken with respect to a vector. This is obtained by vectorizing the appropriate matrix products whereas in this paper the corresponding matrix products are left unchanged.     

Capillary waves past a flat plate in water of finite depth

Free-surface flow past a semi-infinite flat plate in a channelof finite depth is considered. The fluid is assumed to be inviscidand incompressible, and the flow to be two-dimensional and irrotational.Surface tension is included in the dynamic boundary conditionbut the effects of gravity are neglected. It is shown that thereis a three-parameter family of solutions with waves in the farfield and a discontinuity in slope at the separation point.This family includes as particular cases the solutions previouslycomputed by Osborn & Stump (2001, Phys. Fluids, 13, 616–623)and by Andersson & Vanden-Broeck (1996, Proc. R. Soc., 452,1985–1997).     

Block structured preconditioners in tensor form for the all-at-once solution of a finite volume fractional diffusion equation

We propose a tensor structured preconditioner for the tensor train GMRES algorithm (or TT-GMRES for short) to approximate the solution of the all-at-once formulation of time-dependent fractional partial differential equations discretized in time by linear multistep formulas used in boundary value form and in space by finite volumes.Numerical experiments show that the proposed preconditioner is efficient for very large problems and is competitive, in particular with respect to the AMEn algorithm.     

An Expression for the Riemann Tensor of a Submanifold Given by a System of Equations in a Riemannian Space

We derive an expression for the Riemann tensor of a submanifold given implicitly by a system of independent equations in a Riemannian space. In particular, we prove a formula for the internal curvature of a two-surface in a three-dimensional Riemannian space. Some applications of the formula are given.     

Subsonic and supersonic disturbances in the generation of surface waves in a liquid layer adjacent to a high-speed gas-stream

In an effort to shed further light upon the nature of “supersonic” disturbances as distinct from that of ‘subsonic’ disturbances in parallel compressible flows, this paper makes an investigation of the stability characteristics of the surface waves generated in a liquid layer adjacent to a high-speed gas-stream. It turns out that the nature of the surface waves generated in the liquid layer depends markedly upon the type of disturbances present in the high-speed gas-stream. For the case of ‘subsonic’ disturbances it is shown that the energy transfer from the gas stream to the surface waves is contributed predominantly by the Fourier component of the normal gas-pressure force-field in phase with the slope of the wavy surface. For the case of ‘supersonic’ disturbances, this energy transfer is shown to be predominantly due to the component of the pressure-field in phase with the surface-wave displacement and is related to the presence of travelling periodic waves in the gas-stream—this energy transfer is shown to promote always the growth of the surface waves.     

On the propagation of Lamb waves in a sandwich plate made of compressible materials with finite initial strains

The propagation of flexural Lamb waves in a prestrained sandwich plate made from compressible highly elastic materials is investigated within the scope of a piecewise homogeneous body model by utilizing TLTEWISB. The mechanical relations of layer materials are described by a harmonic-type potential, and numerical results are obtained for the first and second vibration modes. According to the results, the influence of problem parameters and of the initial stretching strain along the layers on the wave propagation speed is examined. The asymptotic values of the speed are considered in the cases of short and long wavelengths, and the influence of the initial strains on these asymptotic (limit) values are also analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 231–244, March–April, 2008.     

Composite penalty method of a low order anisotropic nonconforming finite element for the Stokes problem

Composite penalty method of a low order anisotropic nonconforming quadrilateral finite element for the Stokes problem is presented. This method with a large penalty parameter can achieve the same accuracy as the stand method with a small penalty parameter and the convergence rate of this method is two times as that of the standard method under the condition of the same order penalty parameter. The superconvergence for velocity is established as well. The results of this paper are also valid to the most of the known nonconforming finite element methods.     

Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals

We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an $\epsilon$-approximation for the $d$-variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on $d$ and logarithmically on ${\epsilon }^{-1}$. The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on $d$ and logarithmic dependence on ${\epsilon }^{-1}$, we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential $(s,t)$-weak tractability with $max(s,t)\ge 1$. For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential $(s,t)$-weak tractability with $max(s,t)<1$ is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).