  The difficulties in dealing with discontinuities related to a sharp crack are
overcome in the phase-field approach for fracture by modeling the crack as a
diffusive object being described by a continuous field having high gradients.
The discrete crack limit case is approached for a small length-scale parameter
that controls the width of the transition region between the fully broken and
the undamaged phases. From a computational standpoint, this necessitates fine
meshes, at least locally, in order to accurately resolve the phase-field
profile. In the classical approach, phase-field models are computed on a fixed
mesh that is a priori refined in the areas where the crack is expected to
propagate. This on the other hand curbs the convenience of using phase-field
models for unknown crack paths and its ability to handle complex crack
propagation patterns. In this work, we overcome this issue by employing the
multi-level hp-refinement technique that enables a dynamically changing mesh
which in turn allows the refinement to remain local at singularities and high
gradients without problems of hanging nodes. Yet, in case of complex
geometries, mesh generation and in particular local refinement becomes
non-trivial. We address this issue by integrating a two-dimensional phase-field
framework for brittle fracture with the finite cell method (FCM). The FCM based
on high-order finite elements is a non-geometry-conforming discretization
technique wherein the physical domain is embedded into a larger fictitious
domain of simple geometry that can be easily discretized. This facilitates mesh
generation for complex geometries and supports local refinement. Numerical
examples including a comparison to a validation experiment illustrate the
applicability of the multi-level hp-refinement and the FCM in the context of
phase-field simulations.
