A FINITE VOLUME ELEMENT METHOD FOR THERMAL CONVECTION PROBLEMS

Consider the finite volume element method for the thermal convection problem with the infinite Prandtl number. The author uses a conforming piecewise linear function on a fine triangulation for velocity and temperature, and a piecewise constant function on a coarse triangulation for pressure. For general triangulation the optimal order H^1 norm error estimates are given.     

Pseudolinear equivalent constant rigidity concept for analyzing welding residual deformation

To ascertain the extent of deformation due to the thermal cycles caused by welding it calls for solving a complex thermal elasto-plastic problem, which is non-linear and involves plastic deformation of the medium at high temperature varying in both time and space. Analytical solutions turned out to be inadequate. At the same time conventional numerical techniques proved to be highly time consuming and thereby prohibitively expensive in real life situations. The concept of pseudolinear equivalent constant rigidity system was developed in this investigation for thermo-mechanical analysis of plates undergoing variation of rigidity due to a continuously changing temperature profile as is encountered in welding situations. The initial non-linear problem with modulus varying with temperature was transformed into a pseudolinear equivalent system of constant rigidity that was solved by applying linear analysis.     

Guaranteed-accuracy approximation of reachable sets for a linear dynamic system subject to impulse actions

A method is proposed for approximating the reachable set of a dynamic system with a state space dimension no higher than six-eight considered on a finite time interval. The system is governed by linear differential equations with piecewise constant coefficients and impulse actions specified at prescribed times. The method is based on guaranteed-accuracy polyhedral approximations of reachable sets at researcher-specified times. Every approximation is constructed using the preceding one. A procedure is described for choosing parameters of the method that ensure the required accuracy with close-to-minimal time costs.     

An internal-variable theory of thermo-viscoelastic constitutive relations at finite strain

Based on the nonequilibrium thermodynamic theory, a new thermo-viscoelastic relation at finite strain is proposed. Under the assumption that the specific heat at a fixed strain and fixed internal variables can be regarded as a constant, a new expression for the free energy which decouples the mechanical and the thermal effects is derived. Through an analysis of the mesoscopic deformation mechanism of solid polymers, a set of internal variables is introduced, and an internal-variable constitutive theory in thermo-viscoelasticity at finite strain is formulated. An explicit expression of a thermoviscoelastic constitutive relation is obtained for solid polymers in the case where their molecular network has a randomly oriented distribution function at reference configuration. Moreover, the relationship between the relaxation time and the temperature is also discussed. The viscoelastic constitutive theory proposed in reference is only a linear approximation of the present theory.     

Thermoelasticity with second sound—exponential stability in linear and non‐linear 1‐d

We consider linear and non‐linear thermoelastic systems in one space dimension where thermal disturbances are modelled propagating as wave‐like pulses travelling at finite speed. This removal of the physical paradox of infinite propagation speed in the classical theory of thermoelasticity within Fourier's law is achieved using Cattaneo's law for heat conduction. For different boundary conditions, in particular for those arising in pulsed laser heating of solids, the exponential stability of the now purely, but slightly damped, hyperbolic linear system is proved. A comparison with classical hyperbolic–parabolic thermoelasticity is given. For Dirichlet type boundary conditions—rigidly clamped, constant temperature—the global existence of small, smooth solutions and the exponential stability are proved for a non‐linear system. Copyright © 2002 John Wiley & Sons, Ltd.     

由曲率函数和外力场之差支配的凸超曲面的发展

考虑由曲率函数和外力场之差支配的凸超曲面的发展.证明了外力场为常向量场时,初始超曲面的凸性是保持的,且曲率流在有限时间内爆破.对于线性外力场,初始超曲面的凸性保持.而且,若线性常数为负数,则曲率流在有限时间内收敛到一点;若线性常数为正数且初始曲率小于某一与外力场有关的常数,则曲率流光滑地存在于任意有限时间区间,并发散到无穷;若线性常数为正数且初始曲率大于某一与外力场有关的常数,则曲率流在有限时间内爆破.     

有限元近似可计算的误差界

本文研究了有限元近似可计算的误差界，利用“二次插值过渡”方法，获得二维线性、双线性有限元和三维三线性有限元的新的插值常数估计值．理论分析和数值实验表明该结果是有效的，发展了P．Arbenz等人的工作．     

Fundamental solution of generalized thermo-viscoelasticity using the finite element method

The present paper is aimed at an investigation of the temperature, displacement, and stress in a viscoelastic half space of Kelven–Voigt type. The formulation is applied according to three theories of generalized thermoelasticity: Lord–Shulman with one relaxation time, Green–Lindsay with two relaxation times, as well as the coupled theory. The nondimensional governing equations are solved by the finite element method. Numerical results for the temperature distribution, displacement, and thermal stress are represented graphically. Comparisons are made with the results predicted by the CD, L-S, and G-L theories in the presence and absence of the viscoelastic relaxation time.     

