Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 2001"

An Isoperimetric Inequality for Eigenvalues of the Stekloff Problem

Abstract: Let Ω be a bounded smooth domain in ℝn and let 0 = λ1 ≤ λ2 ≤ … denote the eigenvalues of the Stekloff problem: Δu = 0 in Ω and (∥u)/(∥ν) = λi on ∥Ω. We show that , where denotes the second eigenvalue of the Stekloff problem in a ball having the same measure as Ω. The proof is based on a weighted isoperimetric inequality.

A Geometrical Interpretation and Uniform Matrix Formulation of Multibody System Dynamics

Abstract: The purpose of this paper is to associate the multibody dynamics procedures with a geometrical picture involving the concepts of configuration manifolds, tangent vector spaces, and orthogonality of constraint reactions to the constraint surfaces. An unconstrained mechanical system is assigned a free configuration manifold and is treated as a generalized particle on the manifold. The system dynamics is then formulated in the local tangent space to the manifold at the system representation point. Imposed constraints on the system, the tangent space splits into the velocity restricted and velocity admissible subspaces, while the system configuration manifold confines to the holonomic constraint manifold. Based on these geometrical concepts, a uniform vector-matrix formulation is developed. Both holonomic and nonholonomic systems are treated in a unified way, and the dynamic equations are expressible either in generalized velocities or in quasi-velocities. Using a geometrically grounded projection method, compact schemes for obtaining different types of equations of motion and for determination of constraint reactions are provided. Some fresh contributions to the theory of constrained systems are reported. A relationship between the present formulation and the other classical methods of analytical dynamics is shown.

Numerical Computation of a Cauchy Problem for Laplace's Equation

Abstract: This paper investigates the numerical computation of a Cauchy problem for Laplace's equation which is a typical ill-posed problem. By using Green's formula, the problem is transformed to a moment problem. For numerical computation of the moment problem, an error estimation and several numerical examples for verification are presented. Necessary and sufficient conditions for the existence of the solution of the Cauchy problems for Laplace's equation in two-dimension are also given.

Optimal Control of a Phase‐Field Model Using Proper Orthogonal Decomposition

Abstract: The proper orthogonal decomposition (POD) is a procedure to determine a reduced basis for a reduced order model. In this article POD is formulated as a minimization problem in a general Hilbert space setting. The POD-basis functions are given by the solution to the first-order necessary optimality conditions. In this work POD is utilized to solve optimal control problems for a phase-field model. The numerical results are compared with finite element solutions.

Fast solution of optimal control problems in the selective cooling of steel

Abstract: We consider the problem of cooling milled steel profiles at a maximum rate subject to given bounds on the difference of temperatures in prescribed points of the steel profile. This leads to a nonlinear parabolic control problem with pointwise state constraints in a 2D domain. A method of instantaneous control is applied to set up a fast solution technique.

Analysis of Dynamic Behaviour of Viscoelastic Rods Whose Rheological Models Contain Fractional Derivatives of Two Different Orders

Abstract: The problems of longitudinal nonstationary vibrations of a viscoelastic rod of a finite length, the impact of a viscoelastic bar moving along its axis against a rigid barrier, and the stress wave propagation in a semi-infinite viscoelastic rod are investigated. The modified generalized standard linear solid model involving fractional derivatives of two different orders is used as the model describing the viscoelastic properties of the bar's material. The problem is solved by the Laplace integral transformation method, in so doing, as distinct to traditional numerical approaches, the characteristic equation involving fractional powers is not rationalized, but it is solved directly with the fractional powers. The numerical analysis of the enumerated problems is presented. The time dependence of the stress and of the contact stress in the bar corresponding to the first and second problems, respectively, has been obtained and analyzed for various magnitudes of the rheological parameters: the orders of fractional derivatives and the relaxation time, As investigations show, the bar does not adhere to the wall at any magnitudes of the rheological parameters. The asymptotic solutions for the problem of stress wave propagation have been obtained in the vicinity of the wave front and at small magnitudes of time. It is shown that the given model can describe both diffusive and wave phenomena occurring in viscoelastic materials. All is dependent on a relation between the orders of the derivatives standing at the left hand side and right hand side of the rheological equation.

