Corrected Smooth Particle Hydrodynamics Research Articles

A high-efficient accurate coupled mesh-free scheme for 2D/3D space-fractional convection-diffusion/Burgers' problems

This paper first develops a high-efficient accurate mesh-free scheme (CSPH-SFCD) for multi-dimensional time-dependent spatial fractional convection-diffusion (SFCD) equations based on the Riemann-Liouville (RL) derivative in convex domain by coupling a corrected smoothed particle hydrodynamics (CSPH) method with a numerical integral of RL. Subsequently, for the first application, the proposed CSPH-SFCD coupled with up-wind (UW) scheme (CSPH-SFCD-UW) is tentatively extended to investigate the nonlinear shock-wave phenomena in 2D space-fractional Burgers' problem (SFBP) with fractional Neumann boundary at high Reynolds number. The proposed mesh-free scheme for time-dependent SFCD/SFBP with fractional boundary is mainly motived by: (a) the SFCD model with RL fractional calculus is discretized by employing a numerical integral formula and a CSPH scheme, and then a conditionally stable CSPH-SFCD method is first derived; (b) the UW scheme is introduced to stabilize the numerical investigation of convection-dominated SFBP; (c) the fractional Neumann boundary (FNB) condition can be easily enforced in the proposed discretized scheme, and the integer-order NB condition is treated by employing virtual particles interpolation technique; (d) the MPI parallel computing technique is adopted to enhance the computational efficiency. Firstly, the numerical convergence and stability of the proposed scheme are discussed and illustrated by solving several multi-dimensional benchmarks. Secondly, the cases of two-sided and left-sided RL fractional derivatives are also discussed and demonstrated in the numerical tests. The advantages of the proposed method are demonstrated by simulating the 2D/3D SFCD equations in convex domains or under non-uniform point distribution. Finally, the 2D two-component SFBP with FNBC are numerically investigated by using the proposed method, and compared with the FDM results. The complex nonlinear non-local properties governed by SFBP are numerically predicted by the proposed scheme. All the numerical results show that the proposed mesh-free method for time-dependent spatial SFCD problems is high-efficient, flexible and accurate. • A high-efficient accurate CSPH-SFCD scheme is first developed to 2D/3D SFCD equations. • The CSPH-SFCD-UW scheme is first proposed to investigate shock-wave behavior in 2D SFBP. • The FNB condition can be easily enforced in the proposed approximated scheme. • The advantages of the present scheme are shown by numerical tests on convex domain.

Numerical investigation of the flow characteristics of Bingham fluid on a slope with corrected smooth particle hydrodynamics

The Bingham model can effectively describe the flow behavior of viscoplastic fluid. It is important to study the flow characteristics of Bingham fluid to understand the dynamic mechanism of viscous debris flow. In this study, the Bingham fluid flow on a slope is numerically researched using a corrected smooth particle hydrodynamics (CSPH) method based on periodic density re-initialization and artificial stress. First, the accuracy and stability of the improved SPH method are verified by the benchmark problem impacting droplets. Then, the flow characteristics of the Bingham fluid on the slope and the influence of the slope inclination angle on the Bingham fluid movement process are studied with the improved SPH method. The numerical results show that the improved SPH numerical scheme has higher accuracy and better stability and can deal with the complex flow behavior of the unsteady Bingham fluid.

An Interactively Corrected Smoothed Particle Hydrodynamics (IC‐SPH) for Simulating Solute Transport in a Nonuniform Velocity Field

