KR101355997B1 - Sub-particle scale turbulence simulation method and system for SPH fluid, recording medium for the same - Google Patents

Abstract

Disclosed are a sub-particle scale turbulent flow simulation method, system, and recording medium for the SPH fluid. The present invention is a novel SPS turbulence simulation method for an SPH system. In the present invention, a turbulence model is derived by applying scale classification based on the LES technique. In order to obtain a large amount of moving energy, a low pass filter is applied to the analyzed speed. To simulate the effects of SPS dynamics, we solve the strain rate tensor approached in SPH formulations. The SPS noise is then generated and merged for surface deformation according to the analyzed SPS dynamics. By implementing using a GPU, we can effectively simulate detailed SPS turbulence simulations of SPH fluids in three dimensions.

Description

Sub-particle scale turbulence simulation method and system for SPH fluid, recording medium for the same

The present invention relates to a turbulent flow simulation method, a system and a recording medium therefor, and more particularly to a sub-particle scale turbulent flow simulation method, a system for a SPH fluid and a recording medium therefor.

Turbulence of fluids, such as the flow of water from taps and the flow of rivers and waterfalls, is frequently observed in everyday life. Turbulent flows of fluid are usually unstable, irregular, random, and appear on various scales of motion.

Various techniques have been proposed to simulate turbulent flow of such fluids. In particular, the maximum frequency of the velocity field that can be interpreted spatially by the Eurrian-grid or Lagrangian particle introduced for the analysis of turbulent flows is reduced to the large scale at the lower frequencies. In addition, separate modeling techniques have been proposed, which classify velocities with frequencies higher than the maximum frequency of the velocity field (small-scale). However, simulating water turbulence remains a problem that requires high computational costs in small scale velocities analysis. Numerical diffusion is a major cause of the loss of small components during simulation. Adjustment of simulation resolution for small general scenes can prevent unnatural spreading while maintaining a balance between the time required for simulation and visual quality. On the other hand, the Direct Numerical Simulation (DNS) technique for complex or large scale scenes is a computational cost because high resolution grids or very large numbers of particles are used for numerical analysis. , Real-time processing was not possible.

It is an object of the present invention to provide a sub particle scale turbulence simulation method for an SPH fluid.

It is another object of the present invention to provide a sub particle scale turbulent flow simulation system for SPH fluids.

In the sub-particle scale (SPS) turbulence simulation method for mitigating fluid dynamics (SPH) fluid to achieve the above object, a large-scale vortex simulation (LES) technique is applied to the turbulent flow of the SPH fluid so that the large velocity of the SPH fluid Obtaining a momentum equation of the SPH fluid for analyzing the equation, low-pass filtering the momentum equation of the SPH fluid to remove a high frequency component to generate a conversion momentum equation, and the pressure generated in the process of generating the conversion momentum equation A strain rate tensor for calculating a tensor is approximated to calculate the transformed momentum equation, a small velocity of the SPH fluid is generated using the noise function, and the generated small velocity is equal to the analyzed large velocity. Combined to simulate the SPS turbulence.

The momentum equation of the SPH fluid to achieve the above object is based on the assumption that each of the plurality of particles in the turbulent flow of the SPH fluid individually carries a physical quantity including mass, density and pressure.

Where i (i is a natural number) is the particle, v _i is the velocity, p is the pressure, μ is the viscosity coefficient, f _i ^ext is the gravity, user-defined force, or vortices confinement forces And <ρ _i >, <∇ p> and <Δv> are the density field, the pressure field and the viscous force field at position xi, respectively. Symbolizes an interpolation kernel based on an approximation).

The pressure field to achieve the above object is calculated by applying a predictive-corrective incompressible SPH (PCISPH) technique.

Where W _ij = W (x _i (t)-x _j (t)) and pi is the pressure of the particle.

The pressure of the particles to achieve the above object is

(Where ρ ^* _i is the predicted density, σ is the scaling variable, and ρ ₀ is updated by repeatedly solving the prediction correction technique according to the rest density).

The step of generating a conversion momentum equation for achieving the above object is the equation

Low pass filtering is performed using a low pass filter according to k (d) = (1-d ² ) ³ , α is a weight for relaxation and re is an effective radius, typically twice the relaxation radius. The conversion momentum equation is

(Where τ ^s _i is the pressure tensor of the sub-grid scale).

