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Discrete multibody interaction and contact problems and the multiphase interactions such as the sand particles airflow interactions by Aeolian sand transport in the desert are modeled by using the different kernel smoothing lengths in SPH method. Each particle defines a particular kernel smoothing length such as larger smoothing length which is used to calculate continuous homogenous body. Some special smoothing lengths are used to approximate interaction between the discrete particles or objects in contact problems and in different field coupling problem. By introducing the Single Particle Model (SPM) and the Multiparticle Model (MPM), the velocity exchanging phenomena are discussed by using different elastic modules. Some characteristics of the SPM and MPM are evaluated. The results show that the new SPH method can effectively solve different discrete multibody correct contact and multiphase mutual interference problems. Finally, the new SPH numerical computation and simulation process are verified.

Smoothed Particle Hydrodynamics (SPH) is a mesh-free Lagrangian method developed in recent years. Originally, SPH was developed for astrophysics and cosmological applications in three-dimensional open space [

Numerical modeling for discrete multibody interactions and multifield coupling using SPH method still have some challenges. The choice of the influence radius in kernel is very important as it directly affects the accuracy and efficiency of the SPH calculation. If the radius selected is too small for the homogeneous continuous body, the particles inside of influence domain will not be enough. This leads to reduction of the calculation accuracy. On the other hand, if the radius selected is too large for discontinuous discrete multibody, the premature and overdomain evaluation will occur. Also, it cannot reflect the deformation of the locality, which also affects the calculation accuracy. In particular, to evaluate impacts of discontinuous discrete bodies, dynamic contact and multifield coupling problems may also be judged by kernel estimation by changing smoothing length, so there is no need to define special fine contact area or grids. Therefore, the problem becomes how to define a suitable smoothing length in kernel function for interaction of discontinuous discrete bodies, continuous homogenous bodies, and multifield coupling. Most existing adaptive SPH calculations are based on the nearest neighbor approach [

The windblown sand problem in desert regions is one of the typical multiphase interaction problems. In desert regions, the movement of sand caused by wind has been listed as one of the major environmental problems in the world today. The most effective way to study wind-sand multiphase flow is numerical simulation and experimental measurement. Experimental measurement in blown sand physics has been a main research method, and numerical simulation methods can study mechanism and characteristics of wind-sand multiphase flow from a microcosmic perspective. Wind-sand flow is a complex, nonlinear, self-organized air-sand two-phase flow [

However, in the mesh-based methods such as Finite Difference Method (FDM), Finite Element Method (FEM), and Boundary Element Method (BEM), because of mesh tangling and distortion it is hard to track gear transmission process, the trajectory tracking of sand particles, and so forth, especially when calculating large deformation and crack propagation. In this case computationally extensive remeshing is needed in calculation process, which affects accuracy. Coauthors Imin and Geni [

In the SPH method the first step is to approximate a function and its gradient using integrals of kernel function based on interpolation method. Continuous partial differential equation is transformed into integral equations. In the second step, continuous forms of integral equations are discretized into discrete equations using the particle approximation method [

For any function

Using integration by parts, Gauss theorem, and property of kernel function, gradient

SPH particle kernel approximation.

The governing equations for dynamic fluid can be expressed by Navier-Stokes equations (the equations for the rates of change of density, velocity, energy, and position) as follows.

The governing equations for dynamic fluid can be expressed by Navier-Stokes equations (the equations for the rates of change of density, velocity, energy and position) as follows.

Conservation of mass is as follows:

Conservation of momentum is as follows:

Conservation of energy is as follows:

Position equation is as follows:

In the above equations the total stress tensor

So the energy equation can be rewritten as

In solid mechanics, the constitutive model, in general, permits the stress to be a function of strain and strain rate. For the anisotropic shear stress, if the displacements are assumed to be small, the stress rate is proportional to the strain rate through the shear modulus

By substituting the SPH approximations for a function and its derivative using (

In SPH method, the interaction and the contact area are determined by kernel estimation, so there is no need to define special contact areas or particles. As a result, the choice of the influence radius of kernel is very important because it affects the accuracy and efficiency of the calculation directly. When calculating the continuous multibody interaction, such as gear meshing and ball bearing transmission, if the selected radius of kernel is too small, there are not enough particles in the influence domain. This causes reduction of accuracy. On the other hand, if the radius selected is too large, approximation will not reflect locality of the deformation; this also affects accuracy. In addition, when calculating the discrete tiny granular multibody interactions, such as sand particles interactions, if the selected radius of kernel is too large, there are many particles in the influence domain; interaction will occur between two particles before substantive contact. The similar issue occurs in multiphase coupling problems such as the lubrication with inclusion and ball bearing interactions or airflow and sand-flow interactions in the desert. In particular, when a pair of continuous bodies contacts each other, interaction of particles that belong to different bodies will happen near the contact surface and will be taken into account in computation. In this case, if the influence radiuses of all particles are the same, the interaction will occur early between two bodies due to the particles in the kernel radius. This causes that the two bodies always cannot directly contact each other on the surface; hence there remains a gap during the process.

