Optimizing color picture tubes by high-cost nonlinear programming

In this paper we discuss the application of optimization techniques for the design of several parts of the color picture tube, the key component of television sets and computer monitors. These projects have been carried out for Philips Display Components in Eindhoven. Philips developed several computer simulation models of picture tube parts. Designers use these models to simulate the physical behavior of a particular part design. Depending on the amount of detail in the model, the running time of a typical simulation ranges from one up to 10 hours. Tube designers are confronted with the problem of finding settings for a large number of design parameters that are optimal with respect to several simulated tube characteristics. This problem can be modeled as a so-called high-cost nonlinear programming problem. This paper reports on the successful application of our four-step compact model approach to solve this problem. The presented results are based on four projects in which we optimized picture tube parts. Among the realized benefits for Philips are a design improvement of 30% and a time-to-market reduction of 50–60%.     

Waves in an Inhomogeneous Tube with a Flowing Two-Phase Fluid

Results of theoretical and mathematical justification of the problem on a pulsating flow of a two-phase barotropic bubbly fluid enclosed in an elastic semi-infinite cylindrical tube inhomogeneous along its length are presented. Linear one-dimensional equations are used. It is assumed that the tube is rigidly attached to the surrounding medium and therefore its displacement in the axial direction is absent. At infinity, the tube material is assumed to be homogeneous. To describe the pressure, flow rate, and displacement of the fluid, a pulsating pressure is given at the tube end. The problem stated is reduced to a singular Sturm-Liouville boundary-value problem, which in turn is reduced to a Volterra-type integral equation. This equation is solved by the method of successive approximations. By assuming that the corresponding potential is integrable, it is proved that these approximations converge to the exact solution of the problem. It is shown that this assumption also covers the very important practical case of piecewise inhomogeneity. For numerical realization, we consider a homogeneous tube with flowing water containing a small amount of bubbles. The effect of the volume content of bubbles on wave characteristics is revealed. In particular, it is stated that, for the oscillation regime selected, an increased bubble volume content decreases the wave velocity and considerably increases the flow speed (rate).     

Joined dissimilar thin elastic tubes containing steady perfect flow

Two tubes of circular cross-section and of the same radius initially, but composed of different elastic materials, are joined together to form a single straight tube with discontinuous properties. Stretching in the axial direction causes radial displacement that varies along the length of the tube. In a perfect inviscid incompressible flow through the tube of variable cross-section the internal pressure varies as described by Bernoulli's equation, and the variable pressure also causes variation of the radial displacement. The equations of coupled finite deformation and fluid pressure problem can be integrated explicitly (using membrane theory for the tube) for arbitrary material properties, but determination of the integration constants is not trivial. The results are interpreted numerically for Mooney materials. Also considered in this context is the similar problem where two semi-infinite cylindrical membranes of the same material are separated by a cuff of different material. Numerical illustrations are obtained for various upstream velocities. The results obtained here thus solve the problem of steady internal pressure loading of this type in extended dissimilar thin isotropic tubes. The tube will become unstable if the fluid velocity is too large. Applications to engineering structures are possible.     

A bifurcated circular waveguide problem

A rigorous and exact solution is obtained for the problem ofthe radiation of sound from a semi-infinite rigid duct insertedaxially into a larger acoustically lined tube of infinite length.The solution to this problem is obtained by the Wiener-Hopftechnique. The transmission and reflection coefficients, whenthe fundamental mode propagates in the semi-infinite tube, areobtained. The present results could be of use for exhaust design,and as a possible instrument for impedance measurement.     

Stresses in elastic conical tubes of transversely isotropic materials with spherical anisotropies under temperature and force loadings

Analytic solutions are proposed for a number of new problems on determining the state of stress of a transversely-isotropic hollow cone with spherical anisotropy. An exact solution of the problem of the axisymmetric deformation of a long conical tube (or continuous cone) from an elastic transversely-isotropic material with spherical anisotropy subjected to an axial force is obtained in a spherical coordinate system R, , θ, the material axis of symmetry is directed along the spherical radius R. A rigorous solution is given of the problem of the uniform heating of a conical tube of transversely-isotropic material with spherical anisotropy for particular values of Poisson's ratios; the material axis of symmetry is directed along the θ-axis. For arbitrary Poisson's ratios an asymptotic solution is found for the temperature problem for a tube with small conicity.     

