A Common Thermal Network Problem Encountered in Heat Transfer Analysis of Spacecraft

Abstract A widely used discretization method for modeling thermal systems is the thermal network approach. The network approach is derived from energy balance equations and is equivalent to a particular finite difference discretization of the underlying heat-transfer equation. The steady-state problems that arise in the analysis of spacecraft systems using network models are frequently dominated by radiative transfer, which introduces quartic nonlinearities in the equations. Although these systems are routinely encountered, there has not appeared any detailed analysis of these equations. Questions of existence and uniqueness of solutions and numerical methods for solving the systems have never been addressed in any generality. In this paper, general existence and uniqueness properties of the network equations are established. Globally convergent methods for solving the systems are developed and insight into the relative success of existing methods is given. Numerical examples are presented illustrating the methods. The perspective adopted here is also useful in interdisciplinary applications. A simple example involving thermal control is used to demonstrate this.

[1]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[2]  J. J. Moré.  Global convergence of Newton-Gauss-Seidel methods , 1971 .

[3]  Francisco Facchinei,et al.  An Active Set Newton Algorithm for Large-Scale Nonlinear Programs with Box Constraints , 1998, SIAM J. Optim..

[4]  P. Toint,et al.  Global convergence of a class of trust region algorithms for optimization with simple bounds , 1988 .

[5]  J. Yorke,et al.  Finding zeroes of maps: homotopy methods that are constructive with probability one , 1978 .

[6]  J. J. Moré.  Nonlinear generalizations of matrix diagonal dominance with application to Gauss-Seidel iterations. , 1972 .

[7]  Werner C. Rheinboldt,et al.  Methods for solving systems of nonlinear equations , 1987 .

[8]  P. Deuflhard.  A stepsize control for continuation methods and its special application to multiple shooting techniques , 1979 .

[9]  C. K. Krishnaprakas.  Application of accelerated iterative methods for solution of thermal models of spacecraft , 1995 .

[10]  Eugene L. Allgower,et al.  Numerical continuation methods - an introduction , 1990, Springer series in computational mathematics.

[11]  L. Watson.  Numerical linear algebra aspects of globally convergent homotopy methods , 1986 .

[12]  W. Rheinboldt.  On M-functions and their application to nonlinear Gauss-Seidel iterations and to network flows☆ , 1970 .

[13]  B. Cullimore.  SINDA'85/FLUINT - SYSTEMS IMPROVED NUMERICAL DIFFERENCING ANALYZER AND FLUID INTEGRATOR (CONVEX VERSION) , 1994 .

[14]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[15]  H. Keller.  Lectures on Numerical Methods in Bifurcation Problems , 1988 .