  We present an adaptive scheme for isogeometric phase-field modeling, to
perform suitably graded hierarchical refinement and coarsening on both single-
and multi-patch geometries by considering truncated hierarchical spline
constructions which ensures $C^1$ continuity between patches. We apply the
proposed algorithms to the Cahn-Hilliard equation, describing the time-evolving
phase separation processes of immiscible fluids. We first verify the accuracy
of the hierarchical spline scheme by comparing two classical indicators usually
considered in phase-field modeling, for then demonstrating the effectiveness of
the grading strategy in terms of accuracy per degree of freedom. A selection of
numerical examples confirms the performance of the proposed scheme to simulate
standard modes of phase separation using adaptive isogeometric analysis with
smooth THB-spline constructions.