The step of approximating the strain rate tensor to achieve the above object is the pressure tensor (τ ^s _i )

Assuming that the SPS turbulence simulation is performed in a closed space, the linear relationship between the sub-grid scale pressure and the large strain rate is

는 변형 속도 텐서, δ_ij 는 Kronecker 델타) 로 표현되는 맴돌이 점성 가정이 적용되는 단계, 상기 Kronecker 델타가 수학식Where μT is the eddy viscosity,

Is the strain rate tensor, and δ _ij is the eddy viscosity assumption expressed by Kronecker delta), where the Kronecker delta is

The eddy viscosity (μT) is calculated by applying a dynamic sub-particle scale kinetic energy based model

)가 수학식Calculated by the strain rate tensor (

)

)의 계산 비용을 줄이기 위해, 상기 변형 속도 텐서(

)가 수학식And the strain rate tensor (

In order to reduce the computational cost of

)

는 와류 텐서) 로 근사되는 단계를 구비하는 것을 특징으로 한다.(here

Is characterized by having a step approximated by a vortex tensor).

)가 수학식Approximating the strain rate tensor to achieve the above object is the vortex tensor (

)

)에서 ω_i 가 수학식Represented by the vortex tensor (

Ω _i in

)가By the strain rate tensor (

)end

) 로 계산되는 단계를 더 구비하는 것을 특징으로 한다.(here,

It is characterized in that it further comprises a step that is calculated by).

The step of generating a small speed to achieve the above object is to add a small variation to the three-dimensional Perlin noise equation

Where n _i is a three-dimensional perlin noise vector and k is a scaling parameter.

The step of simulating SPS turbulence to achieve the above object is to use a moving average as a reference position to construct a smooth and thin surface from the SPH particles.

The surface reconstruction method is calculated, where λ is the weight for the mean,

The weight λ is

It is characterized by.

Therefore, the sub-particle scale turbulence simulation method, system and recording medium therefor for the SPH fluid of the present invention are based on the LES technique, in order to derive the turbulence model by applying the scale division, and to obtain a large-scale moving energy, Apply a low pass filter to the analyzed speed. To simulate the effects of SPS mechanics, the strain rate tensor approached in SPH formulations is analyzed. The SPS noise is then generated and merged for surface deformation according to the analyzed SPS dynamics. By implementing using a GPU, we can effectively simulate detailed SPS turbulence simulations of SPH fluids in three dimensions.

1 illustrates a sub particle scale turbulence simulation method for an SPH fluid in accordance with an embodiment of the present invention.
FIG. 2 shows a sub particle scale turbulent flow simulation algorithm for the SPH fluid according to FIG. 1.
3 shows an example of an image captured by the simulation according to the SPH turbulence simulation method according to the present invention.
4 shows an example of simulation results according to the SPH turbulence simulation method according to the present invention.
5 shows another example of simulation results according to the SPH turbulence simulation method according to the present invention.

In order to fully understand the present invention, operational advantages of the present invention, and objects achieved by the practice of the present invention, reference should be made to the accompanying drawings and the accompanying drawings which illustrate preferred embodiments of the present invention.

Hereinafter, the present invention will be described in detail with reference to the preferred embodiments of the present invention with reference to the accompanying drawings. However, the present invention can be implemented in various different forms, and is not limited to the embodiments described. In order to clearly describe the present invention, parts that are not related to the description are omitted, and the same reference numerals in the drawings denote the same members.

Throughout the specification, when an element is referred to as &quot; including &quot; an element, it does not exclude other elements unless specifically stated to the contrary. The terms "part", "unit", "module", "block", and the like described in the specification mean units for processing at least one function or operation, And a combination of software.

Recently, various methods have been proposed to reduce the computational cost for simulating turbulence. Some of these methods use a method that combines the modeled small details into a large scale flow that has been pre-interpreted. For example, procedural synthesis methods use a low-resolution grid to analyze the overall flow. And when the overall large flow is analyzed, the missing small details are created by the noise function and combined into the analyzed overall flow.

