Research Advances in Large Deformation Analysis and Applications of the Material Point Method

Abstract

Large deformation analysis is a crucial foundation for studying the nonlinear behavior and progressive damage of materials and structures. Traditional mesh methods often struggle with large-scale mesh distortion when dealing with such issues, which can compromise solution efficiency and accuracy, and in severe cases, even cause computational interruptions. In contrast, the material point method (MPM) employs a dual framework of Lagrangian particles and Eulerian background grids, effectively integrating the advantages of both Lagrangian and Eulerian approaches, thus avoiding mesh distortion and challenges in handling convective terms. Consequently, many researchers are dedicated to developing an MPM for addressing high-speed impact and fluid–structure interaction problems that involve material failure and large deformations. This paper begins by introducing the fundamental theory and contact algorithms of the MPM. It then systematically summarizes the latest advancements and applications of the MPM, including its hybridization and coupling with other algorithms, in simulating various large deformation scenarios such as high-speed impacts, explosions, dynamic cracking, penetration, and fluid–structure interactions. This paper concludes with a summary and a prospective view on future trends. This review highlights the robustness and accuracy of the MPM in tackling large deformation problems, offering valuable insights for the analysis of large deformations and damage evolution in various materials.

1. Introduction

With the rapid development of computer technology, numerical simulation techniques have been widely applied in engineering and scientific research. Common numerical computation methods are generally categorized based on the continuity of the problems they solve into continuous mechanics computation methods and discontinuous mechanics computation methods. The former is typically suitable for solving solid or fluid mechanics scenarios that are described by systems of partial differential equations as a whole, including the finite difference method [1], finite element method [2], finite volume method [3], boundary element method [4], and meshless method [5]; the latter focuses on solving problems involving flow, cracking, large deformation, and other behaviors in block or granular materials, including the discrete element method [6] and discontinuous deformation analysis [7]. Since actual engineering often involves multiphase media, and each material exhibits complex mechanical behaviors such as irregular boundaries, nonlinear constitutive relations, large deformations, cracking, and even fragmentation, scholars have been seeking efficient computation methods beyond continuous and discontinuous mechanics, such as coupled continuous–discontinuous computation models [8,9].

The finite difference method (FDM) primarily employs an Eulerian description, which, while free from grid distortion issues, encounters difficulties with complex geometric shapes, boundary handling, and challenging interface tracking. The finite element method (FEM) and other Lagrangian methods face grid distortion when dealing with large strain problems. Employing erosion algorithms to delete distorted grids can lead to the non-conservation of system mass, while grid re-meshing results in the loss of historical information and increased computational demands. To address the myriad issues associated with grid-based methods, the meshless method emerged. It originated from the smoothed-particle hydrodynamics (SPH) method proposed by Gingold [10], which, although unaffected by grid characteristics and suitable for handling large deformations, still faces issues such as low computational efficiency, complex contact algorithms, and difficult boundary condition handling. In this context, the material point method (MPM) has garnered significant attention due to its advantages in easily handling large deformations, no particle interface crossing, and high computational efficiency.

The MPM traces its origins back to the particle-in-cell (PIC) method proposed by Harlow [11], which was initially applied to fluid dynamics. To address the numerical dissipation issues encountered in PIC, Brackbill et al. [12] developed the fluid implicit particle (FLIP) method based on PIC. In 1994, to extend FLIP to the field of solid mechanics and to resolve the high numerical dissipation problems in PIC, Sulsky [13] optimized FLIP and termed it the MPM. The MPM was first validated for its significant advantages in handling large deformation problems through simulations of rigid body rotation and elastic body collision. Subsequently, the MPM has been widely applied and rapidly developed in the field of computational mechanics. This paper systematically summarizes the applications of the MPM in high-speed impact, explosion, dynamic cracking, impact penetration, and fluid–structure interaction problems, analyzing the unparalleled advantages of this method in dealing with extreme deformation issues.

The structure of this paper is as follows: Section 2 clarifies the fundamental theory, the contact models of the MPM, and the relationship between the MPM and meshless methods; Section 3, Section 4, Section 5, Section 6 and Section 7 provide a review of the application of the MPM in simulating various scenarios involving large deformations, such that Section 3 introduces the application of the MPM in high-speed impact, Section 4 covers its application in explosion, Section 5 discusses its application in dynamic cracking, Section 6 examines its application in impact penetration, and Section 7 explores its application in fluid–structure interaction; Section 8 discusses the existing problems and the possible prospects of the MPM.

2. Methodology of the MPM

The MPM, as a meshless numerical computation method, employs a dual description with Lagrangian material points and Eulerian background grids, thereby avoiding the mesh distortion issues encountered in traditional Lagrangian solutions and overcoming the challenges of non-advective term treatment in Eulerian methods. It is precisely because the MPM combines the strengths of both approaches that it can effectively address problems such as moving material interfaces, multi-material coupling, and large-amplitude flow deformations that arise in grid-based methods. The fundamental concept of the MPM is to discretize a continuum into a set of material points carrying all material information and to establish background grids in the regions where these particles move. The motion of the material points represents the deformation of the object, while the background grids carry no material information and are solely used to solve spatial derivatives and momentum equations. Shape functions serve as the bridge between material points and background grids. In a single time-step calculation, material point information is first mapped onto the background grid through shape functions. Then, spatial derivatives and momentum equations are solved at the background grid nodes. The solved physical quantities are mapped back to the material points through shape functions, and finally, the deformed grid is discarded. In the next time step, a new background grid is used, as shown in Figure 1.

