Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 1999"

On the modeling of dynamic friction phenomena

Abstract: A low-dimensional dynamic model of frictional interactions is proposed by considering microscopic interactions between nominally flat surfaces. Coupling is introduced between the tangential motion and the separation of the surfaces due to the asperity contact between the surfaces. The numerical and analytical predictions of the resulting nonlinear model are found to qualitatively agree with a number of experimentally observed properties of friction. The paper suggests a means for arriving at a friction model in which ad hoc assumptions on the form of functions appear at a more fundamental level in the derivation than in previous modeling work. Indeed, properties of the present model are believed to be generic for models incorporating similar interactions. Ein niedrigdimensionales, dynamisches Modell von Reibungswechselwirkungen wird vorgeschlagen, indem mikroskopische Wechselwirkungen zwischen nominell ebenen Oberflachen in Betracht genommen werden. Eine Kopplung zwischen der tangentialen Bewegung und der Trennung der Oberflachen auf Grund des unebenen Kontaktes zwischen den Oberflachen wird eingefuhrt. Es wird festgestellt, das die numerischen und analytischen Vorhersagen des resultierenden nicht-linearen Modells qualitativ mit einer Reihe von experimentell beobachteten Eigenschaften von Reibung ubereinstimmen. Dieser Artikel zeigt einen moglichen Weg wie ein Reibungsmodell erreicht werden kann, in dem ad hoc Annahmen in der Form der Gleichungen in der Herleitung auf einem mehr fundamentalen Niveau erscheinen als in den bisher entwickelten Modellen. In der Tat erscheinen Eigenschaften des vorliegenden Modells als generisch fur Modelle, die ahnliche Wechselwirkungen berucksichtigen.

Elastic-plastic stresses in rotating discs by von Mises and Tresca

Abstract: The von Mises and Tresca yield criteria are combined with an equilibrium equation to provide the elastic-plastic stress distribution within the discs rotating at high speed. The Tresca criterion provides a closed solution that is traditionally associated with this problem. This paper examines and compares this with an alternative solution from the von Mises criterion. The latter requires that a suitable numerical solution to a govening differential equation is matched to the boundary conditions. For this a Runge-Kutta and a predictor-corrector method are combined to ensure a state of yield and continuity in stress at the interface between the inner core of perfectly plastic material and the outer elastic annulus. A comparison between the two solutions in a hollow disc shows that Tresca advances the elastic-plastic interface further than von Mises for a given speed. Thus, the Tresca prediction to the full y plastic speed is lower. Th e distributions of radial and hoop stress corresponding to the two criteria show only subtle differences within the elastic-plastic speed range for this disc. B y contras t, in a solid disc there is a marked difference in th e radial stress predictions. This alters the e distribution of residual stress and the apparent benefit that can be gained from compressive residual stress when prestressed discs are raised to their operating elastic speeds.

The Invariance of Asymptotic Laws of Linear Stochastic Systems under Discretization

Abstract: The stochastic trapezoidal rule provides the only equidistant discretization scheme from the family of implicit Euler methods (see [12]) which possesses the same asymptotic (stationary) law as underlying continuous time, linear and autonomous stochastic systems with white or coloured noise. This identity holds even when integration time goes to infinity, independent of used integration step size ! Especially, the asymptotic behaviour of first two moments of corresponding probability distributions is rigorously examined and compared in this paper. The coincidence of asymptotic moments is shown for autonomous systems with multiplicative (parametric) and additive noise using fixed point principles and the theory of positive operators. The key result turns out to be useful for adequate implementation of stochastic algorithms applied to numerical solution of autonomous stochastic differential equations. In particular, it has practical importance when accurate long time integration is required such as in the process of estimation of Lyapunov exponents or stationary measures for oscillators in mechanical engineering.

On the Solutions of Electrohydrodynamic Flow in a Circular Cylindrical Conduit

Abstract: This article considers the nonlinear boundary value problem (BVP) for the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit. We present rigorous results concerning the existence and uniqueness of a solution to this BVP for all relevant values of the parameters. We also show that the solution is monotonically decreasing and derive bounds on it in terms of the parameters. In [1] McKee et al. develop perturbation solutions in terms of the parameter governing the nonlinearity of the problem, α. This is done for both large and small values of α. For large α the solutions calculated here are qualitatively different from those calculated in [1]. This stems from the fact that for α large the solutions are O(1/α), not O(1) as proposed in the perturbation expansion used in [1].

