On inequalities of Friedrichs and Babuška-Aziz

The best possible constants in the inequalities of Friedrichs and of Babuka-Aziz are related in a simple manner. This has been shown by Horgan and Payne. Here, we look at the problem from a different point of view considering associated eigenvalue problems in variational form.

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Sommario Le migliori possibili costanti nelle diseguaglianze di Friedrichs e di Babuka-Aziz sono legate in un modo semplice, come mostrato da Horgan e Payne. Qui, il problema è affrontato da un punto di vista differente, e cioè considerando problemi di autovalori in forma variazionale.

     

Two-dimensional discrete element simulations ofice–structure interaction

The paper presents the application of a discrete element technique for the study of the plane strain problem of the interaction between a moving ice sheet and a flexible stationary structure. The discrete element technique accounts for the generation of failure within an initially intact ice sheet. The failure of the ice corresponds to situations where the ice can exhibit combinations of brittle fragmentation and viscoplastic flow. The modelling also accounts for size dependency in the strength of the ice after fragmentation. The inter-fragment interactions are modelled by non-linear constraints which includes Coulomb frictional behaviour. The computational scheme is used to evaluate the time history of the average contact stresses and the distribution of local contact stresses at the ice–structure interface in the fragmentation zone.     

Mittag–Leffler stability of nabla discrete fractional-order dynamic systems

Nonlinear Dynamics - In the present study, the definition of discrete Mittag–Leffler stability is derived to characterize convergence rule of the pseudostates for nabla discrete...     

Stability of quadruped robots’ trajectories subjected to discrete perturbations

In this paper, we study the stability of a mathematical model for trajectory generation of a qua-druped robot. We consider that each movement is composed of two types of primitives: rhythmic and discrete. The discrete primitive is inserted as a perturbation of the purely rhythmic movement. The two primitives are modeled by nonlinear dynamical systems. We adapt the theory developed by Golubitsky et?al. in (Physica D 115: 56?C72, 1998; Buono and Golubitsky in J. Math. Biol. 42:291?C326, 2001) for quadrupeds gaits. We conclude that if the discrete part is inserted in all limbs, with equal values, and as an offset of the rhythmic part, the obtained gait is stable and has the same spatial and spatiotemporal symmetry groups as the purely rhythmic gait, differing only on the value of the offset.     

Supratransmission in discrete one-dimensional lattices with the cubic–quintic nonlinearity

Nonlinear Dynamics - We numerically analyzed the supratransmission phenomenon in the discrete nonlinear Schrödinger equation with the cubic–quintic nonlinearity. It has been reported...     

Bifurcations and chaos in a two-dimensional discrete Hindmarsh–Rose model

In this paper, the dynamics of a two-dimensional discrete Hindmarsh–Rose model is discussed. It is shown that the system undergoes flip bifurcation, Neimark–Sacker bifurcation, and 1:1 resonance by using a center manifold theorem and bifurcation theory. Furthermore, we present the numerical simulations not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, including orbits of period 3, 6, 15, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, 16, quasiperiodic orbits, and chaotic sets. These results obtained in this paper show far richer dynamics of the discrete Hindmarsh–Rose model compared with the corresponding continuous model.     

Hill–Mandel condition and bounds on lower symmetry elastic crystals

Despite advances in contemporary micromechanics, there is a void in the literature on a versatile method for estimating the effective properties of polycrystals comprising of highly anisotropic single crystals belonging to lower symmetry class. Basing on variational principles in elasticity and the Hill–Mandel homogenization condition, we propose a versatile methodology to fill this void. It is demonstrated that the bounds obtained using the Hill–Mandel condition are tighter than the Voigt and Reuss [1], [2] bounds, the Hashin–Shtrikman [3] bounds as well as a recently proposed self-consistent estimate by Kube and Arguelles [4] even for polycrystals with highly anisotropic single crystals.     

Nonlinear dynamics of a non-autonomous network with coupled discrete–continuum oscillators

A network model of a multi-modular floating platform incorporated with a runway structure, viewed as a non-autonomous network with discrete–continuum oscillators, is developed for a general purpose of dynamic analysis. Numerical analysis shows the coupling effect between the two different types of oscillators on various complex dynamics, including sudden leaps, torus motions, beating vibrations, the synergetic effect of phase lock and anti-phase synchronizations. The amplitude death phenomenon, a suppressed weak oscillation state, is studied by using the fundamental solution derived by the averaging method. The parametric domain of the onset of amplitude death is illustrated to show the great significance to the stability design of the floating platform. The effect of the flexural rigidity of the runway on the distribution of amplitude death state is also discussed.     

