Multiple scale meshfree methods for damage fracture and localization

The Reproducing Kernel Particle Method (RKPM), which utilizes the fundamental notions of the convolution theorem, multiresolution analysis and meshfree properties, is reviewed. The multiple-scale RKPMs are then proposed as an alternative to commonly used numerical methods such as the finite element method. The elimination of a mesh, combined with the filtering properties of window functions, makes a particle method suitable for problems with large deformations, high gradients, and localization problems. This class of methods has been applied to shear band problems, and large deformation fracture and damage problems.     

Explicit Reproducing Kernel Particle Methods for large deformation problems

The explicit Reproducing Kernel Particle Method (RKPM) is presented and applied to the simulations of large deformation problems. RKPM is a meshless method which does not need a mesh structure in its formulation. Because of this mesh-free property, RKPM is able to simulate large deformation problems without remeshing which is often required for the mesh-based methods such as the finite element method. The RKPM shape function and its derivatives are constructed by imposing the consistency conditions. An efficient treatment of essential boundary conditions is also proposed for explicit time integration. The Lagrangian method based on the reference configuration is employed for the RKPM simulation of large deformation problems. Several examples of non-linear elastic materials are solved to demonstrate the performance of the method. The numerical experiment for the problem of underwater bubble explosion is also performed using the explicit Lagrangian RKPM formulation. © 1998 John Wiley & Sons, Ltd.     

STRESS POINTS FOR TENSION INSTABILITY IN SPH

In this work, the stress-point approach, which was developed to address tension instability and improve accuracy in Smoothed Particle Hydrodynamics (SPH) methods, is further extended and applied for one-dimensional (1-D) problems. Details of the implementation of the stress-point method are also given. A stability analysis reveals a reduction in the critical time step by a factor of 1/√2 when the stress points are located at the extremes of the SPH particle. An elementary damage law is also introduced into the 1-D formulation. Application to a 1-D impact problem indicates far less oscillation in the pressure at the interface for coarse meshes than with the standard SPH formulation. Damage predictions and backface velocity histories for a bar appear to be quite reasonable as well. In general, applications to elastic and inelastic 1-D problems are very encouraging. The stress-point approach produces stable and accurate results. © 1997 by John Wiley & Sons, Ltd.     

Multiple-scale reproducing kernel particle methods for large deformation problems

Reproducing Kernel Particle Method (RKPM) with a built-in feature of multiresolution analysis is reviewed and applied to large deformation problems. Since the application of multiresolution RKPM to the large deformation problems is still in its early stage of development, we introduce, in this paper, the concept of a multiple-scale measure which is a extension of the linear formulations to nonlinear problems. We also propose an appropriate measure to properly detect the high-scale response of a largely deformed material. Via this technique of multilevel decomposition of a reproducing kernel function, the high-scale component of the measure is used in deriving an adaptive algorithm by simply inserting extra particles. Numerical experiments for non-linear elastic materials are performed to demonstrate the completeness of multiple-scale Reproducing Kernel Particle Method. © 1998 John Wiley & Sons, Ltd.     

Large deformation analysis of rubber based on a reproducing kernel particle method

 A nonlinear formulation of the Reproducing Kernel Particle Method (RKPM) is presented for the large deformation analysis of rubber materials which are considered to be hyperelastic and nearly incompressible. In this approach, the global nodal shape functions derived on␣the basis of RKPM are employed in the Galerkin approximation of the variational equation to formulate the discrete equations of a boundary-value hyperelasticity problem. Existence of a solution in RKPM discretized hyperelasticity problem is discussed. A Lagrange multiplier method and a direct transformation method are presented to impose essential boundary conditions. The characteristics of material and spatial kernel functions are discussed. In the present work, the use of a material kernel function assures reproducing kernel stability under large deformation. Several of numerical examples are presented to study the characteristics of RKPM shape functions and to demonstrate the effectiveness of this method in large deformation analysis. Since the current approach employs global shape functions, the method demonstrates a superior performance to the conventional finite element methods in dealing with large material distortions.     