AbstractThe Smoothed Particle Hydrodynamics (SPH) method is a Lagrangian approach that has been widely used to eliminate numerical dispersion for solving advection‐dispersion equation (ADE) of groundwater solute transport under advection‐dominated situations. It has been found that accuracy of SPH results is severely deteriorated, when particles are irregularly distributed in a model domain with heterogeneous hydraulic conductivity. To resolve this problem, we developed a new approach called Interactively Corrected SPH (IC‐SPH), which is an improved version of the Corrected SPH (C‐SPH) method. IC‐SPH uses an interactively corrected kernel gradient to construct concentration gradients used to solve ADE. This correction is made for each particle by using not only the particle's neighbor particles within the particle's support domain but also the particles within each neighbor particle's support domain. We evaluated IC‐SPH performance in two numerical studies. One considers diffusive transport with an analytical solution, and the other considers advection‐dispersion transport in a heterogeneous field of hydraulic conductivity. For each numerical study, several numerical experiments were conducted using multiple sets of irregularly distributed particles with different levels of particle irregularity. The numerical experiments indicate that, while IC‐SPH is more computationally expensive than SPH and C‐SPH, IC‐SPH produces more accurate ADE solutions, and converges faster to the analytical solution. IC‐SPH is mathematically general, and can be applied to a wide range of problems that require solving ADE.

Modelling non-local elasticity in 1D vibrating rods using Corrected Smoothed Particle Hydrodynamics method

The present study aims to assess the use of the Corrected Smoothed Particle Hydrodynamics (CSPH) method predicting non-local elastic effects in finite deformations. For this purpose, we first recall the discrete and continuum analytical solutions of a 1D bar under longitudinal harmonic vibrations with small scale effects. SPH is an integral-based non-local approach where the non-locality is introduced through the kernel convolution. Contrary to the classical use of the SPH method, our simulations are conducted with values of the smoothing length parameter that may be not infinitesimal and can be related to the characteristic size of the microstructure.The numerical results obtained from the 1D bar simulations under different boundary conditions demonstrate that the CSPH method can capture the non-local effects in dynamic conditions. In particular, it is shown that the numerical results have a good agreement with available discrete and continuum analytical solutions. Moreover, it is observed that CSPH strain-based and stress-based formulations lead to similar responses. We have also simulated hardening effects due to the increase of the vibration amplitude. Finally, we provide a discussion about the influence of finite and infinite support kernel functions.

A local refinement purely meshless scheme for time fractional nonlinear Schrödinger equation in irregular geometry region* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11501495, 51779215, and 11672259), the Postdoctoral Science Foundation of China (Grant Nos. 2015M581869 and 2015T80589), and the Jiangsu Government Scholarship for Overseas Studies, China (Grant No. JS-2017-227).

A local refinement hybrid scheme (LRCSPH-FDM) is proposed to solve the two-dimensional (2D) time fractional nonlinear Schrödinger equation (TF-NLSE) in regularly or irregularly shaped domains, and extends the scheme to predict the quantum mechanical properties governed by the time fractional Gross–Pitaevskii equation (TF-GPE) with the rotating Bose–Einstein condensate. It is the first application of the purely meshless method to the TF-NLSE to the author’s knowledge. The proposed LRCSPH-FDM (which is based on a local refinement corrected SPH method combined with FDM) is derived by using the finite difference scheme (FDM) to discretize the Caputo TF term, followed by using a corrected smoothed particle hydrodynamics (CSPH) scheme continuously without using the kernel derivative to approximate the spatial derivatives. Meanwhile, the local refinement technique is adopted to reduce the numerical error. In numerical simulations, the complex irregular geometry is considered to show the flexibility of the purely meshless particle method and its advantages over the grid-based method. The numerical convergence rate and merits of the proposed LRCSPH-FDM are illustrated by solving several 1D/2D (where 1D stands for one-dimensional) analytical TF-NLSEs in a rectangular region (with regular or irregular particle distribution) or in a region with irregular geometry. The proposed method is then used to predict the complex nonlinear dynamic characters of 2D TF-NLSE/TF-GPE in a complex irregular domain, and the results from the posed method are compared with those from the FDM. All the numerical results show that the present method has a good accuracy and flexible application capacity for the TF-NLSE/GPE in regions of a complex shape.