2.1. Governing Equations

In the MPM, the governing equations are typically derived within the updated Lagrangian framework. When thermal effects are not considered, the governing equations consist of the mass conservation equation, the momentum conservation equation, and the energy conservation equation:

$$\mathrm{Conservation} \mathrm{equation} \mathrm{for} \mathrm{mass}: {\displaystyle \frac{d\rho }{dt}}+\rho {\displaystyle \frac{\alpha v_k}{\alpha x_k}}=0$$

(1)

$$\mathrm{Conservation} \mathrm{equation} \mathrm{for} \mathrm{momentum}: {\displaystyle \frac{\alpha {\sigma }_{ij}}{\alpha x_j}}+\rho b_i=\rho {\ddot{u}}_i$$

(2)

$$\mathrm{Conservation} \mathrm{equation} \mathrm{for} \mathrm{energy}: \rho \dot{e}={\dot{\epsilon }}_{ij}{\sigma }_{ij}$$

(3)

where $\rho$ is the density of the object, $v_k$ is the instantaneous velocity of the object at time $k$, $x_k$ is the spatial coordinate of the object at time $k$; ${\sigma }_{ij}$ is the Cauchy stress, $b_i$ is the force per unit mass acting on the object, ${\ddot{u}}_i$ is the acceleration of the material point in the i-direction; $\dot{e}$ is the internal energy per unit mass of the object, and ${\dot{\epsilon }}_{ij}$ is the strain rate of the object.

Within the solution domain $\Omega$, the momentum equation and the equivalent integral weak form (also known as the virtual work equation) of the given surface force boundary condition ${\Gamma }_t$ are

$$\underset{\Omega }{\int }\rho {\ddot{u}}_i\delta u_{i}dV+\underset{\Omega }{\int }\rho {\sigma }_{ij}^s\delta u_{i,j}dV-\underset{\Omega }{\int }\rho b_i\delta u_{i}dV-\underset{{\Gamma }_t}{\int }\rho {\overline{t}}_i^s\delta u_{i}dA=0$$

(4)

where $\delta u_i$ is the virtual displacement, ${\sigma }_{ij}^s={\sigma }_{ij}/\rho$ is the specific stress, ${\overline{t}}_i^s={\overline{t}}_i/\rho$ is the specific surface traction, $dV$ and $dA$ are the volume and area elements, respectively, and $i$ and $j$ are the spatial coordinate components.

2.2. Discretization of the Governing Equations

As shown in Figure 1, the MPM discretizes the continuum into a series of material points, which carry various physical quantities such as mass, velocity, energy, stress, and strain. Since each material point has a fixed amount of mass, the mass conservation (Equation $\left(1\right)$) for the material points is automatically satisfied. In the discretized format of the MPM, the material density can be expressed as

$$\rho \left(x_i\right)=\sum _{p=1}^{n_p}{m}_p\delta (x_i-x_{ip})$$

(5)

where $n_p$ is the total number of material points, $m_p$ is the mass of the material region where material point $p$ is located, $x_{ip}$ is the coordinate of material point $p$, and $\delta \left(x\right)$ is the Dirac Delta function, which satisfies

$$\underset{-\infty }{\overset{+\infty }{\int }}\delta \left(x\right)dx=1,\delta \left(x\right)=0\left(x\ne 0\right)$$

(6)

Substituting Equation $\left(5\right)$ into the virtual work equation of Equation $\left(4\right)$, we obtain the following summation form:

$$\sum _{p=1}^{n_p}{m}_p{\ddot{u}}_{ip}\delta u_{ip}+\sum _{p=1}^{n_p}{m}_p{\sigma }_{ijp}^s\delta u_{ip,j}-\sum _{p=1}^{n_p}{m}_p{b}_{ip}\delta u_{ip}-\sum _{p=1}^{n_p}{m}_p{\overline{t}}_{ip}^s{h}^{-1}\delta u_{ip}=0$$

(7)

The parameters of a material point, such as displacement $u_{ip}$, virtual displacement $\delta u_{ip}$, and velocity $\delta u_{ip,j}$, can all be obtained by interpolating the corresponding parameters at the background grid nodes $I$ using the interpolation shape functions $N_I\left(x_p\right)$. For example, any variable $r_p$ can be expressed as

$$r_p=\sum _{I=1}^{N_n}{r}_I{N}_I\left(x_p\right)$$

(8)

Substituting the interpolated displacements, virtual displacements, and velocities into the summation form of the virtual work Equation $\left(7\right)$, we obtain the motion equation for node $I$ as follows:

$$p_{iI}=f_{iI}^{int}+f_{iI}^{ext},x_I\notin {\Gamma }_N$$

(9)

where in the equation, $p_{iI}$ represents the momentum of grid node $I$ in the $i$-direction, and $f_{iI}^{int}$ and $f_{iI}^{ext}$ represent the internal and external forces at the grid node, respectively.

2.3. MPM Contact Model

In the standard MPM, a one-to-one mapping function is used between the grid and material points, which automatically satisfies the no-slip contact constraints between different phases of the object and prevents penetration. However, the standard MPM cannot address relative contact sliding and separation between objects. Therefore, many scholars have studied the contact algorithms of the MPM model and proposed corresponding improved methods. Bardenhagen et al. [14] were the first to establish a material point contact algorithm, considering frictional contact. This algorithm prevents interpenetration while permitting separation, sliding, and rolling friction. Node velocities are predicted, and if there is no contact, the contact model is used for correction. The following equation is used to determine whether two objects are in contact:

$$\left({\tilde{v}}_i^{n+1,k}-v_i^{cm}\right)·m_i^k>0$$

(10)

where $i$ is the contact point, $k$ is the normal vector of the object, $v_i^{cm}$ is the centroid velocity, and $m_i^k$ is the mass gradient of the object.

The Bardenhagen contact algorithm uses node-based contact detection and supports only Coulomb friction for frictional contact. Therefore, the resolution provided by the material points and the grid, as well as the MPM discretization method, can affect the accuracy and stability of contact detection. Hu and Chen [15] proposed a sliding/separation contact algorithm based on a multi-grid scheme, placing each material (as shown in Figure 2 with objects I and II) on a separate background grid. However, in their contact algorithm, contact friction is not considered, and the no-penetration condition based on grid nodes does not guarantee the same for different particles. To address this, Huang et al. [16] incorporated the no-penetration condition into the velocity calculation.