A discrete approach for a generalized Beck’s column in parametric resonance

A Galerkin projection based on non-standard bases is conceived to derive an equivalent discrete model of a continuous system under non-conservative forces. The problem of deriving a discrete model capable of describing critical and post-critical scenarios for non-selfadjoint systems is discussed and an heuristic rule for a proper choice of trial functions is given. The procedure is utilized to analyze the effect of non-conservative autonomous and non-autonomous (pulsating) forces acting on a linearly damped Beck’s column involving geometrical nonlinearities. The linear and nonlinear behaviours arising from the analysis of the proposed discrete model are in good agreement with those observed through the unavoidably more involved direct continuous approach. Critical scenarios for the autonomous and non-autonomous cases are investigated and the multiple scales method is applied in order to obtain the bifurcation equations in the neighborhood of a Hopf bifurcation point in primary parametric resonance. A comparison between critical and post-critical continuous and discrete model is performed adopting two control parameters, namely the amplitudes of the static and dynamic components of the forces, playing the role of detuning and bifurcation parameters, respectively.     

Unloading of an elastic–plastic spherical contact under stick contact condition

The unloading process of an elastic–plastic spherical contact under stick contact condition is analyzed for various material properties. The evolution of normal and shear stress distribution at the contact area as well as the residual profile of the sphere and residual von Mises stresses inside the sphere are presented. Empirical expressions for the residual interference and for the evolution of the interference and contact area during the unloading are provided. Good agreement with experimental results is shown.     

Nonlinear supratransmission of multibreathers in discrete nonlinear Schrödinger equation with saturable nonlinearities

We show that the transport of vibrational energy in protein chains modeled by the Discrete Nonlinear Schrödinger equation (DNSE) with saturable nonlinearities can be done through the nonlinear supratransmission phenomenon: we find numerically and semi-analytically threshold amplitudes beyond which the wave propagation takes place within the molecular chains. Subsequently, it is shown that the saturable higher order nonlinearity parameter reduces the supratransmission threshold amplitude. We also prove that the discrete gap multibreathers can be transmitted or supratransmitted according to the frequency belonging to the lower forbidden band gap. More precisely, the discrete gap multibreathers are supratransmitted close to the edge of the lower forbidden band.     

Why do receptor–ligand bonds in cell adhesion cluster into discrete focal-adhesion sites?

Cell adhesion often exhibits the clustering of the receptor–ligand bonds into discrete focal-adhesion sites near the contact edge, thus resembling a rosette shape or a contracting membrane anchored by a small number of peripheral forces. The ligands on the extracellular matrix are immobile, and the receptors in the cell plasma membrane consist of two types: high-affinity integrins (that bond to the substrate ligands and are immobile) and low-affinity integrins (that are mobile and not bonded to the ligands). Thus the adhesion energy density is proportional to the high-affinity integrin density. This paper provides a mechanistic explanation for the clustering/assembling of the receptor–ligand bonds from two main points: (1) the cellular contractile force leads to the density evolution of these two types of integrins, and results into a large high-affinity integrin density near the contact edge and (2) the front of a propagating crack into a decreasing toughness field will be unstable and wavy. From this fracture mechanics perspective, the chemomechanical equilibrium is reached when a small number of patches with large receptor–ligand bond density are anticipated to form at the cell periphery, as opposed to a uniform distribution of bonds on the entire interface. Cohesive fracture simulations show that the de-adhesion force can be significantly enhanced by this nonuniform bond density field, but the de-adhesion force anisotropy due to the substrate elastic anisotropy is significantly reduced.     

Representation of discrete sequences with N-dimensional iterated function systems in tensor form

Iterated Function System (IFS) models have been used to represent discrete sequences where the attractor of the IFS is self-affine or piecewise self-affine in R ² or R ³ (R is the set of real numbers). In this paper, the piecewise hidden-variable fractal model is extended from R ³ to R ⁿ (n is an integer greater than 3), which is called the multi-dimensional piecewise hidden variable fractal model. This new model uses a “mapping partial derivative” and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the hidden variables. Therefore the result is very general. Moreover, the piecewise hidden-variable fractal model in tensor form is more terse than in the usual matrix form.