Application of the method of direct separation of motions to the parametric stabilization of an elastic wire

The paper considers the application of the method of direct separation of motions to the investigation of distributed systems. An approach is proposed which allows one to apply the method directly to the initial equation of motion and to satisfy all boundary conditions, arising for both slow and fast components of motion. The methodology is demonstrated by means of a classical problem concerning the so-called Indian magic rope trick (Blekhman et al. in Selected topics in vibrational mechanics, vol. 11, pp. 139–149, [2004]; Champneys and Fraser in Proc. R. Soc. Lond. A 456:553–570, [2000]; in SIAM J. Appl. Math. 65(1):267–298, [2004]; Fraser and Champneys in Proc. R. Soc. Lond. A 458:1353–1373, [2002]; Galan et al. in J. Sound Vib. 280:359–377, [2005]), in which a wire with an unstable upper vertical position is stabilized due to vertical vibration of its bottom support point. The wire is modeled as a heavy Bernoulli–Euler beam with a vertically vibrating lower end. As a result of the treatment, an explicit formula is obtained for the vibrational correction to the critical flexural stiffness of the nonexcited system.     

Sharp Observability Estimates for Heat Equations

The goal of this article is to derive new estimates for the cost of observability of heat equations. We have developed a new method allowing one to show that when the corresponding wave equation is observable, the heat equation is also observable. This method allows one to describe the explicit dependence of the observability constant on the geometry of the problem (the domain in which the heat process evolves and the observation subdomain). We show that our estimate is sharp in some cases, particularly in one space dimension and in the multi-dimensional radially symmetric case. Our result extends those in Fattorini and Russell (Arch Rational Mech Anal 43:272–292, 1971) to the multi-dimensional setting and improves those available in the literature, namely those by Miller (J Differ Equ 204(1):202–226, 2004; SIAM J Control Optim 45(2):762–772, 2006; Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei (9) Mat Appl 17(4):351–366, 2006) and Tenenbaum and Tucsnak (J Differ Equ 243(1):70–100, 2007). Our approach is based on an explicit representation formula of some solutions of the wave equation in terms of those of the heat equation, in contrast to the standard application of transmutation methods, which uses a reverse representation of the heat solution in terms of the wave one. We shall also explain how our approach applies and yields some new estimates on the cost of observability in the particular case of the unit square observed from one side. We will also comment on the applications of our techniques to controllability properties of heat-type equations.     

A Zonal RANS/LES Approach for Noise Sources Prediction

Hybrid CFD/CAA methods have generally to be used for the numerical simulation of trailing-edge noise (see [9, 20] for instance). This study focuses on the first step of such hybrid methods, which is to predict the unsteady aerodynamic sources by the mean of a 3D unsteady simulation of the flow. Such a simulation is however generally still away from the numerical capabilities of ‘usual’ supercomputers. This paper investigates the use of a zonal LES method (based on the NLDE – Non-Linear Disturbance Equations – technique) for the numerical prediction of the aerodynamic noise sources. This method makes it possible to perform only zonal LES close to the main elements responsible of sound generation, while the overall configuration is only treated by a RANS approach. Attention will be paid to the specific boundary treatment at the interface between the RANS and LES regions. More precisely, the problem of the generation of turbulent inflow conditions for the LES region will be carefully addressed. The method is first assessed in the simulation of a flat plate ended by a blunted trailing-edge, and then applied to the simulation of the flow over a NACA0012 airfoil with blunted trailing-edge.     

A numerical simulation of external heat transfer around turbine blades

External heat transfer prediction is performed in two-dimensional turbine blade cascades using the Reynolds-averaged Navier–Stokes equations. For this purpose, six different turbulence models including the algebraic Baldwin–Lomax (AIAA paper 78-257, 1978), three low-Re k−ɛ models (Chien in AIAA J 20:33–38, 1982; Launder and Sharma in Lett Heat Mass Transf 1(2):131–138, 1974; Biswas and Fukuyama in J Turbomach 116:765–773, 1994), and two k−ω models (Wilcox in AIAA J 32(2):247–255, 1994) are taken into account. The computer code developed employs a finite volume method to solve governing equations based on an explicit time marching approach with capability to simulate subsonic, transonic and supersonic flows. The Roe method is used to decompose the inviscid fluxes and the gradient theorem to decompose viscous fluxes. The performance of different turbulence models in prediction of heat transfer is examined. To do so, the effect of Reynolds and Mach numbers along with the turbulent intensity are taken into account, and the numerical results obtained are compared with the experimental data available.     