Similarly, turbulence energy based models simulate the energy transfer process in a low resolution grid and produce different scales of noise depending on the analyzed kinetic energy distribution. The vortex particle method has the advantage of preserving physical quantities from numerical diffusion and has been widely used to complement the Eulerian grid technique in simulating both turbulence for water and smoke. Although these methods show visually compelling results, they are based on Euler grid systems, so algorithmic modifications are often required to be applied to particle-based simulation systems such as Smoothed Particle Hydrodynamics (SPH). often.

In the field of computational fluid dynamics (CFD), several attempts have been made to simulate turbulence by applying existing turbulence models developed for grid-based approaches to SPH systems. For example, a three-dimensional large eddy simulation (LES) model was applied to simulate the collapse of a column of water (Issa, R., Violeau, D., Laurence, D .: A rst attempt). to adapt 3d large eddy simulation to the smoothed particle hydrodynamics gridless method.In: Proceedings of the International Conference on Computational and Experimental Engineering and Sciences, pp. 87 {94 (2005)). And the SPH-LES model has been proposed to simulate wave propagation near the coast (Shao, S., Gotoh, H .: Simulating coupled motion of progressive wave and oating curtain wall by sph-les model.Coastal Engineering Journal 46, 172 (202 (2004)). A well-known k-ε model has also been applied to simulate turbulent channel flow (Violeau, D .: One and two-equations turbulent closures for smoothed particle hydrodynamics.In: Proceedings of the VIth Int. Conf. Hydroinformatics, pp. 87 (94 (2004)).

However, these methods are often limited to two-dimensional test cases for simple turbulent closed models due to the high computational costs required for complex model simulations in three dimensions.

1 illustrates a sub particle scale turbulence simulation method for an SPH fluid in accordance with an embodiment of the present invention.

In the present invention, the sub-particle scale turbulence simulation method for SPH fluid proposes a new technique based on the LES model to simulate SPS for SPH turbulence in three dimensions. The basic concept of the LES model is to calculate the contribution of large scale flow in momentum transfer and to model the effects of unscaled small scale structures. In the present invention, a low pass filter is used to obtain smooth large scale velocities, and a small model that is not analyzed is simulated using an SPS model.

Referring to Figure 1, the present invention first obtains the momentum equation of the SPH fluid by analyzing the large velocity of the SPH turbulence based on the LES model (S10). Thereafter, a high pass component is removed from the momentum equation of the SPH fluid by applying a low pass filter to the analyzed momentum equation of the SPH fluid (S20). In order to calculate the stress tensor generated in the process of removing the high frequency components from the momentum equation of the SPH fluid, a strain-rate tensor is approximated (S30). When the strain rate tensor is approximated, a large velocity is analyzed, and then a small velocity is generated using a noise function (S40). In addition, the generated small velocity is combined with a pre-analyzed large velocity to simulate a sub-particle scale (SPS) turbulence (S50).

Referring to the step S10 of analyzing the large velocity of the SPH turbulence based on the LES model in FIG. 1, the fluid does not consider the moving particles as a set of moving particles in the SPH turbulence. SPH turbulence is thus considered that each particle carries a physical quantity such as mass, density and pressure. Therefore, the state of the SPH fluid is obtained by approximating the momentum equation as shown in Equation (1).

Where i (i is a natural number) represents a particle, v _i is velocity, p is pressure, μ is the viscosity coefficient, f _i ^ext is gravity, user-defined force, or vortices confinement forces ) To external force. And <ρ _i >, <∇ p> and <Δv> are interpolation kernels based on approximation of density field, pressure force field and viscous force field at position x _i , respectively. interpolation kernel). In the present invention, Muller's SPH method (Muller, M) includes a poly kernel for field interpolation, a spikey kernel for approximating a gradient of physical quantities, and a viscosity kernel for laplacian calculation. ., Charypar, D., Gross, M .: Particle-based fluid simulation for interactive applications.In: Proceedings of the 2003 ACM SIGGRAPH / Eurographics symposium on computer animation, SCA '03, pp. 154 {159. Eurographics Association, Aire -isotropic kernels (la-Ville, Switzerland, Switzerland (2003)).

