Contact algorithms for the material point method in impact and penetration simulation

The inherent no‐slip contact constraint in the standard material point method (MPM) creates a greater penetration resistance. Therefore, the standard MPM was not able to treat the problems involving impact and penetration very well. To overcome these deficiencies, two contact methods for MPM are presented and implemented in our 3D explicit MPM code, MPM3D. In MPM, the impenetrability condition may not satisfied on the redefined regular grid at the beginning of each time step, even if it has been imposed on the deformed grid at the end of last time step. The impenetrability condition between bodies is only imposed on the deformed grid in the first contact method, while it is imposed both on the deformed grid and redefined regular grid in the second contact method. Furthermore, three methods are proposed for impact and penetration simulation to determine the surface normal vectors that satisfy the collinearity conditions at the contact surface. The contact algorithms are verified by modeling the collision of two elastic rings and sphere rolling problems, and then applied to the simulation of penetration of steel ball and perforation of thick plate with a particle failure model. In the simulation of elastic ring collision, the first contact algorithm introduces significant disturbance into the total energy, but the second contact algorithm can obtain the stable solution by using much larger time step. It seems that both contact algorithms give good results for other problems, such as the sphere rolling and the projectile penetration. Copyright © 2010 John Wiley & Sons, Ltd.     

A single energy-based parameter to assess protection capability of debris shields

Protecting spacecraft structures against hypervelocity impacts (HVIs) of space debris, which may cause fatal damage to the spacecraft structures, has received wide attention. In this paper, the numerical simulation of hypervelocity solid–solid impacts is conducted and an energy-based parameter to assess protection capability of debris shields is proposed. To numerically simulate the hypervelocity impact phenomena, a two-dimensional improved smoothed particle hydrodynamics (SPH) method with new particle generation and particle merger techniques is used. The spatial and temporal distributions of the kinetic energy flux density of a debris cloud at the position of the upper surface of a pressure wall are calculated, and the correlation between the kinetic energy of the debris cloud and the deformation and fracture behavior of the pressure wall is discussed. Finally, based on the maximum value of the total kinetic energy of debris cloud per unit area at the position of the upper surface of a pressure wall, an energy-based parameter to assess protection capability of debris shields is proposed.     

Analysis and reduction of quadrature errors in the material point method (MPM)

The material point method (MPM) has demonstrated itself as a computationally effective particle method for solving solid mechanics problems involving large deformations and/or fragmentation of structures, which are sometimes problematic for finite element methods (FEMs). However, similar to most methods that employ mixed Lagrangian (particle) and Eulerian strategies, analysis of the method is not straightforward. The lack of an analysis framework for MPM, as is found in FEMs, makes it challenging to explain anomalies found in its employment and makes it difficult to propose methodology improvements with predictable outcomes. In this paper we present an analysis of the quadrature errors found in the computation of (material) internal force in MPM and use this analysis to direct proposed improvements. In particular, we demonstrate that lack of regularity in the grid functions used for representing the solution to the equations of motion can hamper spatial convergence of the method. We propose the use of a quadratic B‐spline basis for representing solutions on the grid, and we demonstrate computationally and explain theoretically why such a small change can have a significant impact on the reduction in the internal force quadrature error (and corresponding ‘grid crossing error’) often experienced when using MPM. Copyright © 2008 John Wiley & Sons, Ltd.     

SPH and FEM Investigation of Hydrodynamic Impact Problems

Simulation of hydrodynamic impact problems and its effect on surrounding structures, can be considered as a fluid structure coupling problem. The application is mainly used in automotive and aerospace engineering and also in civil engineering. Classical FEM and Finite Volume methods were the main formulations used by engineers to solve these problems. For the last decades, new formulations have been developed for fluid structure coupling applications using mesh free methods as SPH method, (Smooth Particle Hydrodynamic) and DEM (Discrete Element Method). Up to these days very little has been done to compare different methods and assess which one would be more suitable. In this paper the mathematical and numerical implementation of the FEM and SPH formulations for hydrodynamic problem are described. From different simulations, it has been observed that for the SPH method to provide similar results as FEM Lagrangian formulations, the SPH meshing, or SPH particle spacing needs to be finer than FEM mesh. To validate the statement, we perform a simulation of a hydrodynamic impact on an elasto-plastic plate structure. For this simple, the particle spacing of SPH method needs to be at least two times finer than FEM mesh. A contact algorithm is performed at the fluid structure interface for both SPH and FEM formulations. In the paper the efficiency and usefulness of two methods, often used in numerical simulations, are compared.     