In early contact algorithms, the determination of whether two objects are in contact was based on the velocities at common nodes, which could lead to the “early contact” phenomenon (see Figure 3). To prevent contact from occurring before the actual physical contact on the surfaces of the objects, Ma et al. [17] introduced a penalty function to improve contact detection, proposed a geotechnical contact algorithm, and revised the velocity difference in Equation $\left(10\right)$ as follows:

$${\tilde{v}}_i^{n+1,k}-v_i^{cm}=f_p·\left({\tilde{v}}_i^{n+1,k}-v_i^{cm}\right)$$

(11)

$$f_p=1-{\left({\displaystyle \frac{min(∆x,d)}{∆x}}\right)}^{\gamma }$$

(12)

Figure 3. The early contact phenomenon [18].

Figure 3. The early contact phenomenon [18].

where $∆x$ is the size of the background cell; $\gamma$ is the coefficient that controls the smoothness of the smoothing function $f_p$; $d$ is the distance from the contact node to the object’s surface, approximated by traversing all the material points of the object from the contact node, which requires the refinement of both particles and grids for a satisfactory approximation. To address this, Hammerquist and Nairn [19] proposed a more accurate distance formula derived from the extrapolated distance $d^{ext}$:

$${\displaystyle \frac{d}{∆x}}=\left\{\begin{array}{cc}1-2{\left({\displaystyle \frac{-d^{ext}}{1.25∆x}}\right)}^{0.58}& d^{ext}\le 0\\ 2{\left({\displaystyle \frac{-d^{ext}}{1.25∆x}}\right)}^{0.58}-1& d^{ext}>0\end{array}\right.$$

(13)

$$d^{ext}=(x_j-x_i)·n_i$$

(14)

where $x_j$, $x_i$, and $n_i$ represent the extrapolated position of the material point from the contact node, the node position, and the surface normal, respectively. Building upon Equation $\left(13\right)$, Gao et al. [20] modified the smoothing function $f_p$ as follows:

$$f_p=\begin{array}{cc}{\displaystyle \frac{1-\alpha x^{\beta }}{1+\alpha x^{\beta }}}& \alpha ϵ[0,1]\end{array},\beta ϵ[2,+\infty )$$

(15)

where $x=\left|d/∆x\right|$, $\alpha$, and $\beta$ are parameters that adjust the smoothness of the function. Furthermore, Gao et al. compared the Bardenhagen contact algorithm with an improved version by simulating three benchmark scenarios: an elastic cylinder rolling down a slope, a wedge penetrating Mohr–Coulomb soil, and a footing settling on soft ground. They found that the proposed contact algorithm effectively reduced numerical oscillations and yielded more accurate simulation results, as depicted in Figure 4.

Despite the fact that the aforementioned grid-based contact algorithms have been improved and enhanced by many scholars due to their simplicity and efficiency, they inevitably have the following limitations. First, there is a tendency to encounter the “early contact” phenomenon, which is more prevalent in problems with large solution domains. Second, these methods typically rely on mass or volume gradients to estimate the contact normal, which can be inaccurate due to changes in particle position and distribution. Third, in quasi-static situations where velocity changes are negligible, accurately discerning contact is very challenging [21,22].

2.4. Relationship with Meshless Method

As an emerging meshless numerical method, the MPM is typically comparable with SPH, the oldest and most widely used method in the meshless family [23,24,25]. In this section, taking SPH as a representative of typical meshless methods, we explore the similarities and differences between the MPM and SPH.

The MPM and SPH both belong to the category of Lagrangian particle-based numerical methods. They characterize the computational domain using discrete particles, thereby avoiding the mesh distortion issues that can arise in traditional mesh-based methods when simulating large deformations. Both methods are suitable for the numerical simulation of complex physical processes involving large deformations, material fracture, and multiphase flows, and they hold complementary application value in the field of computational mechanics.

Despite similarities in their particle-based descriptions, the MPM and SPH exhibit significant differences in their specific implementations and applications. The MPM employs a “mesh-particle” hybrid strategy, relying on a temporary background mesh for spatial discretization and gradient calculations. Its mesh projection step effectively suppresses numerical oscillations, offering higher computational accuracy and stability when dealing with complex stress states and large deformation problems. It is particularly well-suited for multi-material coupling and solid mechanics issues. In contrast, SPH is a purely meshless method that solves problems entirely based on particle neighborhood interpolation. It has advantages in fluid simulation and parallel computing, especially in handling free-surface flows and wave breaking. However, it is prone to numerical instability and reduced accuracy in regions with uneven particle distribution and high gradients.

3. Application in High-Speed Impact

The structural damage process under high-speed collision often involves phenomena such as the generation and evolution of shock waves, the formation of debris clouds, and damage and destruction. The MPM has distinct advantages in handling fracture, fragmentation, and moving material interfaces, making it highly suitable for solving high-speed impact problems [26,27].

Zhang et al. [28] were the first to validate the feasibility of the MPM in simulating hypervelocity impacts, proposing the explicit material point FEM. The material domain is discretized by finite element grids, with regular background grids defined in areas prone to large deformations. Once within the predefined background grids, nodes are converted into material points, which are then solved by the MPM. They successfully simulated the impact of an elastic rod on a rigid wall, cylinder impact tests, and the impact of spherical projectiles on thin and thick plates. To improve computational efficiency, Lian et al. [29] made improvements by discretizing the material domain into blocks, simulating the dynamic response of a projectile impacting reinforced concrete, where the reinforcement bars were discretized into bar elements, and the concrete was discretized into material points (as shown in Figure 5). Furthermore, all steel reinforcement nodes and concrete particles move within the same single-valued velocity field. The particles at the nodes of the reinforcement elements carry only positional and velocity variables, while the reinforcement elements themselves bear the axial forces. By comparing with the literature, the combined advantages of the FEM and MPM were demonstrated.

To prevent numerical oscillations caused by material points passing through grid boundaries, Bardenhagen et al. [30] proposed the generalized interpolation MPM (GIMP) based on the Petrov–Galerkin method and successfully studied the dynamic response of a rod given an initial velocity impacting a rigid wall. Compared to the MPM simulation results, the solutions obtained by the GIMP are smoother and more continuous. Hu et al. [31] introduced the convected particle domain interpolation technique into the MPM, employing a particle–particle contact algorithm to handle more complex high-speed impact and multi-body contact elastoplastic fracture problems.