An application of the Implicit Function theorem to an energy model of the semiconductor theory

Abstract: In this paper we deal with a mathematical model for the description of heat conduction and carrier transport in semiconductor heterostructures. We solve a coupled system of nonlinear elliptic differential equations consisting of the heat equation with Joule heating as a source, the Poisson equation for the electric field and drift-diffusion equations with temperature dependent coefficients describing the charge and current conservation, subject to general thermal and electrical boundary conditions. We prove the existence and uniqueness of Holder continuous weak solutions near thermodynamic equilibria points using the Implicit Function Theorem. To show the continuous differentiability of maps corresponding to the weak formulation of the problem we use regularity results from the theory of nonsmooth linear elliptic boundary value problems in Sobolev-Campanato spaces.

Numerical Solution of a Singular Integro-Differential Equation

Abstract: Numerical solution is obtained for the singular integro-differential equation with an antisymmetric forcing function f(x) (i.e. f(- x) = - f(x)), with end conditions \phi (- 1) = \phi (1) = 0, by three different methods, the two first of which presented here, produce the solution as accurate as the one obtained by Frankel (see [7]), recently. The convergence of the first method discussed in section 2, is also analysed.

Structural Optimization — The Interaction between Form and Mechanics

Abstract: Usually mechanical laws are applied to determine the structural response, for example deflections and stress state, while loads, boundary conditions, and geometry of the structure, i.e. the topology and the shape, are given. However, the mechanical principles can also be used to determine topology and shape of a structure for a prescribed structural response. This inverse method is called structural optimization. Since structural optimization deals in general with nonlinear and implicit functionals, only numerical methods have a chance to solve application-orientated problems in engineering design. Structural optimization can be distinguished into material, shape, and topology optimization depending on what is varied in the optimization process. The most challenging task is to determine the basic geometrical layout by topology optimization. In particular, recently the so-called material topology optimization of continuous structures has gained substantial interest both by mathematicians as well as engineers. The present contribution tries to consolidate these developments from an engineering point of view. In order to overcome problems of conventional numerical modeling techniques, material topology optimization is extended to geometrically adaptive methods. The adaptive discretization of the geometrical model in topology optimization turns out to be an appropriate way to obtain reliable results and simultaneously to reduce the numerical effort. This is verified by topology optimization problems with stress constraints and considering elastoplastic material behavior. The optimization based on either a linear or a nonlinear structural response leads to completely different results and shows the relevance of an appropriate mechanical model in the optimization process.

Random Eigenvalue Problems for Bending Vibrations of Beams

Abstract: The paper deals with the determination of statistical characteristics of eigenvalues for a class of ordinary differential operators with random coefficients. This problem arises from the computation of eigenfrequencies for the bending vibrations of beams possessing random geometry and material properties. Representations of eigenvalues are found by applying the Ritz method and perturbation results for matrix eigenvalue problems. Approximations of the probability density function and the moments of the random eigenvalues are given by means of expansions in powers of the correlation length of weakly correlated random functions which are used for modelling the random terms. The eigenvalue statistics determined analytically are compared favourably with Monte-Carlo simulations.

Generalized solutions to the equations of compressible Bingham flows

Abstract: A notion of a generalized solution is introduced for the one-dimensional equations of Bingham compressible flows. It makes possible to describe a joint motion of rigid and fluid zones without incorporation of free boundaries corresponding to fluid-rigid interfaces. A global unique solvability is proved. Examples are given to show the formation of fluid and rigid zones.

Right Focal Singular Boundary Value Problems

Abstract: Positive solutions are obtained for the (p, n - p) focal singular boundary value problem (-1) n-p y (n) = φ(t) f(t, y), 0 < t < 1, with y (i) (0) = 0, 0 ≤ i ≤ n - 1, and y (i) (1) = 0, p ≤ i ≤ n - 1. Here f may be singular at y = 0.

Generalized Nonlinear Implicit Quasivariational Inclusion and an Application to Implicit Variational Inequalities

Abstract: In this paper, we introduce and study a new class of generalized nonlinear implicit quasivariational mclusions for set valued mappings. We construct some new iterative algorithms for this generalized nonlinear implicit quasivariational inclasions for set-valued mappings without compactness. We prove the existence of solutions for this generalized non-linear implicit quasivariational inclusions for set-valued mappings without compactness and the convergence of iterative sequences generated by the algorithms. We also give an application to generalized nonlinear implicit variational inequalities.

Modelling finite deformation of polycrystals using local objective frames

Abstract: The formulation of constitutive models at finite strain using local objective frames is applied to solids with microstructure. Priviledged space frames are introduced at both the material level and the micro structure level. For single crystals, the method leads to a model almost equivalent to Mandel's theory. For polycrystals, an explicit concentration rule written in the corotational frame is proposed. Simulations of texture evolutions are then presented.

On the Motion of the Pendulum on an Ellipse

Abstract: In the present study, the motion of a pendulum on an ellipse is considered. The supported point of this pendulum moves on an elliptic path while the end point moves with arbitrary angular displacements. Applying Lagrange's equation, the equation of motion is obtained in terms of a small parameter e. This equation represents a quasilinear system of second order which can be solved in terms of a generalized coordinate φ.