Enhancement of the material point method using B‐spline basis functions

The material point method (MPM) enhanced with B‐spline basis functions, referred to as B‐spline MPM (BSMPM), is developed and demonstrated using representative quasi‐static and dynamic example problems. Smooth B‐spline basis functions could significantly reduce the cell‐crossing error as known for the original MPM. A Gauss quadrature scheme is designed and shown to be able to diminish the quadrature error in the BSMPM analysis of large‐deformation problems for the improved accuracy and convergence, especially with the quadratic B‐splines. Moreover, the increase in the order of the B‐spline basis function is also found to be an effective way to reduce the quadrature error and to improve accuracy and convergence. For plate impact examples, it is demonstrated that the BSMPM outperforms the generalized interpolation material point (GIMP) and convected particle domain interpolation (CPDI) methods in term of the accuracy of representing stress waves. Thus, the BSMPM could become a promising alternative to the MPM, GIMP, and CPDI in solving certain types of transient problems.     

Axisymmetric form of the generalized interpolation material point method

This paper reformulates the axisymmetric form of the material point method (MPM) using generalized interpolation material point (GIMP) methods. The reformulation led to a need for new shape functions and gradients specific for axisymmetry that were not available before. The new shape functions differ most from planar shape functions near the origin where r = 0. A second purpose for this paper was to evaluate the consequences of axisymmetry on a variety MPM extensions that have been developed since the original work on axisymmetric MPM. These extensions included convected particle domain integration (CPDI), traction boundary conditions, explicit cracks, multimaterial mode MPM for contact, thermal conduction, and solvent diffusion. Some examples show that the axisymmetric shape functions work well and are especially crucial near the origin. One real‐world example is given for modeling a cylinder‐penetration problem. Finally, a check list for software development describes all tasks needed to convert 2D planar or 3D codes to include an option for axisymmetric MPM. Copyright © 2014 John Wiley & Sons, Ltd.     

An evaluation of the MPM for simulating dynamic failure with damage diffusion

For dynamic brittle failure, conventional mesh-based methods, such as the finite element method and finite difference method, are handicapped when localized large deformations and subsequent transitions from continuous to discontinuous failure modes occur. To evaluate the potential of the material point method (MPM) in simulating dynamic brittle failure involving different failure modes, the essential features of the MPM are explored for wave and impact problems, and combined wave and diffusion problems are then solved by using the MPM. Through the comparison with the experimental, analytical and numerical data available, it appears that the MPM is a robust tool to simulate multi-physics problems such as dynamic failure under impact.     

基于SPH方法的油箱抗冲击有限元分析

目的验证SPH方法在流固耦合问题上的适用性。方法采用SPH方法(光滑粒子流体动力学方法)模拟无人机油箱抗冲击过程,并与采用CEL方法(耦合欧拉-拉格朗日方法)模拟得到的结果进行对比。结果 2种模拟方法得到的燃油晃荡情况、油箱应力分布及地面支反力结果接近。结论 SPH方法在流固耦合问题上是适用的。     

空间碎片高速撞击的数值模拟方法评述

计算机数值模拟是实现空间碎片撞击效应地面模拟的重要手段之一。撞击速度增加,撞击的物理机制和效应将发生改变,计算机数值模拟方法也应随之丰富和全面。介绍了基于有网格和无网格方法的高速撞击数值模拟发展历程,并针对数值模拟中常用的有限元法和SPH法进行了分析比较,阐述了高速撞击计算机模拟中无网格法的计算优势,并提出量子力学在未来无网格法数值模拟中的可能应用。为空间碎片高速撞击更加真实可靠的数值模拟提供参考。     

Coupling of membrane element with material point method for fluid–membrane interaction problems

This paper proposes a coupled particle–finite element method for fluid–membrane structure interaction problems. The material point method (MPM) is employed to model the fluid flow and the membrane element is used to model the membrane structure. The interaction between the fluid and the membrane structure is handled by a contact method, which is implemented on an Eulerian background grid. Several numerical examples, including membrane sphere interaction, water sphere impact and gas expansion problems, are studied to validate the proposed method. The numerical results show that the proposed method offers advantages of both MPM and finite element method, and it can be used to simulate fluid–membrane interaction problems.     

Numerical simulation of explosively driven metal by material point method

The material point method (MPM) fully takes the advantages of both Lagrangian method and Eulerian method, and can be capable of simulating high explosive explosion problems and impact problems involving large deformation and multi-material interaction of different phases. In this paper, MPM is extended to simulate the explosively driven metal problems, and two typical explosive/metal configurations, open-faced sandwich and flat sandwich, are analyzed in detail using MPM, and numerical results are compared with Gurney solution and its corrections. Based on our MPM results, a new correction to Gurney solution is proposed to account for the lateral effects for flat sandwich configuration. MPM provides a powerful tool for studying the explosively driven metal and other explosive problems.     