Development of a New Cavitation Model based on Bubble Dynamics

Abstract: This paper presents a numerical approach for a cavitation model that bases on a combination of the Volume-of-Fluid technique with a model predicting the growth and collapse process of bubbles. The cavitation model is applied for the simulation of cavitating nozzle flows and cavitating flow over a NACA 0015 hydrofoil and showed it's capability to resolve characteristic effects of cavitation such as the cyclic formation of the cavitation cloud, the formation of the re-entrant jet and the local occurrence of hydrodynamic pressure peaks due to bubble cloud collapse.

Heat Transfer in MHD Visco‐elastic Fluid Flow over a Stretching Surface

Abstract: An analysis has been carried out to study the effect of magnetic field on the visco-elastic fluid flow and heat transfer over a non-isothermal stretching sheet with internal heat generation. The solutions for heat transfer characteristics are evaluated numerically for different parameters such as Prandtl number, magnetic field, suction, visco-elasticity, and the temperature profile. The solutions for the temperature profile, heat transfer characteristics, and their asymptotic limits for large and small Prandtl numbers are obtained in terms of Kummer's function. The important finding is that the effect of visco-elasticity is to decrease temperature profile in the flow field for small values of Prandil number. The temperature profile decreases with the increase of magnetic field.

Asymptotic Behaviour of the Elastic Solution near the Tip of a Crack Situated at a Nonideal Interface

Abstract: The interface crack problem with a nonideal interface described by special transmission conditions is examined. The corresponding modelling boundary value problem is reduced to a system of singular integral equations with moving and fixed point singularities. The existence and uniqueness of the system solution are proved. Possible shapes of an intermediate zone as well as combinations of material parameters are investigated. Asymptotic expansions of the stresses and displacements near the crack tip are found. The obtained results are discussed from a fracture mechanical point of view. Numerical examples concerning the calculation of the stress singularity exponent as well as the generalized SIFs are presented.

Procedures and Models for Aeroservoelastic Analysis and Design

Abstract: The multi-disciplinary area that deals with the interaction of the structural, aerodynamic, and control systems of flight vehicles is called aeroservoelasticity. The various aircraft design aspects which are affected by aeroservoelastic interaction are presented with a common modal-based formulation. The disciplinary and interaction techniques are reviewed and combined for an integrated design optimization scheme where stress, static-aeroelastic, closed-loop flutter, control margins, time response, and continuous gust constraints are treated with a common basic model. The structure is represented in the basic model by a set of low-frequency normal modes of a baseline design. Design changes are adequately addressed without changing the generalized coordinates. Typical difficulties of the modal approach are alleviated by various optional fictitious-mass and modal perturbation techniques. Static modes can be added during the optimization process for better convergence to the optimal solution. Minimum-state rational approximation of the unsteady aerodynamics leads to an efficient state-space model which can be augmented by any combination of linear control components. Physical weighting algorithm is used to improve the aerodynamic approximations and to select modes for truncation or residualization. Linear reduced-size models and the associated analytic sensitivities to design changes facilitate extremely efficient and adequately accurate on-line design sessions. The linear models can be integrated with computational aerodynamics codes for the evaluation of important non-linear aerodynamic effects early in the design process.

Uniqueness and Non‐uniqueness in the Steady Displacement of Two Visco‐plastic Fluids

Abstract: We study steady miscible displacements of two visco-plastic fluids in a long plane channel. If the yield stress of the displacing fluid is less than that of the displaced fluid, uniform static residual layers can be left attached to the walls of the channel as the displacement front propagates steadily. We investigate this steady finger propagation and the problem of finger width selection. The problem is fully two-dimensional, with the two fluids separated by a sharp interface. For a given fixed interface, chosen from a wide class of physically sensible interface shapes, we show that there exists a unique solution. As well as flexibility in the exact shape of the interface, the residual static layer thickness is also non-unique. Typically layer thicknesses h ∈ (h min , h max ) admit a physically sensible static layer solution, where h min and h max are easily computable functions of the dimensionless problem parameters. The dependency of h min and h max on the dimensionless problem parameters is explained and example solutions are computed for different static residual thick-nesses.