Stability and Freezing of Nonlinear Waves in First Order Hyperbolic PDEs

It is a well-known problem to derive nonlinear stability of a traveling wave from the spectral stability of a linearization. In this paper we prove such a result for a large class of hyperbolic systems. To cope with the unknown asymptotic phase, the problem is reformulated as a partial differential algebraic equation for which asymptotic stability becomes usual Lyapunov stability. The stability proof is then based on linear estimates from (Rottmann-Matthes, J Dyn Diff Equat 23:365–393, 2011) and a careful analysis of the nonlinear terms. Moreover, we show that the freezing method (Beyn and Thümmler, SIAM J Appl Dyn Syst 3:85–116, 2004; Rowley et al. Nonlinearity 16:1257–1275, 2003) is well-suited for the long time simulation and numerical approximation of the asymptotic behavior. The theory is illustrated by numerical examples, including a hyperbolic version of the Hodgkin–Huxley equations.     

Elastic Fields for the Ellipsoidal Cavity Problem

Lur’e (Three-dimensional Problem of the Theory of Elasticity. Interscience, New York, 1964, §6.9) presented an approach to solve the problem of an ellipsoidal cavity in a linear, elastic and isotropic medium loaded by uniform principal stresses at infinity. In this paper we show that the approach by Lur’e may have no solution. Derivation mistakes are first pointed out in his (6.9.22), (6.9.23), (6.9.30) and (6.9.31). With the correct expressions, we then prove the coefficient matrix in his (6.9.32) to be singular. Therefore constants A,A ₄,A ₅ may have no solution. The problem lies in the harmonic functions chosen by Lur’e for the Papkovich-Neuber solution. From the solutions obtained by the Eshelby equivalent inclusion method, the present paper derives new Papkovich-Neuber harmonic functions for the ellipsoidal cavity problem.     

An Optimal Condition of Compactness for Elasticity Problems Involving one Directional Reinforcement

This paper deals with the homogenization of a homogeneous elastic medium reinforced by very stiff strips in dimension two. We give a general condition linked to the distribution and the stiffness of the strips, under which the nature of the elasticity problem is preserved in the homogenization process. This condition is sharper than the one used in Briane and Camar-Eddine (J. Math. Pures Appl. 88:483–505, 2007) and is shown to be optimal in the case where the strips are periodically arranged. Indeed, a fourth-order derivative term appears in the limit equation as soon as the condition is no more satisfied. In the periodic case the influence of oscillations in the medium surrounding the strips is also considered. The homogenization method is based both on a two-scale convergence for the strips and the use of suitable oscillating test functions. This allows us to obtain a distributional convergence of two of the three entries of the stress tensor contrary to the Γ-convergence approach of Briane and Camar-Eddine (J. Math. Pures Appl. 88:483–505, 2007).     

A Fractional Model of Continuum Mechanics

Although there has been renewed interest in the use of fractional models in many application areas, in reality fractional analysis has a long and distinguished history and can be traced back to the likes of Leibniz (Letter to L’Hospital, 1695), Liouville (J. éc. Polytech. 13:71, 1832), and Riemann (Gesammelte Werke, p. 62, 1876). Recent publications (Podlubny in Math. Sci. Eng. 198, 1999; Sabatier et al. in Advances in fractional calculus: theoretical developments and applications in physics and engineering, Springer, Berlin, 2007; Das in Functional fractional calculus for system identification and controls, Springer, Berlin, 2007) demonstrate that fractional derivative models have found widespread applications in science and engineering. Late fundamental considerations have led to the introduction of fractional calculus in continuum mechanics in an attempt to develop non-local constitutive relations (Lazopoulos in Mech. Res. Commun. 33:753–757, 2006). Attempts have also been made to model microscopic forces using fractional derivatives (Vazquez in Nonlinear waves: classical and quantum aspects, pp. 129–133, 2004). Our approach in this paper differs from previous theoretical work, in that we develop a general framework directly from the classical continuum mechanics, by defining the laws of motion and the stresses using fractional derivatives. The timeliness and relevance of this work is justified by the surge in interest in applications of fractional order models to biological, physical and economic systems. The aim of the present paper is to lay the foundations for a new non-local model of continuum mechanics based on fractional order derivatives which we will refer to as the fractional model of continuum mechanics. Following the theoretical development, we apply this framework to two one-dimensional model problems: the deformation of an infinite bar subjected to a self-equilibrated load distribution, and the propagation of longitudinal waves in a thin finite bar.     

