4569846498

Multibody systems simulation software 117
referred to as sparse and the ratio of non-zero terms to the total number of
matrix elements is referred to as the sparsity of the matrix. In a program
such as MSC.ADAMS the sparsity of the matrix is typically in the range of
2 to 5% (Wielenga, 1987). The computer solvers that are developed in
multibody systems to solve linear equations can be designed to exploit the
sparsity of the [A] matrix leading to relatively fast solution times. This is
one of the reasons, apart from improvements in computer hardware, that
multibody systems programs can appear to solve quite complex engineering
problems in seconds or minutes. These solution speeds can be quite notable
when compared with other computer software such as finite elements or
computational fluid dynamics programs.
The solution of (3.62) follows an established approach that initially
involves decomposing or factorizing the [A] matrix into the product of a
lower triangular and an upper triangular matrix:
[A] [L][U] (3.63)
For the lower triangular matrix [L] all the elements above the diagonal are
set to zero. For the upper triangular matrix [U] all the elements on and
below the diagonal are set to zero. In the following step a new set of
unknowns [y] is substituted into (3.63) leading to
[L][U][x] [b] (3.64)
[U][x] [y] (3.65)
[L][y] [b] (3.66)
The terms in the [L] and [U] matrices may be obtained by progressive operations
on the [A] matrix where the terms in one row are all factored and
then added or subtracted to the terms in another row. This process can be
demonstrated by considering the expanded [A], [L] and [U] matrices as
shown in (3.67) where to demonstrate the influence of sparsity some of the
terms in the [A] matrix are initially set to zero:
⎡L11
0 0 0 ⎤ ⎡1
U U U
⎢L
21 L22
0 0 ⎥ ⎢
0 1 U U
⎢
⎥ ⎢
⎢L31 L32 L33
0 ⎥ ⎢0 0 1 U
⎢
⎥ ⎢
⎣L41 L42 L43 L44
⎦ ⎣0 0 0 1
12 13 14
23 24
(3.67)
The fact that the [A] matrix is sparse leads to the [L] and [U] matrices also
being sparse with subsequent savings in computer simulation time and memory
storage. If we start by multiplying the rows in [L] into the first column
of [U] we can begin to obtain the constants in the [L] and [U] matrices from
those in the [A] matrix:
L 11 A 11
L 21 A 21
L 31 A 31
L 41 A 41
34
⎤ ⎡A11 A12 0 A14
⎤
⎥ ⎢A
21 0 0 A ⎥
⎥
24
⎢
⎥
⎥ ⎢A31 0 A33
0 ⎥
⎥ ⎢
⎥
⎦ ⎣A41 0 A43 A44
⎦