KR100568564B1 - A real-time simulation and rendering method for fluid effects using mesh-free simulation technique - Google Patents

Abstract

The present invention relates to a fluid effect real-time simulation and rendering method using a non-lattice analysis method and a volume rendering method, and more particularly, to the characteristics and irregular shape of the non-lattice method that can perform numerical analysis without using a grid structure The present invention relates to a method of simulating and rendering fluid motion in real time by combining a volume rendering technique suitable for real-time rendering.

Simulation and rendering method of the present invention is a non-lattice technique that expresses the mathematical correlation between each randomly distributed matter in the calculation domain as a function of distance, and simulates by analyzing the momentum of the fluid containing the free surface. And a volume rendering step of generating volume data using coordinates and velocity components of the quality point obtained in the simulation step, obtaining variables for expressing the three-dimensional effect of the fluid composed of the quality point, and performing rendering.

Grating, Fluid Effects, Simulation, Rendering, SPH, Pre-Integral

Description

A real-time simulation and rendering method for fluid effects using mesh-free simulation technique

1 is an example of a material distribution of a non-lattice model.

2 is a diagram illustrating an influence area and a calculation area.

3A to 3D are graphs showing the analysis of the dam collapse problem by the non-lattice method over time.

4A to 4F are graphs simulating the problem of dams of mixed fluids having different properties over time collapsed and mixed over time.

5A to 5D are graphs for three-dimensional analysis of a free surface flow problem using a non-lattice method.

6A to 6B are diagrams illustrating Phong roughness vectors.

7A to 7D illustrate real-time fluid effects using a non-lattice analysis technique and a volume rendering technique.

8 is a flowchart illustrating an implementation method according to an embodiment of the present invention.

The present invention relates to a fluid effect real-time simulation and rendering method using a non-lattice analysis method and a volume rendering method, and more particularly, to the characteristics and irregular shape of the non-lattice method that can perform numerical analysis without using a grid structure The present invention relates to a method of simulating and rendering fluid motion in real time by combining a volume rendering technique suitable for real-time rendering.

For the realistic representation of real-time fluid effects, computation by physical laws is essential.

In particular, the realistic representation of the motion of fluids with free surfaces is a difficult challenge in the field of computer graphics.

Since the free surface of the fluid moves irregularly over time, tracking its exact shape is not easy and results in a lot of computational load.

Several techniques have been tried to solve the problems related to the flow of fluid in a numerical way.

Finite element method (FEM), finite difference method (FDM), and finite control volume method (FVM) are typical examples, and have been studied for decades and well established.

However, the flow problem with the free surface has not been able to propose breakthrough techniques that combine mathematical rigor and wide applicability despite various attempts.

Although it has been developed in various techniques based on the fixed lattice method and the mobile lattice method, all of these methods compromise on the right line at the expense of one or two of the requirements such as the efficiency of computation or the geometrical rigidity.

The biggest geometrical problems of the flow problem with the free surface are largely tracking the shape of the free surface itself and optimizing the lattice structure of the internal computational area as the flow progresses.

Because the fluid motion in the free surface flow is irregular, rearrangement of the lattice structure is necessary to maintain the optimal surface of the fluid surface and the interior over time, although various methods have been developed and used so far. Optimizing the grid structure always remains a challenge.

On the other hand, the non-lattice method is a technique that can obtain a numerical analysis without using a lattice structure, unlike the conventional finite element method or finite difference method, it is applied to problems such as structural analysis in addition to fluid motion.

In the case of fluid flow analysis, there is a case where the free surface problem is analyzed using the non-lattice method, but there is no case where the result is visualized and visualized as soon as the simulation is performed by combining with the real-time rendering technique.

 The present invention has been made to solve the above problems, and an object of the present invention is to combine the characteristics of the non-lattice method that can perform numerical analysis without using a grid structure and a volume rendering technique suitable for real-time rendering of irregular shapes It provides a method for simulating and rendering fluid motion in real time.

In order to achieve the above object, the present invention is a non-lattice technique for expressing the mathematical correlation between the randomly distributed points in the computational domain as a function of distance, and analyzing and simulating the momentum of the fluid including the free surface; And a volume rendering step of generating volume data using coordinates and velocity components of a material point obtained in the simulation step, and obtaining and rendering variables for expressing a three-dimensional effect of the fluid composed of the material points. To provide.

