  We determine the asymptotic behaviour of eigenvalues of clamped plates under
large compression, by relating this problem to eigenvalues of the Laplacian
with Robin boundary conditions. Using the method of fundamental solutions, we
then carry out a numerical study of the extremal domains for the first
eigenvalue, from which we see that these depend on the value of the
compression, and start developing a boundary structure as this parameter is
increased. The corresponding number of nodal domains of the first eigenfunction
of the extremal domain also increases with the compression.