The SPH and the MPM are both meshless methods, with SPH already widely applied in the field of hypervelocity impacts [32,33,34]. To verify the MPM’s capability in handling such problems, Ma et al. [35] compared the accuracy and efficiency of the MPM and SPH through impact tube, Taylor rod, and debris cloud examples, indicating that SPH has difficulty in easily identifying debris boundaries, while the MPM scheme is more suitable for simulating the impact process at this stage. Culp et al. [36] successfully simulated the fragmentation issues of cylinder explosions, the hypervelocity impacts of copper projectiles on copper targets, and collisions between two steel plates. Zhang et al. [37] simulated the large deformation of 7055 aluminum alloy plates under hypervelocity impacts from aluminum projectiles. Wang et al. [38] investigated the morphology of the debris cloud, the size of perforations in a water-filled aluminum eggshell array structure, the residual velocity of the projectile, the impact energy, and the temperature field.

Molecular dynamics (MD) simulations can track and simulate the dynamic evolution of a system by observing changes in atomic positions, velocities, forces, and other computable parameters [39]. In simulating high-speed impact problems, MD can simulate the mechanical behavior of materials at the atomic scale, but its capabilities are very limited. Su et al. [40] conducted comparative predictions of the impact response in nano-porous solids using both MPM and MD approaches, demonstrating the feasibility of employing the MPM to study high-speed impact behaviors at the nanoscale. The further development of the MPM in conjunction with other numerical methods into a multi-scale approach bridges the gap between the atomic scale and the continuum scale, achieving a reasonable balance between computational efficiency and accuracy [41]. The MPM can simulate macroscopic phenomena but requires an appropriate equation of state (EOS). Therefore, Liu et al. [42] proposed a multi-scale framework combining MD and the MPM, using the EOS data obtained from MD solutions to provide parameters for the MPM. However, this method requires the calculation of thermodynamic properties under different temperature and volume conditions, significantly increasing the computational cost. Kim et al. [43] satisfy the Rankine–Hugoniot conditions by inspecting the state of the system before and after the impact, enabling the direct simulation of compression shock phenomena and directly obtaining thermodynamic properties after applying specific impact velocities.

In simulating the fragmentation phenomenon in high-speed impacts, grid-based methods may experience mesh distortion, which compromises solution accuracy. Although the MPM can evolve structural damage under high-speed collisions, it requires a large number of material points to ensure simulation accuracy, which, to some extent, limits the scale and efficiency of the computations [44]. Due to the extremely high propagation speed of shock waves, the time step must be reduced to ensure numerical stability, which further decreases computational efficiency. Moreover, the MPM may face challenges when multiple physical effects need to be considered, such as thermal effects and phase transitions.

4. Application in Explosion

Explosions involve multi-material coupling at different stages, extreme deformation, damage, and fragmentation, and strong nonlinearity, posing significant difficulties and challenges for numerical simulation methods. It is challenging to simulate the process of gas product dispersion from explosive detonation using grid-based methods. In contrast, the MPM is adept at capturing the physical phenomena produced during the explosion impact process and is a very reliable numerical method for simulating explosions.

Hu et al. [45] were the first to demonstrate that the MPM can effectively simulate the physical processes of explosive detonation and successfully applied it to the impact of concrete–steel, Sedov–Taylor blast waves, and piston–container and piston–gas problems. To avoid numerical fragmentation and numerical noise in the standard MPM, Ma et al. [46] proposed an adaptive MPM, where material points split into two when the cumulative strain in a certain direction exceeds a specified criterion (as shown in Figure 6). By simulating various explosive problems, such as explosively driven flyer plates and shaped charges, it was shown that MPM simulations can fit terminal velocities well and provide the entire deformation process under different conditions. To consider the lateral effects of planar sandwich structures, Lian et al. [47] proposed a new modified Gurney solution that takes into account the lateral effects of planar sandwich structures, successfully simulating the explosion-driven metal problem.

During explosions, interactions between fluids and solids are typically involved. To address this, Wang et al. [48] developed an MPM for simulating the detonation of explosives involving fluid–structure interaction (FSI). Chen et al. [49] applied the MPM to underwater explosion simulations, calculating the shock waves generated during the explosion and observing the diffusion process of the explosion. Li et al. [50] successfully modeled the fragmentation of plates and structural damage to ship hulls subjected to underwater contact explosions by implementing a staggered grid scheme to mitigate cell intersection noise and improve the precision of fluid simulations. The experimental findings were in concordance with the simulation results (see Figure 7).

In explosion simulations, the boundary conditions at the interfaces between solid and fluid regions are often very complex. To effectively handle the issues at the multi-material interfaces, Cui et al. [51] combined the advantages of the FDM and MPM to propose a coupled finite difference MPM (CFDMP). In the contact area, the MPM maps variables from the MPM background grid nodes to the FDM virtual points to provide boundary conditions for the FDM region; simultaneously, the FDM maps variables from the FDM cell center points to the MPM interface grid nodes to provide boundary conditions for the MPM region (see Figure 8). Li et al. [52] simulated the physical process of gas–solid flow in thermite explosions.

To study the entire process, from continuous rock deformation to discontinuous fracturing and the evolution of explosions, Yue et al. [53] proposed a coupling of the MPM and the continuum discontinuous element method (CDEM). The MPM was chosen to model the explosives, simulating the multi-physical phenomena involved in rock blasting, while the CDEM was used to model the rock material, simulating the dynamic fracturing evolution of rocks. The contact model employs a particle–surface/edge contact approach (see Figure 9), and contact is established by meeting the following requirements:

$$\left\{\begin{array}{c}{d}_{mn}\le R_m\\ d_{Bn}\le R_{BC}\\ d_{Cn}\le R_{BC}\end{array}\right.$$

(16)

While the MPM has significant advantages in simulating explosion problems involving large deformations and complex damage processes, it also has some disadvantages: (1) When dealing with high-speed flows, there is the issue of numerical dissipation. (2) The traditional MPM, due to the single-valued nature of the background grid velocity field, cannot accurately handle the dynamic propagation of visible cracks. (3) It is challenging to accurately describe the real-time coupling of chemical reaction kinetics and the mechanical response during the detonation process of explosives. (4) The EOS parameters of the detonation products are sensitive, but there is limited experimental calibration data.

