Effective phase-field modeling of the Cahn–Hilliard equation in 2D and 3D using a quasi-interpolation approach to adaptive isogeometric analysis with THB-splines

Phase separation leads to evolving interfaces that require sufficient spatial resolution to accurately capture their dynamics. We present a multi-dimensional higher-order adaptive isogeometric analysis framework for phase-separation problems, based on a phase-field formulation of the Cahn–Hilliard equation. As basis functions, we employ Truncated Hierarchical B-splines, which form a partition of unity and enable local mesh refinement and coarsening. The adaptive meshing scheme refines the mesh at interfaces and coarsens it in the bulk, with the mesh resolution evolving alongside the solution. Element marking is guided by the solution field, which identifies interface locations, and solution transfer between successive meshes is performed via a quasi-interpolation operator that is naturally parallelizable and efficiently reduces computational cost. The performance of the framework is demonstrated through a spatial convergence study and a series of 2D and 3D numerical examples, showing that locally adaptive meshes accurately track evolving interfaces while reducing the computational effort per iteration compared to uniform tensor-product discretizations.