The fluid flows with large deformation or distortions are common in the free surface flow and fluid structure interaction problems. When the fluid flow accompanies with the large distortions, the Lagrangian method will suffer from a lack of accuracy and stability. In this paper, we present a new Lagrangian discontinuous Galerkin scheme to simulate the compressible flow on the unstructured triangular mesh, which can overcome the large distortions in the Lagrangian algorithm. The new scheme combines the advantage of the discontinuous Galerkin scheme and the adaptive mesh optimization technique. It consists of three phases, Lagrangian phase, remeshing phase and remapping phase. First, the compressible Euler equation in the Lagrangian framework is solved by the discontinuous Galerkin method, where the nodal velocity and numerical fluxes are determined by the nodal solver. Second, the adaptive mesh optimization technique is employed to eliminate the ill-shaped triangular elements caused by the large distortions of fluid flow. The mesh subdivision, mesh simplification and mesh relaxation techniques are adopted to optimize the mesh topology and improve the mesh quality. Third, the physical solutions on the original mesh are projected onto the new remeshed mesh. The scheme can preserve the mass, momentum and energy conversation. Several representative two-dimensional cases are simulated to test the accuracy and robustness of the present model. The scheme can provide a reference for simulating the multi-material compressible flow in applications to the ocean engineering.