On the influence of contact mechanics on friction induced flutter instability

Within this contribution, the general form of perturbation equations for systems of moving elastic bodies is derived and examined with respect to symmetry and definiteness properties. The contact is enforced by a penalty formulation, which is motivated by a constitutive contact model relying on statistical data of the surface. Evaluating Hamilton's Principle and carrying out a linearization about a stationary transport motion, the general perturbation equations are derived. The resulting differential operators are stated and discussed. The findings are exemplified for a rotating cicular Timoshenko beam with frictional contacts. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the acoustics of sonic composites

The primary goal of this paper is to propose an alternative method for obtaining the band structures of the 3D sonic composites without/ with point defects. The point defects are vacancies or foreign interstitial atoms which are supported by the interfaces between the hollow spheres and the matrix. The proposed method is used to simulate a sonic plate composed of an array of acoustic scatterers which are piezoceramic hollow spheres embedded in an epoxy matrix. The scatterers are made from functionally graded materials with radial polarization [1, 2]. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On wellposedness in nonlinear acoustics

Motivated by a medical application from lithotripsy, we study the initial–boundary value problem given by Westervelt equation (1) in a bounded domain Ω. This models the nonlinear evolution of the acoustic pressure u excited at a part Γ₀ of the boundary. Along with the excitation given by Neumann boundary condition as in (1) , we also consider the Dirichlet type of excitation. Whereas shock waves are known to emerge after a sufficiently large time interval for appropriate initial and boundary conditions, we here prove existence and uniqueness as well as stability of a solution u for small data g, u₀ and u₁ or short time T, using a fixed point argument. Moreover we extend the result to the more general model given by the Kuznetsov equation (2) for the acoustic velocity potential ψ. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Damped wave equation in the subcritical case

We study large time asymptotics of small solutions to the Cauchy problem for the one dimensional nonlinear damped wave equation

(1)     

Some problems in acoustics of emulsions

The authors study small vibrations of a mixture (emulsion) of two weakly viscous compressible fluids, construct a macroscopic (homogenized) model of emulsion, and establish convergence (with respect to a small parameter) of solutions of the original boundary value problem for a two-phase fluid to solutions of the corresponding homogenized problem. The paper also describes the results of qualitative analysis of the spectrum of the macroscopic acoustic equation (the dynamical Darcy’s law), as well as typical spectral pictures obtained by numerical experiments.     

Asymptotic study of the elastic seadbed effects in ocean acoustics

A theoretical and asymptotic investigation of the Green' function for the system governing the propagation of time-harmonic acoustic waves in a horizontally stratified ocean with an elastic seabed is presented. Employing the surface Neumann-to-Dirichlet map for the elastic half space, we reduce the problem to an equivalent one in the layer, with a nonlocal boundarycondition at the fluid-bottom interface. The reduced problem is transformedby Hankel transform, to a non-selfadjoint boundary value problem for a second-order ordinary differential equation over the layer depth. The well posedness of this problem is investigated applying analytic Redholm theory for an equivalent Lippmann-Schwinger integral equation. An asymptotic expansionof the transformed nonlocal boundary condition is constructed in the case of a seabed with small shear modulus, and it is used to show that the Green function is a regular perturbation of that one in the case of a fluid bottom.     

Nonlinear acoustics of bounded solid

Reflection and refraction are investigated when the incidental angles of transverse waves are larger than the critical one. An interesting result is obtained that at this angle all the second-order displacements on the stress-free surface have a very strong maximum module and a phase leap; thereupon an appropriate second-order harmonic surface wave could be produced. Project supported by the National Natural Science Foundation of China (Grant No. 19334062).     

Subgraph statistics in subcritical graph classes

Let H be a fixed graph and a subcritical graph class. In this paper we show that the number of occurrences of H (as a subgraph) in a graph in of order n, chosen uniformly at random, follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations [Drmota, Gittenberger, and Morgenbesser, Submitted]. As a case study, we obtain explicit expressions for the number of triangles and cycles of length 4 in the family of series‐parallel graphs. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 631–673, 2017     

Gaussian fluctuations of connectivities in the subcritical regime of percolation

Summary We consider thed-dimensional Bernoulli bond percolation model and prove the following results for allp _c : (1) The leading power-law correction to exponential decay of the connectivity function between the origin and the point (L, 0, ..., 0) isL ^–(d–1)/2 . (2) The correlation length, (p) is real analytic. (3) Conditioned on the existence of a path between the origin and the point (L, 0, ..., 0), the hitting distribution of the cluster in the intermediate planes,x ₁ =qL,0, obeys a multidimensional local limit theorem. Furthermore, for the two-dimensional percolation system, we prove the absence of a roughening transition: For allp>p _c , the finite-volume conditional measures, defined by requiring the existence of a dual path between opposing faces of the boundary, converge—in the infinite-volume limit—to the standard Bernoulli measure.Work supported in part by G.N.A.F.A. (C.N.R.)Work supported in part by NSF Grant No. DMS-88-06552     

