Material Point Method for Impact and Explosion Problems

Introduction The successful DART mission has validated kinetic impact as the most effective method for deflecting or disrupting potentially hazardous asteroids [1]. The impact produced unexpectedly intense material ejection, expanding our understanding of asteroid response to hypervelocity collisions [2]. What insights might this violent fragmentation and ejecta reveal about Dimorphos's material strength and structural composition? Exploring these aspects before the upcoming Hera mission [3] can optimize mission design and interpretation.The artificial impact experiment conducted by the Hayabusa2 mission demonstrated the influence of geological structures on cratering events [4, 5]. Similarly, understanding how the material and structural composition of an asteroid affect kinetic impact efficiency is crucial for planetary defense. To capture the detailed structure of asteroids, here we employ the Material Point Method (MPM), which excels in handling contact and interface problems, for modeling and simulation, aiming to address these critical questions.Method The material point method (MPM) is a well-developed numerical technique in the mechanical industry, but remains underestimated potential in the field of planetary science. MPM discretizes the continuum body into Lagrangian material points and employs a rigidly attached Eulerian background grid at each time step, with the algorithmic flow illustrated in Fig. 1. By combining the strengths of both Lagrangian and Eulerian descriptions, MPM effectively eliminates numerical dissipation and expedites the search for neighbor points, resulting in an efficient, extendable, and robust shock-physics code. This feature makes MPM particularly well-suited for simulating complex geometries and contact interactions, which are common in planetary science applications. To accurately capture the dynamical behavior of materials under hypervelocity impact, a modified pressure-dependent and damage-dependent Lundborg smooth yield model is adopted for the strength model, coupled with a quantitative plasticity correction. The Tillotson equation of state (EOS) is utilized in conjunction with a sound speed modification. Besides, a resolution-independent Grady-Kipp damage model is employed to simulate the fracture process. In the future, we plan to further develop the p-alpha equation to characterize the pore compaction effect.These proposed algorithms and material models have undergone extensive benchmarking and validation, spanning a wide range of parameter spaces, demonstrating their capability to describe the potential dynamic behavior of small celestial body materials [6]. This comprehensive approach, combining state-of-the-art numerical methods and material models, paves the way for a deeper understanding of the dynamic properties of small celestial bodies and their constituent materials.Simulation and Result Constrained by computational resources, we designed simulation scenarios based on China's upcoming kinetic impact mission [7]. A cubic spacecraft (1.5 m in size, 600 kg in mass) impacts a spherical target body (35 m in diameter) at a velocity of 10 km/s. Given the unknown material and structure of the target, we investigated the dynamic responses of small bodies with various structures (monolithic or rubble pile), geometries (spherical or ellipsoidal), and material compositions (rocky or regolith, solid or porous). Except for the solid rocky material with a density of 2700 kg/m3, all other target bodies have equal mass and a bulk density of 1200 kg/m3. Classic basaltic material parameters were used for all materials, except for the regolith, which has lower strength. The ejecta, damage, and velocity distributions after 0.5 s of impact are visually presented in Fig. 2. We further analyzed the momentum enhancement factor and the mass and velocity distributions of the fragments for each case.Discussion and Conclusion The dynamic responses to hypervelocity impacts exhibit significant sensitivity to the target's material and structure, as evidenced by the ejecta intensity, ejection angles, fragment morphology, velocity, and angular velocity. Investigating the relationship between target structure and impact response is crucial for mission design and ejecta avoidance. Moreover, it helps unravel the physical properties of the target body from observational results. This study focused on perpendicular impacts and analyzed the results during the impact fragmentation phase. Future research may delve into the orbital dynamics of the fragments, evaluate the effectiveness of impact defense in mitigating Earth-threatening hazards, and explore the potential of oblique impacts to disrupt the target body through spin-up. These findings may also extend to impacts between asteroids and shed light on phenomena such as binary formation, which contributes to our understanding of the origin and evolution of small celestial bodies in the solar system.Acknowledgment X.Y. and J.L. acknowledge support from the National Natural Science Foundation of China under Grant 12372047. X.Y. acknowledges support from the National Natural Science Foundation of China under Grant 62227901. P.M. acknowledges support from the French space agency CNES and from the French National Centre for Scientific Research (CNRS) through the exploratory research program of the Mission for Transversal and Interdisciplinary Initiatives.

ABSTRACTThe material point method (MPM) aims to avoid mesh distortion problems occurring in the commonly used finite element method (FEM). To achieve this, the continuum is discretized as a set of material points, while the solution of the weak form of the underlying partial differential equations is conducted on an Eulerian background mesh. As this process works similar to FEM, the MPM also displays volumetric locking for near‐incompressible materials. However, this also means that mitigation techniques developed for FEM can be adapted to the MPM. The main difficulty for this process lies in the higher‐continuity basis functions, which are required in MPM to achieve reasonable results. This study first reviews the applicability of different locking mitigation techniques in MPM. Then, a framework for a locking‐free implicit MPM is presented and validated using common tests investigating the locking behavior and stress oscillations. Finally, the presented methods are applied to a viscoelastic material model representing unvulcanized rubber to show the potential of MPM for tire molding simulations.