SPH evaluation of out-of-plane peak force transmitted during a hypervelocity impact

Smooth particle hydrodynamics (SPH) technique is applied to simulate a hypervelocity impact of an aluminum sphere on a simple aluminum target and on a honeycomb structure sandwich panel in order to provide a useful input to Finite Element Model or Statistical Energy Model on evaluating the vibration environment induced by the projectile on the target. The impact velocity range lays between 4 and 5 km/s. Different approaches have been analyzed. At first, the application of SPH technique for direct calculation of the vibration environment is described. Then the calculation of equivalent force impulse is evaluated. Two strategies have been applied: shear stress analysis and momentum calculation using drift velocity measurement. Momentum approach revealed to be the most convenient and reliable method. Results for an aluminum plate and a honeycomb sandwich panel are reported and compared to results of experiments and Finite Element Analysis.     

A precise critical time step formula for the explicit material point method

The material point method (MPM) combines Eulerian method and Lagrangian method and thus both Lagrangian particle position and interaction between neighboring Eulerian grid cells will affect the simulation stability. However, the original critical time step formula in the standard MPM does not reflect the effect of particle position and neighboring cell interaction on stability and overestimates the critical time step so much that the CFL number has to be very small, even smaller than 0.1, to obtain a stable solution at extreme particle positions. Therefore, in many engineering applications, the standard MPM is very expensive due to the small CFL number. In this article, the effect of particle position and neighboring cell interaction on stability of the explicit MPM is studied. An explicit critical time step formula is obtained based on the system eigenvalues in one dimension, and is then extended to two and three dimensions. For extreme deformation problems, the geometric stiffness matrix is taken into consideration which modifies the sound speed of particles in the critical time step formula. Several tests are performed to verify our formula and show a decrease in amount of time steps used for simulation with our formula comparing with the original formula.     

Numerical simulation of explosive welding using the material point method

Explosive welding involves detonation of explosive, interactions of fluid-structure and plastic deformations of metal plates at the instant of the explosion. Conventional mesh-based methods such as the finite element method (FEM) and finite difference method (FDM), are limited in simulation of the explosive welding when mesh distortion and interaction of different materials occur. In order to describe process of the explosive welding and accurately predict physical parameters for the explosive welding, numerical simulation of the explosive welding which involves multi-physical phenomenon is performed by using material point method (MPM). Not only can major physical phenomena of explosion impact be well captured, but also some important technical parameters for the explosive welding can be attained based on the MPM simulation. Through the comparison with the experimental results, it is shown that the MPM is a robust tool in simulation of the explosive welding.     

Hypervelocity impact on CFRP: Testing,material modelling,and numerical simulation

This paper describes the derivation and validation of a numerical material model that predicts the highly dynamic behaviour of CFRP (carbon fibre reinforced plastic) under hypervelocity impact. CFRP is widely used in satellites as face sheet material in CFRP-Al/HC sandwich structures (HC = honeycomb) that can be exposed to space debris. A review of CFRP-Al/HC structures typically used in space was performed. Based on this review, a representative structure in terms of materials and geometry was selected for study in the work described here. An experimental procedure for the characterisation of composite materials is documented by Riedel et al. [ADAMMO – advanced material damage models for numerical simulation codes. ESA CR(P) 4397, EMI report I 75/03, Freiburg; October 31, 2003.]. The test results from the CFRP of the current study allow for the derivation of an experimentally based orthotropic continuum material model data set that is capable of predicting the mechanical behaviour of CFRP under hypervelocity impact. Such a data set was not previously available. In the work by Riedel et al. [Hypervelocity impact damage prediction in composites: part II – experimental investigations and simulations. International Journal of Impact Engineering, 2006;33:670–80.] an orthotropic material data set was used for modelling HVI on AFRP (aramid fibre reinforced plastic), which shows relatively high deformability before failure. The enhancements of the modelling approaches in previous studies [Riedel W, Harwick W, White DM, Clegg RA. ADAMMO – advanced material damage models for numerical simulation codes. ESA CR(P) 4397, EMI report I 75/03, Freiburg; October 31, 2003. Hiermaier S, Riedel W, Hayhurst C, Clegg RA, Wentzel C. AMMHIS – advanced material models for hypervelocity impact simulations. Final report, EMI report E 43/98, ESA CR(P) 4305, Freiburg; July 30, 1999.] necessary to model brittle CFRP are specified. An experimental hypervelocity impact campaign was performed at two different two-stage light gas guns which encompassed both normal and oblique impacts for a range of impact velocities and projectile diameters. Validation of the numerical model is provided through comparison with the experimental results. For that purpose measurements of the visible damage of the face sheets and of the HC core are conducted. In addition, the numerically predicted damage within the CFRP is compared to the delamination areas found in ultrasonic scans.     