An effective pure meshfree method for 1D/2D time fractional convection-diffusion problems on irregular geometry

An easy implemented and effective pure meshfree method is first developed to solve the 1D/2D constant/variable-order time fractional convection-diffusion equation (TF-CDE) on non-regular domain with two boundary conditions in this paper. The proposed method (CSPH-FDM) is derived from that the finite difference scheme (FDM) for Caputo time fractional derivative and a corrected smoothed particle hydrodynamics (CSPH) without kernel derivative for spatial derivatives. In the proposed CSPH-FDM, the high-order spatial derivative is divided into multi first-order derivatives and solved continuously by the CSPH, the Neumann boundary condition can be accurately treated, the two distribution cases of the local refinement and irregular particles or the arbitrary irregular shape domain can be easily and effectively implemented by the CSPH. To demonstrate the validity and numerical convergent order of the proposed method, several 1D/2D analytical examples with local refinement and irregular particles distributions or on complex geometries are first investigated, in which a four-order derivate problem with Neumann boundary is also considered. Subsequently, the CSPH-FDM is extended to predict the solute moving process versus time by two TF-CDEs on different irregular domains and compared with other numerical results. All the numerical results show the flexible application ability and reliability of the present method.

The study on performances of kernel types in solid dynamic problems by smoothed particle hydrodynamics

As the earliest meshless method, smoothed particle hydrodynamics (SPH) has been applied in solid dynamics because of its great potentials in simulating extremely large deformations. In terms of the essence of SPH, it uses a kernel function for numerical approximations. Some studies demonstrate mathematically that the types of the kernel function directly influence the stability and overall accuracy of SPH simulation. However, less attention is paid to the influences derived from kernels, particularly on simulating dynamic solids problems by SPH. In addition, some researches introduce that there are strong relations between SPH kernels and tensile instability which is one of the shortcomings in SPH method, a kind of particles clumping phenomenon probably leading to some unphysical cracks when the solid material is largely stretched or compressed. At present, no method exists to completely avoid these instabilities, although a few corrections like artificial viscosity, artificial stress, corrective SPH or the Godunov-type SPH have been proposed.In this paper, the performances of SPH method with different kernels are studied, including classical types and several new ones proposed recently. Combined with some corrected techniques, the suitability of these kernels in SPH method is discussed in solid dynamic problems like bending deformation of elastic beam and some tests about elastic–plastic deformation in impact problems.

Numerical study of nonlinear Schrödinger equation with high-order split-step corrected smoothed particle hydrodynamics method

To improve the numerical accuracy and computational efficiency of solving high-dimensional nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation by using traditional SPH method, a high-order split-step coupled with a corrected parallel SPH (HSS-CPSPH) method is developed by applying virtual particles to the boundary. The improvements are described as follows. Firstly, the nonlinear Schrödinger equation is divided into linear derivative term and nonlinear term based on the high-order split-step method. Then, the linear derivative term is solved by extending the first-order symmetric SPH method in explicit time integration. Meanwhile, the MPI parallel technique is introduced to enhance the computational efficiency. In this work, the accuracy, convergence and the computational efficiency of the proposed method are tested by solving the NLS equations with the periodic and Dirichlet boundary conditions, and compared with the analytical solutions. Also, the singularity of solitary waves under the periodic boundary condition is accurately obtained using the proposed particle method. Subsequently, the proposed HSS-CPSPH method is used to predict the results of complex two-dimensional and three-dimensioanl GP problems which are compared with other numerical results. The phenomenon of singular sharp angle in the propagation of nonlinear solitary wave and the process of quantum vortex under Bose-Einstein condensates with external rotation are presented accurately.

