Spallation of concrete under dynamic loading has been the hot issue of concern about civil engineering structures and protective engineering. In the present paper, the principle of propagation of stress wave and the induced spallation process along a one-dimensional bar and the RFPA-Dynamics code which considers the heterogeneity of the concrete materials are briefly introduced. In order to numerically investigate the effect of mesh size of numerical model on the computational results, the failure process of concrete with three mesh sizes under dynamic loads was numerically simulated using RFPA-Dynamics code. Numerical simulations show that the failure patterns of concrete with different mesh sizes under dynamic stress waves are different. It is found that for the numerical models with the same mechanical parameters, the smaller mesh size of the numerical model is, the longer propagation of stress wave delay, the smaller the compressive stress is, thus the greater the tensile stress is, and the number of cracks is increasing, in general, rupture is more serious. The accuracy of mesh is little to improve when the number of the grid increases to a certain value.