By using the threshold on the phase-field variable as an error indicator, an adaptive fourth-order phase-field method in the framework of isogeometric analysis is proposed for fracture simulation. The spatial and geometric discretization is based on locally refined non-uniform rational B-splines (LR NURBS), which have the ability for local refinement. The $C^1$ continuous LR NURBS basis function is employed as the fourth-order phase field which requires higher-order derivatives. A hybrid formulation is adopted, and the coupled elasticity and phase-field equations are solved using a staggered method. The simulation starts with a coarse discretization and the spatial discretization is adaptively refined on the fly. The robustness and the reliability of the adaptive fourth-order phase-field model based on isogeometric analysis are demonstrated with a few standard benchmark problems. From the numerical study, it is inferred that the proposed framework yields accurate results without compromising accuracy. Further, the fourth-order phase-field model gives a narrow diffusive region, which reduces the region to be refined that improves the computational efficiency.

以相场变量的阈值作为误差指标，提出了等几何分析框架下的自适应四阶相场方法进行断裂模拟。空间和几何离散化基于具有局部细化能力的局部细化非均匀有理B样条(LR NURBS)。这$C^{\mathrm{1个}}$采用连续的LR NURBS基函数作为需要高阶导数的四阶相场。采用混合公式，并使用交错法求解耦合弹性方程和相场方程。模拟从粗略离散化开始，然后动态自适应地细化空间离散化。通过几个标准基准问题证明了基于等几何分析的自适应四阶相场模型的鲁棒性和可靠性。从数值研究中可以推断，所提出的框架可以在不影响准确性的情况下产生准确的结果。此外，四阶相场模型给出了一个狭窄的扩散区域，减少了需要细化的区域，提高了计算效率。