Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 2007"

Unified Poincaré and Hardy inequalities with sharp constants for convex domains

Abstract: Let 0 be an n-dimensional convex domain, and let v ∈ [0,1/2]. For all f ∈ H 1 0 (Ω) we prove the inequality ∫|∇f| 2 dx ≥ (1 4 - v 2 ) ∫ Ω |f| 2 δ 2 dx + λ 2 v δ 2 0 ∫ Ω |f| 2 dx, where δ = dist(x, ∂Ω), δ 0 = sup δ. The factor λ 2 v , is sharp for all dimensions, λ v being the first positive root of the Lamb type equation J v (λ v ) + 2λ v J' v (λ v ) = 0 for Bessel's functions. In particular, the case v = 0 with λ 0 = 0,940... presents a new sharp form of the Hardy type inequality due to Brezis and Marcus, while in the case v = 1/2 with λ 1/2 = π/2 we obtain a unified proof of an isoperimetric inequality due to Poincare for n = 1, Hersch for n = 2 and Payne and Stakgold for n > 3. A generalization, when the latter integral is replaced by the integral ∫ Ω |f| 2 / δ 2-m dx, m > 0, is proved, too. As a special case, we obtain the sharp inequality ∫ Ω |f| 2 dx ≥ m 2 j 2 1 /m-1 4δ m 0 ∫ Ω |f| 2 δ 2-m dx, where j v is the first positive zero of J v .

Extended non‐linear relations of elastic shells undergoing phase transitions

Abstract: The non-linear theory of elastic shells undergoing phase transitions was proposed by two first authors in J. Elast. 79, 67-86 (2004). In the present paper the theory is extended by taking into account also the elastic strain energy density of the curvilinear phase interface as well as the resultant forces and couples acting along the interface surface curve itself. All shell relations are found from the variational principle of stationary total potential energy. In particular, we derive the extended natural continuity conditions at coherent and/or incoherent surface curves modelling the phase interface. The continuity conditions allow one to establish the final position of the interface surface curve after the phase transition. The results are illustrated by an example of a phase transition in an infinite plate with a central hole.

Numerical solutions of 2-D steady incompressible flow in a driven skewed cavity

Abstract: The benchmark test case for non-orthogonal grid mesh, the "driven skewed cavity flow", first introduced by Demirdzice t al. (5) for skew angles of α =3 0 ◦ and α =4 5 ◦ , is reintroduced with a more variety of skew angles. The benchmark problem has non-orthogonal, skewed grid mesh with skew angle (α). The governing 2-D steady incompressible Navier-Stokes equations in general curvilinear coordinates are solved for the solution of driven skewed cavity flow with non-orthogonal grid mesh using a numerical method which is efficient and stable even at extreme skew angles. Highly accurate numerical solutions of the driven skewed cavity flow, solved using a fine grid (512 × 512) mesh, are presented for Reynolds number of 100 and 1000 for skew angles ranging between 15 ◦ α 165 ◦ . c

Dynamic interactions between parametric pendulum and electro‐dynamical shaker

Abstract: In this paper we investigate the dynamic interactions between a parametric pendulum and an electro-dynamical shaker aiming to explain a weak correlation between the theoretical predictions obtained for a parametric pendulum model and the experimental results for pendulum rig driven by the shaker. In particular, the quasi-periodic rotations were found experimentally to be co-existing with the period one rotations, where no such motion can be predicted for parametric pendulum with an ideal parametric excitation. First the experimental setup including the pendulum, the shaker and data acquisition system is described. Then the electro-dynamical shaker is modelled as two and a half degrees-of-freedom system. Next the parameters of the system have been identified by static and dynamic tests. Finally, comparisons between theoretical and experimental results were made for the pendulum–shaker system and a good agreement was obtained for a wide range of the system parameters. c

On a class of differential equations with left and right fractional derivatives

Abstract: We treat fractional order differential equations that contain left and right Riemann-Liouville fractional derivatives. Such equations arise as the Euler-Lagrange equation in variational principles with fractional derivatives. We find solutions of such equations or construct corresponding integral equations.

Nonlinear solution methods for infinitesimal perfect plasticity

Abstract: We review the classical return algorithm for incremental plasticity in the context of nonlinear programming, and we discuss the algorithmic realization of the SQP method for infinitesimal perfect plasticity. We show that the radial return corresponds to an orthogonal projection onto the convex set of admissible stresses. Inserting this projection into the equilibrium equation results in a semismooth equation which can be solved by a generalized Newton method. Alternatively, an appropriate linearization of the projection is equivalent to the SQP method, which is shown to be more robust as the classical radial return. This is illustrated by a numerical comparison of both methods for a benchmark problem.