On Some Partial Integral Equations Arising in the Mechanics of Solids

Abstract: We study several problems from physics and engineering whose mathematical modelling leads to partial integral equations involving functions of two variables. In particular, we give an explicit description of the essential spectra of the corresponding partial integral operators. This allows us to derive existence, uniqueness, and well-posedness results which are useful for the approximate solution and numerical analysis of the equations involved.

On the Consistency of some Eulerian Strain Rates

Abstract: In the past a variety of objective Eulerian strain rates has been considered. By use of the eigenproject concept it is shown that some of those rates, though being objective, are not mathematically admissible. Moreover, when the deformation rate and the objective strain rate are equalized, only a single strain, namely the Hencky strain remains valid. It is related to a single objective rate, the logarithmic rate.

Transient Motion of a Viscous Liquid with Free Surface in a Cylindrical Container

Abstract: The response of a viscous liquid with a free surface in a cylindrical container for a sudden change of a forced translational excitation x 1 e iQ1t to that of the form x 2 e iQ2t has been determined. The contact line of the liquid surface with the container wall has been treated either as freely slipping or as anchored contact line. The results apply also to the ring-out behavior of the liquid having been subjected to a forced excitation (t < 0), that has been stopped at t = 0. The magnitude of the resulting oscillation and transient time have been determined.

Direct and inverse scattering for point source fields. The penetrable small sphere

Abstract: A point source field is disturbed by the presence of a small penetrable scatterer which is either lossless or lossy. The point generated incident field is normalized in such a way as to be able to recover the relative scattering solutions by plane wave ezcitation, as the location of the source approaches infinity. For the case of a sphere, the low-frequency approximations of the zeroth, the first, and the second order are obtained in closed analytic form for both, the lossy and the lossless case. The scattering amplitude is obtained up to the third order. The scattering, as well as the absorption cross-section are calculated up to the second order. All results recover the case of plane wave incidence as the source recedes to infinity. Detailed parametric analysis shows that if the point source is located approximately four radii away from the spherical scatterer, then the scattering characteristics coincide with those generated from plane wave excitation. Furthermore, the dependence of the cross-sections on the ratio of the mass densities is analyzed. For the inverse scattering problem, we show that the second order approzimation of the scattering cross-section is enough to obtain the position, as well as the radius of an unknown sphere. This is achieved by considering the ezciting point source to be located at five specific places. The inversion algorithm is stable as long as the locations of the excitation points are not too far away from the scatterer. On the other hand if physical parameters are to be recovered from far field data, it seems that plane wave excitation is more promising.

Influence of non-newtonian flow behavior on local hemodynamics

Abstract: The influence of non-Newtonian blood flow in large arteries is studied numerically. The model studies are carried out to demonstrate the shear thinning effect at increased shear rates resulting from the reversible destruction of red blood cell aggregates and to show basic viscoelastic effects. The mathematical models considered use the conservation of momentum and of mass and constitutive relations describing the shear thinning behavior and relations of Jeffreys type (Oldroyd-B model). The numerical approach applies the finite element method. The non-Newtonian inelastic effects are demonstrated in a curved tube model and in an anatomically and physiologically realistic artery bifurcation model. The viscoelastic flow study is carried out using an axisymmetric tube with a local constriction modeling a stenosed blood vessel. The assumption of physiologically realistic shear thinning behavior shows a minor quantitative influence on local flow patterns compared with Newtonian reference flow.

Cahn-Hilliard Description of Secondary Flows of a Viscous Incompressible Fluid in an Unbounded Domain

Abstract: A viscous incompressible fluid under the action of an external force in an infinite strip is considered. We are interested in the situation when the trivial ground state loses stability and a whole sideband of Fourier modes becomes unstable. In order to describe the bifureating solutions a Cahn-Hilliard equation can be derived. We show that the mathematical criteria are satisfied to call the Cahn-Hilliard equation the amplitude or modulation equation of the system. Moreorer, we give a nonlincar diffusive stability result for the marginal stable ground state.

Influence of Variable Viscosity on the Onset of Hydrothermal Waves in Thermocapillary Liquid Bridges

Abstract: The float-zone erystal-growth process is investigated in the framework of the half-zone model. The influence of a temperature-dependent viscosity on the axisymmetric steady thermocapillary flow and its stability with respect to nonaxisymmetric perturbations is examined by means of a linear-stability analysis. The onset of oscillatory convection is studied numerically by a mixed Chebyshev-collocation finite-difference method. Detailed calculations have been carried out for Pr=4 and unit aspect ratio. It is found that the temperature-dependent viscosity decreases the critical Reynolds number for the onset of three-dimensional flow. Since the threshold shift is a function of the azimuthal wavenumber of the neutral mode, the critical mode may depend on the magnitude of the viscosity variation.

