Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 1998"

MHD Unsteady Free Convection Flow past a Vertical Porous Plate

Abstract: The present work is concerned with unsteady 2-dimensional laminar free convection flow of an incompressible, viscous, electrically conducting (Newtonian or polar) fluid through a porous medium bounded by an infinite vertical plane surface of constant temperature. A uniform magnetic field acts perpendicular to the surface which absorbs the fluid with a suction velocity varying periodically with time about a non-zero constant mean value. The equations of conservation of mass, momentum, angular velocity, and energy, which govern the flow and heat transfer problem, are solved analytically using regular perturbation techniques. The effects of material parameters such as Grashof number, Prandtl number, permeability parameter, suction parameter, and magnetic parameter on the velocity, angular velocity, and temperature are discussed. Numerical results are presented graphically and discussed.

Viscoelastic Flow by the Presence of Radiation

Abstract: An analysis of the steady flow of a viscoelastic fluid past an unmoving plate by the presence of radiation is considered. Analytical solutions for the temperature field have been derived and the effect of the radiation parameter on the temperature field is discussed.

Layer‐Adapted Grids for Singular Perturbation Problems

Abstract: In the present paper a survey is given on the application of Shishkin grids to convection-diffusion problems with dominant convection, further some new results and open problems are presented. The practical importance of these simplestructured grids lies in the possibility to resolve layers - the alternative technique of exponential fitting is not always successful. The use of uniformly stable method on a carefully chosen Shishkin mesh often leads to a uniformly convergent method.

Topological index‐calculation of DAEs in circuit simulation

Abstract: In electric circuit simulation we are confronted with highly nonlinear DAEs with low smoothness properties. They may have index 2 but they do not belong to the class of Hessenberg form systems that are well understood. The classic and the charge-oriented modified analysis are shown to lead to the same DAE-index if the circuit models satisfy some natural assumptions. We present a topological criteria for calculating the index. This makes it possible to determine the index also for high-dimensional circuit equation systems.

Bounds on Eigenvalues of Interval Matrices

Abstract: We describe a rectangle in the complex plane enclosing all eigenvalues of an interval matrix. We give theoretical bounds (Theorem 1) that are exact for symmetric or skew-symmetric matrices, and practical bounds (Theorem 2) requiring evaluation of 4 minimal or maximal eigenvalues and 2 spectral radii of symmetric matrices.

Theory of Porous Media — Past and Present

Abstract: Porous solids filled with liquid or gas play an important role in engineering, e.g., in material science, petroleum industry, chemical engineering, and soil mechanics as well as in biomechanics. Although porous media are of considerable practical significance the description of their mechanical and thermodynamical behavior has been unsatisfactory for a long time. The theory to describe the complex thermodynamical behavior of such saturated porous solids has come to certain well-founded conclusions only recently. It is the goal of this paper to show the historical development of the porous media theory, which already started in the eighteenth century, formed in some areas by polemic disputes and tragic events in the lifes of the scientists involved. Furthermore, the current state of the research into this subject is discussed, whereby the state of the development of the material independent basic equations and the constitutive theory is illustrated. For a certain class of models general theorems, such as minimum and maximum problems, are derived and the uniqueness of solutions of boundary value problems is proved.

Analytic Geometry and Singularities of Mechanisms

Abstract: In this article, we analyse the geometrical structure of the set M of configurations of a mechanism. Using Lie groups language and without any hypothesis of regularity, one can attach to this set a structure of an analytic variety. We shall then compute the tangent cone at cach point of this set without introducing any coordinates and we shall deduce the structure of M whatever the singularity may be. Finally, some examples are studied.

Numerical Exploitation of Equivariance

Abstract: Linear operators in equations describing physical problems on a symmetric domain often are also equivariant, which means that they commute with its symmetries, i.e., with the group of orthogonal transformations which leave the domain invariant. Under suitable discretizations the resulting system matrices are also equivariant with respect to a group of permutations. Methods for exploiting this equivariance in the numerical solution of linear systems of equations and eigenvalue problems via symmetry reduction are described. A very significant reduction in computational expense can be obtained in this way. The basic ideas underlying this method and its analysis involve group representation theory. The symmetry reduction method is complicated somewhat by the presence of nodes or elements which remain fixed under some of the symmetries. Two methods (regularization and projection) for handling such situations are described. The former increases the number of unknowns in the symmetry reduced system, the latter does not but needs more overhead. Some examples are given to illustrate this situation. Our methods circumvent the explicit use of symmetry adapted bases, but our methods can also be used to automatically generate such bases if they are needed for some other purpose. A software package has been posted on the internet.