Normal mode analysis of the fully developed free convection flow in a vertical slot with open to capped ends

The fully developed free convection flow in a differentially heated vertical slot with open to capped ends investigated recently by Bühler (Heat Mass Transf 39:631–638, 2003) and Weidman (Heat Mass Transf Online First, February 2006) is revisited in this paper. A new method of solution of the corresponding fourth order boundary value problem, based on its reduction to “normal modes” by a complex matrix similarity transformation is presented. As a byproduct of the method, some invariant relationships involving the heat flux and the shear stress in the flow could be found.     

Application of Rheological Models in Prediction of Turbulent Slurry Flow

The paper deals with fully developed steady turbulent flow of slurry in a circular straight and smooth pipe. The Kaolin slurry consists of very fine solid particles, so the solid particles concentration, and density, and viscosity are assumed to be constant across the pipe. The mathematical model is based on the time averaged momentum equation. The problem of closure was solved by the Launder and Sharma k-ε turbulence model (Launder and Sharma, Lett Heat Mass Transf 1:131–138, 1974) but with a different turbulence damping function. The turbulence damping function, used in the mathematical model in the present paper, is that proposed by Bartosik (1997). The mathematical model uses the apparent viscosity concept and the apparent viscosity was calculated using two- and three-parameter rheological models, namely Bingham and Herschel–Bulkley. The main aim of the paper is to compare measurements and predictions of the frictional head loss and velocity distribution, taking into account two- and three-parameter rheological models, namely Bingham and Herschel–Bulkley, if the Kaolin slurry possesses low, moderate, and high yield stress. Predictions compared with measurements show an observable advantage of the Herschel–Bulkley rheological model over the Bingham model particularly if the bulk velocity decreases.     

The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 1: polycrystalline plasticity

A general set of flow laws and associated variational formulations are constructed for small-deformation rate-independent problems in strain-gradient plasticity. The framework is based on the thermodynamically consistent theory due to Gurtin and Anand (J Mech Phys Solids 53:1624–1649, 2005), and includes as variables a set of microstresses which have both energetic and dissipative components. The flow law is of associative type. It is expressed as a normality law with respect to a convex but otherwise arbitrary yield function, or equivalently in terms of the corresponding dissipation function. Two cases studied are, first, an extension of the classical Hill-Mises or J ₂ flow law and second, a form written as a linear sum of the magnitudes of the plastic strain and strain gradient. This latter form is motivated by work of Evans and Hutchinson (Acta Mater 57:1675–1688, 2009) and Nix and Gao (J Mech Phys Solids 46:411–425, 1998), who show that it leads to superior correspondence with experimental results, at least for particular classes of problems. The corresponding yield function is obtained by a duality argument. The variational problem is based on the flow rule expressed in terms of the dissipation function, and the problem is formulated as a variational inequality in the displacement, plastic strain, and hardening parameter. Dissipative components of the microstresses, which are indeterminate, are absent from the formulation. Existence and uniqueness of solutions are investigated for the generalized Hill-Mises and linear-sum dissipation functions, and for various combinations of defect energy. The conditions for well-posedness of the problem depend critically on the choice of dissipation function, and on the presence or otherwise of a defect energy in the plastic strain or plastic strain gradient, and of internal-variable hardening.     

Inverse dynamics analysis of a general spherical star-triangle parallel manipulator using principle of virtual work

Inverse dynamics of a general model of a spherical star-triangle (SST) parallel manipulator (Enferadi and Akbarzadeh Tootoonchi, Robotica 27:663–676, 2009) is the subject of this paper. This manipulator is of type 3-RRP, has good accuracy and relatively a large workspace which is free of singularities (Enferadi and Akbarzadeh Tootoonchi, Robotica, Revised paper, 2009). First, inverse kinematics utilizing the angle axis representation is solved. Next, velocity and acceleration analysis as well as link Jacobian matrices are obtained in invariant form. Finally, a systematic approach based on the principle of virtual work and the concept of link Jacobian matrices is presented. This method allows elimination of constraint forces and moments at the passive joints from motion equations. It is shown that the dynamics of the manipulator can be reduced to solving a system of three linear equations with three unknowns. Moreover, a computational algorithm for solving the inverse dynamics is developed. Two examples with different trajectories for the moving spherical platform are presented and motor torques are obtained. Results are verified using a commercial dynamics modeling package.     

