Testing the anisotropy of the Universe with the distance duality relation

The distance duality relation (DDR) is valid in Riemannian spacetime. The astronomical data hint that the universe may have certain preferred direction. If the universe is described by anisotropic cosmological models based on Riemannian spacetime, then DDR still valid. If the anisotropy universe is described by other models which are not based on Riemannian spacetime, then DDR is violated. Thus, DDR could be used to test the validity of these anisotropic cosmological models. In this paper, we perform anisotropic DDR parametrization with the dipolar structures. The DDR is tested by comparing the luminosity distance from type-Ia supernovae (Union 2.1 and JLA compilations) and the angular diameter distance from strong gravitational lensing (SL) systems at the same redshift. It is shown that, the DDR is valid with the Union2.1 compilation, while is violated more than 1$\sigma$ confidence level with the JLA compilation. Additionally, we verify the statistical signification of our method with Monte Carlo simulations. Due to the large uncertainty of available data, no strong evidence is found to violate the DDR in the anisotropic models.