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.     

A coupled SDPH–FVM method for gas‐particle multiphase flow: Methodology

This paper describes a novel methodology that combines smoothed discrete particle hydrodynamics (SDPH) and finite volume method (FVM) to enhance the effective performance in solving the problems of gas‐particle multiphase flow. To describe the collision and fluctuation of particles, this method also increases a new parameter, namely, granular temperature, according to the kinetic theory of granular flow. The coupled framework of SDPH–FVM has been established, in which the drag force and pressure gradient act on the SDPH particles and the momentum sources of drag force are added back onto the FVM mesh. The proposed technique is a coupled discrete‐continuum method based on the two‐fluid model. To compute for the discrete phase, its SDPH is developed from smoothed particle hydrodynamics (SPH), in which the properties of SPH are redefined with some new physical quantities added into the traditional SPH parameters, so that it is more beneficial for SDPH in representing the particle characteristics. For the continuum phase, FVM is employed to discretize the continuum flow field on a stationary grid by capturing fluid characteristics. The coupled method exhibits strong efficiency and accuracy in several two‐dimensional numerical simulations. Copyright © 2016 John Wiley & Sons, Ltd.     

An Approach for Robust Design of Reactive Power Metal Mixtures Based on Non-deterministic Micro-scale Shock Simulation

In this paper, we propose a method for the robust design of materials involving processes that are computationally intensive and selectively random. The material system considered is a reactive particle metal mixture (RPMM) composed of aluminum and iron oxide (Al+Fe₂O₃). Shock simulations of discrete energetic particle mixtures are performed to predict the system’s mechanical and thermal behavior that will be used by a designer of the mixture to achieve robust micro-scale reaction initiation. The method used to predict the behavior of the material system is the robust concept exploration method with error margin index (RCEM-EMI). An error margin index is a mathematical construct indicating the location of mean system performance and the spread of this performance considering both variability in design variables and models of the system. Variability in responses of a model may be due to system variation that cannot be easily parameterized in terms of noise factors. Furthermore, lack of data, due to the cost of simulations and experiments, leads to uncertain parameters in empirical models. System response variability and parameter uncertainty in an empirical model are estimated in a computationally efficient manner to formulate the error margin indices, which are then leveraged to search for ranged sets of design specifications. Finally, the RCEM-EMI is illustrated for designing a RPMM.     

High-Fidelity Magnetic Equivalent Circuit Model for an Axisymmetric Electromagnetic Actuator

A computationally inexpensive magnetic equivalent circuit (MEC) improves axisymmetric electromagnet design and modeling tools by accurately capturing fringing and leakage effects. Lumped parameter MEC models are typically less accurate for modeling electromagnetic devices than distributed parameter finite-element models (FEMs). However, MEC models require significantly less computational time to solve than FEMs and therefore lend themselves to applications where solution time is critical, such as in optimization routines, dynamic simulation, or preliminary design. This paper describes how fringing permeances in axisymmetric electromagnetic devices can be derived and then included in a MEC model. Including fringing field effects significantly decreases error in the MEC model, creating a more accurate, or high fidelity, magnetic equivalent circuit (HFMEC). Eighty-nine electromagnets with unique geometries, coil currents, and materials were modeled with MEC, HFMEC, and FEM methods. The axisymmetric HFMEC developed in this work had 67% less average force error and 88% less average flux error compared to traditional MEC results while still being computationally inexpensive to solve.      

A multimass correction for multicomponent fluid flow simulation using smoothed particle hydrodynamics

In multicomponent fluid flow simulations using smoothed particle hydrodynamics, the Lagrangian particles used are mostly of equal mass. This is preferred over multimass particle setup (particles with different values of mass), as it resolves the fluid interfaces comparatively better. But the flip side of using uniform mass particle setup is that it may not be computationally economical in situations with large‐density ratios. Hence, using multimass particle setup is both economical and perhaps inevitable. An attractive feature of multimass particle setup is that it allows uniform resolution in regions with different values of density. To take advantage of the multimass setup, it is therefore imperative to reduce the error associated with its usage. In this work, we present suitable multimass correction terms and assess its effectiveness using the ?h–smooth particle hydrodynamics scheme. Standard benchmark problems, viz, shock tube test, triple‐point shock test, Rayleigh‐Taylor instability, and Kelvin‐Helmholtz instability were solved with multimass particle setup, where significant improvements could be achieved in resolving the associated contact discontinuities.     

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.     

Conservative Taylor least squares reconstruction with application to material point methods

Within the standard material point method (MPM), the spatial errors are partially caused by the direct mapping of material-point data to the background grid. In order to reduce these errors, we introduced a novel technique that combines the least squares method with the Taylor basis functions, called the Taylor least squares (TLS), to reconstruct functions from scattered data while preserving their integrals. The TLS technique locally approximates quantities of interest such as stress and density, and when used with a suitable quadrature rule, it conserves the total mass and linear momentum after transferring the material-point information to the grid. The integration of the technique into MPM, dual domain MPM, and B-spline MPM significantly improves the results of these methods. For the considered examples, the TLS function reconstruction technique resembles the approximation properties of highly accurate spline reconstruction while preserving the physical properties of the standard algorithm.     

A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations

A new algorithm is developed to improve the accuracy and efficiency of the material point method for problems involving extremely large tensile deformations and rotations. In the proposed procedure, particle domains are convected with the material motion more accurately than in the generalized interpolation material point method. This feature is crucial to eliminate instability in extension, which is a common shortcoming of most particle methods. Also, a novel alternative set of grid basis functions is proposed for efficiently calculating nodal force and consistent mass integrals on the grid. Specifically, by taking advantage of initially parallelogram‐shaped particle domains, and treating the deformation gradient as constant over the particle domain, the convected particle domain is a reshaped parallelogram in the deformed configuration. Accordingly, an alternative grid basis function over the particle domain is constructed by a standard 4‐node finite element interpolation on the parallelogram. Effectiveness of the proposed modifications is demonstrated using several large deformation solid mechanics problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Truncation error in mesh‐free particle methods

A truncation error analysis has been developed for the approximation of spatial derivatives in smoothed particle hydrodynamics (SPH) and related first‐order consistent methods such as the first‐order form of the reproducing kernel particle method. Error is shown to depend on both the smoothing length h and the ratio of particle spacing to smoothing length, Δx/h. For uniformly spaced particles in one dimension, analysis shows that as h is reduced while maintaining constant Δx/h, error decays as h² until a limiting discretization error is reached, which is independent of h. If Δx/h is reduced while maintaining constant h (i.e. if the number of neighbours per particle is increased), error decreases at a rate which depends on the kernel function's smoothness. When particles are distributed non‐uniformly, error can grow as h is reduced with constant Δx/h. First‐order consistent methods are shown to remove this divergent behaviour. Numerical experiments confirm the theoretical analysis for one dimension, and indicate that the main results are also true in three dimensions. This investigation highlights the complexity of error behaviour in SPH, and shows that the roles of both h and Δx/h must be considered when choosing particle distributions and smoothing lengths. Copyright © 2005 John Wiley & Sons, Ltd.     

An accelerated,convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics

Convergent and stable domain integration that is also computationally efficient remains a challenge for Galerkin meshfree methods. High order quadrature can achieve stability and optimal convergence, but it is prohibitively expensive for practical use. On the other hand, low order quadrature consumes much less CPU but can yield non‐convergent, unstable solutions. In this work, an accelerated, convergent, and stable nodal integration is developed for the reproducing kernel particle method. A stabilization scheme for nodal integration is proposed based on implicit gradients of the strains at the nodes that offers a computational cost similar to direct nodal integration. The method is also formulated in a variationally consistent manner, so that optimal convergence is achieved. A significant efficiency enhancement over a comparable stable and convergent nodal integration scheme is demonstrated in a complexity analysis and in CPU time studies. A stability analysis is also given, and several examples are provided to demonstrate the effectiveness of the proposed method for both linear and nonlinear problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Comparison study of MPM and SPH in modeling hypervelocity impact problems

Due to the high nonlinearities and extreme large deformation, the hypervelocity impact simulation is a challenging task for numerical methods. Meshfree particle methods, such as the smoothed particle hydrodynamics (SPH) and material point method (MPM), are promising for the simulation of hypervelocity impact problems. In this paper, the material point method is applied to the simulation of hypervelocity impact problems, and a three-dimensional MPM computer code, MPM3D, is developed. The Johnson–Cook material model and Mie–Grüneisen equation of state are implemented. Furthermore, the basic formulations of MPM are compared with SPH, and their performances are compared numerically by using MPM3D and LS-DYNA SPH module.     

Static and free vibrational analysis of beams and plates by differential quadrature method

Summary The new approach for application of boundary conditions in the differential quadrature (DQ) method, proposed earlier by the present authors, is extended to generalized force boundary conditions in two dimensions. A variety of problems is then analyzed by theDQ method with the new approach for application of boundary conditions, such as deflections of beams and circular and rectangular plates under nonuniformly distributed loadings, deflection of a rectangular plate on a Winkler foundation, and buckling and free vibrational analyses of circular plates. It is found that the present method gives good accuracy and is computationally efficient. Exact solutions can be obtained by theDQ method if analytical solutions are polynomials and the method is insensitive to the spacing of grid points for the cases considered.     

Suppressing local particle oscillations in the Hamiltonian particle method for elasticity

The governing equation of elasticity is discretized into motion equations of the particles in a Hamiltonian system. A weighted least‐square method is adopted to evaluate the Green–Lagrange strain. Using a symplectic scheme for the Hamiltonian system, we obtain the property of energy conservation in the discretized calculations. However, local particle oscillations occur, and they excessively decrease low frequency motion. In this study, we propose the use of an artificial potential force to suppress the local oscillations. The accuracy of the model with and without the inclusion of the artificial force is examined by analyzing a cantilever beam and wave propagation. With the inclusion of the artificial force, the local oscillations are reduced while energy conservation is maintained. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Implicit boundary method for finite element analysis using non‐conforming mesh or grid

In the finite element method (FEM), a mesh is used for representing the geometry of the analysis and for representing the test and trial functions by piece‐wise interpolation. Recently, analysis techniques that use structured grids have been developed to avoid the need for a conforming mesh. The boundaries of the analysis domain are represented using implicit equations while a structured grid is used to interpolate functions. Such a method for analysis using structured grids is presented here in which the analysis domain is constructed by Boolean combination of step functions. Implicit equations of the boundary are used in the construction of trial and test functions such that essential boundary conditions are guaranteed to be satisfied. Furthermore, these functions are constructed such that internal elements, through which no boundary passes, have the same stiffness matrix. This approach has been applied to solve linear elastostatic problems and the results are compared with analytical and finite element analysis solutions to show that the method gives solutions that are similar to the FEM in quality but is less computationally expensive for dense mesh/grid and avoids the need for a conforming mesh. Copyright © 2007 John Wiley & Sons, Ltd.     

An improved hybrid particle-element method for hypervelocity impact simulation

An improved hybrid particle-finite element method has been developed for hypervelocity impact simulation. The method combines the general contact-impact capabilities of particle codes with the true Lagrangian kinematics of large strain finite element formulations. Unlike some alternative schemes which couple Lagrangian finite element models with smooth particle hydrodynamics, the present formulation makes no use of slidelines or penalty forces.     

水预湿被爆体降低爆破粉尘机理研究

通过对气流中单个尘粒的运动分析,指出降低爆尘的有效方法是增大粒径.通过对尘流中尘粒受力的分析,得出在干燥环境下作用在尘粒上的主作用力是范德华力;在湿润情况下作用在尘粒上的主作用力是液桥力的结论.指出水预湿被爆体降尘法的机理是:尘粒间的液桥力促使润湿尘流中的尘粒变大,变大的尘粒迅速下降,从而实现降低爆尘目的.     

Performance evaluation of high rate space?time trellis-coded modulation using Gauss?Chebyshev quadrature technique

The performance analysis of high rate space-time trellis-coded modulation (HR-STTCM) using the Gauss-Chebyshev quadrature technique is presented. HR-STTCM is an example of space-time codes that combine the idea used in trellis coded modulation (TCM) design that is signal set expansion and set partitioning into its construction. HR-STTCM construction is based on the concatenation of an outer TCM encoder and inner space-time block code. This paper evaluates the exact pairwise error probability of HR- STTCM based on the Gauss-Chebyshev quadrature formula. Comparison of numerical and simulation results shows that the proposed method is accurate. The method used is shown to be computationally simpler than those in the literature.     

筛分法和激光粒度法联合测定陆源碎屑岩粒度

用筛分法和激光粒度法联合测定陆源碎屑岩粒度,对激光粒度法的分析数据进行了校正,以保证两种方法的分析数据配套衔接。通过对比不同地质年代沉积岩的岩心校正前后概率累积频率曲线,校正后的曲线能更好地表征碎屑岩粒度的分布。对影响马尔文Mastersizer 2000激光粒度仪背景测量的进样器搅拌转速A、泵速B、超声强度C三个因子进行L9(34)正交优化设计,筛选出最优的测量条件是A取600r/min,B取2000r/min,可使背景测量稳定,并得出它们对背景测量影响程度可用B>A>误差>C表示。用仪器自带的9个标准物质校正了粒度标准曲线,粒度标准曲线与仪器标准曲线的加权残差小,在0.795%～2.934%之间。该方法相对标准偏差在0.24%～0.33%之间,适用于各种地质年代陆源碎屑岩的粒度分析。     

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.