## 1 Introduction  
Ceramic materials are extensively used in the aerospace field because they have high melting points, high hardness, low density, superior tribology, and remarkable stability at high temperatures [1]. For instance, they are adopted in gas turbine engines [2], thermal protection materials for nose tips [3], nuclear power generation [4], and the wing leading edges of hypersonic vehicles [5]. Ceramics can be applied to create high-temperature components, most typically thermal barrier coatings, such as turbine blades and combustion chamber linings in aerospace engines [6]. On the other hand, brittle materials, like ceramics, frequently have weak thermal shock resistance, leaving them vulnerable to thermal shock failure [7; 8; 9] in situations with fast temperature changes. Thermal shock failure occurs as a result of temperature gradients inside ceramic materials. Various parts are subjected to different degrees of thermal stress, which causes internal stress concentration and forms cracks. Therefore, it is critical to comprehend the initiation and propagation of cracks in ceramic materials under thermal shock, as well as to evaluate and enhance the thermal shock resistance to ensure the materials' functioning in a high-temperature environment. Scholars have recently used a range of experiments, theories, and numerical approaches to investigate the thermal shock crack behavior of ceramic materials and assess their thermal shock resistance. For example, [PERSON] et al. [10] employed a high-speed imaging technique to analyze the difference in crack growth under cold and thermal shock. [PERSON] et al. [11] examined the ablation mechanisms of Ti\({}_{3}\)SiC\({}_{2}\) ceramic in a 1600 \({}^{\circ}\)C nitrogen plasma flame. [PERSON] et al. [12] investigated the influence of surface brittle-to-ductile transition on the high-temperature thermal shock resistance of Al\({}_{2}\)O\({}_{3}\) ceramics. Ceramic materials are still the subject of a lot of research [13; 14; 15; 16; 17]. However, the initiation and propagation of cracks in ceramic materials under thermal shock are incredibly rapid and complicated. In many cases, only the thermal shock crack pattern and fracture surface following thermal shock can be seen due to the limitations of monitoring technology. Consequently, crack initiation and propagation under thermal shock have been extensively studied using numerical simulation methods (such as the finite element method [18], extended finite element method [19], discrete element method [20], etc.). [PERSON] et al. [21] adopted bond-based peridynamics and FEM to simulate thermal shock cracking in ceramics. [PERSON] et al. [22] carried out simulated analysis and the optimization of SiC ceramics via the laser preheating of RSM. [PERSON] et al. [23] utilized the extended finite element method (XFEM) to forecast the development of cracks in ceramics under various loading conditions. However, the aforementioned numerical simulation methods have several drawbacks in studying the crack propagation behavior of brittle materials, such as crack branching and multi-cracking, especially in three-dimensional issues. The phase-field method, which is based on the energy variational principle [24] of brittle fractures, has lately gained popularity for simulating the fracture of brittle materials. [PERSON] et al. [25] described the crack-cracking process by introducing a continuous phase-field variable (0-1). [PERSON] et al. [26; 27; 28] made significant contributions to the development of phase fields, as well. Additionally, [PERSON] et al. [29; 30] suggested a phase-field regularized cohesive region model that is insensitive to the length-scale parameter based on the phase-field method. In comparison to other models, the phase-field model does not require unique failure criteria to detect when/where cracks nucleate, propagate, branch, merge, etc. It may considerably reduce the computational complexity by simulating the whole solid process, from crack initiation through propagation to fracture.