A two-dimensional mesoscopic numerical method to simulate the failure process of concrete under compression was developed based on the discrete element method by modifying the rigid body-spring model proposed by Nagai et al. In the calculation model, aggregates or aggregate elements inside the concrete were simplified as rigid bodies with regular polygon profiles, which were surrounded by mortar polygons or mortar elements. All of the adjacent elements were connected by springs. According to the random distribution of aggregates, the mesh was generated by using Voronoi diagram method. Plastic behavior after the elastic limit for a spring was considered to set up the constitutive model of the spring, and Mohr-Coulomb criterion was adopted to judge the failure of a spring. Simulation examples show that the proposed method can be used to predict the mechanical behavior of concrete under compression descriptively and quantitatively both for small deformation problems and for larger deformation problems.