Structural Optimization of Thin Elastic Plates: The Three Dimensional Approach

The natural way to find the most compliant design of an elastic plate is to consider the three-dimensional elastic structures which minimize the work of the loading term, and pass to the limit when the thickness of the design region tends to zero. In this paper, we study the asymptotics of such a compliance problem, imposing that the volume fraction remains fixed. No additional topological constraint is assumed on the admissible configurations. We determine the limit problem in different equivalent formulations, and we provide a system of necessary and sufficient optimality conditions. These results were announced in Bouchitté et al. (C. R. Acad. Sci. Paris, Ser. I. 345:713–718, 2007). Furthermore, we investigate the vanishing volume fraction limit, which turns out to be consistent with the results in Bouchitté and Fragalà (Arch. Rat. Mech. Anal. 184:257–284, 2007; SIAM J. Control Optim. 46:1664–1682, 2007). Finally, some explicit computation of optimal plates are given.     

Chaotic behavior of a class of discontinuous dynamical systems of fractional-order

In this paper, the chaos persistence in a class of discontinuous dynamical systems of fractional-order is analyzed. To that end, the initial value problem is first transformed, by using the Filippov regularization (Filippov in Differential Equations with Discontinuous Right-Hand Sides, 1988), into a set-valued problem of fractional-order, then by Cellina’s approximate selection theorem (Aubin and Cellina in Differential Inclusions Set-valued Maps and Viability Theory, 1984; Aubin and Frankowska in Set-valued Analysis, 1990). The problem is approximated into a single-valued fractional-order problem, which is numerically solved by using a numerical scheme proposed by Diethelm et al. (Nonlinear Dyn. 29:3–22, 2002). Two typical examples of systems belonging to this class are analyzed and simulated.     

Long-Time Convergence of an Adaptive Biasing Force Method: The Bi-Channel Case

We present convergence results for an adaptive algorithm to compute free energies, namely the adaptive biasing force (ABF) method (Darve and Pohorille in J Chem Phys 115(20):9169–9183, 2001; Hénin and Chipot in J Chem Phys 121:2904, 2004). The free energy is the effective potential associated to a so-called reaction coordinate ξ(q), where q = (q ₁, … , q _3N ) is the position vector of an N-particle system. Computing free energy differences remains an important challenge in molecular dynamics due to the presence of metastable regions in the potential energy surface. The ABF method uses an on-the-fly estimate of the free energy to bias dynamics and overcome metastability. Using entropy arguments and logarithmic Sobolev inequalities, previous results have shown that the rate of convergence of the ABF method is limited by the metastable features of the canonical measures conditioned to being at fixed values of ξ (Lelièvre et al. in Nonlinearity 21(6):1155–1181, 2008). In this paper, we present an improvement on the existing results in the presence of such metastabilities, which is a generic case encountered in practice. More precisely, we study the so-called bi-channel case, where two channels along the reaction coordinate direction exist between an initial and final state, the channels being separated from each other by a region of very low probability. With hypotheses made on ‘channel-dependent’ conditional measures, we show on a bi-channel model, which we introduce, that the convergence of the ABF method is, in fact, not limited by metastabilities in directions orthogonal to ξ under two crucial assumptions: (i) exchange between the two channels is possible for some values of ξ and (ii) the free energy is a good bias in each channel. This theoretical result supports recent numerical experiments (Minoukadeh et al. in J Chem Theory Comput 6:1008–1017, 2010), where the efficiency of the ABF approach is demonstrated for such a multiple-channel situation.     

Decay Structure for Symmetric Hyperbolic Systems with Non-Symmetric Relaxation and its Application

This paper is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima–Shizuta stability condition formulated in Umeda et al. (Jpn J Appl Math 1:435–457, 1984) and Shizuta and Kawashima (Hokkaido Math J 14:249–275, 1985) and we obtain the asymptotic stability result together with the explicit time-decay rate under that stability condition. However, some physical models which satisfy the stability condition have non-symmetric relaxation term (for example, the Timoshenko system and the Euler–Maxwell system). Moreover, it had been already known that the dissipative structure of such systems is weaker than the standard type and is of the regularity-loss type (see Duan in J Hyperbolic Differ Equ 8:375–413, 2011; Ide et al. in Math Models Meth Appl Sci 18:647–667, 2008; Ide and Kawashima in Math Models Meth Appl Sci 18:1001–1025, 2008; Ueda et al. in SIAM J Math Anal 2012; Ueda and Kawashima in Methods Appl Anal 2012). Therefore our purpose in this paper is to formulate a new structural condition which includes the Kawashima–Shizuta condition, and to analyze the weak dissipative structure for general systems with non-symmetric relaxation.     

