  Aiming at comparing different morphological models of galaxy clusters, we use
two new methods to make a cosmological model-independent test of the
distance-duality (DD) relation. The luminosity distances come from Union2
compilation of Supernovae Type Ia. The angular diameter distances are given by
two cluster models (De Filippis et al. and Bonamente et al.). The advantage of
our methods is that it can reduce statistical errors. Concerning the
morphological hypotheses for cluster models, it is mainly focused on the
comparison between elliptical $\beta$-model and spherical $\beta$-model. The
spherical $\beta$-model is divided into two groups in terms of different
reduction methods of angular diameter distances, i.e. conservative spherical
$\beta$-model and corrected spherical $\beta$-model. Our results show that the
DD relation is consistent with the elliptical $\beta$-model at $1\sigma$
confidence level (CL) for both methods, whereas for almost all spherical
$\beta$-model parameterizations, the DD relation can only be accommodated at
$3\sigma$ CL, particularly for the conservative spherical $\beta$-model. In
order to minimize systematic uncertainties, we also apply the test to the
overlap sample, i.e. the same set of clusters modeled by both De Filippis et
al. and Bonamente et al.. It is found that the DD relation is compatible with
the elliptically modeled overlap sample at $1\sigma$ CL, however for most of
the parameterizations, the DD relation can not be accommodated even at
$3\sigma$ CL for any of the two spherical $\beta$-models. Therefore it is
reasonable that the marked triaxial ellipsoidal model is a better geometrical
hypothesis describing the structure of the galaxy cluster compared with the
spherical $\beta$-model if the DD relation is valid in cosmological
observations.
