Generalized balanced power diagrams for 3D representations of polycrystals

Abstract

Characterizing the grain structure of polycrystalline material is an important task in material science. The present paper introduces the concept of generalized balanced power diagrams as a concise alternative to voxelated mappings. Here, each grain is represented by (measured approximations of) its centre of mass position, its volume and, if available, and by its second-order moments (in the non-equiaxed case). Such parameters may be obtained from 3D X-ray diffraction. As the exact global optimum of our model results from the solution of a suitable linear programme it can be computed quite efficiently. Based on verified real-world measurements, we show that from the few parameters per grain (3, respectively, 6 in 2D and 4, respectively, 10 in 3D) we obtain excellent representations of both equiaxed and non-equiaxed structures. Hence our approach seems to capture the physical principles governing the forming of such polycrystals in the underlying process quite well.

Keywords:

• power diagrams
• generalized balanced power diagrams
• tessellations
• linear programming
• grains
• polycrystals

Acknowledgements

We are grateful to an anonymous referee for pointing out the reference [1]. We further thank Oleg Mishin and Yubin Zhang for providing Data-Set II and Erik Mejdal Lauridsen and Dorte Juul Jensen for valuable discussions.

Additional information

Funding

The Deutsche Forschungsgemeinschaft (DFG) is acknowledged for partial financial support through [grant number GR 993/10-2]. HFP acknowledges the ERC advanced grant ‘diffraction based transmission x-ray microscopy.’ COST Action MP1207 is acknowledged for networking support.