ABSTRACT The phase‐field model is a developing diffusion method for the brittle crack problem in which the fourth‐order model with higher regularity improves the convergence. As well known, the main problem of phase‐field is the extremely high computational cost. In order to reduce the computational burden and accelerate the convergence, we propose an adaptive method on the continuous quadrilateral spline element to simulate the fourth‐order phase‐field model for the brittle fracture. The quadrilateral element QS‐12 has 12 degrees of freedom (DOFs), the function values and two partial derivatives at each vertex. The simplified stiffness matrix computations by the Bézier coefficients further reduce the calculation costs. Hanging nodes of refined hierarchical meshes are handled by the field variable transformation. Then the indicator for the adaptive process is chosen as the phase‐field variable, and hierarchical quadrilateral mesh or the quadtree mesh is used for the local refinement. Meanwhile, a staggered scheme with a hybrid formulation is applied to solve the phase and displacement field equations. Several benchmarks show the accuracy and robustness of the proposed method. Finally, we show the ability to calculate the complex crack behavior by the compression test of a rock‐like plate with double inclined flaws and a plate with random initial nucleation sites.
Liu et al. (2026) studied this question.