The effect of variable properties on the unsteady Couette flow with heat transfer considering the Hall effect

The influence of temperature dependent viscosity and thermal conductivity on the transient Couette flow with heat transfer is studied. An external uniform magnetic field is applied perpendicular to the parallel plates and the Hall effect is taken into consideration. The fluid is acted upon by a constant pressure gradient. The two plates are kept at two constant but different temperatures and the viscous and Joule dissipations are considered in the energy equation. A numerical solution for the governing non-linear equations of motion and the energy equation is obtained. The effect of the Hall term and the temperature dependent viscosity and thermal conductivity on both the velocity and temperature distributions is examined.     

Evolution of hypersurfaces by the mean curvature minus an external force field

In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.     

Self-similar convergence of a shock wave in a heat conducting gas

The problem of the convergence of a spherical shock wave (SW) to the centre, taking into account the thermal conductivity of the gas in front of the SW, is considered within the limits of a proposed approximate model of a heat conducting gas with an infinitely high thermal conductivity and a small temperature gradient, such that the heat flux is finite in a small region in front of the converging SW. In this model, there is a phase transition in the surface of the SW from a perfect gas to another gas with different constant specific heat and the heat outflow. The gas is polytropic and perfect behind the SW. Constraints are derived which are imposed on the self-similarity indices as a function of the adiabatic exponents on the two sides of the SW. In front of the SW, the temperature and density increase without limit. In the general case, a set of self-similar solutions with two self-similarity indices exists but, in the case of strong SW close to the limiting compression, there are two solutions, each of which is completely determined by the motion of the spherical piston causing the self-similar convergence of the SW.     

Effect of random system properties on bending response of thermo-mechanically loaded laminated composite plates

In this paper, effect of random variation in system properties on bending response of geometrically linear laminated composite plates subjected to transverse uniform lateral pressure and thermal loading is examined. System parameters such as the lamina material properties, expansion of thermal coefficients, lamina plate thickness and lateral load are modeled as basic random variables. The basic formulation is based on higher order shear deformation theory to model the system behavior of the composite plate. A C⁰ finite element method in conjunction with the first order perturbation technique procedure developed earlier by authors for the plate subjected to lateral loading is employed to obtain the second order response statistics (mean and variance) of the transverse deflection of the plate. Typical numerical results for the second order statistics of the transverse central deflection of geometrically linear composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and linearly varying temperature distribution are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The performance of the stochastic laminated composite model is demonstrated through comparison of mean transverse central deflection with those results available in literature and standard deviation of the deflection with an independent Monte Carlo simulation.     

Transient analysis of thermo-elastic waves in thick hollow cylinders using a stochastic hybrid numerical method,considering Gaussian mechanical properties

This article attempts to study the stochastic coupled thermo-elasticity of thick hollow cylinders subjected to thermal shock loading considering uncertainty in mechanical properties. The thermo-elastic governing equations based on Green–Naghdi theory (without energy dissipation) are stochastically solved using a hybrid numerical method (combined Galerkin finite element and Newmark finite difference methods). The mechanical properties are considered as random variables with Gaussian distribution, which are generated using Monte Carlo simulation method with various coefficients of variations (COVs). The effects of uncertainty in mechanical properties with various coefficients of variations on thermo-elastic wave propagation are studied in detail. Also, the maximum, mean and variance of temperature, displacement and stresses are illustrated across thickness of cylinder in various times.     

Inverse determination of thermal conductivity using semi-discretization method

In this work a semi-discretization method is presented for the inverse determination of spatially- and temperature-dependent thermal conductivity in a one-dimensional heat conduction domain without internal temperature measurements. The temperature distribution is approximated as a polynomial function of position using boundary data. The derivatives of temperature in the differential heat conduction equation are taken derivative of the approximated temperature function, and the derivative of thermal conductivity is obtained by finite difference technique. The heat conduction equation is then converted into a system of discretized linear equations. The unknown thermal conductivity is estimated by directly solving the linear equations. The numerical procedures do not require prior information of functional form of thermal conductivity. The close agreement between estimated results and exact solutions of the illustrated examples shows the applicability of the proposed method in estimating spatially- and temperature-dependent thermal conductivity in inverse heat conduction problem.     