5. Application in Dynamic Cracking

Crack propagation introduces discontinuities and local gradients. The MPM, not being constrained by the background mesh when simulating crack propagation, can conveniently simulate the dynamic fracturing process of objects and handle the significant topological changes that occur during the fracturing process.

Tan et al. [54], based on the traditional MPM, solved the problem of the energy release rate in dynamic fracture mechanics, using adaptive meshing and a full mass matrix to improve accuracy, and were the first to validate the value of the MPM in studying dynamic crack propagation. Chen et al. [55] further demonstrated the potential of the MPM in simulating dynamic fracture by comparing the FEM solutions and experimental results that simulate dynamic brittle failure. Zhang et al. [56] introduced the characteristic erosion method into the MPM to solve the dynamic fracture problems of brittle materials. Wolper et al. [57] simulated dynamic fracture under large elastoplastic deformation. Xiao et al. [58] analyzed the effects of frictional heating on crack propagation, crack branching, and the fracturing process. To avoid non-physical oscillations and numerical fragmentation, Daphalapurkar et al. [59] utilized a cohesive zone model in the GIMP method to simulate dynamic crack propagation and successfully applied it to the problems of crack propagation in both ductile and brittle materials.

Nairn [60] developed an improved MPM scheme (CRAMP) for simulating cracks with discontinuities in displacement and velocity. In this method, background grid nodes can have multiple velocity fields, allowing adjacent particles to move and deform independently to simulate discontinuities. However, this CRAMP scheme does not have the capability to simulate concurrent crack initiation and propagation nor the ability to propagate cracks to the material’s edge. To address this, Adibaskoro et al. [61] made improvements by introducing additional grids to simulate any number of crack paths (see Figure 10). They defined a ghost tip point for each active crack tip, located on the trajectory of the crack tip and at a distance equal to twice the defined propagation length, thus enabling the simulation of multiple crack initiations, extensions, and coalescences.

Due to the great flexibility of the peridynamics (PD) in simulating dynamic fractures [62,63], and the fact that both PD and the MPM are particle methods that simulate the dynamic processes of particles through explicit time integration, the combination of PD and the MPM is very natural. Lyu et al. [64] simulated plastic fracture. Zeng et al. [65] made improvements by modifying the cutoff radius $\delta$ to $\delta +l_e/2$, thereby eliminating the restrictions of the original PD due to volume integration, and proposed a volume-corrected PD combined with the MPM/GIMP scheme to simulate the evolution of fracturing under transient loading. The PD combined with the MPM scheme requires the pre-segmentation of the simulation subdomains. To overcome this disadvantage, Zeng et al. [56] proposed an adaptive peridynamics MPM, where the simulation region is initially discretized by MPM particles, and during crack growth, the MPM subregions will adaptively switch to PD subregions (see Figure 11), successfully simulating the spallation of a one-dimensional rod, mode I fracture propagation in a pre-cracked plate, mixed-mode crack initiation problems, and the Kalthoff–Winkler experiment.

In the process of dynamic fracture, the thin geometric shapes of small debris and new FSI interfaces are produced, making the reconstruction of FSI interfaces challenging. The immersed boundary method (IBM) has shown significant advantages in simulating FSI with complex boundary geometries [66,67,68]. Ni et al. [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, Kan and Zhang [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 the MPM with the phase-field method. Kakouris and Triantafyllou [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, Peng et al. [75] developed a phase-field MPM crack numerical program based on GPU acceleration. Zeng 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, Zhang 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.

Wang 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.

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.

6. Application in Impact Penetration

The MPM is not only capable of effectively simulating the damage process of target plate materials under impact but also accurately assessing the degree of material damage [79]. With the advantages of easy boundary condition application, no tensile instability issues, and high efficiency in handling large deformation problems, it is particularly well-suited for solving complex problems involving extreme deformations, such as impact penetration.

Sulsky and Schreyer [80] utilized the MPM to obtain numerical solutions for Taylor problems such as the impact penetration of a steel ball into a rigid wall, which matched experimental results and confirmed the significant advantage of the MPM in evolving complex damage behavior at high speeds. Ikkurthi [81] studied the impact penetration of steel shot at different impact velocities into thin boron carbide and Al-6061 plates. Burghardt et al. [82] used the GIMP to simulate the penetration of shaped charge jets into aluminum to enhance numerical stability. However, the inherent no-slip contact of the standard MPM cannot solve penetration problems. Huang et al. [16] proposed two novel contact algorithms and introduced them into the MPM. By simulating large deformation scenarios such as the impact penetration of steel balls, as depicted in Figure 15, it can be observed that the target shape parameter h/D approaches the experimental result with the decrease in the cell size and particle space. This demonstrated that the simulation results obtained with the proposed contact algorithms are closer to the experimental results than those obtained with the standard MPM.

To enhance the accuracy of solving penetration problems with smooth pressure and stress fields, Li et al. [83] proposed a B-spline MPM (cBSMPM) based on the Lagrange multiplier method, which employs B-splines to define the background grid, as shown in Figure 16.

The corresponding basis function for computational grid node $I$ is

$$N_{i,j}^{pq}\left(\xi ,\eta \right)=N_i^p\left(\xi \right)N_j^q\left(\eta \right)$$

(17)

where $N_i^p\left(\xi \right)$ is the B-spline basis function of polynomial degree $p$ for the open uniform knot vector $N={\xi }_0,{\xi }_1,\dots ,{\xi }_{n+p}$, and $N_j^q\left(\eta \right)$ is the B-spline basis function of polynomial degree $q$ for the open uniform knot vector $M={\eta }_0,{\eta }_1,\dots ,{\eta }_{n+q}$. Therefore, the higher the degree of the basis functions, the higher the solution accuracy, as shown in Figure 17. Li et al. demonstrated through the study of high-speed tungsten projectile penetration into thick plates and other impact penetration examples that increasing the degree of B-spline basis functions can improve the solution accuracy of smooth stress/pressure fields for impact and penetration problems.