Muller's SPH technique is an extension of Desbrun's and Gascuel's SPH, based on fluid analysis, diffusion phenomena such as fluid-fluid interaction, bubble generation, spray and air mixing, and flow of fluid through porous materials. Various particles such as were applied. Muller's SPH technique, however, has a problem that it is difficult to enforce incompressibility due to the use of excessive equations of state (EOS) equations. Several methods have been proposed, including weakly compressible SPH (WCSPH) and pressure corrected SPH. In addition, Solenthaler recently proposed a predictive-corrective incompressible SPH (PCISPH) method using a prediction-correction scheme to calculate the pressure intensity (Solenthaler, B., Pajarola, R .: Predictive-corrective incompressible sph.In: ACM SIGGRAPH 2009 papers, SIGGRAPH '09, pp. 40: 1 {40: 6. ACM, New York, NY, USA (2009)). For the efficiency of the simulation, Adams proposed an adaptive sampling method based on the geometrical complexity of the deformation (Adams, B., Pauly, M., Keiser, R., Guibas, LJ: Adaptively sampled particle uids.In: ACM SIGGRAPH 2007 papers, SIGGRAPH '07. ACM, New York, NY, USA (2007)). In addition, among the parallel and real-time simulation methods using the GPU, the SPH kernel is often used because it is easy to implement on the GPU because the SPH kernel interpolates and calculates the physical quantity by referring only to the surrounding SPH particles.

The present invention analyzes SPS turbulence and applies PCISPH to minimize the effects of virtual spurious bouncing motion and numerical error during simulation of SPS turbulence. In contrast to the existing equilibrium (EOS) based approach, the PCISPH method uses the predicted pressure values to approximate the pressure field. Therefore, the pressure field << p> of Equation 1 is approximated as Equation 2.

Where W _ij = W (x _i (t)-x _j (t)), p _i is the pressure of the particles and m is the mass of the particles. In PCISPH, the pressure value for each particle is updated by repeatedly calculating the prediction correction technique according to Equation 3 more than a predetermined number (for example, three times) in order to minimize the prediction error.

Where ρ ^* _i is the predicted density, σ is the scaling parameter, and ρ ₀ is the rest density. In the present invention, sigma is first calculated during testing using randomly selected internal particles, similar to PCISPH. The present invention repeats the predictive-correction step several times for each frame to force the incompressibility of the SPH fluid.

Then, as shown in Figure 1, by applying a low-pass filter to the momentum equation of the SPH fluid of the equation (1) by the equations (2) and (3) to remove the high frequency components (S20).

LES is based on the assumption that, by its nature, smaller homogenous structures in turbulence have more homogeneous. Small movements that cannot be interpreted at the resolution of discretization tools such as the Eulerian grid or Lagrangian particles introduced in the numerical simulation can be modeled according to the above assumptions. Scale separation in LES can be achieved by a spatial filter function, and LES governing equations can be obtained by applying LES filters to operating equations. This filtering operation essentially works like a low pass filter to remove the contribution of high frequency components from the LES model.

The LES filter described above is not limited to a particular filter, and any spatial filter that can effectively mitigate sub-grid scale elements can be used. Monaghan also proposed a mitigation filter in Equation 4 that is similar to the XSPH equation except for the mitigation kernel (Monaghan, JJ: On the problem of penetration in particle methods. J. Comput. Phys. 82, 1 {15 ( DOI 10.1016 / 0021-9991 (89) 90032-6), (Monaghan, JJ: A turbulence model for smoothed particle hydrodynamics.eprint arXiv: 0911.2523 (2009) .DOI http://arxiv.org/abs/0911.2523 v1)).

Where epsilon is a relaxation parameter with a value of 0.8, and G _ij is a kernel that uses symmetrical density to preserve momentum. And m _j represents the mass of the particle j. Equation 4 is Lagrangian Averaged Navier Stokes equations (LANS) (Geurts, B., Holm, D .: Leray and lans-alpha modeling of turbulent mixing.Journal of Turbulence 7 (10), 1 {33 (2006) .URL http http://doc.utwente.nl/66861/.This paper is associated with the focus-issue Cascade Dynamics: Fundamentals and Modeling.Derived by an approximation, linearity according to a constant mean density assumption Conserve both and each momentum. In addition, the computational cost is very low. However, if the value of ε increases greater than 0.3, some of the particles will inadvertently deviate around the boundary of the simulation area and the obstacle.

In order to prevent such a problem, the present invention uses a low pass filter combined with a weighted average to relax the velocity field and suppress high frequency by-products, as shown in Equation 5, instead of the easing filter of Equation 4.

