A numerical prediction scheme of crystal growth on various substrates by using the first-principles calculation was proposed to analyze the epitaxial processes of the piezoelectric thin films, such as the sputtering, the chemical vapor deposition and the molecular beam epitaxy processes. At first, we analyze the epitaxial strain of crystal cluster in piezoelectric thin film, which is caused by the lattice mismatch with the substrate. When the epitaxial strain is introduced in the unit cell of the crystal cluster, the total energy can be calculated by employing the pseudo-potential method based on the density functional theory. Then, a preferred orientation of crystal cluster is selected from the crystal conformations on the substrate which satisfy the structural stability condition. This numerical scheme was applied to BaTiO₃ thin films processes fabricated on various substrates, such as SrTiO₃(110), SrTiO₃(001), MgO(100) and LaTiO₃(001). Numerical results show that our process crystallographic design scheme, which employs the total energy increment of cluster, is efficient tool to evaluate the possibility of thin film crystal growth. Finally, it was confirmed that numerical results of preferred orientations have good correspondence with experimental ones.