Hereinafter, the configuration and operation of the present invention will be described with reference to the accompanying drawings.

As shown in FIG. 8, the implementation sequence according to the present invention includes a step of analyzing and simulating a free surface motion by a non-lattice analysis method (S802), and generating data for volume rendering using the data obtained from the simulation step. It consists of a volume rendering step (S804) for obtaining various variables using the preliminary integration, and an adjustment step (S806) for giving the effect of shading and reflection on the surface of the fluid obtained by volume rendering.

The non-lattice analysis methods include the element-free galerkin method, the meshless local Petrov-Galerkin method, and the point interpolation method.

These techniques express mathematical correlations between randomly existing points as a function of distance without prior information about the connection between each point.

First, the calculation domain will be described in order to explain the implementation method. As shown in FIG. 1, as many points as necessary are distributed inside the calculation domain.

At this time, the distance or distribution density between the material points need not be uniform and may be randomly distributed in the calculation region.

Each material point is called a field node because it has a field variable to perform calculation.

는 위치

의 함수로 수학식 1로 표현된다.Next, when the interpolation of the field variable is described, since the lattice structure is not used in the non-lattice method, the field variable at a given position is given.

Location

It is expressed by Equation 1 as a function of.

은 영향영역(influence domain) 내부의 질점의 총 개수,

는

번째 질점의 필드변수(field variable),

는 이러한 모든 질점들의 속도벡터의 합,

는

번째 질점의 형상함수(shape function)이다.here,

Is the total number of materials in the influence domain,

Is

Field variable of the first quality,

Is the sum of the velocity vectors of all these points,

Is

Shape function of the first material point.

A discretized equation is constructed by using the shape function of each point for an arbitrary point distribution arranged as shown in FIG. 2 and Equation 1, and the discretized equation is applied to the entire point of quality. of equations form a system of equations for the entire computational domain.

Solving this equation through matrix operations yields the distribution of the desired solution, which includes Gaussian elimination, LU decomposition, iterative solver, conjugate gradient solver, and so on. Branch matrices can be used to solve the problem.

The following non-lattice analysis techniques are described, including smooth particle hydrodynamics (SPH) [L. B. lucy, A numerical approach to the testing of the fission hypothesis, The Astron. J. 8 (12), pp. 1013-1024, 1997] will be described.

는 영역(domain)

내에서 수학식 2와 같이 근사화 된다.Field variable of Equation 1

Is the domain

It is approximated as in Equation 2.

는 근사화된 필드변수,

는 가중치함수,

는 영향영역의 크기이고, 상기 가중치함수는 다음의 조건을 충족시켜야 한다.here

Is an approximated field variable,

Is the weight function,

Is the size of the influence area, and the weight function must satisfy the following conditions.

번째 영역

내부에서

first,

Th area

From the inside

번째 영역

외부에서

second,

Th area

From the outside

Third, according to the law of normality

는 거리벡터

의 크기

이 증가함에 따라 단조 감소하는 함수이다.fourth,

Vector street

Size

It is a function that decreases monotonically as it increases.

으로 점근할 때

로 점근한다(여기서

는 디락 델타 함수(Dirac delta function 이다.).fifth,

When asymptotically with

Asymptotic with (where

Is the Dirac delta function.

The weight function may be represented in the form of an exponential function below (Equation 3), a three-dimensional spline (Equation 4), a four-dimensional spline (Equation 5).

이고,

는 영향영역의 최대반경이다.here

ego,

Is the maximum radius of the affected area.

Next, a simulation step (S802) of analyzing free surface flow using the SPH method will be described.

Formulating the momentum equation with the SPH technique can be expressed as follows [J. J. Monaghan, Simulating free surface flows with SPH, J. computational physics, 110, pp. 399-406, 1994], [J. J. Monaghan, A. Kocharyan, SPH simulation of multi-phase flow, Computer physics communications, 87, pp. 225-235, 1995].

은 수학식 6과 같이 표현할 수 있다.That is, the speed from the equation (2)

Can be expressed as in Equation 6.

