An improvement for tensile instability in smoothed particle hydrodynamics

A corrective Smoothed-Particle Method (CSPM) is proposed to address the tensile instability and, boundary deficiency problems that have hampered full exploitation of standard smoothed particle hydrodynamics (SPH). The results from applying this algorithm to the 1-D bar and 2-D plane stress problems are promising. In addition to the advantage of being a gridless Lagrangian approach, improving the above two major obstacles in standard SPH makes it attractive for applications in computational mechanics.     

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.     

Semi-Lagrangian reproducing kernel particle method for fragment-impact problems

Fragment-impact problems exhibit excessive material distortion and complex contact conditions that pose considerable challenges in mesh based numerical methods such as the finite element method (FEM). A semi-Lagrangian reproducing kernel particle method (RKPM) is proposed for fragment-impact modeling to alleviate mesh distortion difficulties associated with the Lagrangian FEM and to minimize the convective transport effect in the Eulerian or Arbitrary Lagrangian Eulerian FEM. A stabilized non-conforming nodal integration with boundary correction for the semi-Lagrangian RKPM is also proposed. Under the framework of semi-Lagrangian RKPM, a kernel contact algorithm is introduced to address multi-body contact. Stability analysis shows that temporal stability of the kernel contact algorithm is related to the velocity gradient between two contacting bodies. The performance of the proposed methods is examined by numerical simulation of penetration processes.     

Modeling mass loading effects in industrial cyclones by a combined Eulerian–Lagrangian approach

Cyclone separation is studied by means of numerical simulations. While the gas flow is modeled by a modified Reynolds stress (RS) model, the behavior of the particles is pictured by a combined Eulerian–Lagrangian approach. A mono-disperse Eulerian particle phase is utilized to account for inter-particle collisions, while the effects of fractional separation and particle-wall collisions are considered by poly-disperse Lagrangian particles. The above particle models interact in two ways. On the one hand, the Lagrangian particles determine the local mean diameter of the substitute Eulerian particle class. On the other hand, especially in regions of high particle concentration, the Eulerian particle phase exerts an additional collisional force onto the Lagrangian particle trajectories. An industrial cyclone is chosen as a test case and the numerical results are evaluated with respect to pressure drop as well as to global and fractional separation efficiency. In this context the influence of the cyclone’s mass loading and wall roughness is highlighted. Simulations indicate that the separation efficiency improves with increasing mass loading until an excess loading is reached while at the same time the pressure drop is reduced. Furthermore, it can be shown that rough walls lead to a reduction of separation efficiency while simultaneously the pressure drop decreases. The simulations results are compared with both an analytic theory of Muschelknautz [Die Berechnung von Zykonabscheidern für Gase. Chem Ing Techn 44, (1+2):63–71, 1972] as well as with real plant measurements.     

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.     

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.     

Simulation of an orifice scrubber performance based on Eulerian/Lagrangian method

A mathematical model based on Eulerian/Lagrangian method has been developed to predict particle collection efficiency from a gas stream in an orifice scrubber. This model takes into account Eulerian approach for particle dispersion, Lagrangian approach for droplet movement and particle-source-in-cell (PSI-CELL) model for calculating droplet concentration distribution. In order to compute fluid velocity profiles, the normal k− turbulent flow model with inclusion of body force due to drag force between fluid and droplets has been used. Experimental data of Taheri et al. [J. Air Pollut. Control Assoc. 23 (11) (1973) 963] have been used to test the results of the mathematical model. The results from the model are in good agreement with the experimental data. After validating the model the effect of operating parameters such as liquid to gas flow rate ratio, gas velocity at orifice opening, and particle diameter were obtained on the collection efficiency.     

Simulation of 2D wedge impacts on water using the SPH–ALE method

Prediction of the forces on pitching boat hulls is a subtle issue, since the underlying phenomena are highly dynamic and unsteady. The solution of a full 3D model of the hull is possible, but comes with a high computational cost; alternatively, it might be simpler to assume the hull as consecutive 2D slices. The traditional CFD approach, with mesh-based algorithms, introduces additional complexity due to mesh motion and interface tracking. By adopting a Lagrangian point of view, the computational algorithm would be substantially simpler, since free surface and moving geometries would be handled easily. In the current work, the smoothed particle hydrodynamics meshless method is used within an arbitrary Lagrangian–Eulerian framework (SPH–ALE), in order to predict the force and motion of a wedge during high-velocity water entry. Various impacts have been simulated, using wedges of different shapes and masses. Two methods of particle refinement have also been examined, in order to increase the accuracy near the point of impact. Results of the method are compared with experimental and numerical results from literature, showing good agreement.     

