Heat Transfer

Computer Simulations and Thermal Design 381

AT C

[][]

=

[]

.

(14.111)

This is the general form of the matrix equation that is solved by computer simulation

tools. If N is the number of nodes in the domain, the coefcient matrix [A] is an

N × N matrix, and [T] and [C] are vectors with N elements.

If [A]

–1

is the inverse of matrix [A] such that [A]

–1

[A] = [I] where [I] is the unity

matrix, the solution of this matrix equation will be

TAC

[]

=

[][]

−1

.

(14.112)

However, obtaining the inverse of a matrix is not a trivial task. There are many

numerical techniques such as Gauss–Seidel and incomplete LU decomposition that

are used to approximate the inverse of a matrix. These methods will not be described

here and the interested readers are referre ...