Addressing heterogeneous nucleation process on materials with complex (superficial) structures is an important issue to understand the thermodynamic principles and forward the engineering applications. Based on classic nucleation theory, we developed a thermodynamic model that is comprised of both the geometry and the surface-structure effects of substrate to represent the process of forming a crystal or drop embryo on a rough concave surface. The model recovers many models developed previously. We extracted two featured factors: ratio factor that indicates the ratio of the critical radius of the embryos forming on the rough concave surface to that on the smooth surface, and shape factor that represents the relative free energy barrier of heterogeneous nucleation of the critical size to that of homogeneous nucleation. Anomalies of both the ratio factor and shape factor were observed for heterogeneous nucleation on the rough concave surface. Our study provides an essential route toward design (or manipulation) of heterogeneous nucleation by structuring the surfaces of the substrate in fractal, through which a number of promising applications in a variety of fields can be envisioned.

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