In this section brief outlines of the so-called
Dual Reciprocity
Method and Multiple Reciprocity Method are given. The interested reader
should consult the referenced papers for full details on the methods.
In section 3 it is shown that the finite element method has the
advantage of resulting in a linear eigenvalue problem whereas
the boundary element method has the advantage of requiring
only a boundary discretization and hence results in much
smaller matrices. In an effort to derive
a new method that retains the main advantages of both of
the original methods, the dual reciprocity method has been developed.
The method originates from a paper by Nardini and Brebbia [23],
in which a similar method is applied to a free vibration problem.
The method is considered further in Ahmad and Banerjee [29] and
was originally adapted for acoustic or Helmholtz problems
in Banerjee et al [15]. The method may be developed
as follows. The solution of the Helmholtz equation (5)
is written as the sum of two components,
j(p) = j0(p) + j1(p) ,
(24)
such that j0 and j1 satisfy the following
equations:
As with the application of the boundary element method in the previous section,
equation (25) may be reformulated as follows,
{ M j0 }S (p) + j0(p) = { L
¶j0
¶n
}S (p) (p Î D) ,
(28)
{ M j0 }S (p) + c(p) j0 (p) = { L
¶j0
¶n
}S (p) (p Î S) ,
(29)
where the operators are defined in a similar way to Lk and Mk
in (16) and (17) but with a corresponding Green's function suitable for
the Laplacian operator.
The application of collocation, as described in subsection 4.2, to the integral equation (29)
reduces it to the following linear system of equations
[ M +
1
2
I ] j0 = Lv0
(30)
where [L]ij={ L zi }S (pj),
[M]ij={ M zi }S (pj),
j0 = [j0(p1), j0(p2), ..., j0(pn)]T,
v0 = [v0(p1), v0(p2), ..., v0(pn)]T.
Note that from (24) and (27) we may write
j = j0 + j1 ,
(31)
v = v0 + v1 ,
(32)
where in (31), (32),
j1 = [j1(p1), j1(p2), ..., j1(pn)]T,
v1 = [v1(p1), v1(p2), ..., v1(pn)]T.
In similarity to the technique of equation (9) in the finite
element method in section 3, j is approximated
in equation (26) as follows
j(p) »
n å
j=1
C(p,pj) F(pj) (p Î D ÈS) .
(33)
so that
j » C F , where
[C]ij = C(pi, pj),
F = [F(p1), F(p2), ..., F(pn)]T.
In all of the papers in which this method is used, the function
C(p,q) is defined as follows
C(p,q) = R - r(p, q)
(34)
where R is the largest distance between any two points on S
and r(p,q) is the distance between the points p
and q.
The substitution of (33) into (26) and further
algebraic manipulation, results in an equation of the form
é ë
D -
^
k
2
E
ù û
^
F
= 0
(35)
where D and E are full matrices.
The solution of the can now be computed using
the QZ algorithm. The representation
of the mode shapes on S can be recovered through the substitution
of the [^(F)] into approximation (33). The
solution in the domain D can then be obtained through the
numerical approximation of (14).
The same method is derived in a slightly different way in Banerjee et al
[15]. Results on the application of this method to acoustic
problems are given in references
[15-17] and [30]. Clearly the success of the method is largely dependent on the
suitability of approximation (34). In Ali et al [17]
several cases are cited when the method above does not work well
and they go on to propose some modifications in the method.
This methods originates from a paper by Nowak and Brebbia [24],
although a similar approach is descibed in Koopman and Benner [31].
The method is based on writing the Green's function as a finite series in
coefficients of -k2
where the G[m] are independent of k.
The substitution of (36) into (22) gives the following
[Ak]ij »
N å
m=1
(-k2)m { M[m] zi }S(pj)+
1
2
dij
(37)
where { M[m] zi }(p) = òS G[m](p,q) zi(q) dSq.
This replaces (23) with the following polynomial eigenvalue problem
[ A[0] + lA[1] + l2 A[2] + ... + lN A[N] ]
~
f
= 0
(38)
where l = -k2 and
[A[m]]ij = { M[m] zi }(pj). The solutions of
(38) are the approximations to the eigenfrequencies and
eigenfunctions.
The approximation of Ak is useful in the first instance
as it gives a polynomial formula for each matrix element, which
is computationally less expensive to evaluate than the
corresponding integral. In the next section we will also
see how a polynomial eigenvalue problem may be reformulated
as a generalised linear eigenvalue problem that can be solved
using a standard method. Methods for solving (38)
are considered in Kamiya and Andoh [19-20]
and Kamiya et al [21-22].