Exact solutions for the unsteady free convection flows on a porous plate with time-dependent heating

Abstract: The unsteady free-convection flows of a viscous and incompressible fluid near a porous infinite vertical plate (or wall) are investigated under an arbitrary time-dependent heating of the plate. Exact solutions of this problem are obtained with the help of Laplace transform technique, when the plate is moving with an arbitrary time-dependent velocity and for special cases of the impulsive and the accelerated heating effects. These solutions are given in closed form for arbitrary Prandtl number of the fluid and for the thermal porous wall with or without suction or injection. The particular cases of the thermal plumes which are responsible for atmospheric pollution are also discussed.

Optimal control of the free boundary in a two‐phase Stefan problem with flow driven by convection

Abstract: We present an optimal control approach for the solidification process of a melt in a container. The process is described by a two phase Stefan problem including flow driven by convection and Lorentz forces. The free boundary (interface between the two phases) is modelled as a graph. We control the evolution of the free boundary using the temperature on the container wall and/or the Lorentz forces. The control goal consists in tracking a prescribed evolution of the free boundary. We achieve this goal by minimizing a appropriate cost functional. The resulting minimization problem is solved numerically by a steepest descent method with step size control, where the gradient of the cost functional is expressed in terms of the adjoint variables. Several numerical examples are presented which illustrate the performance of the method.

On constitutive inequalities in nonlinear theory of elastic shells

Abstract: Constitutive inequalities in general static and dynamic theory of elastic shells undergoing finite deformation are discussed. Constitutive inequalities are well known in continuum mechanics. They express physical or mathematical restrictions for constitutive equations of 3D elastic materials. In this paper we discuss the analogs of the strong ellipticity, Hadamard and Coleman-Noll (GCN-condition) inequalities for nonlinear elastic shells. It is shown that the GCN-condition implies the strong elipticity for shell theory whereas the strong ellipticity is equivalent to the existence conditions of acceleration waves in shell.

Influence of rotary inertia on the fiber dynamics in homogeneous creeping flows

Abstract: The rotation of an inertialess ellipsoidal particle in a Newtonian fluid has been firstly analyzed by Jeffery [1]. He found that in the shear flow the particle rotates such that the end of its axis of symmetry describes a closed periodic orbit. The aim of this paper is to discuss the effects of particle inertia on the rotary motion of a fiber in uniform flow fields. We recall the equations of motion and the constitutive equation for the hydrodynamic moment in the case of a slender fiber. These equations are solved numerically for several flow fields. We demonstrate that for the plane flow fields with dominant vorticity (elliptic and rotational flows) the effect of inertia is the slow particle drift toward the flow plane.

Analytical solution for three-dimensional stresses in a short length FGM hollow cylinder

Abstract: This paper presents the general theoretical analysis of three-dimensional mechanical and thermal stresses for a short hollow cylinder made of functionally graded material. The steady-state temperature, displacements, and stresses distributions are derived due to the general mechanical and thermal boundary conditions, as functions of radial, circumferential and longitudinal directions. The material properties vary continuously along the thickness direction according to power functions of radial direction. The temperature and Navier equations are solved analytically, using the generalized Bessel function and Fourier series. A direct method of solution of Navier equations is presented.

Numerical optimization of the Voith-Schneider® Propeller

Abstract: The Voith-Schneider® Propeller (VSP) is an efficient ship propeller that was invented by Ernst Leo Schneider (1894-1975) in 1927. It is a complex propeller having several parameters determining the behavior of the VSP. One of them is the Blade Steering Curve (BSC) which describes the angle of a propeller blade for a time instant. We investigate the choice of this BSC in order to optimize the efficiency of the VSP. We use numerical optimization and Computational Fluid Dynamics (CFD), in particular the Vortex-Lattice Method for the optimization and Finite Volume Method for validation. We obtain a significant improvement of 4.8%.

On the 2D models of plates and shells including the transversal shear

Abstract: The 2D Kirchhoff‐Love (KL) theory and the Timoshenko‐Reissner (TR) theory for thin shells made of the transversal isotropic homogeneous material are discussed. For the cyclic-symmetric deformations of shells of revolution the asymptotic analysis of strain‐stress states is fulfilled. Two simple linear problems for double-periodic deformations of plates are studied basing on the exact 3D theory and on the 2D approximate theories. From these problems it follows that the KL theory is asymptotically correct because it gives the first term of asymptotic expansion of the 3D solution. The TR theory is asymptotically incorrect. It also gives correctly the first term and incorrectly gives the second term. But if the transversal shear module is comparatively small then this theory gives the main part of the second term. The case of the extremely small shear module is discussed. As an example the multi-layered plate with the alternating hard and soft isotropic layers is studied. Copyright line will be provided by the publisher

Problem of crack perturbation based on level sets and velocities

Abstract: We describe cracks with the help of a given velocity as zero-level sets of a non-negative function satisfying a transport equation. For smooth velocities this description is equivalent to the coordinate transformation of a domain containing the crack inside. Analytical examples of cracks described by smooth as well as discontinuous velocities are presented in 2D and 3D domains. Based on a level-set formulation we consider the crack perturbation problem subject to a non-penetration condition and derive the formula for the shape derivative.