Stress-strain state and stability of polymeric tubes with a honeycomb wall

A problem of determination of the stress-strain state, stability and optimization of the honeycomb structure of a polymeric tube wall under the action of a static force load is considered. As is customary in engineering practice, the ring rigidity of the tube is taken as its main design quantity. The corresponding problem of optimization is formulated. From the condition of provision of the required value of the ring rigidity of a honeycomb polymeric cylindrical shell, we determine the geometric, physicomechanical, and technological parameters. On the basis of the performed calculations, a justification of an optimal geometry (the thickness of the wall, the diameter of a winding tubule, and the number of layers of the tube structure) is presented. Numerical results and their analysis are presented.     

An interpolation problem in tube domains

Some recent results concerning theL-problem of moments in two variables are related via the Fourier-Laplace transform to an interpolation problem in the tube domain over a quadrant inR ². The class of analytic functions for which the interpolation problem is posed is identified with the symbols of all bounded analytic Wiener-Hopf operators acting on theH ²-Hardy space of the tube domain. The extremal solutions of the corresponding truncated problem are computed and the related uniqueness phenomenon is also discussed.     

不可压缩层合橡胶圆管径向膨胀的稳定性分析

研究由两类不可压缩的橡胶材料组成的层合圆柱形管道,在内表面受到突加的径向压力作用时的膨胀机理．建立了问题的数学模型;利用材料的不可压缩条件、边界条件以及圆管的径向位移和径向应力的连续性条件将相应的控制方程约化为一个二阶非线性常微分方程,并得到了该方程的首次积分．给出了管道拟静态膨胀和动态膨胀的定性分析,特别地,结合数值算例讨论了材料参数、结构参数以及径向压力对管道径向膨胀和非线性周期振动的影响．     

A model of liquid level measurements

A model of measuring the level of a viscous incompressible liquid in a tank as based on the liquid level in a measuring tube is investigated. The tank is in the field of gravity, and the tank liquid level varies according to some law. As a result, a Dirichlet boundary value problem for a nonlinear integrodifferential equation of parabolic type is obtained. A global existence and uniqueness theorem is proved for a weak solution of the problem. In the case of a tank level decreasing linearly with time, it is shown numerically that the liquid level in the measuring tube oscillates with a decaying amplitude with respect to the tank level.     

Inflation-Extension and Eversion of a Tube of Incompressible Isotropic Elastic Material

Adopting a form of strain-energy function for highly deformableincompressible elastic materials recently proposed by Ogden,we study in this paper two deformations of a circular cylindricaltube. In the first deformation the tube is inflated by increasingits internal radius by an assigned factor, and in the secondthe tube is turned inside out. In each case it is assumed thatthe tube is free of longitudinal constraint to the extent thatthe normal tractions on the end-faces have zero resultant. Inflationand eversion are therefore accompanied by axial shortening orelongation. Detailed numerical results are presented and comparedwith the limited experimental data to hand. Consideration isalso given to the existence and uniqueness of deformations whichpreserve the shape of the tube and comply with the assumed boundaryconditions. Restrictions are obtained on the material constantsappearing in the strain-energy function which ensure the existenceof precisely one such solution to the problem of combined inflationand extension, and sufficient conditions for the existence ofat least one solution to the eversion problem are also found.These requirements agree with criteria for "physically realisticresponse" which have been established elsewhere.     

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to time consuming simulations, we propose a local time stepping algorithm. We also use high order time extensions applying the Adams-Bashforth time integration technique as well as the second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: Sod’s shock tube problem, Lax’s shock tube problem and the Shu-Osher test problem.     

A Simultaneous Tube of the Mean Value of Multiple Observations in Multivariate Regression

Moscow University Computational Mathematics and Cybernetics - The problem of constructing a simultaneous confidence tube of the mean value of multiple responses in a multivariate linear normal...     