On study of nonclassical problems of fracture and failure mechanics and related mechanisms

Nonclassical problems of fracture and failure mechanics that have been analyzed by the author and his collaborators at the S. P. Timoshenko Institute of Mechanics (Kiev, National Academy of Sciences of Ukraine) during the past forty years are considered in brief. The results of the analysis are presented in a form that would be quite informative for the majority of experts interested in various fundamental and applied aspects of fracture and failure problems including the identification of related mechanisms. This paper was prepared on invitation of the Editorial Board of the journal “Annals. The European Academy of Sciences” and may be considered as an Extended Pascal Medal Lecture (The 2007 Blaise Pascal Medal in Materials Sciences of the EAS) This is an updated edition of the author’s lecture prepared at the invitation of the Annals—The European Academy of Sciences Magazine on the occasion of awarding him the 2007 Blaise Pascal Medal in Materials Sciences by the EAS. The author’s speech at the award ceremony at the General Assembly of the Academy has already been published in International Applied Mechanics [75]. The electronic version of the paper in Annals has been prepared; this issue of Annals is to be published as a book. The paper includes an additional section and extended list of references [41–99]. Published in Prikladnaya Mekhanika, Vol. 45, No. 1, pp. 3–40, January 2009.     

Global Classical Solutions of the Relativistic Vlasov–Darwin System with Small Cauchy Data: The Generalized Variables Approach

We show that a smooth, small enough Cauchy datum launches a unique classical solution of the relativistic Vlasov–Darwin (RVD) system globally in time. A similar result is claimed in Seehafer (Commun Math Sci 6:749–769, 2008) following the work in Pallard (Int Mat Res Not 57191:1–31, 2006). Our proof does not require estimates derived from the conservation of the total energy, nor those previously given on the transversal component of the electric field. These estimates are crucial in the references cited above. Instead, we exploit the formulation of the RVD system in terms of the generalized space and momentum variables. By doing so, we produce a simple a priori estimate on the transversal component of the electric field. We widen the functional space required for the Cauchy datum to extend the solution globally in time, and we improve decay estimates given in Seehafer (2008) on the electromagnetic field and its space derivatives. Our method extends the constructive proof presented in Rein (Handbook of differential equations: evolutionary equations, vol 3. Elsevier, Amsterdam, 2007) to solve the Cauchy problem for the Vlasov–Poisson system with a small initial datum.     

Adhesive Frictionless Contact Between an Elastic Isotropic Half-Space and a Rigid Axi-Symmetric Punch

In this paper we consider the problem of adhesive frictionless contact of an elastic half-space by an axi-symmetric punch. We obtain integral equations that define the tractions and displacements normal to the surface of the half-space, as well as the size of the contact regions, for the cases of circular and annular contact regions. The novelty of our approach resides in the use of Betti’s reciprocity theorem to impose equilibrium, and of Abel transforms to either solve or substantially simplify the resulting integral equations. Additionally, the radii that define the annular or circular contact region are defined as local minimizers of the function obtained by evaluating the potential energy at the equilibrium solutions for each pair of radii. With this approach, we rather easily recover Sneddon’s formulas (Sneddon, Int. J. Eng. Sci., 3(1):47–57, 1965) for circular contact regions. For the annular contact region, we obtain a new integral equation that defines the inverse Abel transform of the surface normal displacement. We solve this equation numerically for two particular punches: a flat annular punch, and a concave punch.     

Non-linear filtering for a dust-perturbed two-body model

In standard textbooks on classical mechanics, the two-body central forcing problem is formulated as a system of the coupled non-linear second-order deterministic differential equations. Uncertainties, introduced by the astronomical ‘dust’, are not assumed in the orbit dynamics. The dust population produces an additional random force on the orbiting particle. This work is a continuation of the paper (Sharma and Parthasarathy, Proc. R. Soc. A: Math. Phys. Eng. Sci. 463:979–1003, [2007]) in which the authors developed and analyzed the dust-perturbed two-body model, which accounts for the dust perturbation felt by the orbiting particle. The theory of the dust-perturbed stochastic system was developed using the Fokker–Planck equation. This paper discusses the problem of realizing non-linear stochastic filters for estimating the states of the dust-perturbed planar two-body stochastic system, especially from noisy observations. This paper utilizes the Kushner’s theory of non-linear filtering, which involves stochastic observation term in the evolution of conditional probability density, for deriving the stochastic evolutions of the conditional mean and conditional covariance. The effectiveness of the non-linear filters of this paper is examined on the basis of their ability to preserve the perturbation effect, less random fluctuations in the mean trajectory and stability characteristics in the mean and variance trajectories. Most notably, this paper reveals the efficacy of the second-order approximate Kushner filter for the estimation procedure in contrast to the first-order approximate filter. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed in this paper.