WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

whereas if a is increased material 2 is dominating and for lower values of a material 1 is
dominating. If m A ¼ 1 (e.g. a taut string) we have a ¼ 1=bo1 which shows that material 1 (the
lighter material) is always dominant in the optimized design for this special case.
3. The 2D scalar case
We now consider the more complex problem of the 2D scalar case.
3.1. Model
The 2D scalar time-reduced wave equation (Helmholtz equation) is given by
= T ðAðx; yÞ=wÞþo 2 Bðx; yÞw ¼ 0, (17)
where the problem-dependent material coefficients A and B can now vary in the 2D plane ðx; yÞ.
As in the 1D case we apply a standard FEM discretization, which leads to the discrete eigenvalue
problem stated in Eqs. (2)–(3). The element matrices are in the 2D case given by
Z
ke ¼ ð ONÞ T Z
ON dV; me ¼ N T N dV, (18)
where
V e
" #
O ¼ q=qx 0
0 q=qy
V e
. (19)
Also in the 2D case we may study different structural vibration problems by changing the two
coefficients A and B. Letting A ¼ 1 and B ¼ r=T enables us to analyze the membrane problem
where rðx; yÞ is the density and T is the uniform tension (force per area). Alternatively with
A ¼ E=ð2ð1 þ nÞÞ, where Eðx; yÞ is Young’s modulus and nðx; yÞ is Poisson’s ratio, and with B ¼
rðx; yÞ being the density, Eq. (19) governs out-of-plane shear vibrations of a thick elastic body.
3.2. Optimization
ARTICLE IN PRESS
J.S. Jensen, N.L. Pedersen / Journal of Sound and Vibration 289 (2006) 967–986 977
When we optimize a 2D domain with respect to maximizing the gap between eigenfrequencies
there are a number of extra difficulties we must deal with. The primary source of the difficulties is
the possibility of multiple eigenfrequencies. The multiple eigenfrequencies can be calculated
without difficulty using, e.g. the subspace iteration method [20].
The objective for the optimization is as in the 1D case given by
maximize J ¼ onþ1 on, (20)
where the gap between the eigenfrequency of order n þ 1 and n is maximized.
If the eigenfrequencies of order n þ 1 and n are both distinct eigenpairs, with squared
eigenfrequencies o 2 nþ1 and o2 n and corresponding eigenvectors f nþ1 and f n, no problems arise and
we use the objective (20) directly since the sensitivities of the squared eigenfrequency with respect