Linear Programming with Inexact Data is NP‐Hard

Abstract: We prove that the problem of checking existence of optimal solutions to all linear programming problems whose data range in prescribed intervals is NP-hard.

Downwind Gauß-Seidel-Smoothing for Convection-Dominated Problems

Abstract: Due to the vanishing ellipticity, multigrid methods for convection-dominated problems tend to loose their good convergence properties. With a suited discretisation, an appropriate ordering of the unknowns and, if necessary, a special block partitioning, the smoothing property of the Gaus-Seidel iteration and therefore good convergence can be regained. This paper deals with such an ordering strategy and block partitionings.

The Saint-Venant Problem for Monoclinic Piezoelectric Cylinders

Abstract: The Saint-Venant problem is formulated in the framework of the linear theory of piezoelectricity for homogeneous, monoclinic, piezoelectric cylinders. A relaxed version of this problem is stated and is proved to be well-posed under Clebsch-type assumptions. The general solution of the relaxed problem is explicitly given, up to the solution of a two-dimensional Dirichlet-type problem involving two scalar fields, and is decomposed into five “fundamental” solutions. Finally, a Saint-Venant principle, related to the present relaxed formulation of the piezoelectric Saint-Venant problem, is established. In particular, the exponential decay along the cylinder axis of the free energy stored into a piezoelectric cylinder, subject to self-equilibrated loads on one base only, is proved.

The Principles of d'Alembert, Jourdain, and Gauss in Nonsmooth Dynamics Part I: Scleronomic Multibody Systems

Abstract: Die vorliegende Arbeit behandelt die Berechnung der Beschleunigungen in starren Mehrkorpersystemen, wenn diese dem Einflus mengenwertiger Kraftgesetze ausgesetzt sind. Die Kraftgesetze werden uber nicht-glatte Potentialfunktionen berucksichtigt und uber deren generalisiertes Differential dargestellt. Die dadurch entstehenden Punkt-Mengen-Abbildungen beinhalten neben glatten Kraftkennlinien auch Krafte aus zweiseitigen Bindungen sowie Kombinationen aus beiden wie einseitige Bindungen, Trockenreibung oder vorgespannte Federn mit Spiel. Stose werden nicht behandelt. Die klassischen Prinzipe von d'Alembert, Jourdain und Gauss werden mit Hilfe variationeller Ungleichungen verallgemeinert. Es wird ein streng konvexes Minimierungsproblem fur die unbekannten Beschleunigungen des Systems aufgestellt, das in der klassischen Mechanik als das Prinzip des kleinsten Zwangs bekannt ist. The paper treats the evaluation of the accelerations in rigid multibody systems which are subjected to set-valued force interactions. The interaction laws may be represented by non-smooth potential functions, and then derived through generalized differentiation. The resulting multifunctions contain the cases of smooth force characteristics, bilateral constraints, as well as combinations of them like unilateral constraints, dry friction, or prestressed springs with play. Impacts are excluded. A generalization of the classical principles of d'Alembert, Jourdain, and Gauss in terms of variational inequalities will be given. A strictly convex minimization problem depending on the unknown accelerations of the system will be stated, known in classical mechanics as the Principle of Least Constraints.

Anwendungen von Tensorfunktionen in der Kontinuumsmechanik anisotroper Materialien

Abstract: During the last three decades much effort has been devoted to the elaboration of phenomenological theories describing the relation between force and deformation in bodies of materials which do not obey either the linear laws of the classical theories of elasticity or the hydrodynamics of viscous fluids. Such problems will play a central role for mathematicians, physicists, and engineers also in the future [1]. - Material laws and constitutive theories are the fundamental bases for describing the mechanical behaviour of materials under multi-axial states of stress involving actual boundary conditions. In solving such complex problems, the tensor function theory has become a powerful tool. This paper will provide a short survey of some recent advances in the mathematical modelling of materials behaviour including anisotropy and damage. The mechanical behaviour of anisotropic solids (materials with orientated internal structures, produced by forming processes and manufacturing procedures, or induced by permanent deformation) requires a suitable mathematical modelling. The properties of tensor functions with several argument tensors constitute a rational basis for a consistent mathematical modelling of complex material behaviour, This paper presents certain principles, methods, and recent successful applications of tensor functions in solid mechanics. The rules of specifying irreducible sets of tensor invariants, and tensor generators of material tensors of rank two and four are also discussed. Furthermore, it is very important to determine the scalar coefficients in constitutive and evolutional equations as functions of the integrity basis and experimental data. It is explained in detail that these coefficients can be determinded by using tensorial interpolation methods. Some examples for practical use are discussed. Finally, we have carried out our own experiments in order to examine the validity of the mathematical modelling. - Like applications in solid mechanics, tensor functions also play a significant role in mathematical modelling in fluid mechanics. This paper, however, is restricted to the mechanical behaviour of solids.

