Constitutive Theories for Nonlocal Micropolar Continua with Implicity and with Multiple Interactions

In this paper the constitutive theory for nonlocal micropolar continuawhich was proposed by A.C.Eringen is extended to the cases for nonlocalmicropolar continua with implicity and with multiple interactions.Herenonlocal micropolar thermoelastic solids with implicity and with multipleinteractions are cited as instances to illustrate the procedure for the estab-lishment of their constitutive theories as well as two relevant theoremsconcerning the constitutive theories for those solids are given.     

On Laminar Bouudary Layers with Suction

In this paper,we obtain the asymptotic solution of the general equa-tion for laminar boundary-layer flows with suction.Formulae for calcu-lating the displacement thickness,momentum thickness,and skin frictionare then derived,Furthermore,the problem of determining the separationpoint is dealt with.Finally,as a numerical example,we compute certaincharacteristic boundary-layer parameters for the case of uniform flow overa flat plate with constant suction.Our numerical results obtained are ingood agreement with those of Iglisch.     

Fibre-reinforced composites with fibre-bending stiffness under azimuthal shear – Comparison of simulation results with analytical solutions

In Ref. [1], Spencer and Soldatos proposed an enhanced modelling approach for fibre-reinforced composites which accounts for the fibre-bending stiffness in addition to the directional dependency induced by the fibres. Although analytical solutions for simple geometries have been derived over the past years, often subject to specific assumptions such as small deformation kinematics, the application to more general and non-academic boundary value problems is desirable. Motivated by the latter, the numerical solution of the general system of partial differential equations by means of a multi-field finite element approach is proposed in Ref. [2] and the principal model properties are studied for a specific form of the elastic energy potential. In the present contribution a comparison of the numerical solution by means of the multi-field finite element approach against the analytical solution is presented for the azimuthal shear deformation of a tube-like structure. To this end, the general deformation pattern and especially the distribution of the stress and couple stress tensor are taken into account. We find that, although the analytical solution is derived subject to the assumption of small deformations, whereas the numerical solution is based on the finite strain counterpart of the theory, the simulation results are quasi identical, which verifies the numerical framework proposed.     

Buckling of Columns with Movable Boundaries∗

The Euler buckling problem of a slender tubular column subject to its own weight, tension or compression exerted at its top, and internal and external variable static fluid pressure is studied. This problem finds many applications in drilling and production risers, mining risers, hydraulic columns, and legs of Tension Leg Platforms. The supports at the upper end of the column are considered movable to properly simulate drill ships or platforms that support such columns. The corresponding eigenvalue problem is comprised of a fourth-order differential equation with variable coefficients and four homogeneous boundary conditions. Two methods of solution derived in previous work are implemented numerically. The first solution is expressed in terms of Airy functions of the first and second kind and the second in terms of power series. The combined results of the two methods yield the critical buckling curves over the entire domain of practical interest. The critical curves are plotted in the plane of the two loading variables for the first six buckling modes for four different sets of boundary conditions. The results reveal the dependence of the asymptotic behavior of the critical curves for long columns on the boundary conditions.     

Substructures’ coupling with nonlinear connecting elements

Nonlinear Dynamics - The analysis of complex structures is often very challenging since reliable data can only be obtained if the underlying model represents properly the real case. Thus,...     

Hyperbolic Conservation Laws¶with Umbilic Degeneracy

We continue to study hyperbolic systems of conservation laws with umbilic degeneracy. We further extend our compactness framework established earlier to other canonical classes of quadratic flux systems with an isolated umbilic point. With the aid of this compactness framework, we establish the compactness of solution operators and the long-time behavior of entropy solutions in L ^∞ with large initial data, and we prove the convergence of the viscosity method, as well as the Lax-Friedrichs scheme and the Godunov scheme, for a canonical class of nonlinear hyperbolic systems with umbilic degeneracy.     

The Continuous Coagulation-Fragmentation¶Equations with Diffusion

Existence of global weak solutions to the continuous coagulation-fragmentation equations with diffusion is investigated when the kinetic coefficients satisfy a detailed balance condition or the coagulation coefficient enjoys a monotonicity condition. Our approach relies on weak and strong compactness methods in L ¹ in the spirit of the DiPerna-Lions theory for the Boltzmann equation. Under the detailed balance condition the large-time behaviour is also studied.     

