Moving‐least‐squares‐particle hydrodynamics—I. Consistency and stability

The Smooth‐Particle‐Hydrodynamics (SPH) method is derived in a novel manner by means of a Galerkin approximation applied to the Lagrangian equations of continuum mechanics as in the finite‐element method. This derivation is modified to replace the SPH interpolant with the Moving‐Least‐Squares (MLS) interpolant of Lancaster and Saulkaskas, and define a new particle volume which ensures thermodynamic compatibility. A variable‐rank modification of the MLS interpolants which retains their desirable summation properties is introduced to remove the singularities that occur when divergent flow reduces the number of neighbours of a particle to less than the minimum required. A surprise benefit of the Galerkin SPH derivation is a theoretical justification of a common ad hoc technique for variable‐h SPH. The new MLSPH method is conservative if an anti‐symmetric quadrature rule for the stiffness matrix elements can be supplied. In this paper, a simple one‐point collocation rule is used to retain similarity with SPH, leading to a non‐conservative method. Several examples document how MLSPH renders dramatic improvements due to the linear consistency of its gradients on three canonical difficulties of the SPH method: spurious boundary effects, erroneous rates of strain and rotation and tension instability. Two of these examples are non‐linear Lagrangian patch tests with analytic solutions with which MLSPH agrees almost exactly. The examples also show that MLSPH is not absolutely stable if the problems are run to very long times. A linear stability analysis explains both why it is more stable than SPH and not yet absolutely stable and an argument is made that for realistic dynamic problems MLSPH is stable enough. The notion of coherent particles, for which the numerical stability is identical to the physical stability, is introduced. The new method is easily retrofitted into a generic SPH code and some observations on performance are made. Copyright © 1999 John Wiley & Sons, Ltd.     

A Lagrangian gradient smoothing method for solid‐flow problems using simplicial mesh

A novel Lagrangian gradient smoothing method (L‐GSM) is developed to solve “solid‐flow” (flow media with material strength) problems governed by Lagrangian form of Navier‐Stokes equations. It is a particle‐like method, similar to the smoothed particle hydrodynamics (SPH) method but without the so‐called tensile instability that exists in the SPH since its birth. The L‐GSM uses gradient smoothing technique to approximate the gradient of the field variables, based on the standard GSM that was found working well with Euler grids for general fluids. The Delaunay triangulation algorithm is adopted to update the connectivity of the particles, so that supporting neighboring particles can be determined for accurate gradient approximations. Special techniques are also devised for treatments of 3 types of boundaries: no‐slip solid boundary, free‐surface boundary, and periodical boundary. An advanced GSM operation for better consistency condition is then developed. Tensile stability condition of L‐GSM is investigated through the von Neumann stability analysis as well as numerical tests. The proposed L‐GSM is validated by using benchmarking examples of incompressible flows, including the Couette flow, Poiseuille flow, and 2D shear‐driven cavity. It is then applied to solve a practical problem of solid flows: the natural failure process of soil and the resultant soil flows. The numerical results are compared with theoretical solutions, experimental data, and other numerical results by SPH and FDM to evaluate further L‐GSM performance. It shows that the L‐GSM scheme can give a very accurate result for all these examples. Both the theoretical analysis and the numerical testing results demonstrate that the proposed L‐GSM approach restores first‐order accuracy unconditionally and does not suffer from the tensile instability. It is also shown that the L‐GSM is much more computational efficient compared with SPH, especially when a large number of particles are employed in simulation.     

Research and Improvement on the Accuracy of Discontinuous Smoothed Particle Hydrodynamics (DSPH) Method

Discontinuous smoothed particle hydrodynamics (DSPH) method based on traditional SPH method, which can be used to simulate discontinuous physics problems near interface or boundary. Previous works showed that DSPH method has a good application prospect [Xu et al, 2013], but further verification and improvement are demanded. In this paper, we investigate the accuracy of DSPH method by some numerical models. Moreover, to improve the accuracy of DSPH method, first order and second order multidimensional RDSPH methods are proposed by following the idea of restoring particle consistency in SPH (RSPH) method which has shown good results in the improvement of particle consistency and accuracy for non-uniform particles. This restoring particle consistency in DSPH (RDSPH) method has the advantages from both RSPH method and DSPH method. In addition, the accuracy of RDSPH methods near the interface, boundary and in non-uniform interior region are tested in one-dimensional and twodimensional spaces.     

On boundary conditions in the element-free Galerkin method

 Accurate imposition of essential boundary conditions in the Element Free Galerkin (EFG) method often presents difficulties because the Moving Least Squares (MLS) interpolants, used in this method, lack the delta function property of the usual finite element or boundary element method shape functions. A simple and logical strategy, for alleviating the above problem, is proposed in this paper. A discrete norm is typically minimized in the EFG method in order to obtain certain variable coefficients. The strategy proposed in this work involves a new definition of this discrete norm. This new strategy works very well in all the numerical examples, for 2-D potential problems, that are presented here. In addition to the discussion of boundary conditions, some recommendations are also made in this paper regarding strategies for refinements in order to improve the accuracy of numerical solutions from the EFG method.     

Improved element-free Galerkin method for two-dimensional potential problems

Potential difficulties arise in connection with various physical and engineering problems in which the functions satisfy a given partial differential equation and particular boundary conditions. These problems are independent of time and involve only space coordinates, as in Poisson's equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, they usually cannot be solved with analytical solutions. The element-free Galerkin (EFG) method is a meshless method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by using the weighted orthogonal basis function to construct the MLS interpolants, we derive the formulae of an improved EFG (IEFG) method for two-dimensional potential problems. There are fewer coefficients in the improved MLS (IMLS) approximation than in the MLS approximation, and in the IEFG method fewer nodes are selected in the entire domain than in the conventional EFG method. Hence, the IEFG method should result in a higher computing speed.     

Moving least‐square interpolants in the hybrid particle method

The hybrid particle method (HPM) is a particle‐based method for the solution of high‐speed dynamic structural problems. In the current formulation of the HPM, a moving least‐squares (MLS) interpolant is used to compute the derivatives of stress and velocity components. Compared with the use of the MLS interpolant at interior particles, the boundary particles require two additional treatments in order to compute the derivatives accurately. These are the rotation of the local co‐ordinate system and the imposition of boundary constraints, respectively. In this paper, it is first shown that the derivatives found by the MLS interpolant based on a complete polynomial are indifferent to the orientation of the co‐ordinate system. Secondly, it is shown that imposing boundary constraints is equivalent to employing ghost particles with proper values assigned at these particles. The latter can further be viewed as placing the boundary particle in the centre of a neighbourhood that is formed jointly by the original neighbouring particles and the ghost particles. The benefit of providing a symmetric or a full circle of neighbouring points is revealed by examining the error terms generated in approximating the derivatives of a Taylor polynomial by using a linear‐polynomial‐based MLS interpolant. Symmetric boundaries have mostly been treated by using ghost particles in various versions of the available particle methods that are based on the strong form of the conservation equations. In light of the equivalence of the respective treatments of imposing boundary constraints and adding ghost particles, an alternative treatment for symmetry boundaries is proposed that involves imposing only the symmetry boundary constraints for the HPM. Numerical results are presented to demonstrate the validity of the proposed approach for symmetric boundaries in an axisymmetric impact problem. Copyright © 2005 John Wiley & Sons, Ltd.     

The SPH equations for fluids

Existing smoothed particle hydrodynamics (SPH) formulations for simulating continuous fluids have errors that may be divergent and it has been known for some time that the SPH equations do not satisfy low‐order polynomial completeness conditions. Here SPH equations are derived that have convergent error terms and a correction method is presented for enforcing low‐order polynomial completeness irrespective of how many completeness conditions are required. Discretization is achieved through division of the model domain, in its initial state, into sub‐domains that have Lagrangian boundaries. It is shown that boundary integrals appearing in one derivation of the SPH equations may be treated as a convergent error. In simulations of basic fluid flows convergence and zeroth‐order completeness are demonstrated, but significant instabilities and a failure to conserve energy are observed. Copyright © 2009 John Wiley & Sons, Ltd.     

A semi-implicit stabilized particle Galerkin method for incompressible free surface flow simulations

A new particle Galerkin method is introduced to solve the Naiver-Stokes equations in a Lagrangian fashion. The present method aims to suppress key numerical instabilities observed in the strong form Lagrangian particle methods such as smoothed particle hydrodynamics (SPH), incompressible SPH, and moving particle semi-implicit for incompressible free surface flow simulations. It is well-known that strong form Lagrangian particle methods usually rely on ad hoc particle stabilization techniques based on particle shifting, artificial viscosity, or density-invariant condition due to some formulation inconsistency issues. In the present method, we introduce a momentum-consistent velocity smoothing algorithm which is used to combine with the second-order rotational incremental pressure-correction scheme to stabilize the pressure field as well as to enforce the consistency of Neumann boundary condition. To further impose slip-free or nonslip boundary conditions for the fluid flow, a penalty method which is free of ghost or dummy particles is developed. Finally, a particle insertion-deletion adaptive scheme is proposed when the violent fluid flow is considered. Four numerical examples are studied to validate the accuracy and stability of the present method.     

Simulation of high velocity concrete fragmentation using SPH/MLSPH

The simulation of concrete fragmentation under explosive loading by a meshfree Lagrangian method, the smooth particle hydrodynamics method (SPH) is described. Two improvements regarding the completeness of the SPH‐method are examined, first a normalization developed by Johnson and Beissel (NSPH) and second a moving least square (MLS) approach as modified by Scheffer (MLSPH). The SPH‐Code is implemented in FORTRAN 90 and parallelized with MPI. A macroscopic constitutive law with isotropic damage for fracture and fragmentation for concrete is implemented in the SPH‐Code. It is shown that the SPH‐method is able to simulate the fracture and fragmentation of concrete slabs under contact detonation. The numerical results from the different SPH‐methods are compared with the data from tests. The good agreement between calculation and experiment suggests that the SPH‐program can predict the correct maximum pressure as well as the damage of the concrete slabs. Finally the fragment distributions of the tests and the numerical calculations are compared. Copyright © 2003 John Wiley & Sons, Ltd.     

The boundary node method for three‐dimensional problems in potential theory

The boundary node method (BNM) is developed in this paper for solving potential problems in three dimensions. The BNM represents a coupling between boundary integral equations (BIE) and moving least‐squares (MLS) interpolants. The main idea here is to retain the dimensionality advantage of the former and the meshless attribute of the later. This results in decoupling of the ‘mesh’ and the interpolation procedure for the field variables. A general BNM computer code for 3‐D potential problems has been developed. Several parameters involved in the BNM need to be chosen carefully for a successful implementation of the method. An in‐depth and systematic study has been carried out in this paper in order to better understand the effects of various parameters on the performance of the method. Numerical results for spheres and cubes, subjected to different types of boundary conditions, are extremely encouraging. Copyright © 2000 John Wiley & Sons, Ltd.     

An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions

The smoothed particle hydrodynamics (SPH) method has proven useful for modeling large deformation of fluids including fluids with stress‐free surfaces. Because of the Lagrangian nature of the method, it is well suited to address the thermal evolution of these free surface flows. Boundary conditions at the interface of the fluid with a solid wall are usually enforced through the use of boundary particles. However, applying conditions at free surfaces, in particular gradient boundary conditions, can be problematic with traditional SPH formulations due to the degradation of the gradient approximation in these regions. Compounding this difficulty is that traditional approximations of the Laplacian operator suffer a similar degradation near free surfaces. A new SPH formulation of the Laplacian operator is presented, which improves the accuracy near free surface boundaries. This new form is based on a gradient approximation commonly used in thermal, viscous, and pressure projection problems, but includes higher‐order terms in the appropriate Taylor series. Comparisons with other approximations of second‐order derivatives are given. The discretization is tested by solving steady‐state and transient problems of thermal diffusion using the Backward Euler method with a GMRES solver. Boundary conditions are imposed through an augmented matrix. Copyright © 2008 John Wiley & Sons, Ltd.     

Analyzing three-dimensional potential problems with the improved element-free Galerkin method

The potential problem is one of the most important partial differential equations in engineering mathematics. A potential problem is a function that satisfies a given partial differential equation and particular boundary conditions. It is independent of time and involves only space coordinates, as in Poisson’s equation or the Laplace equation with Dirichlet, Neumann, or mixed conditions. When potential problems are very complex, both in their field variable variation and boundary conditions, they usually cannot be solved by analytical solutions. The element-free Galerkin (EFG) method is a promising method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by employing improved moving least-squares (IMLS) approximation, we derive the formulas for an improved element-free Galerkin (IEFG) method for three-dimensional potential problems. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method, the IEFG method should result in a higher computing speed.     

SPH with the multiple boundary tangent method

In this article, we present an improved solid boundary treatment formulation for the smoothed particle hydrodynamics (SPH) method. Benchmark simulations using previously reported boundary treatments can suffer from particle penetration and may produce results that numerically blow up near solid boundaries. As well, current SPH boundary approaches do not properly treat curved boundaries in complicated flow domains. These drawbacks have been remedied in a new boundary treatment method presented in this article, called the multiple boundary tangent (MBT) approach. In this article we present two important benchmark problems to validate the developed algorithm and show that the multiple boundary tangent treatment produces results that agree with known numerical and experimental solutions. The two benchmark problems chosen are the lid‐driven cavity problem, and flow over a cylinder. The SPH solutions using the MBT approach and the results from literature are in very good agreement. These solutions involved solid boundaries, but the approach presented herein should be extendable to time‐evolving, free‐surface boundaries. Copyright © 2008 John Wiley & Sons, Ltd.     

