Abstract: The general expressions of constitutive tensors of the arbitrary invisibility cloak are derived based on the linear coordinatetransformation theory, and then some full-wave simulations based on the COMSOL Multiphysics validate the constitutive tensors.Lossy cloaks are also analyzed based on the simulation results. It is believed that this study provides a feasible way to fulfill theconstruction of 2D cloaks.