Where k (d) = (1-d ² ) ³ , α is a weight for relaxation and r _e is an effective radius that is typically twice the relaxation radius.

  When the low pass filter of Equation 5 is applied to the momentum equation of the SPH fluid of Equation 1, Equation 1 may be converted to Equation 6.

As in Equation 6, the filtered momentum equation describes the temporal and spatial evolution of the large kinetic energy transfer scale. At this time, a pressure tensor τ ^s _i of a sub-grid scale is derived after filtering by non-linearity of the convection.

The strain-rate tensor is approximated to calculate the induced pressure tensor (S30).

In the filtered momentum equation shown in Equation 6, the pressure tensor τ ^s _i is calculated as Equation 7 to reflect the effects of energy transfer and subgrid scale waves. M _j in Equation 7 represents the mass of the particle j.

However, the expression of the pressure tensor (τ ^s _i ) of Equation 7 in the momentum equation of Equation 6 may reduce the reliability of the SPH using a symmetric formulation. That is, since the pressure tensor τ ^s _i of Equation 7 includes components that have not been analyzed, an assumption about closure is required to convert them into interpretable equations.

To this end, the present invention applies a eddy viscosity assumption that proposes a linear relationship between the sub-grid scale pressure and the large strain rate represented by Equation (8).

는 변형 속도 텐서이며 δ_ij 는 수학식 9로 계산되는 Kronecker 델타이다. 그리고

는 tensor notation으로서 변형속도 텐서

의 행렬에서 대각 요소의 합(

)을 의미한다.Where μ _T is the eddy viscosity,

Is the strain rate tensor and δ _ij is the Kronecker delta calculated by equation (9). And

Is the tensor notation and the strain tensor

Sum of diagonal elements in matrix of (

).

The eddy viscosity (μ _T ) is calculated using Equation 10 by applying the Smagorinsky model.

로서 주어진다. Smagorinsky 모델은 저비용으로 쉽게 구현할 수 있지만, C_S는 유일한 값으로 정의되지 않으며, 매개 변수가 없는 공간과 시간에서는 동적으로 전파되지 않는다는 단점으로 인해 복잡한 난류에 대해서는 적합하지 않다. 이에 본 발명에서는 맴돌이 점성(μ_T)을 수학식 11과 같이 동적 서브- 입자 스케일 운동 에너지 기반의 모델(dynamic sub-particle scale kinetic energy based model)을 적용하여 계산한다.Where the Smagorinsky constant C _S is 0.12, Δ is the particle spacing, and the local strain rate is

. The Smagorinsky model is easy to implement at low cost, but C _S is not defined as a unique value and is not suitable for complex turbulence due to the disadvantage that it does not propagate dynamically in parameterless space and time. In the present invention, the eddy viscosity (μ _T ) is calculated by applying a dynamic sub-particle scale kinetic energy based model as shown in Equation (11).

)는 수학식 12와 같이 주어질 수 있다.In Equation 11, the difference in the kinetic energy between the flow of fluid and the relaxed particle velocity is replaced by the local strain rate. According to equation (11), the strain rate tensor (

) May be given by Equation 12.

)를 분석하기 위한 계산 비용은 무시할 수 없는 수준이다. 변형 속도 텐서(

)의 각 요소는 속도 구배(velocity gradient)의 계산을 필요로 하므로, 직접 계산을 위해 전체적으로 적어도 6번 이상의 보간(interpolation)이 요구된다. 이에 본 발명에서는 수학식 13을 활용하여 변형 속도 텐서(

)를 근사한다.However, the strain rate tensor of equation (12)

The computational cost for analyzing) is insignificant. Strain rate tensor (

Each element of) requires the calculation of velocity gradients, and therefore requires at least six interpolations in total for direct calculations. In the present invention, the strain rate tensor (

Approximate).

는 회전 또는 와류 텐서이다. 와류 텐서(

)는 스큐 - 시메트릭(skew-symmetric) 행렬과 와류 벡터 구성 요소로 수학식 14와 같이 표현될 수 있다.here

Is a rotating or vortex tensor. Vortex Tensor

) May be represented by the skew-symmetric matrix and the vortex vector component as shown in Equation 14.

Here, ω _i is easily calculated by the equation (15).