SPH方法在剪切式碰撞能量吸收器中的应用

剪切式碰撞能量吸收器是提高汽车被动安全性的一种实用新型设计。建立了剪切式碰撞能量吸收器数值分析模型,采用SPH方法数值模拟了碰撞吸能器的碰撞过程,研究了其碰撞吸能特性,并通过不同碰撞速度的台车碰撞试验进行了试验验证研究。数值模拟结果同试验结果相符,表明SPH方法在碰撞吸能器性能研究中是行之有效的数值计算方法。同时分析了在碰撞吸能器的复杂碰撞情况下,SPH方法相比有限元法的优势。     

Remarks on tension instability of Eulerian and Lagrangian corrected smooth particle hydrodynamics (CSPH) methods

The paper discusses the problem of tension instability of particle‐based methods such as smooth particle hydrodynamics (SPH) or corrected SPH (CSPH). It is shown that tension instability is a property of a continuum where the stress tensor is isotropic and the value of the pressure is a function of the density or volume ratio. The paper will show that, for this material model, the non‐linear continuum equations fail to satisfy the stability condition in the presence of tension. Consequently, any discretization of this continuum will result in negative eigenvalues in the tangent stiffness matrix that will lead to instabilities in the time integration process. An important exception is the 1‐D case where the continuum becomes stable but SPH or CSPH can still exhibit negative eigenvalues. The paper will show that these negative eigenvalues can be eliminated if a Lagrangian formulation is used whereby all derivatives are referred to a fixed reference configuration. The resulting formulation maintains the momentum preservation properties of its Eulerian equivalent. Finally a simple 1‐D wave propagation example will be used to demonstrate that a stable solution can be obtained using Lagrangian CSPH without the need for any artificial viscosity. Copyright © 2001 John Wiley & Sons, Ltd.     

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。     

Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations

Smooth particle hydrodynamics (SPH) is a robust and conceptually simple method which suffers from unsatisfactory performance due to lack of consistency. The kernel function can be corrected to enforce the consistency conditions and improve the accuracy. For simplicity in this paper the SPH method with the corrected kernel is referred to as corrected smooth particle hydrodynamics (CSPH). The numerical solutions of CSPH can be further improved by introducing an integration correction which also enables the method to pass patch tests. It is also shown that the nodal integration of this corrected SPH method suffers from spurious singular modes. This spatial instability results from under integration of the weak form, and it is treated by a least‐squares stabilization procedure which is discussed in detail in Section 4. The effects of the stabilization and improvement in the accuracy are illustrated via examples. Further, the application of CSPH method to metal‐forming simulations is discussed by formulating the governing equation associated with the process. Finally, the numerical examples showing the effectiveness of the method in simulating metal‐forming problems are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Reproducing kernel particle methods for structural dynamics

This paper explores a Reproducing Kernel Particle Method (RKPM) which incorporates several attractive features. The emphasis is away from classical mesh generated elements in favour of a mesh free system which only requires a set of nodes or particles in space. Using a Gaussian function or a cubic spline function, flexible window functions are implemented to provide refinement in the solution process. It also creates the ability to analyse a specific frequency range in dynamic problems reducing the computer time required. This advantage is achieved through an increase in the critical time step when the frequency range is low and a large window is used. The stability of the window function as well as the critical time step formula are investigated to provide insight into RKPMs. The predictions of the theories are confirmed through numerical experiments by performing reconstructions of given functions and solving elastic and elastic–plastic one-dimensional (1-D) bar problems for both small and large deformation as well as three 2-D large deformation non-linear elastic problems. Numerical and theoretical results show the proposed reproducing kernel interpolation functions satisfy the consistency conditions and the critical time step prediction; furthermore, the RKPM provides better stability than Smooth Particle Hydrodynamics (SPH) methods. In contrast with what has been reported in SPH literature, we do not find any tensile instability with RKPMs.     

Analyzing interaction between coplanar square cracks using an efficient boundary element‐free method

This paper examines the interaction between coplanar square cracks by combining the moving least‐squares (MLS) approximation and the derived boundary integral equation (BIE). A new traction BIE involving only the Cauchy singular kernels is derived by applying integration by parts to the traditional boundary integral formulation. The new traction BIE can be directly applied to a crack surface and no displacement BIE is necessary because all crack boundary conditions (both upper and lower ones) are incorporated. A boundary element‐free method is then developed by combining the derived BIE and MLS approximation, in which the crack opening displacement is first expressed as the product of weight functions and the characteristic terms, and the unknown weight is approximated with the MLS approximation. The efficiency of the developed method is tested for isotropic and transversely isotropic media. The interaction between two and three coplanar square cracks in isotropic elastic body is numerically studied and the case of any number of coplanar square cracks is deduced and discussed. Copyright © 2012 John Wiley & Sons, Ltd.