The Far-Field Equations in Linear Elasticity — an Inversion Scheme

Abstract: In this paper the far-field equations in linear elasticity for the rigid body and the cavity are considered. The direct scattering problem is formulated as a dyadic one. This imbedding of the vector problem for the displacement field into a dyadic field is enforced by the dyadic nature of the free space Green's function. Assuming that the incident field is produced by a superposition of plane dyadic incident waves it is proved that the scattered field is also expressed as the superposition of the corresponding scattered fields. A pair of integral equations of the first kind which hold independently of the boundary conditions are constructed in the far-field region. The properties of the Herglotz functions are used to derive solvability conditions and to build approximate far-field equations. Having this theoretical framework, approximate far-field equations are derived for a specific incidence which generates as far-field patterns simple known functions. An inversion scheme is proposed based on the unboundedness for the solutions of these approximate “far-field equations” and the support of the body is found by noting that the solutions of the integral equations are not bounded as the point of the location of the fundamental solution approaches the boundary of the scatterer from interioir points. It is also pointed that it is sufficient to recover the support of the body if only one approximate “far-field equation” is used. The case of the rigid sphere is considered to illuminate the unboundedness property on the boundary.

Influence of Fluctuating Surface Temperature and Concentration on Natural Convection Flow from a Vertical Flat Plate

Abstract: A linearized theory is used to investigate how a free double-diffusive boundary layer flow is affected by small amplitude temporal variations in the surface temperature and species concentration. The mean temperature and the mean species concentration are assumed to vary as a power n of the distance measured from the leading edge. Three distinct methods, namely, a perturbation method for low frequencies, an asymptotic series expansion for high frequencies, and a finite difference method for intermediate frequencies, are used. Calculations have been carried out for a wide range of parameters in order to examine the results obtained from the three methods. Comparisons are made in. terms of the amplitudes and phases of the surface. heat transfer and surface mass transfer. It has been found that the amplitudes and phase angles predicted by perturbation theory and the asymptotic method are in good agreement with the finite difference computations.

On a Mathematical Model of Bars with Variable Rectangular Cross-sections

Abstract: Der Vekuasche Ansatz [1] des Aufbaus von Platten- und Schalentheorien, bei dem die Felder der Verschiebungen, der Deformationen und der Spannungen des dreidimensionalen Modells der linearen Elastizitatstheorie in Fourier-Legendresche Reihen nach der Veranderlichen der Dicke entwickelt werden, wird zum Aufbau einer Stabtheorie verallgemeinert. Diese Grosen werden hierbei in doppelte Fourier-Legendresche Reihen nach den Veranderlichen Dicke und Breite entwickelt. Sodann werden alle auser den ersten (N3 + 1) (N2 + 1), N3, N2 = 0, 1, … , Gliedern vernachlassigt. Eine solche Naherung des dreidimensionalen Modells durch ein eindimensionales Modell heist (N3, N2)-Approximation. Die Frage der Wohlgestelltheit der Anfangs- und Randwertprobleme wird untersucht. Der Fall, in dem der veranderliche Querschnitt zu einem Intervall oder einem Punkt entartet, wird auch betrachtet. Solche Stabe heisen zugespitzte Stabe (s. auch [2]). We generalize an idea of I. Vekua [1] who, in order to construct a theory of plates and shells, expands the fields of displacements, strains, and stresses of the three-dimensional theory of linear elasticity into orthogonal Fourier-Legendre series with respect to the variable thickness, to a bar model. In the bar model all above-mentioned quantities are expanded into orthogonal double Fourier-Legendre series with respect to the variables thickness and width of the bar, and then all but the first (N3 + 1) (N2 + 1), N3, N2 = 0, 1, …, terms are neglected. This case is called (N3, N2) approximation. The question of well-posedness of the initial and boundary value problems is investigated. The cases in which a variable cross-section degenerates to a segment of a straight line or into a point are also considered. Such bars are called cusped bars (see also [2]).

Sufficient condition of basis stability of an interval linear programming problem

Abstract: The contribution deals with basis stability of an interval linear programming (ILP) problem. We consider the coefficients of the matrix, right-hand side values and cost vector values to vary independently in the given intervals. We present the new necessary and sufficient condition, and the sufficient condition of basis stability of ILP problem.

On three-dimensional microcrack density distribution

Abstract: This paper considers tensorial representations of several microerack distribution functions due to tensile and compressive principal stresses in brittle materials in the framework of continuum mechanics. The common framework for deriving the damage tensors of different order from any density function is suggested. Second and fourth order damage tensors are derived for Dirac-. truncated Gauss-, and trigonometrical (cos 2 -) microcrack distributions using harmonic Fourier-like series. Each distribution is investigated under different combinations of tensile and compressive principal stresses for three-dimensional load cases. It is emphasized that only the trigonometrical distribution yields a spherical crack density surface for the fourth order tensor approximation under three equal principal stresses.