)는 수학식 16과 같이 계산된다.Similarly, the present invention computes the velocity gradient using an existing kernel based on interpolation by the matrix representation. Finally, strain rate tensor (

) Is calculated as in Equation 16.

이다.here

to be.

While SPS turbulence models can exhibit global dissipative effects from small to large, tensors can directly reproduce sub-particle velocities and local energy dynamics such as interactions between unanalyzed scales. none.

Accordingly, the small speed generation step S40 of FIG. 1 is performed.

3D Perlin noise (Perlin, K., Neyret, F .: Flow noise (2001) to generate small velocity) URL http://www-evasion.imag.fr/Publications/2001/PN01 ) Was tested. The test proceeded by first observing the effects of perturbation on the shape of the kernel function, using the generated three-dimensional small velocity. In this test, the interpolation kernel shows the distribution of physical quantities such as density and speed, and it is expected that the effect of SPS speed will modify the local distribution of the kernel, but the actual test results show that errors accumulated during fusion Make the system unstable.

Therefore, in the present invention, the subtle fluctuation is further coupled to the speeds in the analyzed speeds as shown in Equation 17.

Where n _i is the generated three-dimensional perlin noise vector and k is the scaling parameter. The noise vector n _i shows an excellent effect in the part of the region with low density. The effect of noise can be neglected to scale elements that are not large. On the other hand, too large a parameter value may cause it to move out of the simulation area during the simulation. In order to reduce by-products, the present invention can use a simple attraction model to apply surface tension force.

Afterwards, the small velocity generated is combined with the large velocity analyzed beforehand to simulate the SPS turbulence (S50).

To construct the SPH fluid surface from the SPH particles, the present invention applies the surface reconstruction method calculated by Equation 18 using the moving average as the reference position.

Λ is a weight for the mean, and is defined as in Equation 19.

Here, the kernel k and the effect radius r _e are the same as in Equation 5. When the adjacent particles a sufficient number (N _i) is present, the surface reconstruction method of Equation (17) can reconstruct a very smooth surface due to the effect (smoothing effect) is relaxed in the moving average formulation (moving average formulation) . In the present invention, a test as shown in Equation 20 is first performed to add SPS scale details directly to a kernel function.

Where noise1D is a simple noise function according to Perlin noise that returns a value ranging from 1 to -1, and k is a scale parameter with a range of 0.2-0.4 in the test according to the present invention. However, the perturbation details from noise are easily mitigated by the moving average. In addition, the mean perturbation represents high frequency accompaniments that have no temporal and spatial coherence during the simulation. Similarly, test results from adding one-dimensional noise to φ (x) often show unnatural changes in the surface mesh.

Therefore, the present invention directly modifies the position of the particles using Equation 21.

Where x _i is the position of the particle. The amount modified is proportional to the magnitude of the SPS force associated with the effect of SPS dynamics on SPS surface deformation. In order to prevent excessive correction, in the present invention, the upper limit value d _max is determined in advance.

FIG. 2 shows a sub particle scale turbulent flow simulation algorithm for the SPH fluid according to FIG. 1.

)를 획득하기 위해 로우 패스 필터를 적용한다. 이후 힘들 (F^v, F^g, F^ext)과 변형 속도 텐서(

)를 계산한다. 그리고 계산된 힘들과 변형 속도 텐서(

)를 이용하여 SPS 힘(F^SPS)를 계산하고, SPS 노이즈(noise(x_i))를 생성 및 입자 위치(x_i)를 수정한다. 이후 PCISPH를 이용하여 속도(v_i)를 업데이트하여 시뮬레이션을 수행한다. 도2 의 알고리즘에 대한 상세한 설명은 상기한 도1 에 대한 설명과 같으므로 이하 생략한다.
In FIG. 2 i represents particles and N _i represents particles adjacent to particle i. Referring to Fig. 2, the sub-particle scale turbulence simulation algorithm for the SPH fluid according to the present invention first finds the particle N _i adjacent to the i th particle i, and then relaxes the velocity for all the particles i.

Apply a low pass filter to get Afterwards the forces (F ^v , F ^g , F ^ext ) and the strain rate tensor (

). And the calculated forces and strain rate tensors (

) To force SPS ^(SPS calculate F) and, SPS noise (noise (x _i) modifying a) the particle generation and the location (x _i) used. Then, the speed (v _i ) is updated using PCISPH to perform the simulation. Detailed description of the algorithm of FIG. 2 is the same as that of FIG.