Large scale parallelisation of the material point method with multiple GPUs

Assessing and improving strong-shock accuracy in the material point method

The material point method (MPM) represents an alternative discretization method for numerical simulations. It aims to combine the benefits of a Lagrangian representation of bodies and an Eulerian numerical solution approach. Therefore, especially at high material deformations the method is not prone to mesh distortions such as the finite element method (FEM). For this reason, the MPM is used to a great extent for modeling granular materials as in geo-mechanics. However, high deformations occur in many industrial processes on metallic materials. The Split-Hopkinson-Pressure-Bar (SHPB) experiment is used to characterize material properties at high deformation rates. Although widely used, this experiment is not yet standardized and shows a variety of sensitivities, e.g. to friction. Inter alia for this reason, simulations are conducted with the experiment to allow for a better evaluation of the measured data. The purpose of this work from an engineering point of view is to analyze the performance of the MPM on an SHPB experiment. In order to validate the experimental results for the material characterization under dynamic loading conditions we introduce frictional contact. We use arbitrary tri-linear brick domains in a 3D CPDI1 scheme, instead of originally used parallelepipeds. This allows for a more flexible geometry approximation using standard meshes. The results of the method are analyzed with respect to discretization sensitivity and discussed in the context of the experimental results for a 42CrMo4 steel. We were able to show that the method is capable to reproduce the SHPB experiment. Additionally the method shows convergency in the results with finer discretizations. Thus, the MPM has underlined its importance as an alternative simulation technique for problems with high deformation.

GeoTaichi: A Taichi-powered high-performance numerical simulator for multiscale geophysical problems

A generalized Hellinger-Reissner variational principle and its PFEM formulation for dynamic analysis of saturated porous media

Finite element-smoothed particle hydrodynamics adaptive method in simulating debris cloud

In this study, a material point method (MPM) framework is presented to investigate hydroelastic vibration characteristics of elastic structures submerged into fluid. To analyze the coupled fluid-structure problem, MPM discretization is formed in such a way that the structure is assumed to be linearly elastic material while the surrounding fluid is assumed to be weakly compressible fluid utilizing an artificial equation of state. The dynamic analysis is conducted by applying an impulsive force to the structure in time domain. Then, hydrodynamic characteristics, i.e. natural frequencies, are determined by applying Fast Fourier Transform (FFT). The developed MPM framework is utilized to investigate the hydroelastic vibration analysis of a cantilever beam fully submerged into fluid.

Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems

In order to study the fragmentation of projectile and ejection of debris clouds caused by hypervelocity impacting mesh bumper, simulation of aluminum sphere projectile hypervelocity normal impacting aluminum mesh bumper was practiced with SPH arithmetic of LS-DYNA soft. The diameter of projectile was 4mm. Impact velocities of aluminum spheres were varied between 2.2km/s and 6.2km/s. The impact angle was 0°. The relationship between the debris clouds characteristic of projectile and the impact position on aluminum mesh bumper was studied. The effect on fragmentation of projectile from different combination mode of aluminum mesh bumper was analyzed. The results showed that the morphologies of the debris cloud varied with the impact position when a projectile impacted the mesh bumper. The debris clouds as palpus was found, and some local kinetic energy concentrated appeared in the debris clouds. Debris clouds distribution was more uniform when projectile impacted wire across point on the mesh bumper. Debris clouds had more diffuse area and less residual kinetic energy when mesh bumper was combined with interleaving mode. Mesh bumper combined with interleaving mode was helpful in enhancing the protection performance of shields.

This work developed a method for calculating the Jones-Wilkins-Lee (JWL) equation of state (EOS) for the detonation products of aluminized explosives based on hexanitrohexaazaisowurtzitane (CL-20). On the basis of previous studies, chemical reaction equations, Hess’ law and Avogadro’s law were all incorporated in this process because of the non-ideal detonations of aluminized explosives. The EOS parameters were determined for explosive formulations comprising pure CL-20 as well as aluminized versions and compared to literature reports. Calculated pressure-specific volume curves were in good agreement with prior studies. A 50 mm cylinder test using a CL-20/Al/wax composition with a 75/20/5 mass ratio was conducted and the calculated EOS parameters were used to simulate the expansion of the cylinder wall. The experimental and simulation plots were found to agree within 5%, which confirms the validity of this method with regard to engineering analyses.

Shock initiation performance of NTO-based polymer bonded explosive

Many equations of state (EOS) for detonation products have been proposed and used. Some of them are in analytical form and some in tabular form. The most popular is the Jones-Wilkins-Lee (JWL) EOS. One of the main parameters of a product's EOS is the so-called adiabatic gamma along its main isentrope (γs). For JWL EOSs γs(V) varies in a nonmonotonic way. Going down from the CJ point along the main isentrope, it first increases to create a hump, and then, as V goes to infinity, gamma decreases to perfect gas-like behavior with gamma around 1.3. But according to Davis [1], γs(V) should decrease monotonically with V. Accordingly, in this article we investigate the following: (1) Is the hump in γs(V) necessary? and (2) Is it possible to construct a product's EOS with a monotonic γs(V) that is consistent with experimental data? We find that (1) it is possible to construct a product's EOS without a hump in γs(V); and (2) without a hump in γs(V) there are not enough degrees of freedom to reproduce cylinder test data.

An arbitrary Lagrangian-Eulerian (ALE) approach is used to model the orthogonal metal cutting in a steady state situation. The thermomechanical model includes the effects of elasticity, plasticity, strain rate, large strains and friction with heat generated between the tool and the chip. The ALE formulation can combine the advantages of both the Eulerian and Lagrangian approaches in a single description. Particularly, problems linked to the free surface in a Eulerian description and those linked to severe mesh distortions in a Lagrangian one can be solved by this formulation. The ALE governing equations are briefly reviewed in this paper; finite element and finite volume methods are used for the discretization of the conservation equations and an explicit time integration is adopted. Only the steady state solution is required; the ALE formulation is exploited to update the free and the contact surfaces. The model predicts the thermomechanical quantities, the chip geometry and the cutting forces from the cutting data and the material and friction parameters. Cutting experiments were performed with 42CD4 steel and comparisons of experimental tool forces and chip geometry with the numerical results are presented.