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Dendrites are one of the most widely observed patterns in nature and occuracross a wide spectrum of physical phenomena. In solidification and growthpatterns in metals and crystals, the multi-level branching structures ofdendrites pose a modeling challenge, and a full resolution of these structuresis computationally demanding. In the literature, theoretical models ofdendritic formation and evolution, essentially as extensions of the classicalmoving boundary Stefan problem exist. Much of this understanding is from theanalysis of dendrites occurring during the solidification of metallic alloys.Motivated by the problem of modeling microstructure evolution from liquid meltsof pure metals and alloys during MAM, we developed a comprehensive numericalframework for modeling a large variety of dendritic structures that arerelevant to metal solidification. In this work, we present a numericalframework encompassing the modeling of Stefan problem formulations relevant todendritic evolution using a phase-field approach and a finite element methodimplementation. Using this framework, we model numerous complex dendriticmorphologies that are physically relevant to the solidification of pure meltsand binary alloys. The distinguishing aspects of this work are - a unifiedtreatment of both pure metals and alloys; novel numerical error estimates ofdendritic tip velocity; and the convergence of error for the primal fields oftemperature and the order parameter with respect to numerical discretization.To the best of our knowledge, this is a first-of-its-kind study of numericalconvergence of the phase-field equations of dendritic growth in a finiteelement method setting. Further, we modeled various types of physicallyrelevant dendritic solidification patterns in 2D and 3D computational domains.

Modeling of dendritic solidification and numerical analysis of the phase-field approach to model complex morphologies in alloys