Predicting the damage on a target plate produced by hypervelocity impact using a decoupled finite particle method

Hypervelocity impact (HVI) is associated with large deformations of structures, phase transitions of materials and scattered debris cloud. It is a great challenge to accurately model the HVI process and predict the damage produced by HVI for conventional numerical methods. In this paper, a recently developed corrective smoothed particle hydrodynamics (SPH) method, decoupled finite particle method (DFPM), is extended to model HVI problems. Validation examples show that DFPM is as flexible as SPH while having much better performance in improving accuracy and removing tensile instability. DFPM is also very attractive for modeling problems with extremely disordered particle distribution (e.g., HVI) as no matrix inversion is required. DFPM is then applied to model the penetration of a sphere on a target plate at various impact velocities and impact angles. It is found that as the impact velocity increases, the hole size increases accordingly and maintains basically constant in a steady state after the impact velocity reaches a critical value. Based on extensive DFPM simulation results, novel formulae are given for predicting the hole size produced by normal and oblique HVIs, which can reproduce experimental data and shows better performance than existing empirical formulae in a wider range of impact velocity.

A decoupled finite particle method for modeling incompressible flows with free surfaces

• A decoupled finite particle method (DFPM) is developed, which avoids solving the corrective matrix equations. • DFPM is an improved SPH method with much better accuracy than conventional SPH. • Particle distribution and the selection of smoothing function/length have little influence on DFPM simulation results. • DFPM is more stable and more efficient than other improved SPH methods with matrix inversion. • DFPM is effective in modeling incompressible flows with free surfaces. Smoothed particle hydrodynamics (SPH) is a meshfree Lagrangian particle method, and it has been applied to different areas in engineering and sciences. One concern of the conventional SPH is its low accuracy due to particle inconsistency, which hinders the further methodology development. The finite particle method (FPM) restores the particle consistency in the conventional SPH and thus significantly improves the computational accuracy. However, as pointwise corrective matrix inversion is necessary, FPM may encounter instability problems for highly disordered particle distribution. In this paper, through Taylor series analyses with integration approximation and assuming diagonal dominance of the resultant corrective matrix, a new meshfree particle approximation method, decoupled FPM (DFPM), is developed. DFPM is a corrective SPH method, and is flexible, cost-effective and easy in coding with better computational accuracy. It is very attractive for modeling problems with extremely disordered particle distribution as no matrix inversion is required. One- and two-dimensional numerical tests with different kernel functions, smoothing lengths and particle distributions are conducted. It is demonstrated that DFPM has much better accuracy than conventional SPH, while particle distribution and the selection of smoothing function and smoothing length have little influence on DFPM simulation results. DFPM is further applied to model incompressible flows including Poiseuille flow, Couette flow, shear cavity and liquid sloshing. It is shown that DFPM is as accurate as FPM while as flexible as SPH, and it is very attractive in modeling incompressible flows with possible free surfaces.

On the kernel and particle consistency in smoothed particle hydrodynamics

The problem of consistency of smoothed particle hydrodynamics (SPH) has demanded considerable attention in the past few years due to the ever increasing number of applications of the method in many areas of science and engineering. A loss of consistency leads to an inevitable loss of approximation accuracy. In this paper, we revisit the issue of SPH kernel and particle consistency and demonstrate that SPH has a limiting second-order convergence rate. Numerical experiments with suitably chosen test functions validate this conclusion. In particular, we find that when using the root mean square error as a model evaluation statistics, well-known corrective SPH schemes, which were thought to converge to second, or even higher order, are actually first-order accurate, or at best close to second order. We also find that observing the joint limit when N→∞, h→0, and n→∞, as was recently proposed by Zhu et al., where N is the total number of particles, h is the smoothing length, and n is the number of neighbor particles, standard SPH restores full C0 particle consistency for both the estimates of the function and its derivatives and becomes insensitive to particle disorder.