圆柱形长杆超高速正碰撞薄板结构破碎效应

超高速碰撞多层板结构破碎效应研究对空间碎片防护及动能武器毁伤效应研究有着重要意义。采用ANSYS/AUTODYN程序的SPH方法,对超高速碰撞碎片云的形成过程进行了数值模拟,某典型时刻一次及二次碎片云形貌的数值模拟结果与实验结果吻合较好,验证了计算方法和模型参数的正确性。在此基础上采用数值模拟方法,对钨合金、轧制均质装甲(Rolled Homogeneous Armor,RHA)及LY12铝三种材料的圆柱形弹体超高速碰撞薄板的破碎规律进行了研究,基于量纲分析方法得出了弹体破碎长度随弹靶材料特性、弹靶尺寸及初始撞击速度变化的关系式。并研究了钨合金及RHA两种材料的长杆弹对八层RHA板结构的超高速碰撞效应。     

Hybrid particle methods in frictionless impact‐contact problems

The dual particle dynamic (DPD) methods which employ two sets of particles have been demonstrated to have better accuracy and stability than the co‐locational particle methods, such as the smooth particle hydrodynamics (SPH). The hybrid particle method (HPM) is an extension of the DPD method. Besides the advantages of the DPD method, the HPM possesses features which better facilitate the simulation of large deformations. This paper presents the continued development of the HPM for the numerical solution of two‐dimensional frictionless contact problems. The interface contact force algorithm which employs a modified kinematic constraints method is used to determine the contact tractions. In this method, both the impenetrability condition and the traction condition are simultaneously enforced. In the original kinematic constraints method, only the former condition is satisfied. A new formulation to find stress derivatives at stress‐free corners by imposing stress‐free boundary conditions is also developed. The results for 1‐D and 2‐D contact problems indicate good accuracy for the contact formulation as well as the corner treatment when compared to analytical solutions and explicit finite element results using the commercial code LS‐DYNA. Copyright © 2004 John Wiley & Sons, Ltd.     

Modeling soft granular materials

Soft-grain materials such as clays and other colloidal pastes share the common feature of being composed of grains that can undergo large deformations without rupture. For the simulation of such materials, we present two alternative methods: (1) an implicit formulation of the material point method (MPM), in which each grain is discretized as a collection of material points, and (2) the bonded particle model (BPM), in which each soft grain is modeled as an aggregate of rigid particles using the contact dynamics method. In the MPM, a linear elastic behavior is used for the grains. In order to allow the aggregates in the BPM to deform without breaking, we use long-range center-to-center attraction forces between the primary particles belonging to each grain together with steric repulsion at their contact points. We show that these interactions lead to a plastic behavior of the grains. Using both methods, we analyze the uniaxial compaction of 2D soft granular packings. This process is nonlinear and involves both grain rearrangements and large deformations. High packing fractions beyond the jamming state are reached as a result of grain shape change for both methods. We discuss the stress-strain and volume change behavior as well as the evolution of the connectivity of the grains. Similar textures are observed at large deformations although the BPM requires higher stress than the MPM to reach the same level of packing fraction.     

Analysis and simulation of hypervelocity gouging impacts for a high speed sled test

An analysis and simulation of the gouging impact phenomenon which occurs at the Holloman Air Force Base High Speed Test Track (HHSTT) during hypervelocity impact testing is presented. Simulations of the sled/rail interactions were conducted using the hydrocode, CTH. These simulations utilize the most accurate and validated material models for the sled shoes (VascoMax 300) and rail (1080 steel) – which were recently developed. Sled shoe impacts with the rail were evaluated using various geometries possible in the field. The conditions leading to hypervelocity gouging were identified, as well as the condition which resulted in rail wear. The CTH simulations match results observed in the field extremely well. Recommendations are made, based on the latest material models and simulations, which should significantly reduce the occurrences of hypervelocity gouging at the HHSTT.     

A study of hypervelocity impact on cryogenic materials

This paper reports a result of hypervelocity impact experiments on cryogenically cooled aluminum alloys and a composite material. Experiments are carried out on a target palate at 122 K. Aluminum spheres at 1.95 km/s in 50 kPa air were impinged against the target plate at cryogenic temperature and the result was compared with room temperature target plates. Hypervelocity impact (HVI) processes were visualized with shadowgraph arrangement and recorded with high-speed video camera and to ensure the temperature dependence we compared HVI tests with metal target plates with AUTODYN 2D and SPH numerical simulations. We found that cryogenic impacts created slight differences of impact damage from room temperature ones, i.e., the shape and averaged diameters of HVI crater holes were less at cryogenic impacts.