WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1610 J. S. JENSEN
The following procedure used to obtain bi is limited to the case where the number of degrees
of freedom Nd is higher than the chosen number of expansion terms N. As mentioned previously,
this is a practical necessity when dealing with large finite element models and hence poses no
real limitation on the method. With this assumption the equation system in Equation (24) is overdetermined
and can be used to find a common set of bi coefficients for all degrees of freedom in
the model. This approach differs from many previous implementations (e.g. [5]) where additional
equations of the form in (24) are created by matching terms of order N+2 up to 2N .
Equation (24) can be written as
⎛ ⎞
b1
⎜ ⎟
⎜ b2 ⎟
⎜ ⎟
[uN uN−1 ··· u1] ⎜ ⎟ =−uN+1
(25)
⎜ ⎟
⎝ . ⎠
and is solved by finding the least-squares solution with the use of the pseudoinverse matrix [17]:
⎛ ⎞ ⎡
b1 u
⎜ ⎟ ⎢
⎜ b2 ⎟ ⎢
⎜ ⎟ ⎢
⎜ ⎟ =−⎢
⎜ ⎟ ⎢
⎝ . ⎠ ⎢
⎣
+
1
u +
⎤
⎥
2 ⎥ uN+1
(26)
. ⎥
⎦
bN
u + N
in which u + i denotes a row vector in the pseudoinverse matrix. The pseudoinverse matrix can in
this case be explicitly computed as
⎡
u
⎢
⎣
+
1
u +
⎤ ⎡
u
⎥ ⎢
2 ⎥ ⎢
⎥ ⎢
⎥ = Q ⎢
. ⎥ ⎢
⎦ ⎣
∗ N
u ∗ ⎤
⎥
N−1 ⎥
(27)
⎥
. ⎦
in which Q = P −1 and
⎡
⎢
P = ⎢
⎣
u + N
u ∗ N
u ∗ N−1
.
u ∗ 1
bN
u ∗ 1
⎤
⎥ [uN
⎥
⎦
uN−1 ... u1] (28)
with u ∗ i denoting the conjugate transpose of the ui vector. The procedure is justified by numerical
experiments showing that P is of full rank so that the matrix inversion is possible.
However, P turns out to be ill-conditioned if N is large setting a limit to the order of approximation
that can be constructed. As pointed out in [10], this problem can be circumvented by using the
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme