Contents
Dedication i
About the Authors vii
Preface ix
1. Introduction
1.1 Lagrangian Methods 1
1.2 Eulerian Methods 3
1.3 Hybrid Methods 4
1.3.1 Arbitrary EulerianLagrangian Method and Its Variations 4
1.3.2 Particle-In-Cell Method and Its Variations 5
1.3.3 Material Point Method 6
1.4 Meshfree Methods 7
2. Governing Equations
2.1 Description of Motion 11
2.2 Deformation Gradient 14
2.3 Rate of Deformation 16
2.4 Cauchy Stress 17
2.5 Jaumann Stress Rate 19
2.6 Updated Lagrangian Formulation 20
2.6.1 Reynolds Transport Theorem 20
2.6.2 Conservation of Mass 21
2.6.3 Conservation of Linear Momentum 22
2.6.4 Conservation of Energy 23
2.6.5 Governing Equations 24
2.7 Weak Form of the Updated Lagrangian Formulation 25
2.8 Shock Wave 27
2.8.1 RankineHugoniot Equations 27
2.8.2 Artificial Bulk Viscosity 29
2.9 Detonation Wave 32
2.9.1 CJ Detonation Model 32
2.9.2 ZND Detonation Model 35
iii
iv Contents
3. The Material Point Method
3.1 Material Point Discretization 37
3.1.1 Lagrangian Phase 38
3.1.2 Convective Phase 41
3.2 Explicit Material Point Method 42
3.2.1 Explicit Time Integration 43
3.2.2 Explicit MPM Scheme 47
3.2.3 Qualitative Demonstration 53
3.2.4 Comparison Between MPM and FEM 56
3.3 Contact Method 59
3.3.1 Boundary Conditions at Contact Surface 60
3.3.2 Contact Detection 62
3.3.3 Contact Force 64
3.3.4 Numerical Algorithm for Contact Method 67
3.4 Generalized Interpolation MPM and Other Improvements 68
3.4.1 Contiguous Particle GIMP 71
3.4.2 Uniform GIMP 72
3.4.3 Convected Particle Domain Interpolation 74
3.4.4 Dual Domain Material Point Method 75
3.4.5 Spline Grid Shape Function 76
3.5 Adaptive Material Point Method 77
3.5.1 Particle Adaptive Split 77
3.5.2 Adaptive Computational Grid 79
3.6 Non-reflecting Boundary 87
3.7 Incompressible Material Point Method 88
3.7.1 Momentum Equation of Fluid 89
3.7.2 Operator Splitting 89
3.7.3 Pressure Poisson Equations 90
3.7.4 Pressure Boundary Conditions 91
3.7.5 Velocity Update 92
3.8 Implicit Material Point Method 93
3.8.1 Implicit Time Integration 95
3.8.2 Solution of a System of Nonlinear Equations 95
3.8.3 The Jacobian of Grid Nodal Internal Force 97
3.8.4 Solution of a Linearized System of Equations 99
4. Computer Implementation of the MPM
4.1 Execution of the MPM3D-F90 104
4.2 Input Data File Format of the MPM3D-F90 105
4.2.1 Unit 105
4.2.2 Keywords 105
4.2.3 Global Information 106
4.2.4 Material Model 106
4.2.5 Background Grid 108
4.2.6 Solution Scheme 109
4.2.7 Results Output 109
Contents v
4.2.8 Bodies 110
4.2.9 Load 111
4.2.10 An Example of Input Data File 111
4.3 Source Files of the MPM3D-F90 113
4.4 Free Format Input 113
4.5 MPM Data Encapsulation 115
4.5.1 Particle Data 115
4.5.2 Grid Data 116
4.5.3 Data Input 119
4.5.4 Data Output 120
4.6 Main Subroutines 121
4.7 Numerical Examples 137
4.7.1 TNT Slab Detonation 137
4.7.2 Taylor Bar Impact 138
4.7.3 Perforation of a Thick Plate 139
4.7.4 Failure of Soil Slope 141
5. Coupling of the MPM with FEM
5.1 Explicit Finite Element Method 143
5.1.1 Finite Element Discretization 143
5.1.2 The FEM Formulation in Matrix Form 145
5.1.3 Hexahedron Element 147
5.1.4 Numerical Algorithm for an Explicit FEM 155
5.2 Hybrid FEM and MPM 156
5.3 Coupled FEM and MPM 162
5.3.1 Global Search 164
5.3.2 Local Search 165
5.3.3 Contact Force 166
5.4 Adaptive FEMP Method 168
5.4.1 Discretization Scheme 168
5.4.2 Conversion Algorithm 169
5.4.3 Coupling Between Remaining Elements and Particles 170
6. Constitutive Models
6.1 Stress Update 175
6.2 Strength Models 178
6.2.1 Elastic Model 178
6.2.2 Elastoplastic Models 179
6.2.3 Return Mapping Algorithm 182
6.2.4 J2 Flow Theory 188
6.2.5 Pressure-Dependent Elastoplasticity 196
6.2.6 Newtonian Fluid 205
6.2.7 High Explosive 205
6.3 Equation of State 207
6.3.1 Polytropic Process 207
vi Contents
6.3.2 Nearly Incompressible Fluid 208
6.3.3 Linear Polynomial 209
6.3.4 JWL 210
6.3.5 MieGrneisen 211
6.4 Failure Models 213
6.4.1 Effective Plastic Strain Failure Model 214
6.4.2 Hydrostatic Tensile Failure Model 214
6.4.3 Maximum PrincipalShear Stress Failure Model 214
6.4.4 Maximum PrincipalShear Strain Failure Model 214
6.4.5 Effective Strain Failure Model 215
6.5 Computer Implementation of Material Models 215
6.5.1 Module MaterialData 215
6.5.2 Module MaterialModel 217
7. Multiscale MPM
7.1 Governing Equations at Different Scales 222
7.2 Solution Scheme for Concurrent Simulations 224
7.2.1 Preprocessor 224
7.2.2 Central Processing Unit 224
7.3 Interfacial Treatment 227
7.4 Demonstration 228
8. Applications of the MPM
8.1 Fracture Evolution 231
8.2 Impact 236
8.3 Explosion 242
8.4 FluidStructureSolid Interaction 247
8.5 Multiscale Simulation 251
8.6 Biomechanics Problems 259
8.7 Other Problems with Extreme Deformations 261
Bibliography 265
Index 277
內容試閱：
PrefaceSimulation-based Engineering Science SBES has become the third pillar ofmodern science and technology, a peer alongside theory and physical experiment[1]. Computer modeling and simulation are now an indispensable tool forresolving a multitude of scientific and technological problems we are facing [2].To model and simulate those extreme loading events such as hypervelocityimpact, penetration, blast, machining, transient crack propagation and multiphasesolidliquidgas interactions involving failure evolution, however, howto effectively describe localized large deformations, the transition from continuousto discontinuous failure modes, and fragmentation remains a challengingtask.Both Lagrangian and Eulerian approaches have been used in SBES to tackledifferent kinds of extreme events. Lagrangian methods have a computationalgrid embedded and deformed with the material [3,4]. As a result, material interfacescan be easily tracked, and history-dependent constitutive models canbe readily implemented. However, Lagrangian methods suffer from the difficultiesassociated with grid distortion and element entanglement, which makeLagrangian methods unsuitable for solving problems involving localized largedeformation, fragmentation, melting and vaporization. By contrast, in Eulerianmethods, the computational grid is fixed in space, and mass flows through thegrid. There is no difficulty associated with grid distortion and element entanglementin Eulerian methods so that they can easily solve the problems involvingextreme deformation, fragmentation, melting and vaporization. However,special procedures are required to identify the material interfaces and historydependency,which are very computationally intensive as compared with Lagrangianmethods.To take advantage of both Eulerian and Lagrangian methods while avoidingthe shortcomings of each, the Material Point Method MPM has evolved overmore than twenty years since its first journal paper was published in 1994 [5].The MPM is an extension of the particle-in-cell PIC method in computationalfluid dynamics to computational solid dynamics, formulated using theweak formulation and including the history-dependency of constitutive models.ixx PrefaceIt discretizes a continuum body into a set of material points particles movingthrough an Eulerian background grid. Hence, the MPM is a continuum-basedparticle method. The particles carry all material properties such as mass, velocity,stress, strain and state variables so that it is easy to track material interfacesand to implement history-dependent constitutive models. As the equations ofmotion are solved on the Eulerian background grid, there is no grid distortionor element entanglement, which makes the MPM robust in dealing with varioustypes of extreme loading events.After providing the necessary background information, this book describesthe fundamental theory, implementation and application of the MPM as well asits recent extensions. It contains eight chapters. Chapter 1 briefly introducesthe basic ideas and features of the Lagrangian methods, Eulerian methods,hybrid methods and meshfree methods, respectively. Chapter 2 reviews the Lagrangianand Eulerian descriptions of deformation and motion, as well as thestrain and stress measures in large deformation theory. The governing equationsof motion in an updated Lagrangian framework are given. Based on theupdated Lagrangian description, Chapter 3 establishes the MPM formulationby discretizing a continuum body into a set of particles. Both explicit and implicitformulations are presented. The Generalized Interpolation Material PointGIMP method, contact algorithm, adaptive MPM, incompressible MPM andnon-reflection boundary are discussed in detail. The computer implementationof theMPMand corresponding source codes are described in Chapter 4 based onour open source MPM code, MPM3D-F90. A users guide and several numericalexamples of the MPM3D-F90 code are also presented, for which the input datafiles can be downloaded from our web site： http：mpm3d.comdyn.cn. Chapter 5first reviews the explicit finite element method, and then presents the materialpoint finite element method, coupled material point finite element method, adaptivematerial point finite element method and hybrid material point finite elementmethod as developed in the Computational Dynamics Lab of the School ofAerospace Engineering at Tsinghua University. Chapter 6 discusses the constitutivemodels which describe different types of material behaviors, with a focuson the extreme events. The computer implementation of these constitutive modelsis specified in detail, and corresponding source codes are provided. Chapter 7introduces a multiscale MPM that could couple discrete forcing functions asused in molecular dynamics with constitutive models as used in the continuousapproaches in a single computational domain. The mapping and remapping processin the MPM could effectively coarse-grain fine details. Chapter 8 describesthe applications of the MPM and its extensions in those extreme events suchas transient crack propagation, impactpenetration, blast, fluidstructure interaction,and biomechanical responses to extreme loading.Preface xiThe most materials of this book were based on our MPM book in Chinese[6] with significant extensions and revisions. Zhen Chen added Sect. 3.2.3and Chapter 7 while Yan Liu drafted Chapter 8. The remaining chapters weredrafted by Xiong Zhang. Xiong Zhang and Zhen Chen have revised the wholebook.Finally, the first author wishes to acknowledge his students, S. Ma, P. Huang,Z.T. Ma, Y.P. Lian, H.K. Wang, W.W. Gong, S.Z. Zhou, P.F. Yang, X.X. Cui,P. Liu, Y.T. Zhang, X.J. Wang, Z.X. Hu, J.G. Li, Z.P. Chen and F. Zhang,for their contributions to the algorithm development and programming relatedto the book. Especially, Tams Benedek who implemented a subroutine inMPM3D-F90 to output simulation results to ParaView [7] for postprocessingwhen he worked on his master thesis at Tsinghua University.
Chapter 1IntroductionContents1.1 Lagrangian Methods 1 1.3.2 Particle-In-Cell Method1.2 Eulerian Methods 3 and Its Variations 51.3 Hybrid Methods 4 1.3.3 Material Point Method 61.3.1 Arbitrary 1.4 Meshfree Methods 7EulerianLagrangianMethod and Its Variations 4
Simulation-based Engineering Science SBES [2] is the third pillar of the modern science and engineering, a peer alongside theory and physical experi-ment [1]. Compared with physical experiment, SBES has the advantages of low cost, safety, and ef.ciency in solving various kinds of challenging problems. To better simulate those extreme events such as hypervelocity impact, penetration, blast, crack propagation, and multi-phase solidliquidgas interactions involv-ing failure evolution, yet effectively discretize localized large deformation, the transition among different types of failure modes and fragmentation remains a very dif.cult task. Based on the way how deformation and motion are described, existing spatial discretization methods can be classi.ed into Lagrangian, Eule-rian, and hybrid ones, respectively.
1.1 LAGRANGIAN METHODSIn Lagrangian methods the computational grid is embedded and deformed with the material. Since there is no advection between the grid and material, no ad-vection term appears in the governing equations, which signi.cantly simpli.es the solution process. The mass of each material element keeps constant during the solution process, but the element volume varies due to element deformation. Lagrangian methods have the following advantages：1. They are conceptually more simple and ef.cient than Eulerian methods. Be-cause there is no advection term that describes the mass .ow across element boundaries, the conservation equations for mass, momentum, and energy are simple in form, and can be ef.ciently solved.FIGURE 1.1 Lagrangian grid.2. Element boundaries coincide with the material interfaces during the solution process so that it is easy to impose boundary conditions and to track material interfaces.
3. Since Lagrangian methods track the .ow of individual masses, it is easy to
implement history-dependent constitutive models. Fig. 1.1 shows a typical Lagrangian grid which is embedded and deformed with the material. Severe element distortion results in signi.cant errors in numerical solution, and even leads to a negative element volume or area which would cause abnormal termination of the computation. To obtain a stable solution with an ex-plicit time integration scheme, the time step must be smaller than a critical time step which is controlled by the minimum characteristic length of all elements in the grid. Because severe element distortion would signi.cantly decrease the characteristic element length, the time step in a Lagrangian calculation could become smaller and smaller, and .nally approach zero, which makes the com-putation impossible to be completed. To complete a Lagrangian computation for an extreme loading case, a distorted grid must be remeshed and its result must be interpolated to the remeshed grid. The remesh or rezone technique has been successfully used in solving many 1D and 2D problems, but rezoning a com-plicated 3D material domain is still a challenging task. For a history-dependent material, the history variables are also required to be interpolated from the old grid to the new grid, which may further cause numerical error in stress calcula-tion.Another way to eliminate the element distortion is to use the erosion tech-nique, which simply deletes the heavily distorted elements. An element is considered to be heavily distorted if its equivalent plastic strain exceeds a user-de.ned erosion strain value, or the critical time step size is less than a prescribed value. Introducing element erosion can resolve some of the issues related to the severe element distortion, but also introduce new issues. The global system will lose both mass and energy, which can severely affect the simulation out-come. Furthermore, the erosion technique cannot model the formation process of debris cloud and its interaction with other panels in hypervelocity impact simulation.
Many Lagrangian codes have been developed, as shown in the open lit-erature. The HEMP [8] was developed in the early 1960s by Wilkins at the Lawrence Livermore National Laboratory. The HEMP was an explicit La-grangian .nite-difference code that could handle large strains, elasticplastic .ow, wave propagation, and sliding interfaces. The EPIC code [9] was an ex-plicit Lagrangian .nite element code developed in the 1970s by Johnson. Both the rezoning and erosion techniques were employed in the EPIC to simulate high velocity impact and blast problems. The PRONTO3D code [10] was a 3D tran-sient solid dynamics code developed at the Sandia National Laboratory for an-alyzing large deformations of highly nonlinear materials subjected to extremely high strain rates. This code was based on an explicit .nite element formulation, and had been coupled with the smoothed particle hydrodynamics SPH method through a contact-like algorithm [11]. The DYNA2D and DYNA3D codes were developed in the 1970s at the Lawrence Livermore National Laboratory as ex-plicit Lagrangian .nite element codes and were successfully commercialized [1214].
1.2 EULERIAN METHODSFor problems in which a material domain could become heavily distorted or different materials are mixed, an Eulerian method is more appropriate. In Eu-lerian methods, the computational grid is .xed in space and does not move with the material such that the material .ows through the grid, as shown in Fig. 1.2.There is no element distortion in Eulerian methods, but the physical vari-ables, such as mass, momentum, and energy, advect between adjacent elements across their interface. The volume of each element keeps constant during the simulation, but its density varies due to the advection of mass. Eulerian meth-ods are suited for modeling large deformations of materials so that most of computational .uid dynamics codes and early hydrocodes for impact and blast simulation employ Eulerian methods.Eulerian methods only calculate the material quantities advected between elements without explicitly and accurately determining the position of material interface and free surface so that they are quite awkward in following deforming material interfaces and moving boundaries. Signi.cant efforts have been made to develop interface reconstruction methods.HELP Hydrodynamic plus ELastic Plastic [15], developed by Walsh and Hageman in the 1960s, is a multi-material Eulerian .nite difference program for compressible .uid and elasticplastic .ows. To treat the material interface or free surface, massless tracer particles are used, which de.ne the surface po-sition and move across the Eulerian grid. CTH [16] is an Eulerian .nite volume code developed at Sandia National Laboratories to model multi-dimensional, multi-material, large deformation, and strong shock wave physics. The CTH code employs a two-step Eulerian solution scheme, a Lagrangian step in which the cells distort to follow the material motion, and a remesh step where the distorted cells are mapped back to the Eulerian mesh. Material interfaces are re-constructed using the Sandia Modi.ed Youngs Reconstruction Algorithm. The CTH has adaptive mesh re.nement and uses second-order accurate numerical methods to reduce numerical dispersion and dissipation. It is still under devel-opment at Sandia National Laboratories [17].The Zapotec developed at Sandia National Laboratories is a framework that tightly couples the CTH and PRONTO codes [18,19]. In a Zapotec analysis, both CTH and PRONTO are run concurrently. For a given time step, the Zapotec maps the current con.guration of a Lagrangian body onto the .xed Eulerian mesh. Any overlapping Lagrangian material is inserted into the Eulerian mesh with the updated mesh data passed back to the CTH. After that the external loading on the Lagrangian material surfaces is determined from the stress state in the Eulerian mesh. These loads are passed back to PRONTO as a set of ex-ternal nodal forces. After the coupled treatment is completed, both CTH and PRONTO are run independently over the next time step.
1.3 HYBRID METHODSBoth purely Lagrangian and purely Eulerian methods possess different short-comings and advantages so that it is desirable to .nd new approaches to take advantage of both methods to better tackle challenging problems. The arbitrary LagrangianEulerian ALE method [20] and the particle-in-cell PIC method [21,22] are two representatives.1.3.1 Arbitrary EulerianLagrangian Method and Its VariationsThe ALE method was .rst proposed in the .nite difference and .nite volume context [23,24], and was subsequently adopted in the .nite element contextFIGURE 1.3 ALE grid.[2527]. The mixed EulerianLagrangian method [24] involves the Eulerian set-up with respect to one dimension and the Lagrangian one to the other di-mension which corresponds to the direction of .uid .ow. The coupled Eulerian Lagrangian [23] code employs an Eulerian mesh for the entire region and Lagrangian meshes for the subregions of .uids with nonstationary boundaries approximated by Lagrangian lines.In the above methods, the computational mesh may be moved with the ma-terial in Lagrangian manner, or be held .xed in Eulerian manner, or be moved independently of material deformation to optimize element shapes and to de-scribe the boundaries accurately [20], as shown in Fig. 1.3. Because these methods offer great .exibility in moving the computational mesh, they can han-dle a much greater distortion of the material than a Lagrangian method, with a higher resolution than that afforded by an Eulerian method. However, the con-vective terms still pose some problems. Furthermore, designing an ef.cient and effective mesh-moving algorithm for complicated 3D problems remains a chal-lenging task.
1.3.2 Particle-In-Cell Method and Its VariationsThe PIC method was proposed and developed at Los Alamos National Labora-tory by Harlow in the late 1950s [21,22,28]. PIC makes use of both Lagrangian and Eulerian descriptions, namely, the .uid is discretized as a set of Lagrangian particles that carry material position, mass, and species information, but the computational mesh is a uniform Eulerian one. A computational cycle is di-vided into two phases, a Lagrangian phase and an Eulerian remap or rezone phase. In the Lagrangian phase, all the variables, including the mesh coordi-nates and the particle positions, are advanced. In the Eulerian phase, the mesh is mapped back into its original con.guration, leaving the particles at their new locations. This process can also be viewed in a time splitting way, namely, the Lagrangian phase updates the quantities by all the processes except for advec-tion, while the Eulerian phase moves the particles and accomplishes all of the advective .uxing [29].As a variation of the PIC method, the marker-and-cell MAC method was developed by Harlow and Welch [30,31] to treat incompressible and free surface .ows. In the MAC method, particles are used as markers to de.ne the location of the free-surface, and the Poisson equation for the pressure is solved to treat the .uid incompressibility. The MAC method was the .rst successful technique for simulating incompressible .ows [32].The original version of PIC is not a fully Lagrangian particle method be-cause only the material position, mass, and species information is carried by the particles, while the remaining quantities are still stored in the computa-tional grid. The transfer of information between the particles and the underlying grid leads to signi.cant numerical diffusion. There are two strategies to reduce the numerical diffusion, namely, second-order accuracy advection scheme [33] and fully Lagrangian particle method. Brackbill et al. developed a fully La-grangian particle method, FLuid-Implicit-Particle FLIP method [34,35],in which each particle carries all of the properties of the .uid, including mo-mentum and energy. FLIP preserves the ability of the original PIC to resolve contact discontinuities, but eliminates the major source of numerical diffu-sion.
1.3.3 Material Point MethodWhen working on the penetration problems in the early 1990, Zhen Chen and his former PhD advisor, Buck Schreyer, faced a challenging task to improve the computational .delity and ef.ciency of the .nite element method FEM, due to its limitation in the required use of a pin-hole in the mesh design. In a semi-nar at University of New Mexico, Deborah Sulsky presented the advances of the PIC method, based on her collaborative research on computational .uid dynam-ics with the scientists at Los Alamos National Laboratory. Since the particle motion in .uid is similar to the penetrators motion in solid from the view-point of hardsoft body interaction, Sulskys seminar opened Chens eyes to a new direction of research so that he initiated an interdisciplinary discussion. In collaboration with Sandia National Laboratories, the team of three folks with diversi.ed tastes then started to combine computational .uid dynamics with computational solid dynamics to develop a continuum-based particle method with its .rst journal paper published in 1994 [5], which was later named as the Material Point Method MPM. Over the last two decades, many research teams in the world have further developed the MPM and combined the MPM with other numerical methods for multiphase, multiphysics, and multiscale simula-tions to advance SBES.The MPM is an extension of the FLIP method from computational .uid dynamics to computational solid dynamics with two key differences. First, the constitutive equations are solved at the particles material points rather than the grid cell centers such that the MPM can readily model history-dependent materials. Second, the MPM is formulated in the weak form consistent with the FEM so that the FEM and MPM could be effectively combined together [3641] for large-scale simulations.The MPM is a fully Lagrangian particle method which utilizes the advan-tages of both Eulerian and Lagrangian methods. As compared with Eulerian methods, the numerical dissipation normally associated with a Eulerian method is eliminated, while the complete deformation history of material points are tracked. Compared with Lagrangian methods, mesh distortion and element en-tanglement are avoided. Therefore, the MPM has demonstrated obvious ad-vantages in tackling those extreme events such as impact, blast, penetration, perforation, machining, fragmentation, and multi-phase interaction involving failure evolution, as demonstrated in Chapter 8.
1.4 MESHFREE METHODSIn addition to the evolution of the MPM, different types of meshfree and particle methods for improved spatial discretization in different problems have also been proposed and developed in the SBES community [42]. Since all these meshfree and particle methods do not use a rigid mesh connectivity compared with the conventional mesh-based methods such as the FEM, they have been applied to many challenging problems of current interests such as impactcontact, local-ization, crack propagation, penetration, perforation, and fragmentation. Never-theless, many of the meshfree methods suffer from higher computational costs, and the accuracy of some meshfree methods is still dependent on the node reg-ularities to some extent.Smoothed particle hydrodynamics SPH [4346] is one of the earliest mesh-free Lagrangian particle methods. The SPH was .rst proposed by Lucy [43] and Gingold and Monaghan [44] in 1977 to solve astrophysical problems in the 3D open space, and has been extensively studied and extended to solid and .uid dynamics problems with large deformations. The SPH and its improved ver-sions have been successfully applied to the hypervelocity impact simulations, and become some of the most popular meshfree methods in this area. Because of their good performance, several commercial softwares, such as AUTODYN, PAM-CRASH, and LS-DYNA, have incorporated the SPH into their solvers. However, the SPH is limited in simulating multiphase interactions involving failure evolution.
Ma et al. [47] compared the basic formulation and features of the MPM with SPH from the following aspects： neighbor searching, approximation functions, consistency of shape functions, tensile instability, time integration, boundary conditions, and contact algorithm. A comparative study showed that the MPM possesses many prominent features. The formulation of the MPM is simple and similar to the traditional FEM. The time consuming neighbor searching, which is compulsory in most meshfree methods, is not required in the MPM. The MPM shape functions exactly satisfy the constant and linear consistency. The MPM avoids tensile instability that is annoying in the SPH. The bound-ary conditions can be applied in the MPM as easily as in the FEM, and the contact algorithm can be ef.ciently implemented whose cost is linear in the number of material points involved. Because the same regular computational grid can be used in all time steps, the time step keeps constant in the MPM sim-ulations. Numerical studies have showed that the computational ef.ciency and stability of our MPM3D code are much higher than those of LS-DYNA SPH module.Fig. 1.4 compares the CPU time per step as used by LS-DYNA SPH mod-ule and our MPM3D code in the simulation of the translation motion of a cubic block [48]. It demonstrates that the CPU time per step used by both methods in-creases linearly with the increase of number of particles, but the rate of increase of the SPH is much higher than that of the MPM.Ma et al. [47] also investigated the accuracy and ef.ciency of the SPH and MPM by simulating the impact of a copper cylinder to a rigid wall with an im-pact velocity of 190 ms. In the SPH simulations, the constant associated with the smoothing length was set to 1.2 SPH1 and 1.4 SPH2, respectively. The value of 1.2 is the default value used in LS-DYNA, and a larger value will in-crease the computational time but may improve the result with more neighbors for each particle. Fig. 1.5 compares the .nal con.gurations of the bar obtained
c MPM3D [47].by the SPH and MPM, which shows that the SPH algorithm suffers from nu-merical fracture due to tensile instability. Enlarging the smoothing length can alleviate the numerical fracture, but particle clumps may still exist. Furthermore, enlarging the smoothing length increases the time step size, which raises the one-step computational cost signi.cantly.
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