Topological classification of affine mappings from ℝ<Superscript>2</Superscript> into ℝ<Superscript>2</Superscript>

We investigate affine mappings from ℝ² into ℝ² and establish necessary and sufficient conditions for the topological conjugacy of these mappings. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 472–480, October–December, 2008.     

The Navier-Stokes Equations in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with Linearly Growing Initial Data

Consider the equations of Navier-Stokes on ⁿ with initial data U₀ of the form U₀(x)=u₀(x)–Mx, where M is an n×n matrix with constant real entries and u₀ L^p(ⁿ). It is shown that under these assumptions the equations of Navier-Stokes admit a unique local solution in L^p(ⁿ). Moreover, if ||e^tM||1 for all t0, then this mild solution is even analytic in x. This is surprising since the underlying semigroup of Ornstein-Uhlenbeck type is not analytic, in contrast to the Stokes semigroup.Acknowledgement It is our pleasure to thank G. METAFUNE, E. PRIOLA and A. RHANDI for fruitful discussions on the Ornstein-Uhlenbeck     

Entropy Solutions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">∞</Emphasis></Superscript> for the Euler Equations in Nonlinear Elastodynamics and Related Equations

A compactness framework is established for approximate solutions to the Euler equations in one-dimensional nonlinear elastodynamics by identifying new properties of the Lax entropies, especially the higher order terms in the Lax entropy expansions, and by developing ways to employ these new properties in the method of compensated compactness. Then this framework is applied to establish the existence, compactness, and decay of entropy solutions in L for the Euler equations in nonlinear elastodynamics with a more general stress-strain relation than those for the previous existence results. This compactness framework is further applied to solving the Euler equations of conservation laws of mass, momentum, and energy for a class of thermoelastic media, and the equations of motion of viscoelastic media with memory.     

Two-Phase Problems for Linear Elliptic Operators with Variable Coefficients: Lipschitz Free Boundaries are <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">1,γ</Emphasis></Superscript>

We prove C ^1, regularity of Lipschitz free boundaries of two-phase problems for linear elliptic operators with Hölder continuous coefficients.     

On Local Uniqueness of Weak Solutions to the Navier–Stokes System with <Emphasis Type="Italic">BMO</Emphasis><Superscript>−1</Superscript> Initial Datum

We address the problem of local uniqueness of weak solutions to the Navier–Stokes system, with the initial datum in a subspace of . The existence and uniqueness of local mild solutions has been proven by Koch and Tataru (Adv Math 157:22–35, 2001). We present a necessary and sufficient condition for two weak solutions to evolve from the same initial datum, and for weak solutions to be mild.      

Chaos control and synchronization of the Φ<Superscript>6</Superscript>-Van der Pol system driven by external and parametric excitations

The dynamical behavior of the Φ⁶-Van der Pol system subjected to both external and parametric excitation is investigated. The effect of parametric excitation amplitude on the routes to chaos is studied by numerical analysis. It is found that the probability of chaos happening increases along with the parametric excitation amplitude increases while the external excitation amplitude fixed. Based on the invariance principle of differential equations, the system is lead to desirable periodic orbit or chaotic state (synchronization) with different control techniques. Numerical simulations are provided to validate the proposed method.     

Measurement of Strain Localization Resulting from Monotonic and Cyclic Loading at 650 <Superscript>∘</Superscript>C in Nickel Base Superalloys

A methodology is presented for the use of the oxide scale that develops in polycrystalline Ni-base superalloys at service temperature, as a speckle pattern for μm-scale resolution strain measurements. Quantitative assessment of the heterogeneous strain field at the grain scale is performed by high-resolution SEM digital image correlation under monotonic and cyclic loading in polycrystalline Ni-base superalloys up to 650 ^°C. In the René 88DT superalloy, strain localization is observed near twin boundaries during low cycle fatigue (LCF) at intermediate temperatures, correlating with activation of {111} 〈110〉 and {111} 〈112〉 slip systems. A strong correlation between the microstructural configuration that promotes strain localization during monotonic loading and crack initiation at 650 ^°C in low cycle fatigue was observed.     

Patterns and heat transfer in Rayleigh-Bénard thermal gravitational convection in helium (<Superscript>3</Superscript>He) near the critical point

Thermal gravitational convection in a bottom-heated layer of near-critical ³He is considered. The range of criteria determining the convection parameters beyond the stability threshold is discussed. The specific features of 2D and 3D supercritical structures, the adiabatic compression effect, and heat transfer are considered.     

Synchronization of unified chaotic system by sliding mode/mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control

This paper deals with the synchronization of uncertain unified chaotic system in the presence of two kinds of disturbances, white noise and bounded power signal. A sliding mode controller (SMC) is established to guarantee the sliding motion. Moreover, a proportional-integral (PI) switching surface is used to determine the performance of the system in the sliding motion. Also, by using a mixed H ₂/H _∞ approach, the effect of external disturbances on the sliding motion is reduced. The necessary parameters of constructing controller and switching surface are found via semidefinite programming (SDP) which can be solved effectively by a standard software. Finally, a numerical simulation is presented to show the effectiveness of the proposed method.     