A possibility of realizing the Landau—Hopf scenario in the problem of tube oscillations under the action of a fluid flow

Theoretical and Mathematical Physics - We consider a boundary-value problem for a quasilinear partial differential equation describing tube oscillations under the action of a fluid flow. We show...     

A Cauchy–Kowalevsky Theorem for Overdetermined Systems of Nonlinear Partial Differential Equations and Geometric Applications

We consider a Cauchy problem for an overdetermined system of nonlinear partial differential equations, and give necessary and sufficient conditions for solvability of this Cauchy problem for all data. As an application, we find all real tube hypersurfaces in complex space whose Levi number is maximal.     

钢管订购与运输问题三的数学模型与灵敏度分析

本文针对 2 0 0 0年全国大学生数学建模竞赛 B题——钢管订购与运输问题的问题 ( 3) ,建立了数学模型 ,给出了该数学模型的精确求解 .然后对问题 ( 1 )与问题 ( 3)给出了灵敏度分析 .     

钢管订购与运输问题一的数学模型与求解

本文针对 2 0 0 0年全国大学生数学建模竞赛 B题—钢管订购与运输问题的问题 ,建立了数学模型 ,并给出了该数学模型的精确求解     

Degenerate cases of stability loss of an elastic fluid-conveying tube

The two-dimensional non-linear dynamics of a liquid-filled tube is considered. The tube is clamped at the upper end, a point mass is fixed to its free lower end and laterally it is supported by two springs. The uniform flow velocity of the fluid, the end mass, the spring constant and the vertical position of the springs are considered as the distinguished parameters of the problem. A linear stability analysis shows that the (degenerate) case of a Takens-Bogdanov-Hopf bifurcation exists, which is associated with a high frequency flutter movement superimposed on a low frequency flutter around a statically buckled state of the tube. We account for this degenerate case by indicating the parameter regime necessary for its occurence and and give the bifurcation diagram for the trivial equilibrium position of the tube. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Alternative configurations for cutting machines in a tube cutting mill

In a steel tube mill where an endless stream of steel tube is supplied from a manufacturing facility, trim waste is never made regardless of cutting patterns used and the standard cutting stock problem seems meaningless. Therefore, the continuous stock cutting problem with setup is introduced to minimize the sum of cutting time and pattern changing time to meet the given demand. We propose a new configuration of cutting machines to achieve higher production efficiency, namely the open-ended configuration as opposed to the traditional closed-ended configuration, thereby two variants of the problem are defined. We propose linear formulations for both problems using binary expansion of the number of pieces of different types in a pattern. Furthermore, we define the time for pattern change as a linear function of the number of knives used in the pattern to be more realistic. Computational studies suggest that the open-ended cutting machine may improve the production time by up to 44% and that our linear formulations are more efficient than the existing ones.     

An exact finite deformation elasto-plastic solution for the outside-in free eversion problem of a tube of elastic linear-hardening material

An exact solution is obtained in this paper for the elasto-plasticoutside-in free eversion problem of a tube of elastic linear-hardeningmaterial using a tensorial formulation. The solution is basedon a finite-strain version of Hencky's deformation theory, thevon Mises yield criterion, and the assumptions of volume incompressibilityand axial length constancy. All expressions for the stress,strain distributions and the eversion load are derived in anexplicit form. In addition, with both the linear-elastic andstrain-hardening-plastic responses of the material being includedand with the thickness effect of the tube being incorporated,this solution provides a rigorous and complete theoretical analysisof the elasto-plastic eversion problem, unlike existing solutions.Two specific solutions are also presented as limiting casesof the solution. Also provided are some numerical results andthe related observations to show quantitatively applicationsof the solution.     

Axially symmetric wave motion of a viscous liquid in a linearly viscoelastic tube

The problem of the propagation of progressive waves in a tube made of a linearly viscoelastic material which encloses a viscous Newtonian liquid is examined. For numerical calculations, it is proposed that the behavior of the tube wall material be described by the Voigt model. Dispersion curves are constructed for this case.S. M. Kirov Azerbaidzhan State University, Baku. Translated from Mekhanika Polimerov, No. 2, pp. 317–321, March–April, 1976.