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.     

无网格MPM法在冲击载荷问题中的应用

物质点法MPM(MaterialPointMethod)是无网格方法之一,它是在质点网格法(PIC)基础上发展而来的一种新数值方法,它利用了欧拉法和拉格朗日法两者的优点,仅简单地划分背景网格,物质点信息通过形函数映射到背景网格节点,然后根据相应的控制方程和本构关系,计算物质点在冲击载荷下的应力和应变,通过物质点来跟踪材料体的变形和破损,而在整个计算过程中背景网格始终固定不变,避免了重新划分网格。系统地介绍了MPM法的基本理论,同时给出了使用MPM法来解决冲击载荷问题的两个具体计算实例。     

Stress integration and mesh refinement for large deformation in geomechanics

This paper first discusses alternative stress integration schemes in numerical solutions to large‐ deformation problems in hardening materials. Three common numerical methods, i.e. the total‐Lagrangian (TL), the updated‐Lagrangian (UL) and the arbitrary Lagrangian–Eulerian (ALE) methods, are discussed. The UL and the ALE methods are further complicated with three different stress integration schemes. The objectivity of these schemes is discussed. The ALE method presented in this paper is based on the operator‐split technique where the analysis is carried out in two steps; an UL step followed by an Eulerian step. This paper also introduces a new method for mesh refinement in the ALE method. Using the known displacements at domain boundaries and material interfaces as prescribed displacements, the problem is re‐analysed by assuming linear elasticity and the deformed mesh resulting from such an analysis is then used as the new mesh in the second step of the ALE method. It is shown that this repeated elastic analysis is actually more efficient than mesh generation and it can be used for general cases regardless of problem dimension and problem topology. The relative performance of the TL, UL and ALE methods is investigated through the analyses of some classic geotechnical problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A fast high‐resolution algorithm for linear convection problems: particle transport method

The paper is devoted to a novel explicit technique, the particle transport method (PTM), for solving linear convection problems. While being a Lagrangian (characteristic based) method, PTM has the advantage of Eulerian methods to represent the solution on a fixed mesh. The proposed approach belongs to the class of monotone high‐resolution numerical schemes, possesses the property of unconditional stability and works with structured and unstructured meshes. It is also demonstrated that the method has a linear computational complexity. The performance of the presented algorithm is tested on one‐ and two‐dimensional benchmark problems. The numerical results confirm that the method has the 2nd‐order spatial accuracy and can be significantly faster than the grid‐based methods of the same order. Copyright © 2006 John Wiley & Sons, Ltd.     

Source terms in Eulerian–Lagrangian contaminant transport simulation

Standard Eulerian treatment of source terms in Eulerian–Lagrangian numerical simulations results in poor performance at higher Courant numbers. To regain the customary high accuracy of Eulerian–Lagrangian methods under these conditions, a Lagrangian treatment of source terms is needed. It is also important to include the effects of fluid sources as well as contaminant sources. A new Lagrangian source formulation is presented, which has been implemented in a finite element simulator for contaminant transport in rivers and estuaries. Test problems demonstrate the high accuracy of the technique under a range of conditions, and its applicability to general multi‐dimensional problems and unstructured grids. Copyright © 2004 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.     

Mesh motion techniques for the ALE formulation in 3D large deformation problems

Efficient mesh motion techniques are a key issue to achieve satisfactory results in the arbitrary Lagrangian–Eulerian (ALE) finite element formulation when simulating large deformation problems such as metal‐forming. In the updated Lagrangian (UL) formulation, mesh and material movement are attached and an excessive mesh distortion usually appears. By uncoupling mesh movement from material movement, the ALE formulation can relocate the mesh to avoid distortion. To facilitate the calculation process, the ALE operator is split into two steps at each analysis time step: UL step (where deformation due to loading is calculated without convective terms) and Eulerian step (where mesh motion is applied). In this work, mesh motion is performed by new nodal relocation methods, developed for eight‐node hexahedral elements, which can move internal and boundary nodes, improving and concentrating the mesh in critical zones. After mesh motion, data is transferred from the UL mesh to the relocated mesh using an expansion of stresses in a Taylor's series. Two numerical applications are presented, comparing results of UL and ALE formulation with results found in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