Görtler Vortices with System Rotation

The steady primary instability of Görtler vortices developing along a curved Blasius boundary layer subject to spanwise system rotation is analysed through linear and nonlinear approaches, to clarify issues of vortex growth and wavelength selection, and to pave the way to further secondary instability studies.A linear marching stability analysis is carried out for a range of rotation numbers, to yield the (predictable) result that positive rotation, that is rotation in the sense of the basic flow, enhances the vortex development, while negative rotation dampens the vortices. Comparisons are also made with local, nonparallel linear stability results (Zebib and Bottaro, 1993) to demonstrate how the local theory overestimates vortex growth. The linear marching code is then used as a tool to predict wavelength selection of vortices, based on a criterion of maximum linear amplification.Nonlinear finite volume numerical simulations are performed for a series of spanwise wave numbers and rotation numbers. It is shown that energy growths of linear marching solutions coincide with those of nonlinear spatially developing flows up to fairly large disturbance amplitudes. The perturbation energy saturates at some downstream position at a level which seems to be independent of rotation, but that increases with the spanwise wavelength. Nonlinear simulations performed in a long (along the span) cross section, under conditions of random inflow disturbances, demonstrate that: (i) vortices are randomly spaced and at different stages of growth in each cross section; (ii) upright vortices are the exception in a universe of irregular structures; (iii) the average nonlinear wavelengths for different inlet random noises are close to those of maximum growth from the linear theory; (iv) perturbation energies decrease initially in a linear filtering phase (which does not depend on rotation, but is a function of the inlet noise distribution) until coherent patches of vorticity near the wall emerge and can be amplified by the instability mechanism; (v) the linear filter represents the receptivity of the flow: any random noise, no matter how strong, organizes itself linearly before subsequent growth can take place; (vi) the Görtler number, by itself, is not sufficient to define the state of development of a vortical flow, but should be coupled to a receptivity parameter; (vii) randomly excited Görtler vortices resemble and scale like coherent structures of turbulent boundary layers.A.Z. has been supported, during his stay at EPFL, by an ERCOFTAC Visitor Grant. A.B. acknowledges the Swiss National Fund, Grant No. 21-36035.92, for travel support associated with this research. This work was also supported by the Swedish Board of Technical Development (NUTEK), the Swedish Technical-Scientific Council (TFR), and an ERCOFTAC Visitor Grant, through which the stay of B.G.B.K. at the EPFL was made possible. Cray-2 computing time for this research was generously provided by the Service Informatique Centrale of EPFL.     

Rotating Brinkman–Lapwood Convection with Modulation

The purpose of this article is to analyze, theoretically, the effect of modulation on rotating Brinkman–Lapwood convection, i.e., buoyancy-driven convection in a sparse porous medium subjected to rotation. Darcy–Brinkman momentum equation with Coriolis term has been used to describe the flow. The system is considered rotating about an axis with non-uniform rotation speed. In particular, we assume that the rotation speed is varying sinusoidally with time. A linear stability analysis has been performed to find the critical Rayleigh number in modulated case. The effect of modulated rotation speed is found to have a stabilizing effect on the onset of convection for different values of modulation frequency and the other physical parameters involved.     

The Strongest Structures with Macrostiffness Constraints∗

An example of a truss structure with limited global stiffness, measured by some parameter k, is discussed. The optimal structural design problem of maximizing the external load intensity, with axial stresses in the rods of the truss remaining within an allowable range, is well known and leads to a statically determinate solution with uniformly loaded elements. However, if one takes into account the additional constraint on stiffness k, the problem is open and the solution leads, in general, to a redundant truss. The paper presents this problem and proposes its application to optimal design of supporting structures     

Maxwell’s Equations with Vector Hysteresis

Electromagnetic processes in magnetic materials are described by Maxwells equations. In ferrimagnetic insulators, assuming that D = E, we have the equationIn ferromagnetic metals, neglecting displacement currents and assuming Ohms law, we instead getAlternatively, under quasi-stationary conditions, for either material we can also deal with the magnetostatic equations:(Here f_ext and J_ext are prescribed time-dependent fields.) In any of these settings, the dependence of M on H is represented by a constitutive law accounting for hysteresis: M= (H), being a vector extension of the relay model. This is characterized by a rectangular hysteresis loop in a prescribed x-dependent direction, and accounts for high anisotropy and nonhomogeneity. The discontinuity in this constitutive relation corresponds to the possible occurrence of free boundaries.Weak formulations are provided for Cauchy problems associated with the above equations; existence of a solution is proved via approximation by time-discretization, derivation of energy-type estimates, and passage to the limit. An analogous representation is given for hysteresis in the dependence of P on E in ferroelectric materials. A model accounting for coupled ferrimagnetic and ferroelectric hysteresis is considered, too.Acknowledgement This research was partly supported by the project Free boundary problems in applied sciences of Italian M.I.U.R.. I gratefully acknowledge the useful suggestions from the reviewers.     

Fluid–structure interactions with applications to biology

正Fluid–structure interactions(FSI)are encountered by almost all animals and plants that live in fluid environments,e.g.flying insects/birds,swimming fish,blood cells,seeds,leaves,and trees.The interactions of living species with fluids are commonplace in our daily life and have attracted wide attention for a long time.However,they are still puzzling today for their intrinsic complexities,which bring major challenges in experimental,theoretical,and numerical studies.     