Unsteady PTV velocity field past an airfoil solved with DNS: Part 2. Validation and application at Reynolds numbers up to <Emphasis Type="Italic">Re</Emphasis> ≤ 10<Superscript>4</Superscript>

Hybrid unsteady-flow simulation combining particle tracking velocimetry (PTV) and direct numerical simulation (DNS) is introduced in the series of two papers. Particle velocities on a laser-light sheet acquired with time-resolved PTV in a water tunnel are supplied to two-dimensional DNS with time intervals corresponding to the frame rate of the PTV. Hybrid velocity fields then approach those representing the PTV data in the course of time, and the reconstructed velocity fields satisfy the governing equations with the resolution comparable to numerical simulation. In part 2, by extending the capabilities of the hybrid simulation to higher Reynolds numbers, we simulate flows past the NACA0012 airfoil over ranges of Reynolds numbers (Re ≤ 10⁴) and angles of attack (−5° ≤ α ≤ 20°) and validate the proposed technique by comparing with experimental results in terms of the lift and drag coefficients. We also compare the results with unsteady Reynolds-averaged Navier–Stokes (URANS) simulation in two-dimensions and show the advantages of the hybrid simulation against two-dimensional URANS.     

A New Approach to Counterexamples to <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> Estimates: Korn’s Inequality,Geometric Rigidity,and Regularity for Gradients of Separately Convex Functions

The derivation of counterexamples to L¹ estimates can be reduced to a geometric decomposition procedure along rank-one lines in matrix space. We illustrate this concept in two concrete applications. Firstly, we recover a celebrated, and rather complex, counterexample by Ornstein, proving the failure of Korns inequality, and of the corresponding geometrically nonlinear rigidity result, in L¹. Secondly, we construct a function f:² which is separately convex but whose gradient is not in BV_loc, in the sense that the mixed derivative ²f/x₁x₂ is not a bounded measure.Acknowledgement We thank BERND KIRCHHEIM for bringing the question of regularity of separately convex functions to our attention. This work was partially supported by the EU Research Training Network Hyperbolic and Kinetic Equations, contract HPRN-CT-2002-00282, and by the Deutsche Forschungsgemeinschaft through the Schwerpunktprogramm 1095 Analysis, Modeling and Simulation of Multiscale Problems.     

Unsteady PTV velocity field past an airfoil solved with DNS: Part 1. Algorithm of hybrid simulation and hybrid velocity field at <Emphasis Type="Italic">Re</Emphasis> ≈ 10<Superscript>3</Superscript>

We develop a hybrid unsteady-flow simulation technique combining direct numerical simulation (DNS) and particle tracking velocimetry (PTV) and demonstrate its capabilities by investigating flows past an airfoil. We rectify instantaneous PTV velocity fields in a least-squares sense so that they satisfy the equation of continuity, and feed them to the DNS by equating the computational time step with the frame rate of the time-resolved PTV system. As a result, we can reconstruct unsteady velocity fields that satisfy the governing equations based on experimental data, with the resolution comparable to numerical simulation. In addition, unsteady pressure distribution can be solved simultaneously. In this study, particle velocities are acquired on a laser-light sheet in a water tunnel, and unsteady flow fields are reconstructed with the hybrid algorithm solving the incompressible Navier–Stokes equations in two dimensions. By performing the hybrid simulation, we investigate nominally two-dimensional flows past the NACA0012 airfoil at low Reynolds numbers. In part 1, we introduce the algorithm of the proposed technique and discuss the characteristics of hybrid velocity fields. In particular, we focus on a vortex shedding phenomenon under a deep stall condition (α = 15°) at Reynolds numbers of Re = 1000 and 1300, and compare the hybrid velocity fields with those computed with two-dimensional DNS. In part 2, the extension to higher Reynolds numbers is considered. The accuracy of the hybrid simulation is evaluated by comparing with independent experimental results at various angles of attack and Reynolds numbers up to Re = 10⁴. The capabilities of the hybrid simulation are also compared with two-dimensional unsteady Reynolds-Averaged Navier–Stokes (URANS) solutions in part 2. In the first part of these twin papers, we demonstrate that the hybrid velocity field approaches the PTV velocity field over time. We find that intensive alternate vortex shedding past the airfoil, which is predicted by the two-dimensional DNS, is substantially suppressed in the hybrid simulation and the resultant flow field is similar to the PTV velocity field, which is projection of the three-dimensional velocity field on the streamwise plane. We attempt to identify the motion that originates three-dimensional flow patterns by highlighting the deviation of the PTV velocity field from the two-dimensional governing equations at each snapshot. The results indicate that the intensive spots of the deviation appear in the regions in which three-dimensional instabilities are induced in the shear layer separated from the pressure side.     

Roughness effects in low-<Emphasis Type="Italic"> Re</Emphasis><Subscript><Emphasis Type="Italic"> θ</Emphasis></Subscript> open-channel turbulent boundary layers

This paper reports velocity measurements obtained on a smooth and two geometrically different types of rough surfaces in an open channel. The measurements were obtained using a laser-Doppler anemometer. The recent boundary layer theory proposed by George and Castillo (1997) and conventional scaling laws are used to analyze the data. The present flow shows a strong structural similarity to a canonical turbulent boundary layer in the inner layer. The results demonstrate that surface roughness increases the wake parameter. Surface roughness also enhances the levels of turbulence intensities, Reynolds shear stress and triple correlations over most of the boundary layer, but decreases the stress anisotropy.     