绝对节点坐标列式流体单元建模方法及在液体晃动问题中的研究

计算流体力学的建模方法主要采用欧拉描述,而多体系统动力学的建模方法主要采用拉格朗日描述。与欧拉描述关注于流过空间固定点或固定体积上的流场状态不同,绝对节点坐标列式流体单元采用拉格朗日描述,能够跟踪流体物质点,建立流体与多体系统的统一描述。该文在绝对节点坐标列式流体单元方法基础上,提出和完善单元建模理论,并使用绝对节点坐标列式流体单元实现了对流体系统的建模,并首次将绝对节点坐标列式流体单元应用于液体晃动分析,初步验证了理论的正确性和可行性。仿真结果表明,在单元数量较少情况下,绝对节点坐标列式流体单元可以满足晃动计算需求。     

Numerical simulation of friction stir welding by natural element methods

In this work we address the problem of numerically simulating the Friction Stir Welding process. Due to the special characteristics of this welding method (i.e., high speed of the rotating pin, very large deformations, etc.) finite element methods (FEM) encounter several difficulties. While Lagrangian simulations suffer from mesh distortion, Eulerian or Arbitrary Lagrangian Eulerian (ALE) ones still have difficulties due to the treatment of convective terms, the treatment of the advancing pin, and many others. Meshless methods somewhat alleviate these problems, allowing for an updated Lagrangian framework in the simulation. Accuracy is not affected by mesh distortion (and hence the name meshless), but the price to pay is the computational cost, higher than in the FEM. The method used here, the Natural Element Method (NEM), presents some interesting characteristics, such as the ease of imposition of essential boundary conditions and coupling with FEM codes. Even more, since the method is formulated in a Lagrangian setting, it is possible to track the evolution of any material point during the process and also to simulate the Friction Stir Welding (FSW) of two slabs of different materials. The examples shown in this paper cover some of the difficulties related with the simulation of the FSW process: very large deformations, complex nonlinear and strongly coupled thermomechanical behaviour of the material and mixing of different materials.     

A PARTICLE TRACKING TECHNIQUE FOR THE LAGRANGIAN–EULERIAN FINITE ELEMENT METHOD IN MULTI-DIMENSIONS

This paper presents a multi-dimensional particle tracking technique for applying the Lagrangian–Eulerian finite element method to solve transport equations in transient-state simulations. In the Lagrangian– Eulerian approach, the advection term is handled in the Lagrangian step so that the associated numerical errors can be considerably reduced. It is important to have an adequate particle tracking technique for computing advection accurately in the Lagrangian step. The particle tracking technique presented here is designed to trace fictitious particles in the real-world flow field where the flow velocity is either measured or computed at a limited number of locations. The technique, named ‘in-element’ particle tracking, traces fictitious particles on an element-by-element basis. Given a velocity field, a fictitious particle is traced one element by one element until either a boundary is encountered or the available time is completely consumed. For the tracking within an element, the element is divided into a desired number of subelements with the interpolated velocity computed at all nodes of the subelements. A fictitious particle, thus, is traced one subelement by one subelement within the element. The desired number of subelements can be determined based on the complexity of the flow field being considered. The more complicated the flow field is, the more subelements are needed to achieve accurate particle tracking results. A single-velocity approach can be used to efficiently perform particle tracking in a smooth flow field, while an average-velocity approach can be employed to increase the tracking accuracy for more complex flow fields.     

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     

Direct numerical simulation of particulate flows with heat transfer in a rotating cylindrical cavity

The purpose of this work was the direct numerical simulation of heat and fluid flow by granular mixing in a horizontal rotating kiln. To model particle behaviour and the heat and fluid flow in the drum, we solve the mass conservation, momentum and energy conservation equations directly on a fixed Eulerian grid for the whole domain including particles. At the same time the particle dynamics and their collisions are solved on a Lagrangian grid for each particle. To calculate the heat transfer inside the particles we use two models: the first is the direct solution of the energy conservation equation in the Lagrangian and Eulerian space, and the second is our so-called linear model that assumes homogeneous distribution of the temperature inside each particle. Numerical simulations showed that, if the thermal diffusivity of the gas phase significantly exceeds the same parameter of the particles, the linear model overpredicts the heating rate of the particles. The influence of the particle size and the angular velocity of the drum on the heating rates of particles is studied and discussed.