On new phase inclusions in elastic solids

Abstract: Two-phase deformation of an elastic body is studied by a small strain approach. Phase transformations of martensite type are implied. An arbitrary anisotropy of the phases and arbitrary transformation eigenstrains are taken into account. An additional thermodynamic condition on the equilibrium interface and a configurational force are presented in a form relating an effective strain or stress on one side of the interface and the normal to the interface. The characteristic features and the shape of the equilibrium ellipsoidal and cylindrical new phase inclusions are studied. Then the difference between the equilibrium and optimal ellipsoidal inclusions is discussed. A necessary stability conditions is proved. An expression of the configurational force acting on the interface of an ellipsoidal inclusion is derived in a form convenient for the analysis. An example of the analytical examination of the behavior of the inclusion not far from the equilibrium is given.

Isospectral families of high-order systems†

Abstract: Earlier work of the authors concerning the generation of isospectral families of second order (vibrating) systems is generalized to higher-order systems (with no spectrum at infinity). Results and techniques are developed first for systems without symmetries, then with Hermitian symmetry and, finally, with palindromic symmetry. The construction of linearizations which retain such symmetries is discussed. In both cases, the notion of strictly isospectral families of systems is introduced - implying that properties of both the spectrum and the sign-characteristic are preserved. Open questions remain in the case of strictly isospectral families of palindromic systems. Intimate connections between Hermitian and unitary systems are discussed in an Appendix.

Buckling and post‐buckling of a nonlinearly elastic column

Abstract: The classical problem of buckling of an inextensible elastic column, under the action of a compressive force is examined The column is made of nonlinearly elastic material for which the stress-strain relation is represented by the Ludwick constitutive law An approximative formula for determination of the force at immediate post-buckling is given Further post-buckling solutions are obtained for different values of the nonlinearity parameter by numerical integration using the Runge-Kutta-Fehlberg algorithm, and are presented in non-dimensional diagrams It is shown that no bifurcation point is found in the case of nonlinearly elastic column

Vibration of generalized double well oscillators

Abstract: We have applied the Melnikov criterion to examine a global homoclinic bifurcation and transition to chaos in a case of a double well dynamical system with a nonlinear fractional damping term and external excitation. The usual double well Duffing potential having a negative square term and positive quartic term has been generalized to a double well potential with a negative square term and a positive one with an arbitrary real exponent q > 2. We have also used a fractional damping term with an arbitrary power p applied to velocity which enables one to cover a wide range of realistic damping factors: from dry friction p ! 0 to turbulent resistance phenomena p = 2. Using perturbation methods we have found a critical forcing amplitude µc above which the system may behave chaotically. Our results show that the vibrating system is less stable in transition to chaos for smaller p satisfying an exponential scaling low. The critical amplit ude µc as an exponential function of p. The analytical results have been illustrated by numerical simulations using standard nonlinear tools such as Poincare maps and the maximal Lyapunov exponent. As usual for chosen system parameters we have identified a chaotic motion above the critical Melnikov ampl itude µc. Copyright line will be provided by the publisher

Solving variational problems in image processing via projections – a common view on TV denoising and wavelet shrinkage

Abstract: In this paper we investigate a special class of minimization problems for image denoising and deblurring. We focus on problems with positively one homogeneous penalty terms and show that the minimizer can be given in terms of projections. Especially we apply the results to wavelet shrinkage and TV denoising. Furthermore we present an application to deblurring with a TV penalty term. The approach presented here provides a common view on TV denoising and wavelet shrinkage.

Local modelling of sea surface topography from (geostrophic) ocean flow

Abstract: Calculating locally without boundary correction would lead to errors near the boundary. To avoid these Gibbs phenomenona we additionally consider the boundary integral of the corresponding region on the sphere which occurs in the integral formula of the solution. For reasons of simplicity we discuss a spherical cap first, that means we consider a continuously differentiable (regular) boundary curve. In a second step we concentrate on a more complicated domain with a non continuously differentiable boundary curve, namely a rectangular region. It will turn out that the boundary integral provides a major part for stabilizing and reconstructing the approximation of the solution in our multiscale procedure.