Simulation of polymer filling process by an improved smoothed particle hydrodynamics method based on higher-order Taylor expansion

In this work, an improved smoothed particle hydrodynamics (SPH) method based on higher order Taylor expansion (CSPH_HT) is proposed and tentatively applied to the filling process of the viscoelastic fluid. Owing to the disadvantages of the traditional SPH method and the presented corrected SPH methods, the CSPH_HT method based on the higher order Taylor expansion is proposed and described in detail. In order to illustrate the validity and merits of the CSPH_HT method, two benchmark problems are simulated and discussed. The numerical results show that the proposed CSPH_HT method has higher numerical accuracy and better stability. Subsequently, the proposed improved SPH method is extended to simulate the filling processes of the viscoelastic fluid in the ring-shaped mold, for the purpose of exhibiting the capacity of the proposed method. The extended pom-pom (XPP) model and finitely extensible nonlinear elastic-Peterlin (FENE-P) model fluid are all considered in this case, in which the viscoelastic fluid flow and extra stress are shown. The differences in fluid flow between the XPP model and FENE-P model are also discussed. Finally, the filling process of the viscoelastic fluid in the square mold with single inlet or two inlets are tentatively simulated. The differences between the filling process of XPP fluid and the filling process of FENE-P fluid are shown, and the influences of the parameters of the mold on the flow are analyzed. Especially, the influences of locations and sizes of two inlets on the filling process of viscoelastic fluid are illustrated. The XPP model fluid and FENE-P model fluid show different characteristics in the filling process and small change of the size of the mold can lead to obvious change of the flow.

A corrected smoothed particle hydrodynamics approach to solve the non-isothermal non-Newtonian viscous fluid flow problems

In this paper, a corrected smoothed particle hydrodynamics (SPH) method is proposed to solve the problems of non-isothermal non-Newtonian viscous fluid. The proposed particle method is based on the corrected kernel derivative scheme under no kernel derivative and incompressible conditions, which possesses higher accuracy and better stability than the traditional SPH method. Meanwhile, a temperature-discretization scheme is deduced by the concept of SPH method for the purpose of precisely describing the evolutionary process of the temperature field. Reliability of the corrected SPH method for simulating the non-Newtonian viscous fluid flow is demonstrated by simulating the isothermal Poiseuille flow and the jet fluid of filling process; and the validity and accuracy of the proposed SPH discrete scheme in a temperature model for solving the non-isothermal fluid flow are tested by solving the non-isothermal Couette flow and 4:1 contraction flow. Subsequently, the proposed corrected SPH method combined with the SPH temperature-discretization scheme is tentatively extended to include the simulation of the non-isothermal non-Newtonian viscous free-surface flows in the ring-shaped and C-shaped cavities. Especially, the convergence of numerical simulations is analyzed, and the influences of heat flow parameters on the temperature and fluid flow at different positions are discussed.

A new particle method for simulating breakup of liquid jets

A corrected smoothed particle hydrodynamics (CSPH) method for simulating two-phase flows including surface tension is presented. The effects of the instability in the compressional regime and particle deficiency are suppressed by adopting a new smoothing kernel and its gradient corrections. The insensitivity to the compressional instability of the adopted method is confirmed by numerical tests. The method is validated and calibrated through a series of standard numerical tests and showed quantitative agreements. The method is extended to two-dimensional jet breakup problems and provided good comparison to the theoretical and experimental results. In particular, the numerical model exhibits the transition from the jetting to dripping when the Weber number is close to its theoretical critical value.

A corrected smoothed particle hydrodynamics method for solving transient viscoelastic fluid flows

In this work, a corrected smoothed particle hydrodynamics (CSPH) method is proposed and extended to the numerical simulation of transient viscoelastic fluid flows due to that its approximation accuracy in solving the Navier–Stokes equations is higher than that of the smoothed particle hydrodynamics (SPH) method, especially near the boundary of the domain. The CSPH approach comes with the idea of combining the SPH approximation for the interior particles with the modified smoothed particle hydrodynamics (MSPH) method for the exterior particles, this is because that the later method has higher accuracy than the SPH method although it also needs more computational cost. In order to show the validity of CSPH method to simulate unsteady viscoelastic flows problems, the planar shear flow problems, including transient Poiseuille, Couette flow and transient combined Poiseuille and Couette flow for the Oldroyd-B fluid are solved and compared with the analytical and SPH results. Subsequently, the general viscoelastic fluid based on the eXtended Pom–Pom (XPP) model is numerically investigated and the viscoelastic free surface phenomena of impacting drop are simulated by the CSPH for its extended application and the purpose of illustrating the ability of the proposed method. The numerical results are presented and compared with available solutions, which shows a very good agreement. All the numerical results show the higher accuracy and better stability of the CSPH than the SPH, especially for larger Weissenberg numbers.