An Example for the Orbital Stability of Permanently Disturbed Non‐smooth Motions

Abstract: Developing a mathematical model of a mechanical system with friction and impact leads to problems regarding the integration of systems with variable structure. The main point of interest is the influence of permanent disturbances (both of physical and numerical nature) on the orbital stability of the non-disturbed solution. It is shown that both types of disturbances have similar effects and that, depending on the type of solution, different stability limits exist.

Subgrid stabilization of Galerkin approximations of monotone operators

Abstract: This paper presents a stabilized Galerkin technique for approximating monotone linear operators in a Hilbert space. The key idea consists in introducing an approximation space that is broken up into resolved scales and subgrid scales so that the bilinear form associated with the problem satisfies a uniform inf-sup condition with respect to this decomposition. An optimal Galerkin approximation is obtained by introducing an artificial diffusion on the subgrid scales.

On the Autoconvolution Equation and Total Variation Constraints

Abstract: This paper is concerned with the numerical analysis of the autoconvolution equation x * = = y icted to the interval [0, 1]. We present a discrete constrained least squares approach and prove its convergence in L p (0, 1), 1 ≤ p < ∞, where the regularization is based on a prescribed bound for the total variation of admissible solutions. This approach includes the case of non-smooth solutions possessing jumps. Moreover, an adaptation to the Sobolev space H 1 (0, 1) is added. A numerical case study concerning the determination of non-monotone smooth and non-smooth functions x from the autoconvolution equation with noisy data y completes the paper.

Nonnodal Condensation of Eigenvalue Problems

Abstract: We generalize the Guyan condensation of large symmetric eigenvalue problems to allow general degrees of freedom to be maste r variables. On one hand useful information from other condensation methods (such as Component Mode Synthesis) thus can be incorporated into the method. On the other hand this opens the way to iterative refinement of eigen-vector approximations. Convergence of such a procedure follows from the result, that one step of (static) condensation is equivalent to one step of inverse subspace iteration. A short outlook on several applications is included.

Fluid structure interaction in blood flow problems

Abstract: In this note, we consider some issues relevant for the application of Computational Fluid Dynamics to biomedical vascular investigations. More precisely, models for blood and vessel walls are briefly addressed together with some preliminary results in the numerical study of the fluid-structure interaction problem arising when investigating blood flow in compliant walls.

Exact Solution for Time-Dependent Diffraction of Plane Waves by Semi-Infinite Soft/Hard Wedges and Half-Planes

Abstract: At time t = 0 a plane wave front G(t - x cos θ - y sin θ) with incident angle θ strikes the diffracting edge (0, 0) of a wedge Γ with the opening angle β between the positive x-axis Γ 0 and a second half-line Γ β drawn from the origin. Homogeneous boundary conditions are posed on the outer banks of the wedge, Dirichlet on the upper half-line Γ + β : β+0, Neumann on the lower positive x-axis Γ - 0 : 2π - 0. For zero-initial data, the explicit total wave field solution in the exterior domain sector, β < a < 2π, is obtained from formulae for the half-plane (β = 0) derived by the author [31] as a convolution with respect to time with G.

Numerical Analysis of the Closed Osmometer Problem

Abstract: The closed osmometer problem, formulated by RUBINSTEIN and studied by RUBINSTEIN and MARTUZANS [5], is solved analytically and numerically. The model consists of a diffusion equation on an unknown domain and an equation describing the motion of a membrane that constitutes the boundary of the domain. In particular, the 1D case is studied where the second equation reduces to an ODE which expresses the equilibrium of osmotic and mechanical pressures. The initial-boundary value problem for the model equations is shown to be well-posed in appropriate spaces by a constructive technique. We compare results obtained by direct implementation of the iteration used in the proof with numerical solutions calculated by a semi-discretization method.

On the flow around a circular cylinder. Part II: Turbulent regime

Abstract: A concise review about the properties of the turbulent flow (260 < Re < 107) around a circular cylinder is presented. Some open problems of this flow are outlined.

Flow and heat transfer on a continuous flat surface moving in a parallel free stream with variable fluid properties

Abstract: Boundary layer flow and heat transfer on a continuous flat surface moving in a parallel free stream with variable fluid properties are investigated. The similarity solution is used to transform the problem under consideration into a boundary value problem of coupled ordinary differential equations. Numerical results are carried out for various values of the dimensionless parameters of the problem. The results have demonstrated that the assumption of constant properties may introduce severe errors in the prediction of surface friction factor and heat transfer rate. For the same Reynolds numbers, Prandtl numbers, heating parameter, temperature exponents for viscosity and thermal conductivity parameters, and the same velocity difference |U w - U∞|, larger skin friction, and heat transfer coefficient results for U w > U∞ than for U w < U∞.