On the Stokes Paradox for Power-Law Fluids

Abstract: We study a purely viscous flow of a non-Newtonian fluid obeying the power-law in an exterior domain. We prove that for pseudo-plastic fluids the Stokes paradox does not take place, while for the dilatant fluids it takes place in any space dimension n if the flow index is larger or equal to n.

Compressible Potential Flows with Free Boundaries. Part I: Vibrocapillary Equilibria

Abstract: Various variational formulations describing nonstationary compressible fluid flows are considered. In particular, for high-frequency excitations a variationally based approximation frame is deduced which may explain experimentally observed phenomena.

Some Remarks on Quadrature Formulas for Refinable Functions and Wavelets

Abstract: This paper is concerned with the efficient computation of integrals of (smooth) functions against refinable functions and wavelets, respectively. We derive quadrature formulas of Gauss-type using these functions as weight functions. The methods are tested for several model problems and possible practical applications are discussed. In der vorliegenden Arbeit beschaftigen wir uns mit der effizienten Berechnung von Integralen (glatter) Funktionen gegen verfeinerbare Funktionen und Wavelets. Diese Funktionen werden als Gewichtsfunktionen fur Gauss-Quadratur Formeln gewahlt. Die entwickelten Quadraturformeln werden an mehreren Modellproblemen getestet und praktische Anwendungen werden diskutiert.

Free Terms and Compatibility Conditions for 3D Hypersingular Boundary Integral Equations

Abstract: In this paper two basic issues concerning hypersingular boundary integral equations (HBIE's) for three-dimensional problems are addressed. Firstly, a new general method for the evaluation of all free-term coefficients is presented. Secondly, the so-called Tricomi-Mikhlin compatibility conditions at non-smooth bounday points are proved. In both cases, the analysis is performed in the parametric space and with deep recourse to differential geometry. The final formulas are quite simple, Numerical results are provided to test these theoretical findings.

Universal wall‐boundary conditions for turbulence‐transport models

Abstract: In the industrial design process of fluids engineering devices, the use of numerical simulation is of ever increasing importance. The predictive quality of such simulations is often governed by the representation of turbulence. Virtually all industrial simulations mimic the influence of turbulence by a closure model based on transport equations for statistical turbulence properties. Besides the derivation of such transport-equation models, the adequate formulation of wall-boundary conditions has come into the focus of attention. Conventional boundary conditions rely on the validity of specific flow conditions pertaining to the wall-shear stress and the resolution properties of the computational grid in the wall-adjacent region. Since the shear stress is part of the simulation result, this approach—strictly speaking—requires the anticipation of the solution. Moreover, it significantly affects the efficiency and flexibility of the simulation due to the associated mesh constraints. The principal aim of this research is the development of a universal boundary condition. Examples included show encouraging results for attached and separated flows.

The Surface Temperature of a Halfplane Heated by Friction and Cooled by Convection

Abstract: Based on earlier works by the authors an analytical expression for the surface temperature of a halfplane heated by fast moving rolling/sliding contact and cooled by convection on the surface of the halfplane is presented by applying the Laplace transformation technique and the solution of a Volterra integral equation of the second kind. Furthermore, the influence of heat conduction in the rolling/sliding direction is investigated. Aufbauend auf fruhere Arbeiten der Autoren wird ein analytischer Ausdruck fur die Oberflachentemperatur einer Halbebene angegeben, die in einer rasch entlang der Oberflache bewegten Roll- oder Gleitkontaktzone aufgeheizt wird und durch freie Konvektion auserhalb der Kontaktzone gekuhlt wird. Dazu wird die Laplacetransformationstechnik herangezogen und eine Volterra-Integralgleichung zweiter Art fur die Oberflachentemperatur analytisch gelost. Der Einflus der Warmeleitung in Richtung der Roll-/Gleitrichtung wird zusatzlich abgeschatzt.

Green Function for a Stokes Flow near a Porous Slab

Abstract: The Green function of the Stokes equations for the creeping flow of a viscous fluid near a porous slab is calculated analytically. The flow in the porous slab is represented by Darcy equations and the boundary condition on the interface between the fluid and the porous medium is a slip condition proposed by Beavers and Joseph (1967) on the basis of experimental results. An alternative simpler condition in which a no-slip condition applies on the fluid side is also used for comparison. Streamlines then are calculated numerically and represented for various values of the parameters: porosity, slip length, thickness of the porous slab.

