  We present adaptive finite element simulations of dendritic and eutectic
solidification in binary and ternary alloys. The computations are based on a
recently formulated phase-field model that is especially appropriate for
modelling non-isothermal solidification in multicomponent multiphase systems.
In this approach, a set of governing equations for the phase-field variables,
for the concentrations of the alloy components and for the temperature has to
be solved numerically, ensuring local entropy production and the conservation
of mass and inner energy. To efficiently perform numerical simulations, we
developed a numerical scheme to solve the governing equations using a finite
element method on an adaptive non-uniform mesh with highest resolution in the
regions of the phase boundaries. Simulation results of the solidification in
ternary Ni$_{60}$Cu$_{40-x}$Cr$_{x}$ alloys are presented investigating the
influence of the alloy composition on the growth morphology and on the growth
velocity. A morphology diagram is obtained that shows a transition from a
dendritic to a globular structure with increasing Cr concentrations.
Furthermore, we comment on 2D and 3D simulations of binary eutectic phase
transformations. Regular oscillatory growth structures are observed combined
with a topological change of the matrix phase in 3D. An outlook for the
application of our methods to describe AlCu eutectics is given.