For penetration problems involving rigid materials, only certain areas experience significant deformation, and the material deformation is minimal at the initial moment. To investigate the damage effects of a projectile on ballistic gelatin, Chen et al. [84] introduced a contact transition criterion in the adaptive finite element MPM to convert contacting gelatin finite elements into MPM particles, with the coupling between the steel ball and gelatin being achieved on the MPM background grid. They successfully simulated the physical process of a steel ball penetrating ballistic gelatin at a high speed. At the beginning of the calculation, both the steel ball and gelatin are finite element bodies (as shown in Figure 18). The results indicate that the deformation of the ball is limited, while a local area of the gelatin experiences significant deformation, and the remaining areas of the gelatin have limited deformation.

Although the MPM effectively overcomes some limitations of grid-based methods by combining material points with a background grid, it requires updating the background grid at every time step when dealing with penetration problems. This frequent updating of the background grid not only has the potential to become a bottleneck in the computational process but also can lead to the accumulation of numerical errors and a significant increase in the demand for computational resources. Moreover, for penetration problems involving thermomechanical and chemical coupling, the current MPM framework may still have interpolation errors in the transfer of multi-field data.

7. Application in Fluid–Structure Interaction

The FSI refers to the physical phenomena that occur when one or more solid structures interact with internal or surrounding fluid flow [85] (see Figure 19). Since the MPM algorithm can calculate fluids and solids within the same solution domain and has significant advantages in simulating large deformations and the fragmentation of fluids, many scholars have attempted to use it to address FSI problems.

York et al. [87] were the first to validate the feasibility of the MPM in FSI analysis. Hu et al. [88] introduced the adaptive mesh refinement technique into the MPM and successfully simulated complex dynamic FSI problems in aerospace applications, further confirming the capability of the MPM in handling flows with complex geometries, large deformations, and FSI issues. However, the MPM solver has high requirements for computer performance. To efficiently and accurately simulate large-scale FSI problems, Sun et al. [89] immersed MPM particles into the MMALE grid, achieving a strong coupling between the MMALE method and the MPM. They introduced a virtual velocity field near the FSI interface to update the stress of solid particles and proposed a local subdomain smoothing method to accelerate the remapping phase, which involves only the calculation of grid distortion regions. Finally, they validated the efficiency of this model in solving multiphase flow and FSI through examples. Li et al. [90] devised a hybrid data-driven framework supporting the popular and powerful MPM. They applied neural networks to the MPM, achieving numerical acceleration while maintaining physical accuracy. This approach enables the efficient and accurate simulations of fluid–solid interactions in a data-driven manner.

The multiphase MPM has been widely applied to simulate the interaction between porous soil and pore water, mainly based on two primary formulations. The first is the single-point formulation [91,92], where material points carry all the information of both soil and pore water (see Figure 20), and the positions of material points are updated with the movement of the solid phase. Yerro et al. [93] proposed a three-phase single-point MPM, treating soil as a medium integrated with solid, liquid, and gas phases. Wang et al. [94] enhanced the three-phase single-point MPM with a GPU acceleration framework and introduced cubic B-spline shape functions to improve numerical stability. To improve computational efficiency, Ceccato et al. [95] neglected the gas phase, considering liquid acceleration and volume fraction gradients. Wang et al. [96] developed a dynamic grid algorithm to optimize memory usage.

The second is the two-point formulation [98,99], which can use two sets of material points to represent the solid and liquid phases, respectively (see Figure 21), thus easily ensuring mass conservation for both phases and capturing their interactions. Yao et al. [100] were the first to extend the two-phase two-point MPM to the dynamic analysis of porous media, using two sets of material points to represent the deformation of solids and pore fluids. Bandara and Soga [101] developed a fully coupled MPM formulation that considers the relative acceleration between solids and fluids. To minimize numerical oscillations, Liu et al. [102] employed the GIMP for spatial discretization and developed a three-dimensional two-phase two-point MPM code, successfully simulating fully saturated seepage failure. Building on Liu’s work, Zhan et al. [103] developed a two-phase two-point MPM suitable for solving large deformations in both saturated and unsaturated soils. However, the presence of two sets of material points increases computational costs and requires the careful modeling of the interface, which limits the practical application of the MPM.

In FSI problems, it is common for the solid region to undergo small deformations while the fluid region experiences large deformations. Therefore, some scholars have coupled the FEM and MPM to solve FSI problems. For example, Lian et al. [104] proposed the coupled finite element MPM (CFEMP). When both MPM and FEM objects contribute to the same grid node, it indicates that the two objects are in contact at the grid node (see Figure 22). The feasibility of this method was verified by simulating the FSI model of a water column passing through an elastic barrier. The CFEMP contact algorithm requires consistent meshing between the FEM domain and the MPM domain, which can lead to excessive meshing in the FEM domain. To address this issue, Chen et al. [105] employed a particle-to-surface contact algorithm and introduced a Coulomb friction model. Cheon and Kim [106] used distributed interactive nodes on the finite element contact surface. Chen et al. [107] introduced the v-p formulation into the CFEMP algorithm to solve numerical oscillation and volume locking issues in FSI problems. However, the CFEMP method still faces some challenges. For instance, when the contact surface is severely deformed, contact penetration is likely to occur. When contact forces act on each pair of contacting bodies, it is difficult to satisfy the contact conditions for all contact pairs simultaneously. To this end, Song et al. [108] proposed an improved local search method, a set of contact forces that satisfy the contact conditions for each contact pair, and successfully simulated the impact of a wedge entering water.

To address the issues of a traditional MPM in solving FSI, such as the difficulty in imposing boundary conditions and low computational efficiency, Li et al. [109] proposed the immersed finite element MPM. In this method, the interaction between the MPM and FEM objects is handled using an immersed interface approach, with weighted tracking points set on the FEM object to efficiently track the FSI interface, as shown in Figure 23.