3 shows an example of an image captured by the simulation according to the SPH turbulence simulation method according to the present invention.

In FIG. 3, the simulation is performed in the order of the upper left to the lower right, and the size of the calculated SPS force (F ^SPS ) for each particle is visually represented by color. In FIG. 3, the weak SPS force (F ^SPS ) is represented in blue, and the large SPS force (F ^SPS ) is represented in red.

4 shows an example of simulation results according to the SPH turbulence simulation method according to the present invention.

In FIG. 4, the top image represents a dam destruction simulation, and the bottom image represents a simulation of water flow when there is an obstacle. In each of the upper and lower images, (a) represents an existing SPH simulation image, (b) represents an SPH simulation image for SPS turbulence, and (c) represents an image rendered (b).

Parameters for surface reconstruction can be set to represent smooth and thin surfaces, and parameters for incompressibility can be set by testing the water scene in a steady state. The simulation of FIG. 4 was performed in a cube-shaped domain. The dam destruction simulation shown at the top of FIG. 4 used 44,000 particles, and the water flow simulation at the bottom used 16,000 particles. As can be seen in Figure 4, a noticeable effect occurs in the region of high deformation.

The splashing motion of the water flow simulation is different from the existing SPH simulation shown in (a). The difference between the analyzed velocity and the relaxed velocity in the water splash is greater than in the laminar flow region. Since the simulation model according to the invention assumes that this difference reflects the kinetic energy dissipated from the large flow, the larger turbulent motion of the particles can be judged as valid. Although scattered energy is not directly used to speed up the bouncing particles, it can be an approximate measure of for turbulence in a particular region.

5 shows another example of simulation results according to the SPH turbulence simulation method according to the present invention.

In Fig. 5, the relationship between the noise generated by simulating a scene having both large and small surface deformations and the SPS dynamics is represented. In the present invention, the shape of the fluid is determined by mechanics in the SPS. In FIG. 5, both laminar and turbulent flows exist in one scene. Small details on the surface of the crashing wave are easily visible in the rendered image. In contrast, less detail is visible in the region of laminar flow. Using the approximation according to the invention, further modifications can be added in an optional manner.

Table 1 shows the time taken to perform the simulations of FIGS. 4 and 5.

All simulations were run on a workstation with an Intel Xeon CPU, 8G of RAM, and an NVIDIA GTX 480 graphics card. Table 1 does not include offline rendering times.

4 and 5 have been implemented using the Compute Unified Device Architecture (CUDA) to maximize the use of parallel computation. The most frequent and time-consuming process in simulation is to search for the nearest particle. For efficiency, particle hashing can be applied to auxiliary grids by a radix sort algorithm. According to the present invention, simulations can be obtained at high frame rates by accelerations due to parallel processing. In addition, the SPS model and noise generation process can be implemented in the GPU to cope with the high computational cost of turbulence simulation. We can also implement a marching cube algorithm with CUDA to reduce the total cost of the simulation cycle.

The device according to the invention can be embodied as computer readable code on a computer readable recording medium. A computer-readable recording medium includes all kinds of recording apparatuses in which data that can be read by a computer system is stored. Examples of the recording medium include a ROM, a RAM, a CD-ROM, a magnetic tape, a floppy disk, an optical data storage device, and the like, and a carrier wave (for example, transmission via the Internet). The computer readable recording medium can also be distributed over network coupled computer systems so that the computer readable code is stored and executed in a distributed fashion.

While the present invention has been particularly shown and described with reference to exemplary embodiments thereof, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art.

Therefore, the true technical protection scope of the present invention will be defined by the technical spirit of the appended claims.

