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Poisson Voronoi (PV) tessellations as artificial microstructures are widely used in investigations of material deformation behaviors. However, a PV structure usually describes a relative homogeneous field. This work presents a simple numerical method for generating 2D/3D artificial microstructures based on hierarchical PV tessellations. If grains/particles of a phase cover a large size span, the concept of “artificial phases” can be used to create a more realistic size distribution. From case to case, detailed microstructural features cannot be directly achieved by commercial or free softwares, but they are necessary for a deep or thorough study of the material deformation behavior. PV tessellations created in our process can fulfill individual requirements from material designs. Another reason to use PV tessellations is due to the limited experimental data. Concerning the application of PV microstructures, four examples are given. The FE models and results will be presented in consecutive works, i.e. “part II: applications”.