Advances in multiple scale kernel particle methods

A novel approach to multiresolution analysis based on reproducing kernel particle methods (RKPM) and wavelets is presented. The concepts of reproducing conditions, discrete convolutions, and multiple scale analysis are described. By means of a newly proposed semidiscrete Fourier analysis, RKPM is further elaborated in the frequency domain, and the interpolation estimate and the convergence of Galerkin solutions are given. The elimination of a mesh, combined with the properties of the dilation and translation of a window function, multiresolution analysis, wavelet-based error estimators, and edge detection brings about a new generation of hp adaptive methods. In addition, this class of multiple scale reproducing kernel particle methods is particularly suitable for problems with large deformations, high gradients, and high modal density. The current application areas of RKPM include structural acoustics, structural dynamics, elastic-plastic deformation, computational fluid dynamics and hyperelasticity.Dedicated to the 10th anniversary of Computational Mechanics     

A Runge–Kutta–Chebyshev SPH algorithm for elastodynamics

A stable and accurate Smoothed Particle Hydrodynamics (SPH) method is proposed for solving elastodynamics in solid mechanics. The SPH method is mesh-free, and it promises to overcome most of disadvantages of the traditional finite element techniques. The absence of a mesh makes the SPH method very attractive for those problems involving large deformations, moving boundaries and crack propagation. However, the conventional SPH method still has significant limitations that prevent its acceptance among researchers and engineers, namely the stability and computational costs. In approximating unsteady problems using the SPH method, attention should be given to the choice of time integration schemes as accuracy and efficiency of the SPH solution may be limited by the timesteps used in the simulation. This study presents an attempt to reconstruct an unconditionally stable SPH method for elastodynamics. To achieve this objective we implement an explicit Runge–Kutta Chebyshev scheme with extended stages in the SPH method. This time stepping scheme adds in a natural way a stabilizing stage to the conventional Runge–Kutta method using the Chebyshev polynomials. Numerical results are shown for several test problems in elastodynamics. For the considered elastic regimes, the obtained results demonstrate the ability of our new algorithm to better maintain the shape of the solution in the presence of shocks.     

基于罚函数SPH新方法的水模拟充型过程的数值分析

同传统的网格法相比,光滑粒子流体动力学(SPH)方法非常适合于求解大变形和自由表面流动问题.阐述了SPH理论及其应用,并用罚函数处理流体与壁面的相互作用,以解决传统SPH本质边界条件不易施加的问题.对水模拟的允型过程实验进行数值分析,并和文献实验结果以及传统SPH进行对比,最终表明仿真结果与实验非常吻合,比传统SPH方...     

B spline-based method for 2-D large deformation analysis

In this paper, quadratic cardinal B spline functions are used for solution of 2-D large deformation problems. Because the B spline functions are directly used in function approximation, no meshes are needed and the mesh distortion issues in nonlinear analyses are avoided in this method. Using the B spline functions, the solution formulations based on total Lagrangian (TL) approach for two dimensional large deformation problems are established. The numerical examples of 2-D large deformation problems indicate that the B spline method is effective and stable for solving complicated problems.     

基于SPH方法的剪切流驱动液滴在固体表面变形运动数值模拟研究

光滑粒子流体动力学(SPH)方法是纯拉格朗日粒子方法,可以有效避免网格法在模拟大变形过程中带来的网格扭曲等缺陷,适合模拟含大变形的剪切流驱动液滴在固体表面变形运动过程。在基于CSF模型的表面张力SPH方法基础上,采用新的边界处理方式和界面法向修正方法,引入Brackbill提出的壁面附着力边界条件处理方法,得到了含壁面附着力边界条件的表面张力算法。基于新方法模拟了剪切流驱动液滴在固体表面变形运动过程并与实验结果和VOF方法模拟结果进行了对比验证。结果表明：该方法在处理壁面附着力问题时精度较高,稳定性较好,适合处理工程中剪切流驱动液滴在固体表面变形运动问题。     

Numerical study via total Lagrangian smoothed particle hydrodynamics on chip formation in micro cutting

Numerical simulation is an effective approach in studying cutting mechanism. The widely used methods for cutting simulation include finite element analysis and molecular dynamics. However, there exist some intrinsic shortcomings when using a mesh-based formulation, and the capable scale of molecular dynamics is extremely small. In contrast, smoothed particle hydrodynamics (SPH) is a candidate to combine the advantages of them. It is a particle method which is suitable for simulating the large deformation process, and is formulated based on continuum mechanics so that large scale problems can be handled in principle. As a result, SPH has also become a main way for the cutting simulation. Since some issues arise while using conventional SPH to handle solid materials, the total Lagrangian smoothed particle hydrodynamics (TLSPH) is developed. But instabilities would still occur during the cutting, which is a critical issue to resolve. This paper studies the effects of TLSPH settings and cutting model parameters on the numerical instability, as well as the chip formation process. Plastic deformation, stress field and cutting forces are analyzed as well. It shows that the hourglass coefficient, critical pairwise deformation and time step are three important parameters to control the stability of the simulation, and a strategy on how to adjust them is provided.The full text can be downloaded at https://link.springer.com/article/10.1007/s40436-020-00297-z     

A quasi‐linear reproducing kernel particle method

Reproducing kernel particle method (RKPM) has been applied to many large deformation problems. RKPM relies on polynomial reproducing conditions to yield desired accuracy and convergence properties but requires appropriate kernel support coverage of neighboring nodes to ensure kernel stability. This kernel stability condition is difficult to achieve for problems with large particle motion such as the fragment‐impact processes that commonly exist in extreme events. A new reproducing kernel formulation with ‘quasi‐linear’ reproducing conditions is introduced. In this approach, the first‐order polynomial reproducing conditions are approximately enforced to yield a nonsingular moment matrix. With proper error control of the first‐order completeness, nearly second‐order convergence rate in L2 norm can be achieved while maintaining kernel stability. The effectiveness of this quasi‐linear RKPM formulation is demonstrated by modeling several extremely large deformation and fragment‐impact problems. Copyright © 2016 John Wiley & Sons, Ltd.     

弹性力学静力问题的SPH方法

光滑质点流体动力学方法(Smoothed Particle Hydrodynamics,SPH)是纯Lagrangian方法,可用于模拟流体或固体的静动力学问题。不需网格系统即可进行空间导数计算,可避免Lagrangian网格在处理结构变形计算时的缠结和扭曲问题。但经典SPH方法计算二阶以上导数时易引起计算失败。该文提出一种改进的SPH方法,既可避免二阶导数的计算失败,又可提高二阶导数的精度。据此计算了均布荷载作用下两端固结梁的变形问题。经与ANSYS计算结果比较,该方法的计算足够精确。虽以弹性力学小变形问题为例,但结论可推广到大变形情形。     

Multiresolution reproducing kernel particle methods

Reproducing Kernel Particle Methods (RKPM) with a built-in feature of multiresolution analysis are addressed. Some fundamental concepts such as reproducing conditions, and correction function are constructed to systematize the framework of RKPM. In particular, Fourier analysis, as a tool, is exploited to further elaborate RKPM in the frequency domain. Furthermore, we address error estimation and convergence properties. We present several applications which confirm the widespread applicability of multiresolution RKPM.     

A stabilized Runge–Kutta,Taylor smoothed particle hydrodynamics algorithm for large deformation problems in dynamics

Numerical simulation of large deformation and failure problems present a series of difficulties when solved using mesh based methods. Meshless methods present an interesting alternative that has been explored in the past years by researchers. Here we propose a Runge–Kutta Taylor SPH model based on formulating the dynamic problem as a set of first‐order PDEs. Two sets of nodes are used for time steps n and n + 1 ∕ 2, resulting on avoiding the classical tensile instability of some other SPH formulations. To improve the accuracy and stability of the algorithm, the Taylor expansion in time of the advective terms is combined with a Runge–Kutta integration of the sources. Finally, as boundaries change during the process, a free surface detection algorithm is introduced. Copyright © 2012 John Wiley & Sons, Ltd.     

基于SPH方法的低韦伯数下三维液滴碰撞聚合与反弹数值模拟研究

光滑粒子流体动力学方法（SPH方法）作为纯拉格朗日粒子方法,可以有效避免网格法在模拟大变形过程中带来的网格扭曲等缺陷,适合模拟含大变形的液滴碰撞聚合与反弹过程。该文基于Ott和Schnetter提出的修正SPH方法,利用有限差分与SPH一阶导数相结合的方法处理粘性项中的二阶导数问题,进行Couette流算例验证,数值解...     