The more appropriate method to solve this problem is to introduce different kernel radiuses. So in this paper different kernel radiuses are defined by choosing different smoothing lengths to compute multibody and multiphase interaction problems. The basic idea of this method is to choose large smoothed length for evaluating the continuous body and to choose small one for evaluating the discrete body as shown in Figures

Different behaviors of different smoothing length.

Continuous body

Discrete body

The most frequently used kernel function in the SPH method is the cubic B-spline kernel [

Figures

Cubic B-spline kernel function in different smoothing length.

Kernel function

Derivative of kernel function

The different kernel radiuses are defined by choosing different smoothing lengths to calculate continuous multibody interaction problem as shown in Figure

Selection of influence radius for continuous and discrete body in multibody and multiphase interaction problems.

Multibodies interactions

Multiphase interactions

For two different phases the following expression is used to evaluate of airflow field as fluid phase and sand-flow field as granular phase interaction. In this study to compute multiphase flow problem such as sand-driving flow as shown in Figure

For validation of this new method, different kernel smoothing lengths with different elastic modules and different particle models are used. The two kinds of influence radius are explored, one is original model by setting

The cubic B-spline for kernel function is used in approximation with ^{3} and the elastic modulus is set as

SPM and MPM for Impact Velocity Exchange.

SPM

MPM square model

Figure

The velocity exchanges of SPM and MPM with the same smoothing length (

MPM square model

Figure

The velocity exchanges of SPM and MPM with different smoothing lengths (

SPM

MPM square model

It is obviously seen from Figures

Figures

The velocity exchanges of SPM with original and new SPH method (

Original SPH method

New SPH method

The velocity exchanges of MPM model with original and new SPH method (

Original SPH method

New SPH method

Coauthors Imin and Geni [

The two kinds of models for wind-sand multiphase flow numerical simulation are generated as the flat sand bed surface model and the small dune models as shown in Figures

Velocity vector of sand in the process of takeoff.

Step = 20

Step = 5400

As shown in Figure

Figure

Changes of velocity and position of sand particles at top of sand bed in the process of takeoff.

Step = 0~12600

Step = 18000

Step = 21900

The initial state of calculated domain of sand flow simulation with small dune.

While the airflow acts on the sand particles, the sand also brings certain reactive force to the airflow. This reactive force causes the airflow velocity to change near the grains of sand. Changes of airflow velocity caused large differences in velocity and direction of sand particles at the same layer; see Figure

Figures

Figures

The velocity distribution of airflow.

Figure

The distribution of sand structure.

In this paper two different models are introduced to evaluate the velocity exchanges during the impact process. It is shown that in the SPM the velocity exchange during the impact process is very close to the one in the MPM square model, so from perspective of velocity exchange on impact the single particle shape is close to multiparticle square shape and it represents the perfect inelastic impact phenomenon. However, the two objects A and B in the SPM and MPM influenced each other at the very early stage of the process; this may cause some error in interaction for discrete bodies.

The different kernel radiuses are defined by choosing different smoothing lengths to approximate continuous and discrete multibody interactions as well as multiphase interactions. The new SPH kernel smoothing length is represented and modeled for evaluating the multibody correct contact and multiphase interaction problems.

By using the new SPH kernel smoothing length, the results show that the velocity is exchanged nearly to half in the SPM, but in the MPM square model there are some differences. In this case the velocity of object A is smaller than object B. Additionally; the two objects (A and B) do not stick together. This represents an elastic collision. The other benefit of this new method by using the different smoothing length is that the exact interaction between two objects A and B is improved and realized in both SPM and MPM for discrete bodies.

The velocity exchanges between two objects A and B are increased with increasing elastic module in SPM and MPM in both original and new SPH methods. The new SPH method gives more perfect elastic collision behavior and represents the correct impact process. So the SPM represents a perfect inelastic while the MPM square model represents a week elastic characteristics. In real particles such as sand or inclusion it always shows a strong elasticity when impacting each other. If the MPM is used in the SPH method, the numerical simulation has high time and space complexity due to the increase of the particle number. It is clear that by increasing the elasticity of the SPM it can increase the perfect elastic impact characteristics. This will bring great convenience to solve the multibody and multiphase coupling such as tiny sand-particles impact and air-sand two-phase interaction with relatively lower time and space complexity.

The new SPH method is applied to the evaluation of sand-air multiphase interactions. The two kinds of models for wind-sand multiphase flow numerical simulation are applied as the flat sand bed surface model and the small dune models with the new SPH method. The results give a good agreement with real sand-air flow phenomenon and represent air-to-air, air-to-sand, and sand-to-sand multiphase interactions.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors wish to acknowledge the support of the China natural science foundation (no. 51075346) and National Key Basic Research and Development Program (973 Program no. 2011CB706600).