MHD Three‐Dimensional Couette Flow with Transpiration Cooling

Abstract: An analysis of the Couette flow between two horizontal parallel porous flat plates of an electrically conducting, viscous, incompressible fluid is presented. The stationary plate is subjected to a transverse sinusoidal injection of the fluid and its corresponding removal by constant suction through the other plate in uniform motion. The flow becomes three-dimensional due to such an injection velocity. A magnetic field of uniform strength is also applied normal to the planes of the plates. The effects of the injection/suction velocity and the magnetic field on the flow field, skin friction, and heat transfer are reported.

On Meshless Collocation Approximations of Conservation Laws: Preliminary Investigations on Positive Schemes and Dissipation Models

Abstract: We consider meshless collocation, methods for the numerical solution of transport processes described by hyperbolic conservation laws. The future goal is the construction of a robust and reliable meshfree discretization method for the equations of gas dynamics in complex geometrics. In this paper we start with the simplest scalar model problems and analyze basic problems occurring in the grid-free approach. A topological condition on clouds of points is derived and several possible versions of a generalized Lax-Friedrichs scheme are discussed with respect to their numerical dissipation. A moving least-squares approach is followed to construct a positive discretization for solutions with shocks which is thoroughly analyzed and applied to test problems.

On a Class of Singularly Perturbed Parabolic Equations

Abstract: A method of matched asymptotic expansions has been used to construct an n-term uniformly valid approximate solution for un initiol value problem of a bnear singularly pertarted parabolic equation exhibiting an internal layer behavior. It is shown that each internal layer function caused by a non-smooth initial data can be described by an n-th iteroted integral of the complementary error function.

Bifurcations and Vortex Formation in the Ginzburg—Landau Equations

Abstract: Bifurcations from the normal to superconducting state are investigated for the two-dimensional Ginzburg-Landau system modelling a superconductor near the critical temperature T c . Nucleation of vortices is shown under periodic boundary conditions imposed on three observables, the local density of superconducting electrons, the local supercurrent density, and the microscopic magnetic field. This is demonstrated by rigorous analysis showing that the zeros of the eigenfunction involved in the bifurcation process (the so-called vortices) remain preserved in the solution during the bifurcation process for all values of the applied magnetic field H (used as the bifurcation parameter) near a critical value H c2 (H < H c2 ). The method is based on a factorization of the order parameter as the product of the eigenfunction and a bounded function. In particular, the solution describes the Abrikosov vortex state. Classical elliptic functions of Weierstrass are used to perform a singular perturbation analysis which employs well-known complex-analytic tools for the inhomogeneous Cauchy-Riemann equations.

The Meso-Shape Functions for the Meso-Structural Models of Wavy-Plates

Abstract: In this study, the determination of meso-shape functions for the meso-structural model of a wavy-plates is investigated.

Neglect of the Fluid Extra Stresses in Volumetrically Coupled Solid-Fluid Problems

Abstract: Usually the fluid flow through porous media is described by Darcy 's law or generalizations of it. In this contribution, Darcy 's law is derived from the theory of porous media. To archive this, the fluid extra stress (frictional stress) is neglected in comparision with the momentum exchange (drag force) between the fluid phase and the solid skeleton of a two-phase model. This common assumption is motivated by the results obtained from a microscopic capillary model in combination with a dimensional analysis of the continuum mechanical model. This analysis shows that the influence of the fluid extra stress decreases with increasing number of capillary tubes per area element.

Stabilization of Gyroscopic Systems

Abstract: Wir betrachten die Stabilitat gyroskopischer Systeme der Form d 2 u/dt 2 + 2ξG du/dt + (W - ξ 2 K) u = 0 mit G T = -G, W T = W und K > 0 und fur grosse Werte des Parameters ξ > 0. Eine Charakterisierung derjemigen Systeme wird gegeben, die (bei ξ → ∞) stabil unter Storungen sind. Ausserdem werden Resultate uber strenge Stabilitat hergeleitet. Die Berechnung von Stabilitatsintervallen der Art (ξ cr , ∞) wird diskutiert. This paper concerns the stability of gyroscopic systems of the form d 2 u/dt 2 + 2ξG du/dt + (W - ξ 2 K) u = 0, with G T = -G, W T = W, and K > 0, and for large values of the parameter (frequency of rotation) ξ > 0. A complete characterization is given of those limit systems (as ξ → ∞) which are stable after perturbation. Results are also derived concerning strong stability (i.e., when there is stability for the system and all neighbouring systems of the same type) and for computing an upper stability region (ξ cr . ∞).