Two scale meshfree method for the adaptivity of 3-D stress concentration problems

 In this study, a meshfree method called Reproducing Kernel Particle Method (RKPM) with an inherent characteristic of multi-resolution is modified to develop structural analysis algorithm using two scales. The shape function of RKPM is decomposed into two scales, high and low. The two scale decomposition is incorporated into linear elastic formulation to obtain high and low scale components of von Mises stresses. The advantage of using this algorithm is that the high scale component of von Mises stress indicates the high stress gradient regions without posteriori estimation. This algorithm is applied to the analysis of 2- and 3-dimensional stress concentration problems. It is important to note that the two scale analysis method has been applied to 3-dimensional stress concentration problem for the very first time. Also, the possibility of applying this algorithm to adaptive refinement technique is studied. The proposed method is verified by analyzing typical 2- and 3-dimensional linear elastic stress concentration problems. The results show that the algorithm can effectively locate the high stress concentration regions and can be utilized as an efficient indicator for the adaptive refinement technique. Received 10 January 2000     

Automatic monitoring of element shape quality in 2-D and 3-D computational mesh dynamics

One of the major problems in fluid–structure interaction using the arbitrary Lagrangian Eulerian approach lies in the area of dynamic mesh generation. For accurate fluid-dynamic computations, meshes must be generated at each time step. The fluid mesh must be regenerated in the deformed fluid domain in order to account for the displacements of the elastic body computed by the structural dynamics solver. In the elasticity-based computational dynamic mesh procedure, the fluid mesh is modeled as a pseudo-elastic solid the deformation of which is based on the displacement boundary conditions, resulting from the solution of the computational structural dynamics problem. This approach has a distinct advantage over other mesh-movement algorithms in that it is a very general, physically based approach that can be applied to both structured and unstructured meshes. The major drawback of the linear elastostatic solver is that it does not guarantee the absence of severe element distortion. This paper describes a novel mesh-movement procedure for mesh quality control of 2-D and 3-D dynamic meshes based on solving a pseudo-nonlinear elastostatic problem. An inexpensive distortion measure for different types of elements is introduced and used for controlling the element shape quality. The mesh-movement procedure is illustrated with several examples (large-displacement and free-boundary problems) that highlight its advantages in terms of performance, mesh quality, and robustness. It is believed that the resulting scheme will result in a more economical simulation of the motion of complex geometry, 3-D elastic bodies immersed in temporally and spatially evolving flows. Received 20 April 2000     

Simulation and verification of particle flow with an elastic collision by the immersed edge-based smoothed finite element method

The immersed edge-based smoothed finite element method (IES-FEM) is proposed for the study of elastic collision particulate flow. Particle collision becomes more realistic by using the penalty function and the hyperelastic constitutive model. The effects of grid resolution and Reynolds numbers on particle terminal velocity and drag coefficient are discussed to verify the calculation accuracy and stability. Single-particle collisions with the bottom and side walls are analyzed and experimentally verified. Results show that the calculation error of IES-FEM is less than 0.6% when the fluid grid size is 0.5 times the particle mesh size and the time step is 10^–4 s. Particle drag coefficient and flow characteristics agree well with the published models and experiment results. To demonstrate the capabilities of IES-FEM in complex elastic particle systems, the collision and rebound of multiple particles are determined, including the drafting–kissing–tumbling of two circular particles; the chase, collision, and deformation of rectangular particles; and the repeated formation and separation of particle clusters. This work extends the application of IES-FEM in particle-resolved direct numerical simulation methods, which will provide an optional tool for future elastic blood cell flow and collision.     

3D adaptive RKPM method for contact problems with elastic–plastic dynamic large deformation

The adaptive procedure of reproducing kernel particle method (RKPM) for 3D contact problems with elastic–plastic dynamic large deformation is presented. In this study, a modified cell energy error (MCEE) estimate model is constructed to capture the high gradients of stresses behavior in large deformation. Refinement particles with a new proper refinement function are inserted into the high error distribution regions. A domain decomposition method is proposed to determine the support domain size for nodes. A collocation formulation is used in the discretization of the boundary integral of the contact constraint equations formulated by a penalty method. By the use of a particle-to-segment contact algorithm, the contact constraints are imposed directly on the new added contact nodes, consequently the contact forces and their associated stiffness matrices are formulated at the nodal coordinate. For verification of the simulation results, a general benchmark test is applied to justify the accuracy and efficiency of the adaptive RKPM method. Several numerical examples are provided to illustrate the effectiveness and robustness of the suggested approach.