To avoid volumetric locking in the MPM, Lei et al. [110] introduced the $B^{¯}$ method into the MPM-FEM model, discussing the impact of an inclined seabed on seepage resistance, soil flow patterns, lateral response, stress distribution, and failure mechanisms. Considering that fluid compressibility can cause the reflection of sound waves at boundaries, Lei et al. [111] proposed an incompressible MPM and FEM coupling model to study the hydrodynamic characteristics of a triangular semi-submerged platform. This coupling scheme requires determining the specific locations of fluids and solids and classifying MPM background grid cells and solid finite elements, as shown in Figure 24. Evolving such complex FSI engineering typically requires significant computational costs, and there are no guarantees of accuracy.

Non-slip boundary conditions can be conveniently implemented on the fluid–solid interface using the lattice Boltzmann method (LBM), which can simulate fluid flow problems simply and efficiently. Therefore, the LBM has been successfully applied to multiphase flows [112,113], flow within porous media [114,115], and turbulence [116,117], among other issues. Based on this, Liu et al. [118] proposed a coupled lattice Boltzmann-MPM (LBMPM) model, in which the LBM simulates fluid flow and the MPM simulates the dynamic large deformation of solids. Liu et al. [119] expanded on this by using the LBM to handle binary fluid systems with large differences in density and viscosity, employing four sets of spatial discretization (see Figure 25), where Lagrangian marker points act as a bridge between multiphase flows and solids.

The coupling of the LBM and MPM can effectively capture the interactions between fluids and solids and even perform multiphase flow analysis at the pore scale. However, this coupled algorithm has low computational efficiency, especially in 3D cases.

Debris flows, such as landslides and mudslides, often involve large deformations and interactions between fluids, solids, and particles. Analyzing these mixed flows using only continuous medium methods or discontinuous methods is inappropriate. To address this, many scholars have developed MPM-DEM coupling methods for continuous–discontinuous mechanics problems involving fluid–solid coupling. The MPM is used to handle the large deformations of solid and liquid phases, while the DEM captures the motion of particles. Ren et al. [120] simulated landslides involving fluids, soil, and rocks by converting the MPM points into DEM particles and calculating the contact forces between material points and the DEM particles within a unified DEM framework, thereby determining the contact between the MPM objects and DEM particles, as shown in Figure 26. To improve computational efficiency, Li et al. [121] proposed a scheme to convert material points in contact with DEM particles into virtual DEM particles, modeling soil with fewer particles and larger particle diameters. Liang et al. [122] used a two-phase u-v-p MPM to solve the fluid–solid coupling problems in saturated porous media and proposed a semi-implicit integration scheme based on a splitting algorithm. Yu et al. [123] developed a three-phase MPM-DEM coupling method to simulate multiphase, multi-physical field, and multi-scale problems in frozen and thawed granular media, where the MPM solver uses a semi-implicit staggered integration scheme, allowing for larger time steps and providing better computational efficiency for the DEM. Li et al. [124] proposed a CPU-GPU parallel solution scheme.

Despite the development of various MPM models and their coupling schemes, which have achieved certain application results in FSI analysis, there are still some difficulties in practical applications: (1) The interpenetration and entanglement between solid and fluid regions make the boundary between the two vague, which poses a challenge in determining precise boundary conditions, especially when dealing with fluid–solid coupling seepage problems in porous media at the microscale. (2) In the process of fluid–solid coupling analysis, it is necessary to transfer key data, such as displacement, velocity, and pressure, between different grid systems. However, different interpolation techniques may lead to discontinuities in data at the coupling interface, thereby affecting the accuracy of the simulation results. (3) When dealing with large-scale FSI problems, the MPM requires tracking and updating the positions and velocities of a large number of material points at each time step, which inevitably increases the computational cost. (4) It is challenging to embed the constitutive equations of shear-thinning/shear-thickening fluids, such as blood and lava. The coupling simulation of these non-Newtonian fluids with solids is prone to large errors.

8. Conclusions and Outlook

The MPM is an efficient, stable, and easily implementable new approach that has received widespread attention in the field of computational mechanics since its inception. Through continuous research and development by scholars at home and abroad, the MPM has been widely applied in various fields and has yielded a multitude of research outcomes. This article provides a systematic review of the advancements in the MPM and its diverse coupling schemes as applied to scenarios like high-speed impact, explosion, dynamic cracking, impact penetration, and FSI.

Table 1. Summary of main variants and developmental history of MPM.

