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