# Research Advances in Large Deformation Analysis and Applications of the Material Point Method
## 5 Application in Dynamic Cracking
[69] developed the immersed boundary MPM to simulate both fluid-structure coupling and dynamic fracture problems. In this approach, the MPM solver is used to simulate the extreme deformation problems of solid structures, and the Lagrangian-based continuous IBM simulates FSI, thereby strictly ensuring the boundary velocity conditions at each time step without the need to reconstruct the FSI interface, which improves solution efficiency to some extent. The multi-material ALE (MMALE) method does not require tracking complex material interfaces and can efficiently simulate the compressible flows of multiple materials [70]. Therefore, [PERSON] and [PERSON] [71] proposed the immersed ALE multi-material MPM for FSI problems involving multi-material fluid flow and extreme structural deformation accompanied by crack and fracture propagation. In this method, the IBM immerses MPM particles in the MMALE grid; that is, the solid region is immersed in the fluid region, and the space occupied by the solid domain is filled with virtual fluid, as shown in Figure 12.  
The phase-field method has addressed the issues of explicit crack propagation and crack initiation in the MPM. Consequently, some researchers have attempted to combine  
Figure 11: Region division of adaptive algorithm [63]. Figure 12: The spatial discretization of the IALEMPM [71]. the MPM with the phase-field method. [PERSON] and [PERSON] [72; 73; 74] were the first to introduce the phase-field model into the MPM to simulate crack propagation in brittle materials. Subsequently, they proposed an anisotropic phase-field MPM that considers the anisotropic surface energy to address the simulation of brittle fracture in anisotropic media. To improve solution efficiency, [PERSON] et al. [75] developed a phase-field MPM crack numerical program based on GPU acceleration. [PERSON] et al. [76] proposed an explicit phase-field MPM to handle dynamic crack branching and problems with cracks under velocity boundary conditions. In the simulation of dynamic fracture in hyperelastic materials, [PERSON] et al. [77] introduced an explicit phase-field total Lagrangian MPM, extended the particle contact algorithm to handle self-contact situations (see Figure 13), and demonstrated the effectiveness of the method by simulating ring collisions, the compression of hyperelastic blocks, and the impact of metal balls on soft membranes. [PERSON] et al. [78] proposed an explicit finite element material point coupled with the phase-field model (PF-FEMPM). They compared it with the phase-field FEM and the phase-field MPM, as shown in Figure 14, and found that the elastic strain energy and crack dissipation energy exhibited significant consistency with previous research findings. This approach effectively addresses challenges related to imposing boundary conditions and the cell-crossing instability issue encountered in the phase-field MPM. Figure 14: Evolution of dynamic crack propagation calculated by the PF-FEMPM method at times (**a**) \(t=35\) μs, (**b**) \(t=50\) μs, (**c**) \(t=65\) μs, and (**d**) \(t=80\) μs, (**e**) simulated by PF-FEM at \(t=80\) μs and (**f**) simulated by PF-MPM at \(t=80\) μs [78]. Figure 13: The self-contact identification for the phase-field total Lagrangian MPM [77]. The aforementioned MPM schemes require an accurate description of the position of the crack surface and its normal vector, and additional relationships are needed to link the initiation of the crack with its propagation speed and direction, which increases the complexity of simulating dynamic crack propagation. Furthermore, the computational efficiency of the PF-FEMPM scheme is lower than that of the phase-field MPM, and it focuses on studying the propagation of cracks in solids under conditions of small deformation.