EXTENDED ABSTRACT: In this talk, some mathematical approaches for microstructure evolution of Ni-based superalloys are introduced. Understanding microstructure and estimating property related coefffcients have been a long standing demand for developing advanced materials. To achieve goals, we employ the phase-ffeld system [1], consisting of Cahn-Hilliard and Allen-Cahn equations, as the governing functions. In this system, the identification for material properties is considered as an inverse problem of reconstructing the energy potential functional from the system by using the phase ffeld data of certain time points. Theoretically, we deduced the local well-posedness of the phase-field system based on the work [2] and implicit function theorem in Banach spaces, while the uniqueness of recovering the nonlinear energy potential functions are provided through the higher order linearization technique. Numerically, two efffcient algorithms are proposed based on certain prior knowledge of the nonlinear potentials. For a single phase-ffeld equation, an inversion algorithm based on ETD method [3] is proposed to estimating the potential by the ffeld data of two times, while for the phase-ffeld system, similar result can be retrieved by our proposed optimization method combining Adam optimizer. In addition, a novel deep learning framework named PFWNN which bases on the weak forms is constructed to recover the potential of phase-ffeld equations in the absence of prior knowledge. Other microstructure prediction approaches based on data driven methods with laboratory data are also shown in this talk.

Keywords: Single crystal superalloys; Phase-field model; Inverse problem; Optimization; Neural network;

REFERENCES:

[1] Y. Wang, D. Banerjee, C.-C. Su and A. Khachaturyan, Acta Materialia, 46, (1998) 2983–3001. [2] A. Brunk, H. Egger and O. Habrich, Inverse Problems, 39, (2022). [3] Z. Fu and J. Yang, J. Comput. Phys, 454, (2022) 110943.