Feedforward tracking control for non‐uniform Timoshenko beam models: combining differential flatness, modal analysis, and FEM

Abstract: This paper considers the feedforward tracking control problem for flexible structures with spatially varying geometrical and material parameters. The design approach is illustrated for a Timoshenko beam model and is based on the Riesz spectral properties of the system operator, which allow to determine the modal system representation. This serves as the starting point for the derivation of an infinite-dimensional inverse system description in terms of a so-called flat output parameterizing modal states and input. Convergence of the resulting parameterizations can be ensured by appropriate motion planning. The feedforward control design methodology is further complemented to compute appropriate control commands directly from finite element approximations of the considered structure. As a result, feedforward tracking control design for flexible structures with complex geometry, boundary conditions, or material parameters can be performed within a general framework. Simulation scenarios for feedforward boundary control of a clamped-free Timoshenko beam model with variable system parameters illustrate the usefulness of the approach.

Wax diffusivity under given thermal gradient: a mathematical model

Abstract: In this paper we describe how to obtain wax diffusivity and solubility in a saturated crude oil using the measurements of solid wax deposit in the experimental apparatus known as cold finger. Assuming that migration of dissolved wax is primarily driven by thermal gradients, a mathematical model is derived relating the deposit growth rate to the above mentioned quantities. We investigate the case in which the oil is not agitated. Comparisons with available experimental data are performed and possible sources of errors are discussed.

On the stability of quasi-static paths for finite dimensional elastic-plastic systems with hardening

Abstract: In this paper we prove the stability of quasi-static paths of finite dimensional mechanical systems that have an elastic-plastic behavior with linear hardening. The concept of stability of quasi-static paths used here is essentially a continuity property relatively to the size of the initial perturbations (as in Lyapunov stability) and to the smallness of the rate of application of the external forces (which plays here the role of the small parameter in singular perturbation problems). The discussion of stability is preceded by the presentation of mathematical formulations (plus existence and uniqueness results) for those dynamic and quasi-static problems, in a form that is convenient for the subsequent discussion of stability.

Non-Newtonian Flow due to Noncoaxially Rotations of a Disk and a Fluid at Infinity

Abstract: The flow produced by a rotating disk in a second-order fluid which at infinity is itself in a rigid body rotation around another axis of rotation, parallel but not coincident with that of the disk is considered. It is shown that the equations of motion have an exact solution that may be obtained in general numerically. The expressions of the force and the torque exerted by the fluid on the disk are given. A special case in which rotations are with nearly the same angular velocities is discussed. Betrachtet wird die Stromung, die von einer rotierenden Scheibe in einer Flussigkeit zweiter Ordnung erzeugt wird, welche im Unendlichen starr um eine der Scheibenachse parallele, nicht mit dieser zusammenfallenden Achse rotiert. Es wird gezeigt, das die Bewegungsgleichungen eine exakte Losung haben, die man im allgemeinen auf numerischem Wege erhalten kann. Ausdrucke fur die Kraft und das Drehmoment, welche die Flussigkeit auf die Scheibe ausubt, werden angegeben. Es wird ein Spezialfall diskutiert, in welchem beide Rotationen fast die gleiche Winkelgeschwindigkeit haben.

Non local balance laws in traffic models and crystal growth

Abstract: This note addresses the well posedness of Temple systems with non local sources The resulting theorem holds globally in time and without requiring any smallness of the initial data Its scope comprises models for traffic flow and for crystal growth

On control of shear flow of an upper convected Maxwell fluid

Abstract: We consider parallel shear flow of an upper convected Maxwell fluid. A control is available in the form of a body force. The system of equations governing the velocity and shear stress is equivalent to a damped linear wave equation, and well known results imply that it is controllable if the time interval is sufficiently long for waves to propagate back and forth across the uncontrolled region. On the other hand, the issue of controlling normal stresses is far more difficult. If the control is available in the entire flow region, then normal stresses can be controlled, subject to a necessary positive definiteness constraint. On the other hand, if the control is confined to a subinterval, then the normal stresses are not controllable.

Asymptotic and discrete concepts for optimal control in radiative transfer

Abstract: Optimal control problems for the radiative transfer equation and for approximate models are considered. Following the approach first discretize, then optimize, the discrete SP N approximations are for the first time derived exactly and used for the study of optimal control based on reduced order models. Moreover, combining asymptotic analysis and the adjoint calculus yields diffusion-type approximations for the adjoint radiative transport equation in the spirit of the approach first optimize, then discretize.