WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1004
O. Sigmund and J. S. Jensen
where A and ,u are Lame's coefficients, p is the material density, and u {uvw}T
the displacement vector.
In the following we assume that any variation in the material parameters occurs in
the (x, y)-plane only; thus we have A = A(x, y), ,L = -u(x, y), and p = p(x, y). Further,
we restrict the analysis by only considering waves that propagate in the (x, y)-plane,
so that Ou/Oz = 0. Equations (2.1)-(2.3) can then be split into two coupled in-plane
equations (governing the longitudinal and transverse modes) and an out-of-plane
equation (the acoustic modes):
Pt2 = a
(2/,+A) a + -
Pt2 -Ox O2 x A Dy +3ya((,9 y ay y x
Q ^ay (~Lh)
v 9u\
(2.4)
P2 = , + au)) + a
(2 + A) ?v + A (2.5)
9t2 ax\ 9x 9y ) 9yy 19Y Ox
02w a ( 9w\ ( 9Ow\
P9t2 Ax Ax tO ) +y ay (2.6)
/ DyOxDy
Equations (2.4)-(2.6) can be solved with the appropriate boundary conditions
applied. For the design of waveguides we need to be able to simulate travelling waves,
and for that purpose we introduce absorbing boundary conditions. In the scalar case,
for normal incident waves on a flat boundary, the absorbing boundary condition can
be written as (e.g. Krenk 2002)
9w 1 8w
+- = 0, (27)
On c Ot
where n is the outward-pointing normal vector and c = /-/p is the local wave speed.
Further analysis is now split into two separate parts. First, the material problem is
addressed by analysing base cells that are repeated infinitely, and then the structural
problem is considered for finite-dimensional media with external loading. In both
cases, equations (2.4)-(2.6), with the boundary conditions (2.7), are solved using a
standard FEM procedure using four-noded bilinear quadrilateral elements for the
discretization.
(a) The material problem
For the material problem, the wave equation may be solved as an eigenvalue prob-
lem for an infinitely periodic structure. We may solve the global problem by analys-
ing the smallest repetitive unit, the base cell Y. We assume that the modes can be
described by the expression
u(y, k) = u(y)eikTYeit, (2.8)
where i is a Y-periodic displacement field, y = (yl, Y2) are the local cell coordinates
and k is the plane wave vector. For k = 0, the solution mode u(y) will be Y-periodic,
for k = r, the solution mode will be 2Y-periodic and for other k, the solution modes
can take any kind of periodicity in the plane. This kind of modelling is based on the
so-called Floquet-Bloch wave theory (Mathews & Walker 1964; Kittel 1986).
Inserting (2.8) into either (2.4), (2.5) or (2.6), dropping the tilde, and converting
to FEM notation yields
(K(k) -2M) = 0, (2.9)
Phil. Trans. R. Soc. Lond. A (2003)