The point of departure for the particle-based discrete element simulations is the generation of disk or sphere packing of interest domain. In this paper, based on a triangular mesh, two geometric packing algorithms are proposed for 2D problems. New disk(s) are obtained element-wise by solving system of equations, which can be easily derived according to the local topological relationships. In this way, the adjacent disks are guaranteed to be in contact with each other exactly. Related issues are discussed in detail, such as boundary treatment for good definition of the boundaries, elimination of incorrect overlaps, random perturbation for avoiding over-homogeneous packing. The main advantages of these two algorithms are the high efficiency and applicability for arbitrary domains. Furthermore, with a refined mesh, a refined packing can be easily obtained, which is a potential alternative for multiscale simulations. The proposed methods are verified by several examples. (C) 2020 Elsevier B.V. All rights reserved.