Efficient Solution of Spacecraft Thermal Models Using Preconditioned Conjugate Gradient Methods

Abstract

The thermal mathematical models ofspacecraft areusually constructed using a lumped parameternetwork formulation. Thegoverning equations under the steady-state condition are a system of nonlinearalgebraicequations, often large and sparse. Generally, the quadratic convergent Newton’ s method is used to solve nonlinear systems, in which a linear system with a Jacobian coefe cient matrix is solved at each iteration. It is necessary to solve this linear system in an efe cient and economical way, exploiting the sparsity of the coefe cient matrix in storage as well as in computation. The conjugate gradient method with appropriate preconditioners is a very successful tool to solve large, sparse, linear systems. Numerical experiments were conducted to investigate the effectiveness of preconditioned conjugate gradient methods for large spacecraft thermal problems. The resultsare compared with thoseofsparseelimination methodsand thesuccessive overrelaxation method. Ithasbeen found thattheconjugate gradient method with the symmetric successive overrelaxation preconditioner is a simple and efe cient method to solve large-order problems. Nomenclature A = area,m 2 Fa = albedo load factor Fe = Earth-shine load factor I = solar constant, W/m 2 K = conduction exchange factor, W/K L = Cholesky factor for a positive dee nite symmetric matrix M = preconditioning matrix n = total number of nodes; number of equations; order of the matrix P = internal power dissipation, W R = radiation exchange factor, m 2 T = temperature, K ® = absorptance = emittance µ = solar incidence angle ·.A/ = spectral condition number of the matrix A Ω = Earth’ s albedo factor or ree ectance Ω.A/ = spectral radius of the matrix A ae = Stefan‐Boltzmann constant, 5.67E-8 W/m 2 -K 4 ! = relaxation parameter Subscripts i = node under consideration j = node thermally coupled to node i opt = optimum s = space node Superscripts T = transpose i1 = inverse * = exact solution