Transmutation of non-local boundary conditions in ocean acoustics

We employ a transmutation technique to construct a new non-local boundary condition for the paraxial approximation in ocean acoustics. The transmutation operator introduced by De Santo and Polyanskii, when applied to the Helmholtz equation governing the acoustic pressure in the water column, leads to the so-called parabolic equation of Fock and Tappert. This transmutation operator acting on the N-D map at the water–bottom interface yields an intermediate non-local boundary condition for the parabolic equation which eliminates the backscattering terms in the Schwartz kernel of the N-D map. The kernel of the intermediate condition is approximated by a uniform stationary phase formula taking account of the possible coalescence of the brach points of the integrand with the stationary points of the phase, and it leads to a non-local boundary condition of Volterra-type for the parabolic equation. This condition is quite different than similar conditions derived by other approximations, in that the kernel of the Volterra operator is smooth, the smoothing effect coming from the fact that the horizontal range coordinate is scaled with the relative refraction index between the water column and the bottom.     

Derivation of the equations of nonisothermal acoustics in elastic porous media

We consider the problem of the joint motion of a thermoelastic solid skeleton and a viscous thermofluid in pores, when the physical process lasts for a few dozens of seconds. These problems arise in describing the propagation of acoustic waves. We rigorously derive the homogenized equations (i.e., the equations not containing fast oscillatory coefficients) which are different types of nonclassical acoustic equations depending on relations between the physical parameters and the homogenized heat equation. The proofs are based on Nguetseng’s two-scale convergence method.     

On the simulation source technique for exterior problems in acoustics

For the solution of the exterior Dirichlet problem for the Helmholtz equation we propose to seek the solution in the form of an acoustic single-layer potential with a distribution extended over an internal surface. This leads to an ill-posed integral equation of the first kind which can be approximately solved by the Tikhonov regularization technique.     

On solving structural acoustics problems in time-domain

The structural acoustics problem is formulated as a hyperbolic system of conservation laws which leads to an abstract Cauchy problem in common Hilbert space settings. The Cauchy problem is approximated by using high-order, multi-stage Taylor-Galerkin methods which provide high-order temporal accuracy and unconditional stability on arbitrary (unstructured) finite element grids. The formulation is extended to problems posed on unbounded domains by introduction of iterative radiation boundary conditions. The proposed approaches are shown to produce very good results for the test cases considered.     

Uniqueness and nonuniqueness of the solution of boundary-contact acoustics problems

One establishes uniqueness theorems and one gives examples of nonuniqueness for a series of boundary-contact acoustics problems, i.e. diffraction problems under boundary conditions, described by a higher order operator and vanishing at isolated points.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 21–32, 1983.     

Relaxation of multiple integrals in subcritical Sobolev spaces

The study of lowersemicontinuity properties in W^1·p of multiple integrals with quasiconvex integrands withq growth is undertaken forp >q ? 1. In the case where the integrand is not quasiconvex, the absolutely continuous part of the relaxed functional is analyzed.     

Estimates for the p-Laplacian-type operator in the subcritical Sobolev exponent range

We obtain Calderón–Zygmund-type gradient estimates below the duality exponent of very weak solutions to p-Laplacian-type elliptic equations with non-divergence datum by providing an analytic approach without using the fractional maximal function of order 1 for the non-divergence datum.     

Investigation of the dispersion equation for an acoustics problem in a borehole

One gives the results of the analytic and numerical analysis of the roots of the dispersion equation for a model of a log borehole. One investigates the effect of the elastic parameters and of the densities on the dispersion and the damping of the waves in the case of radial oscillations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 89–94, 1983.     

Uniform Decay properties of a model in structural acoustics

We investigate decay properties for a system of coupled partial differential equations which model the interaction between acoustic waves in a cavity and the walls of the cavity. In this system a wave equation is coupled to a structurally damped plate or beam equation. The underlying semigroup for this system is not uniformly stable, but when the system is appropriately restricted we obtain some uniform stability. We present two results of this type. For the first result, we assume that the initial wave data is zero, and the initial plate or beam data is in the natural energy space; then the corresponding solution to system decays uniformly to zero. For the second result, we assume that the initial condition is in the natural energy space and the control function is L²(0,∞) (in time) into the control space; then the beam displacement and velocity are both L²(0,∞) into a space with two spatial derivatives.     

The evolution of subcritical Achlioptas processes

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. AlthouSgh the evolution of such ‘local’ modifications of the Erd?s–Rényi random graph process has received considerable attention during the last decade, so far only rather simple rules are well understood. Indeed, the main focus has been on ‘bounded‐size’ rules, where all component sizes larger than some constant B are treated the same way, and for more complex rules very few rigorous results are known. In this paper we study Achlioptas processes given by (unbounded) size rules such as the sum and product rules. Using a variant of the neighbourhood exploration process and branching process arguments, we show that certain key statistics are tightly concentrated at least until the susceptibility (the expected size of the component containing a randomly chosen vertex) diverges. Our convergence result is most likely best possible for certain generalized Achlioptas processes: in the later evolution the number of vertices in small components may not be concentrated. Furthermore, we believe that for a large class of rules the critical time where the susceptibility ‘blows up’ coincides with the percolation threshold. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 174–203, 2015