Time-dependent plane wave diffraction by a half-plane : Explicit solution for Rawlins' mixed initial boundary value problem

Abstract: This paper deals with the diffraction of a time-dependent plane wave field G(t - xcosθ - ysinθ) governed by the two-dimensional wave equation and striking the edge of the half-plane Σ: x > 0, y = 0 at time t = 0 with some incident angle θ. The explicit solution formula for the total wave field is derived as a convolution with respect to time for homogeneous initial data and homogeneous boundary conditions. Dirichlet on the upper, Neumann on the lower bank of Σ. The Cagniard de Hoop method [1] is seen to be applicable due to the Wiener-Hopf solution of the corresponding stationary Rawlins problem [13] obtained in [16] by generalized L 2 -factorization of its piece-wise continuous Fourier matrix symbol relative to the real line on the basis of [17]. This approach is also inspired by the attempt to solve transient half-plane problems via spectral theory ([2;16]). The method to prove the limiting absorption principle for the pure Dirichlet problem in [2] (by deforming integral paths [9]) has intrinsic anologies to the Cagniard de Hoop method used here.

Analysis of Membrane Locking in hp FEM for a Cylindrical Shell

Abstract: In this paper we analyze the performance of the hp-Finite Element Method for a cylindrical shell problem. Our theoretical investigations show that the hp approximation converges exponentially, provided that boundary layers stemming from the edge effect are resolved. The numerical results illustrate the mesh independence of the exponential convergence of the hp-FEM.

Identification of Weakly Singular Memory Kernels in Viscoelasticity

Abstract: Inverse problems for the identification of memory kernels in the linear theory of viscoelasticity with constitutive stress-strain-relation of Boltzmann type are dealt with in the case of weakly singular kernels in the space L p , and of continuous kernels with power singularity at zero. The problems are reduced to nonlinear Volterra integral equations of convolution type for which by the method of contraction with weighted norms global existence, uniqueness, and stability of solutions are proved.

An Inverse Sturm‐Liouville Problem by Three Spectra

Abstract: It is shown that the potential of the Sturm-Liouville equation on interval [0,a] may be restored by the spectra of three boundary problems generated by the equation on the intervals [0,a], [0, 1/2a] and [1/2a,a], respectively. The algorithm of construction is given as well as the sufficient conditions for three sequences of real numbers to be the spectra of the mentioned boundary problems. The problem on [0,a] describes small vibrations of a smooth string with fixed ends. The problems on the half-intervals describe vibrations of the same string clamped at the point of equilibrium.

Optimal control of multibody systems in minimal coordinates

Abstract: The present paper deals with numerical methods for optimal control of mechanical multibody systems, such as many industrial robots, whose dynamical behavior can be described in minimal coordinates by a system of semi-explicit second order differential equations: M(q(t))q(t) = u(t) + h(q(t), q(t), t), 0 ⩽ t ⩽ tf, where q denotes the state variables, u the control variables, and M(q) the positive definite mass matrix. Numerical methods are addressed for computing an approximation of the optimal control u*(t), 0 ⩽ t ⩽ tf, which steers the system from an initial to a final position minimizing a performance index, such as time or energy, subject to bounds or nonlinear constraints on q, q, and u. A method is investigated in more detail which is based on piecewise polynomial approximations of state variables and utilizes the structure of the dynamical equations as well as the structure in the resulting large and sparse, nonlinearly constrained optimization problems. Results for an industrial robot with six joints demonstrate that tailored optimization methods are very well suited for fast off-line optimization of robot trajectories.

On the Computation of Lamé Functions, of Eigenvalues and Eigenfunctions of Some Potential Operators

Abstract: Eigenvalue problems for the double-layer potential operator and the electrostatic integral operator are of considerable interest in some models for permanent magnetization of compact bodies. For the case that the underlying surface is either a sphere, a spheroid or a triazial ellipsoid, explicit expressions for eigenvalues and eigenfunctions are well known. For the ellipsoid, these quantities are given in terms of Lame functions and surface ellipsoidal harmonics. Since only few Lame functions can be written in closed form numerical methods are required. We provide an effective scheme for the computation these functions.

Locally Supported Wavelets on the Sphere

Abstract: We construct explicitly wavelets on the sphere that provide a locally supported and stable basis for the Sobolev spaces H2,0 ⩽ s < 1. We get at hand at fast wavelet transform with almost optimal complexity. This basis can be easily implemented in numerical schemes. We apply the wavelet transform to singularity detection and data compression. This contribution summarizes the results of [1].

