David H. Y. Yen

Green's functions, temperature and heat flux in the rectangle

Abstract Steady heat conduction in the rectangle is treated with the method of Greens functions. Single-sum series for the Greens functions are reported in terms of exponentials which have better numerical properties than hyperbolic functions. Series expressions for temperature and heat flux caused by spatially uniform effects are presented. The numerical convergence of these series is improved, in some cases by a factor of 1000, by replacing slowly converging portions of the series with fully summed forms. This work is motivated by high-accuracy verification of finite-difference and finite-element codes.

METHODOLOGY TO GENERATE ACCURATE SOLUTIONS FOR VERIFICATION IN TRANSIENT THREE-DIMENSIONAL HEAT CONDUCTION

This article describes the development of accurate solutions for transient three-dimensional conductive heat transfer in Cartesian coordinates for a parallelepiped which is homogeneous and has constant thermal properties. The intended use of these solutions is for verification of numerical computer programs which are used for solving transient heat conduction problems. Verification is a process to ensure that a computer code is free of errors and accurately solves the mathematical equations. The exact solutions presented in this article can have any combination of boundary conditions of specified temperature, prescribed heat flux, or imposed convection coefficient and ambient temperature on the surfaces of the parallelepiped. Additionally, spatially uniform nonzero initial condition and internal energy generation are treated. The methodology to obtain the analytical solutions and sample calculations are presented.

Solution of an initial-boundary value problem for heat conduction in a parallelepiped by time partitioning

Abstract An initial-boundary value problem for transient heat conduction in a rectangular parallelepiped is studied. Solutions for the temperature and heat flux are represented as integrals involving the Greens function (GF), the initial and boundary data, and volumetric energy generation. Use of the usual GF obtained by separation of variables leads to slowly convergent series. To circumvent this difficulty, the dummy time interval of integration is partitioned into a short time and a long time subintervals where the GFs are approximated by their small and large time representations. This paper deals with the analysis and implementation of this time partitioning method.

On the non-linear response of an elastic string to a moving load

Abstract The non-linear response of an elastic string to a moving load is studied, using a special perturbation method. Solutions are obtained that are valid throughout a particular neighborhood of the critical speed of the linear theory. The solutions show how the transitions from subcritical responses to supercritical responses, and vice versa , take place near the critical speed.

On the normal modes of non-linear dual-mass systems

Abstract The motions of a class of non-linear dual-mass systems are investigated. A new characterization of non-linear normal modes in terms of solutions of initial value problems is developed that leads to interesting properties both on the existence of such non-linear normal modes and their continuous dependence upon the energies.

Green's functions and three-dimensional steady-state heat-conduction problems in a two-layered composite

A boundary-value problem for steady-state heat conduction in a three-dimensional, two-layered composite is studied. The method of Greens function is used in the study. Greens functions are constructed as double sums in terms of eigenfunctions in two of the three directions. The eigenfunctions in the direction orthogonal to the layers are unconventional and must be defined appropriately. The use of different forms of the Greens functions leads to different representations of the solutions as double sums with different convergence characteristics and it is shown that the method of Greens functions is superior to the classical method of separation of variables.

Influence Functions for the Infinite and Semi-Infinite Strip

Influence functions appropriate for the boundary-element method for the Laplace equation are given for the infinite and semi-infinite strip. The method of Greens functions is used to produce single-sum series for the influence functions, which reflect the domain shape and the boundary conditions. Boundary conditions of type 1, 2, and 3 are treated. Series convergence is improved by identifying slowly converging terms and replacing them with fully summed polynomial forms. Numerical examples are given

A note on the nonlinear response of an elastic beam on a foundation to a moving load

Abstract The nonlinear response of an elastic beam to a moving transverse load is studied, using a special perturbation method. Solutions are obtained that remain valid throughout some neighborhood of the critical speed of the linear beam theory. It is found that in general, depending upon the type of dominant nonlinearity in the beam, either the subcritical response or the supercritical response may be continued up to the critical speed and even beyond. The solutions also show how the transitions from a subcritical response to a supercritical response and vice versa take place near the critical speed.

Steady-state temperature distribution for infinite region outside a partially heated cylinder

Abstract The steady-state conduction problem of the temperature distribution in the region outside an infinite cylinder of finite radius a is considered. At the surface r = a, between z = −c and c, there is a constant heat flux and the remainder of the surface is insulated (r and z are cylindrical coordinates). An integral representation for the exact solution is derived by the Fourier transform method, from which numerical results are obtained. Several approximate solutions are also given and analyzed in light of the exact solution The solution is a basic one in heat conduction and has direct application to a number of problems such as underground storage of heat-producing radioactive waste; drilling of holes in teeth, rock, oil-bearing soil, etc., and underground freezing. Another important application is as a new influence function for the surface element method of solution of diffusion problems.

Magnetohydrodynamic Mach Cones

The problem of general fast‐hyperbolic magnetohydrodynamic flows past a point source is considered. Questions on the geometry of the magnetohydrodynamic Mach cones and on the nature of the singularities of the disturbances carried on such cones are studied.