Author	Method	Key Innovations	Limitations
Sulsky et al. (1994) [13]	Standard MPM	Firstly extended the FLIP method to solid mechanics, establishing a coupling framework between material points and background grids.	Significant numerical dissipation. Incomplete contact algorithms.
Hu and Chen (2003) [15]	multi-mesh MPM	Assigning independent background grids to different materials to address material interface issues.	Neglecting friction; inability to ensure non-penetration conditions at the particle level.
Bardenhagen and Kober (2004) [30]	GIMP	Introducing generalized interpolation shape functions to reduce grid-crossing noise.	The computational cost is slightly higher than that of the standard MPM.
Zhang et al. (2006) [28]	explicit material point FEM	Automatically converting FEM nodes to MPM particles in highly deformable regions.	The treatment of the coupling interface is complex.
Ma et al. (2009) [46]	adaptive MPM	Automatically splitting material points based on strain thresholds to improve simulation accuracy.	Particle splitting increases computational load, necessitating the optimization of multiple local background grids.
Lian et al. (2011) [104]	CFEMP	Coupling finite element truss elements with the material point method on a unified background grid to enhance computational efficiency and accuracy.	Consistent meshing between the FEM domain and the MPM domain is required, which may lead to over-meshing in the FEM domain.
Liu et al. (2013) [42]	multi-scale framework combining MD and MPM	Driving MPM macroscopic simulations with EOS data calculated from MD.	A large amount of MD state equation data needs to be precomputed.
Cui et al. (2014) [51]	CFDMP	More precise treatment of explosive multi-material interfaces.	The introduction of virtual points at the interface leads to interpolation errors, which necessitate precise boundary conditions.
Kakouris and Triantafyllou (2017) [72]	phase-field MPM	When combined with the phase-field fracture model, explicit crack tracking is not required.	The computational workload is significantly increased.
Ni et al. (2020) [69]	immersed boundary MPM	Incorporating the IBM avoids the need for FSI interface reconstruction.	The introduction of virtual fluid incurs additional computational costs.
Liu et al. (2020) [118]	LBMPM	Efficient simulation of large-deformation fluid–solid interaction without mesh reconstruction is achieved.	The computational efficiency in 3D is low.
Chen et al. (2020) [84]	adaptive finite element material point method	Dynamically converting finite elements to material points effectively avoids mesh distortion in extreme deformation problems.	The threshold for conversion needs to be set empirically.
Zeng et al. (2022) [63]	adaptive peridynamics MPM	Dynamically switching between MPM and peridynamics regions enables multi-scale fracture simulation.	It requires a preset threshold for conversion.
Yue et al. (2022) [53]	coupling of MPM and CDEM	The particle–surface contact model accurately simulates the entire process of rock transitioning from continuous deformation to fracture during blasting.	The contact criterion requires iterating through all material points.
Kan and Zhang (2022) [71]	IALEMPM	The contact criterion requires iterating through all material points.	The algorithm implementation is complex.
Li et al. (2022) [109]	immersed finite element material point	Incorporating weighted tracking points avoids the need for FSI interface reconstruction.	The computational cost is high, and it is difficult to apply boundary conditions.
Li et al. (2024) [83]	cBSMPM	Using B-spline background grids yields a smooth stress field.	Higher-order basis functions increase the computational workload.
Wang et al. (2025) [78]	PF-FEMPM	Mitigate mesh-crossing instabilities.	The efficiency is lower than that of the pure MPM phase-field method.

summarizes the main variants and developmental history of MPM, while

Table 2. Advantages and disadvantages of MPM in large deformation applications.

Application	Advantages	Disadvantages
High-speed impact	Effectively captures the physical phenomena of shock wave generation, debris cloud formation, and material damage; possesses robust handling capabilities for moving material interfaces and multi-material coupling issues.	The computational efficiency in regions of small deformation is inferior to the finite element method, necessitating the use of coupling methods (such as the CFEMP) to enhance efficiency; it struggles to accurately capture very fine details such as intricate fragmentation structures and vortex flow features.
Explosions	Has good adaptability for problems involving high pressure, high-speed flow, and solid deformation caused by explosions; capable of simulating the significant deformation, phase changes, and flow induced by blasts.	In handling high-speed flows, numerical dissipation and grid-crossing errors may occur, necessitating improvements with the GIMP method.
Dynamic cracking	Not constrained by the background mesh; it conveniently handles the significant topological changes during crack propagation; it can be combined with phase-field methods to further enhance simulation accuracy.	The precision of stress at the crack tip is dependent on the density of material points, and additional relationships are required to link the initiation of cracks with their propagation speed and direction, which increases computational costs and the complexity of the simulation.
Penetration	Effectively simulates the damage process of target plate materials under penetration; boundary conditions are readily applied, with no issues of tensile instability, offering high computational efficiency.	Complex contact determination necessitates the optimization of multiple local background grids.
Fluid–structure interaction	Uniformly handles fluid–solid interfaces (such as the coupling of explosive gases with structures); it can also be coupled with other methods (such as FEM, LBM, DEM, etc.) to enhance the capability of dealing with complex fluid–solid coupling problems.	Multiphase coupling computations are resource-intensive; applying boundary conditions is challenging; contact penetration is likely to occur when solid regions undergo small deformations while fluid regions experience large deformations.

further outlines the strengths and weaknesses of the MPM in these large deformation application areas. Although the MPM has achieved significant development, future attention and in-depth research are still needed in the following areas:

(1)

Enhancing the computational efficiency of the MPM. Current developments in the MPM are primarily based on explicit time integration; although implicit integration schemes can improve the computational efficiency of the MPM in large deformation analysis, the computational cost can significantly increase when dealing with large-scale physical problems. Therefore, future research could focus on optimizing the distribution and quantity of material points, developing a three-dimensional MPM computational architecture with GPU parallel acceleration, and exploring intelligent MPM computation methods combined with neural networks.

(2)

Addressing the grid dependency of the MPM. Similarly to the FEM, the simulation results of the MPM are highly dependent on the background grid used, and in cases where strain localization phenomena are significant, pathological grid dependency may occur. Therefore, generating an appropriate grid is a time-consuming and challenging task, especially when dealing with problems of complex shapes and moving boundaries. It is recommended to introduce non-local grid regularization methods, dynamic grid methods, and overlapping grid methods to obtain objective and stable MPM solutions.

(3)

Refining boundary detection algorithms. In MPM computations, even when using a fixed background grid, grid nodes may not coincide perfectly with domain boundaries, which can lead to inaccurate boundary representation. Thus, the use of machine learning-assisted detection can be considered, such as employing CNNs to identify boundary features.

(4)

Expanding the MPM’s multi-field coupling capabilities. While enhancing MPM’s mechanical computation capabilities, it is essential to further expand the application scope of this method in multi-physical field problems, leveraging the MPM’s advantages in solving other physical fields such as heat transfer and electromagnetism. This will significantly promote the resolution of the complex engineering computation problems currently being faced.

(5)

In the field of fluid dynamics, there is a need to develop an efficient MPM coupling algorithm that can accurately solve problems involving highly compressible fluids, as well as to develop an MPM model capable of simulating high-viscosity non-Newtonian fluids.

In summary, the further exploration of the MPM in more domains and more in-depth research and applications is warranted. This article takes the application of the MPM in handling large deformation problems as an example to organize related research, aiming to continuously deepen the theoretical connotations of this method, provide more practical basis for researchers conducting material numerical studies, and offer valuable technical references for expanding the application range of this method.