Analytical Mechanics. Addendum I: Inequality Constraints with Elastic Impacts. The Convex Case

Abstract: The aim of the present paper is to present some new results in Analytical Mechanics when impacts may occur We prove certain equivalent forms for d'Alembert's principle including the velocity discontinuity, and then we show that the theorem of stationary action still holds as an inequality The consideration of the variation of the unknown impact time implies certain new variational expressions in inequality form for the impact problem

Residual Stresses in an Elastic-Plastic Solid Shaft with Fixed Ends after Previous Rotation

Abstract: The subject of this paper is the distribution of the residual stresses in an elastic-perfectly plastic solid cylinder with fixed ends after previous rotation. Special attention is paid to the occurrence of secondary plastic flow during unloading. All the results are based on Tresca yield condition and the flow rule associated with it.

On the Computation of the Matrixk-th Root

Abstract: In this paper we derive a family of iterative methods for computing the k-th root and the inverse k-th root of a given matrix. We will show that the methods are locally convergent. The methods are analyzed and their numerical stability is investigated.

Generation of Difference and Error Formulae of Arbitrary Consistency Order on an Unstructured Grid

Abstract: For the application of the finite difference method (FDM) on an unstructured finite element grid we want to generate difference formulae and corresponding error estimates of arbitrary consistency order. The problem is to find grid points with sufficient information for the determination of the interpolation polynominals, good error estimate and optimal structure of the resulting linear system of equations of the FDM. A sophisticated algorithm for the selection of good and the exclusion of bad grid points is presented. For the generation of the formulae the principle of the influence polynomials is used.

Unbounded Solutions in Non‐Smooth Dynamical Systems at Resonance

Abstract: We consider periodically forced non-smooth dynamical systems at resonance, described by differential inclusions, and we show that analogously to the case of ODEs all solutions are unbounded in the (x)-phase plane, if the first Fourier coefficient of the forcing is large compared to a certain quantity related to the nonlinearity.

Der hydraulische Mechanismus des Spinnenbeines und seine Anwendung für technische Probleme

Abstract: Das Verstehen der Funktionsweise biologischer Systeme und ihre Modellierung ist fur die Erarbeitung neuer Wirkprinzipien nach biologischem Vorbild eine wesentliche Grundlage. Dafur leistet die vorliegende Arbeit einen Beitrag. Die Resultate der Arbeit zeigen, das Modellierung und Simulation des Bewegungsmechanismus des Spinnenbeines auf der Basis eines Starrkorpermodells unter der Betrachtung des hydraulischen Antriebs moglich ist. Die Ubertragung der Konstruktions- und Wirkungsprinzipien aus dem biologischen Bereich auf die technischen Systeme (auch Mikrosysteme) ist anschaulich am Beispiel eines Greifers demonstriert worden. Dabei soll jedoch nur die grundsatzliche Moglichkeit der Realisierung eines Greifers nach dem Vorbild des Spinnenbeines geschildert werden, ohne auf konstruktive Belange tiefer einzugehen. Dies konnte Gegenstand einer weiteren Publikation sein. To understand the function of biological systems and their mathematical description is important in order to find new principles for technical applications. In this paper, the possibility of the description and the simulation of a Spider Leg Locomotion System using a Rigid Body System with hydraulic drive is presented. The transformation of the biological and design principles to technical systems including microsystems is demonstrated on a gripper. There, however, only the principal possibility to realize a gripper on the model of a spider leg is presented without going into more details how to design such a gripper. This might be the subject of a forthcoming paper.

Effect of Thermocapillarity on the Production of a Conducting Thin Film in the Presence of a Transverse Magnetic Field

Abstract: The gradual development of a thin liquid film on the surface of a rotating disk is studied analytically when the thermocapillary force is in action in the presence of a transverse magnetic field. It is found that the acting thermocapillary force has a profound effect in enhancing the thinning rate of the film even in the presence of large Hartmann number M. A physical explanation of this result is provided. Large amounts of fluid are depleted in a small span of time when the Hartmann number M is small.

A Nonlinear Parabolic Initial Boundary Value Problem Modelling the Continuous Casting of Steel

Abstract: In this paper the dependence of the solution of a nonlinear parabolic initial boundary value problem modelling the continuous casting of steel on the problem parameters, especially on the heat transfer function entering into the boundary conditions, is investigated. After proving the well-posedness of the problem it is possible to show that the solution is continuously Frechet differentiable with respect to the heat transfer function under appropriate smoothness assumptions on the problem parameters. In the case of a piecewise smooth heat transfer function, which is practically relevant, uniqueness and existence of a weak solution of the problem and a qualitative stability result are proved.