Non-Darcy free convection flow over a horizontal cylinder in a saturated porous medium with variable viscosity,thermal conductivity and mass diffusivity

In this paper, the effects of variable viscosity and thermal conductivity on coupled heat and mass transfer by free convection about a permeable horizontal cylinder embedded in porous media using Ergun mode are studied. The fluid viscosity and thermal conductivity and are assumed to vary as a linear function of temperature while the mass diffusion is assumed to vary as linear function of concentration. The surface of the horizontal cylinder is maintained at a uniform wall temperature and a uniform wall concentration. The transformed governing equations are obtained and solved by using the implicit finite difference method. Numerical results for dimensionless temperature and concentration profiles as well as Nusselt and Sherwood numbers are presented for various values of parameters namely, Ergun number, transpiration parameter, Rayleigh and Lewis numbers and buoyancy ratio parameter.     

Transformation of surface waves in a liquid with local changes in depth

The propagation of a pulse on the surface of a liquid of finite depth is studied when the depth decreases over a finite interval between liquids with constant depths to the left and right. The decrease in depth is specified by a parabolic function and the pulse, which increases sharply in time and then decays, is turned on at the initial time some distance to the right of the section with a variable depth. A Laplace transform method is used to solve the corresponding initial value-boundary value problem and this makes it possible to obtain a solution in hypergeometric functions in the transform space. In the limiting case of a linear variation in the depth, a numerical inversion of the Laplace transform is used to construct solutions which are analyzed for various geometric parameters and at different times. Institute of Hydromechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 131–142, 1999.     

Discretized Energy Minimization in a Wave Guuide with point Sources

An anti-noise problem on a finite time interval is solved by minimization of a quadratic functional on the Hilbert space of square integrable controls. To this end, the one-dimensional wave equation with point sources and pointwise reflecting boundary conditions is decomposed into a system for the two propagating components of waves. Wellposedness of this system is proved for a class of data that includes piecewise linear initial conditions and piecewise constant forcing functions. It is shown that for such data the optimal piecewise constant control is the solution of a sparse linear system. Methods for its computational treatment are presented as well as examples of their applicability. The convergence of discrete approximations to the general optimization problem is demonstrated by finite element methods.     

On a nonstandard problem for heat conduction in a cylinder

Decay bounds are derived for the solution of a heat conduction problem in a semi-infinite cylinder when the lateral surface is held at zero temperature, a nonzero temperature is prescribed on the finite base, and the temperature at time T is prescribed to be a constant multiple of the temperature at initial time. Both energy and pointwise decay bounds are computed for a range of values of the constant multiple. Such problems were originally introduced as a means of stabilizing the backward-in-time problem for the heat equation.     

Stochastic heat conduction analysis of a functionally graded annular disc with spatially random heat transfer coefficients

The mean and variance of the temperature are analytically obtained in a functionally graded annular disc with spatially random heat transfer coefficients (HTCs) on the upper and lower surfaces. This annular disc has arbitrary variations in the HTCs (i.e., arbitrary thermal interaction with the surroundings) and gradient material composition only along the radial direction and is subjected to deterministic axisymmetrical heating at the lateral surfaces. The stochastic temperature field is analysed by considering the annular disc to be multilayered with spatially constant material properties and spatially constant but random HTCs in each layer. A type of integral transform method and a perturbation method are employed in order to obtain the analytical solutions for the statistics. The correlation coefficients of the random HTCs are expressed in the form of a linear function with respect to the radial distance as a non-homogeneous random field of discrete space. Numerical calculations are performed for functionally graded annular discs composed of stainless steel and ceramic, which comprise two types of material composition distributions. The effects of the magnitude of the means of HTCs, volume fraction distributions of the constitutive materials and correlation strengths of the HTCs on the standard deviation of the temperature are discussed.     

Modelling the Temperature Time‐dependent Speed of Mean Reversion in the Context of Weather Derivatives Pricing

In this paper, in the context of an Ornstein–Uhlenbeck temperature process, we use neural networks to examine the time dependence of the speed of the mean reversion parameter α of the process. We estimate non‐parametrically with a neural network a model of the temperature process and then compute the derivative of the network output w.r.t. the network input, in order to obtain a series of daily values for α. To our knowledge, this is the first time that this has been done, and it gives us a much better insight into the temperature dynamics and temperature derivative pricing. Our results indicate strong time dependence in the daily values of α, and no seasonal patterns. This is important, since in all relevant studies performed thus far, α was assumed to be constant. Furthermore, the residuals of the neural network provide a better fit to the normal distribution when compared with the residuals of the classic linear models used in the context of temperature modelling (where α is constant). It follows that by setting the mean reversion parameter to be a function of time we improve the accuracy of the pricing of the temperature derivatives. Finally, we provide the pricing equations for temperature futures, when α is time dependent.