Numerical modelling of fracture of particulate composites using SPH method

Simplicity of mesh generation and robustness against mesh entanglement during large deformations are key attractive features of particle based methods. These features can be exploited in number of engineering problems where traditional techniques suffer due to aforementioned limitations. Numerical modelling of particulate composites is one of such ideal engineering applications where particle based methods can be effectively used due to their simplicity and robustness. Complicated geometrical configurations of particulate composites obtained from techniques such as scanning electron microscopy (SEM) can be easily converted to particle based mesh without loosing much information. This enables more accurate analysis of the chosen composite materials. Therefore, a smooth particle hydrodynamics (SPH) based numerical technique is developed here to investigate the mechanical properties and evolution of debonding process in particulate composites. To perform the numerical study, a Lagrangian corrected SPH (CSPH) method is presented together with an appropriate numerical model for treating material interface discontinuity within the particulate composites. The material interface discontinuity is enforced using an innovative method which combines penalty formulation with a bilinear interface cohesive model for SPH method. The proposed SPH methodology is used in a number of numerical examples involving composite materials and related interface problems. The effect of penalty value on the interface model and of the smoothing length of the SPH method are also analysed during these simulations. The results illustrate the effectiveness, robustness and potential of the developed methodology. It is concluded that the proposed numerical techniques can be easily and effectively applied to simulate multi-phase composites with various interface conditions and, can provide useful information regarding the inherent mechanism of damage evolution and fracture of particulate or fibre reinforced composites.

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

AbstractThe 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.

Ring upsetting simulation by the meshless method of corrected smooth particle hydrodynamics

Based on the flow formulation for slightly compressible rigid-plastic materials, the process of the axis symmetrical ring upsetting was simulated by the meshless method of corrected smooth particle hydrodynamics (CSPH). The proposal research was focused on a wide range of theoretical and numerical subjects, such as discretization procedures, numerical integration of the system matrices derived from the weak form, geometrical update of the deforming work piece and treatment of the frictional contact interface between the work piece and the die. The overall presentation is supported by upsetting experimental parts resulting from the compression of rings under laboratory conditions. Comparisons of the deformation shape of simulation of CSPH and experiment showed that the agreement between the prediction and experimental results is good.

NUMERICAL SIMULATION OF METAL FORMING PROCESSES BY CORRECTED SMOOTH PARTICLE HYDRODYNAMICS MESHLESS METHODS

Based on the flow formulation of slightly compressible rigid plastic materials a code of bar and rod plane-strain upsetting in metal forming is developed, in which corrected smooth particle hydrodynamics (CSPH) meshless method is employed. A simple method to smoothing technique of calculating volume strain rate is proposed to improve the calculating of stress field. The results of velocity field and stress field by CSPH, finite element method and experiment of pure commercial aluminum and A16060 materials upsetting are analyzed. It is shown that the large deformation in metal forming processes can be solved by CSPH. The method of the CSPH will be a good measure to solve the complicated metal forming further.

Simulation of Bulk Metal Forming by Means of Finite Integral Mesh Free Methods

This paper draws from the fundamentals of the finite integral mesh free representation methods and of the rigid-plastic formulation for slightly compressible materials to the main aspects of computer implementation and modelling of bulk-metal forming processes. Fundamental research and development is based on the reproducing kernel particle method (RKPM) and the corrected smoothed particle hydrodynamics (CSPH) method. Special emphasis is placed on a wide range of theoretical and numerical subjects such as discretization procedures, numerical integration of the system matrices derived from the weak form, relief of volumetric locking, geometrical update of the deforming workpiece, and treatment of the frictional contact interface between the workpiece and tooling.