Neural network ℋ<Subscript>∞</Subscript> chaos synchronization

This paper proposes a new neural network ?_∞ synchronization (NNHS) scheme for unknown chaotic systems. In the proposed framework, a dynamic neural network is constructed as an alternative to approximate the chaotic system. Based on this neural network and linear matrix inequality (LMI) formulation, the NNHS controller and the learning law are presented to reduce the effect of disturbance to an ?_∞ norm constraint. It is shown that finding the NNHS controller and the learning law can be transformed into the LMI problem and solved using the convex optimization method. A numerical example is presented to demonstrate the validity of the proposed NNHS scheme.     

A C<Superscript>0</Superscript>-type zig–zag theory and finite element for laminated composite and sandwich plates with general configurations

In order to conveniently develop C⁰ continuous element for the accurate analysis of laminated composite and sandwich plates with general configurations, this paper develops a C⁰-type zig–zag theory in which the interlaminar continuity of transverse shear stresses is a priori satisfied and the number of unknowns is independent of the number of layers. The present theory is applicable not only to the cross-ply but also to the angle-ply laminated composite and sandwich plates. On the premise of retaining the merit of previous zig–zag theories, the derivatives of transverse displacement have been taken out from the displacement fields. Therefore, based on the proposed zig–zag theory, it is very easy to construct the C⁰ continuous element. To assess the performance of the proposed model, the classical quadratic six-node triangular element with seven degrees of freedom at each node is presented for the static analysis of laminated composite and sandwich plates. The typical examples are taken into account to assess the performance of finite element based on the proposed zig–zag theory by comparing the present results with the three-dimensional elasticity solutions. Numerical results show that the present model can produce the more accurate deformations and stresses compared with the previous zig–zag theories.     

Multivalued penalty method for evolution variational inequalities with <Emphasis Type="Italic">λ</Emphasis><Subscript>0</Subscript>-pseudomonotone multivalued maps

We consider evolution variational inequalities with λ ₀-pseudomonotone maps. The main properties of these maps are investigated. By using the finite-difference method, we prove the property of strong solvability for the class of evolution variational inequalities with λ ₀-pseudomonotone maps. Using the penalty method for multivalued maps, we show the existence of weak solutions of evolution variational inequalities on closed convex sets. The class of multivalued penalty operators is constructed. We also consider a model example to illustrate this theory.     

On the convergence of successive Galerkin approximation for nonlinear output feedback <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control

Although the formulation of the nonlinear theory of H _∞ control has been well developed, solving the Hamilton–Jacobi–Isaacs equation remains a challenge and is the major bottleneck for practical application of the theory. Several numerical methods have been proposed for its solution. In this paper, results on convergence and stability for a successive Galerkin approximation approach for nonlinear H _∞ control via output feedback are presented. An example is presented illustrating the application of the algorithm.     

The Singular Set of <Emphasis Type="Italic">ω</Emphasis>-minima

We consider ω-minima of convex variational integrals in the vectorial case n,N≥2, and we provide estimates for the Hausdorff dimension of their singular sets.     

Elastic properties of Al<Subscript>2</Subscript>O<Subscript>3</Subscript>–NiAl: a modified version of Hashin–Shtrikman bounds

The modified nonlinear relations for the estimation of elastic constants of Al₂O₃–NiAl composite material are developed. The concept of microstructure and interconnectivity of phases at the interface is used. Hashin–Shtrikman relations are described in their actual form and modified version of Hashin–Shtrikman relations for bulk and shear moduli are discussed. These relations for elastic and mechanical properties are applied mainly for Al₂O₃–NiAl composite material. Theoretical predictions using modified relations are compared with Hashin–Shtrikman bounds and experimental results of elastic properties for Al₂O₃–NiAl matrix-inclusion-based composite. It is found that the predicted values of elastic and mechanical properties using modified relations are quite close to the experimental results.     

Stability and implementable ℋ<Subscript>∞</Subscript> filters for singular systems with nonlinear perturbations

In this paper, we investigate the problem of designing ℋ_∞ filter for a class of continuous-time uncertain singular systems with nonlinear perturbations, which can be realized in practice. The perturbation is a time-varying function of the system state and satisfies a Lipschitz constraint. The design objective is to guarantee that a prescribed upper bound on an ℋ_∞ performance of the robust filter is attained for all possible energy-bounded input disturbances and all admissible uncertainties and which can be implemented on-line to get a good replica of the state. We first establish sufficient condition for the existence and uniqueness of solution to the singular system connected with the normal filter. Using a linear matrix inequality (LMI) format, we then provide a sufficient condition for the asymptotic stability of the realizable ℋ_∞ filter. Then by means of a convex analysis procedure the filter gain matrices are derived and an important special case is readily deduced. Finally, a numerical example is presented to illustrate the theoretical developments.