Claims (11)

A computer-implemented sub-particle scale (SPS) turbulent flow simulation method for mitigating fluid dynamics (SPH) fluid, the computer comprising:
A large-scale vortex simulation (LES) technique is applied to the turbulent flow of the SPH fluid so that Eulerian-grid and Lagrangian-particles in the SPH fluid are lower than the maximum frequency of the velocity field. Obtaining a momentum equation of the SPH fluid for analyzing a large velocity with velocities of the velocity;
Low pass filtering the momentum equations of the SPH fluid to remove high frequency components to generate transformed momentum equations;
Calculating the transformed momentum equation by approximating the strain velocity tensor to the incident velocity and the vortex tensor to calculate the pressure tensor generated during the generation of the transformed momentum equation;
Calculating a small velocity having a frequency of a velocity field higher than the large velocity in the SPH fluid using a noise function; And
Combining the calculated small velocity with the analyzed large velocity to simulate SPS turbulence,
The momentum equation of the SPH fluid is
Equation assuming that each of the plurality of particles in the turbulent flow of the SPH fluid individually carries a physical quantity including mass, density and pressure

Where i (i is a natural number) is the particle, vi is the velocity, p is the pressure, μ is the viscosity coefficient, fiext is the gravity, user-defined force, or vortices confinement forces And <ρi>, <∇ p> and <Δv> represent the kernel of the density field, pressure force field and viscous force field at position xi, respectively. Symbolizes based approximation.)
SPS turbulence simulation method for SPH fluid, characterized in that approximated by.
delete The method of claim 1 wherein the pressure field is
Equation by applying predictive-corrective incompressible SPH (PCISPH) technique

(Where W _ij = W (x _i (t)-x _j (t)), p _i is the pressure of the particle, m is the mass of the particle)
SPS turbulence simulation method for SPH fluid, characterized in that approximated by.
The method of claim 3, wherein the pressure of the particles
Equation

(Where ρ ^* _i is the predicted density, σ is the scaling variable, and ρ ₀ is the rest density)
SPS turbulence simulation method for SPH fluid, characterized in that to repeat the prediction correction method according to the predetermined number of times to minimize the error for the predicted density.
5. The method of claim 4, wherein generating the transformed momentum equation is
Equation

Where k (d) = (1-d ² ) ³ and d is

And, α is the weight for the mitigation, r _e is typically two times the effective radius of the relaxation radius)
Performs low pass filtering using a low pass filter according to
The conversion momentum equation is
Equation

(Where τ ^s _i is the pressure tensor of the sub-grid scale)
SPS turbulence simulation method for SPH fluid, characterized in that calculated by.
6. The method of claim 5, wherein calculating the transformed momentum equation
The pressure tensor (τ ^s _i ) is expressed as

Where m _j is the mass of the particle j)
Acquiring with;
Assuming that the SPS turbulence simulation is performed in a closed space, the linear relationship between the sub-grid scale pressure and the first scale strain rate is

Where μ _T is the eddy viscosity,

Is the strain rate tensor, δ _ij is the Kronecker delta,

Is the tensor notation and the strain tensor

Sum of diagonal elements in matrix of (

Means.)
Applying a eddy viscous assumption expressed as;
The Kronecker Delta Equation

The eddy viscosity (μ _T ) is calculated by applying a dynamic sub-particle scale kinetic energy based model

Calculating as;
Strain rate tensor (

) Equation

Obtaining with; And
Strain rate tensor (

In order to reduce the computational cost of

) Equation

(here

Vortex tensor)
SPS turbulence simulation method for SPH fluid, characterized in that it comprises the step of approximating.
7. The method of claim 6, wherein calculating the transformed momentum equation is
The strain rate tensor (

Expressed in a matrix fashion

Wow

By applying
The strain rate tensor (

)

(here,

,

),
The strain rate tensor (based on the converted equation)

SPS turbulence simulation method for SPH fluid, characterized in that it further comprises the step of calculating).
8. The method of claim 7, wherein calculating the small velocity
Equation by adding fine variation to 3D Perlin noise

Where n _i is a three-dimensional perlin noise vector and k is a scaling parameter
SPS turbulence simulation method for SPH fluid, characterized in that for calculating the small velocity.
9. The method of claim 8, wherein simulating the SPS turbulence
Equation using a moving average as a reference position to construct the surface of the SPH fluid from the SPH particles

Where λ is the weight for the mean
Apply the surface reconstruction method calculated by
The weight λ is

SPS turbulence simulation method for SPH fluid, characterized in that defined as.
10. The computer readable medium according to any one of claims 1 to 3, wherein program instructions for simulating the SPS turbulence are recorded using the SPS turbulence simulation method for the SPH fluid.
10. A computer system according to any one of claims 1 and 3 to 9, comprising a display unit for driving a program in which the SPS turbulence simulation method for the SPH fluid is recorded to display and store the simulation results.

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