Painlevé paradox during oblique impact with friction

In analyses using non-smooth dynamics, oblique impact of rough bodies in an unsymmetrical configuration can result in self-locking or “jam” at the sliding contact if the coefficient of friction is sufficiently large; this has been termed, Painlevé’s paradox. In the range of configurations and coefficients of friction where Painlevé’s paradox occurs, analyses based on rigid body dynamics give results indicating that either there are multiple solutions or the solution is nonexistent. This conundrum has been resolved by considering that the contact has small normal and tangential compliance which is representative of deformability in a local region around the contact point. An analysis using a hybrid model which includes local compliance of the contact region has calculated the time-dependent changes in relative motion of colliding bodies for a range of incident angles of obliquity, tan^?1[?V₁(0)/V₃(0)] where V₁(0)and V₃(0) are the incident tangential and normal relative velocities at the contact point, respectively. The paradox is shown to result from a negative relative acceleration of the contact points during an initial period of sliding – a negative acceleration that is inconsistent with the assumption of rigid-body contact.     

A chaotic system with Hölder continuity

This paper presents a new chaotic system with infinitely many equilibria. The new system contains two system parameters and a nonlinear term which does not satisfy Lipschitz continuity but does satisfy $\frac{1}{2}$ -Hölder continuity condition. The complicated dynamics are studied through theoretical analysis and numerical simulation. Synchronization for two identical systems by a piecewise linear feedback controller is investigated based on Lyapunov stability criteria.     

Are waves with negative spatial damping unstable?

Conventional plane harmonic waves decay in direction of propagation, but unconventional harmonic waves grow in the direction of propagation. While a single unconventional wave cannot be a solution to a physically meaningful boundary value problem, these waves may have an essential contribution to the overall solution of a problem as long as this is a superposition of unconventional and conventional waves. A fourth order diffusion equation with proper thermodynamic structure, and the Burnett equations of rarefied gas dynamics exhibit conventional and unconventional waves. Steady state oscillating boundary value problems are considered to discuss the interplay of conventional and unconventional waves. Results show that as long as the second law of thermodynamics is valid, unconventional waves may contribute to the overall solution, which, however is dominated by conventional waves, and behaves as these.     

Reaction–Diffusion Equations with Nonlinear Boundary Delay

We consider dissipative scalar reaction–diffusion equations that include the ones of the form u _t–u=f(u(t)), subjected to boundary conditions that include small delays, that is, we consider boundary conditions of the form u/n _a=g(u(t), u(t–r)). We show the global existence and uniqueness of solutions in a convenient fractional power space, and furthermore, we show that, for r sufficiently small, all bounded solutions are asymptotic to the set of equilibria as t tends to infinity.     

Risk-sensitive reinforcement learning algorithms with generalized average criterion

A new algorithm is proposed, which immolates the optimality of control policies potentially to obtain the robusticity of solutions. The robusticity of solutions maybe becomes a very important property for a learning system when there exists non-matching between theory models and practical physical system, or the practical system is not static, or the availability of a control action changes along with the variety of time. The main contribution is that a set of approximation algorithms and their convergence results are given. A generalized average operator instead of the general optimal operator max (or min) is applied to study a class of important learning algorithms, dynamic programming algorithms, and discuss their convergences from theoretic point of view. The purpose for this research is to improve the robusticity of reinforcement learning algorithms theoretically.     

Dirichlet problem with Φ-Laplacian and mixed singularities

Assume that A₁, A₂ ⊂ ℝ are closed intervals containing 0, ϕ is an increasing odd homeomorphism with ϕ (ℝ) = ℝ, and T ∈ (0, ∞). We study a singular Dirichlet problem of the form {fx080-01} and prove the existence of its smooth solution satisfying the conditions {fx080-02}, where ƒ satisfies the Carathéodory conditions on the set (0, T) × D and can have time singularities at t = 0 and t = T and space singularities at x = 0, y = 0. Published in Neliniini Kolyvannya, Vol. 11, No. 1, pp. 81–95, January–March, 2007.     

Homogeneous diffusion in ℝ with power-like nonlinear diffusivity

We study the nonnegative solutions of the initial-value problem u_t=(u^r|u_x|^p-1u_x)_x,u(x, 0)L ¹(), where p>0, r+p>0. The local velocity of propagation of the solutions is identified as V = -v_x| v_x|^p-1 where v =cu (with r +p - 1)/p and c (r +p/(r +p- 1)) is the nonlinear potential. Our main result is the a priori estimate (v_x|v_x|^p-1)_x-