[en] It has been well known that addition of rare earth elements was indispensable for production of ductile cast iron. The addition reduces solidification shrinkage. However there are still ambiguities in understanding the mechanism and predicting the casting defects. A possible explanation for the reduction of shrinkage is that the addition of rare earth promotes expansion of castings due to the graphite formation, which is related to cooling rate. In this study, the effect of rare earth addition was considered as a function of cooling rate and solidification shrinkage rate was determined by the macro-scale simulation. In the meso-scale and the micro-scale simulations, distribution of graphite nodules was compared with the experimental results. The multi-scale simulation result well reproduced the experimental results. (paper)