는

의 질량을 가지고

에 존재하는 질점

의 속도, 그리고

는 밀도이다.here

Is

Take the mass of

Present in

Speed, and

Is the density.

The momentum equation for the mass motion is as shown in Equation 7, and Equation 7 can be replaced as shown in Equation 8.

는 중력 등의 체적력을 의미하고, 점성항

는 수학식 9와 같이 주어진다.here Is a viscous term,

Means the gravitational force such as gravity,

Is given by Equation (9).

The continuous equation of the fluid (a change in density per unit time) is expressed as in Equation 10.

Equation 10 can be rewritten as in Equation 11 using the SPH technique.

In the case of a flow colliding with a solid wall surface, the simulations are performed assuming that the materials at the boundary surface collide with the wall surface to obtain a repulsive force in the form of Lennard-Jones, as shown in Equation 12 below.

In this way, the flow variables (speed, acceleration, position, density, etc.) are obtained through the SPH method through step S802.

In order to illustrate the validity of the non-lattice free surface flow analysis technique described above, a total of 1681 materials were used, and 105 fixed materials were assigned to the walls to analyze the dam collapse problem as shown in FIGS. 3A to 3D. Got.

In addition, as a result of simulating the problem that the dams of the fluids having different properties from each other collapsed and mixed, the graphs shown in FIGS. 4A to 4F were obtained.

In the conventional flow analysis, it is difficult to easily solve such a problem of fluid mixing. However, when the SPH method according to the present invention is interpreted, each fluid point represents a specific volume of the fluid. This is easily simulated.

In addition, the free surface flow problem can be analyzed three-dimensionally by using a non-lattice method as shown in FIGS. 5A to 5D.

Next, the volume rendering step (S804) will be described. A process of generating volume data from particle simulation is described, and then a method of volume rendering by obtaining variables using pre-integration will be described.

Quality points that move freely in space basically take the form of point data with coordinates and velocity components.

To apply this data to volume rendering, we first subdivide the space into regular unit grids.

Then, the density of particles contained per unit volume is obtained for the space corresponding to each lattice.

This density value can be thought of as the fluid content data of the voxel representing the unit volume. Therefore, if the real numbers from 0 to 1 are properly proportional to the number of particles, volume rendering is achieved. Data is created.

Next, the basic volume rendering equation is given by the following form using pre-integration.

는 공간 좌표(space coordinate),

는 연속적인 스칼라 장(continuous scalar field),

는 최대 거리(maximum distance),

는 시점으로부터의 거리,

는 기본 칼라,

는 감쇄도이다.here

Is the space coordinate,

Is a continuous scalar field,

Is the maximum distance,

Is the distance from the point of view,

Is the base collar,

Is the degree of attenuation.

이라고 할 때, 수학식 13의 지수함수(

) 내부는 수학식 14와 같이 근사화 된다.

In this case, the exponential function of Equation 13 (

The inside is approximated by Equation (14).

번째 광선 세그먼트(ray segment)의 불투명도(opacity)는 수학식 15와 같이 근사화 된다.finally

The opacity of the first ray segment is approximated by Equation 15.

번째 광선 세그먼트(ray segment)에서 방출된 컬러는

로 근사화 된다.

The color emitted from the first ray segment

Is approximated.

As a result, the integration of Equation 13 is arranged as in Equation 16.

Therefore, the back-to-front compositing algorithm is calculated as shown in Equation 17.

는

로 교체될 수 있으므로, 수학식 17은 수학식 18과 같이 된다.Also,

Is

Equation 17 can be replaced by Equation 18.

Using Equation 18, the integral of Equation 13 is expressed in a more general form as shown in Equation 19.

In this case, the back-front synthesis algorithm corresponding to Equation 17 is calculated as Equation 20.

가 되고, 세그먼트 뒤쪽의 스칼라 값은

가 된다.For the i-th segment, the scalar value in front of the segment is

, And the scalar value after the segment

Becomes

Using this value, the opacity of the i-th segment is approximated as shown in equation (21).

Where w is the integral parameter.

는

의 함수가 됨을 알 수 있고, 따라서,

는 수학식 22와 같이 계산된다.here

Is

Is a function of, therefore,

Is calculated as shown in Equation 22.

를 정의하면,

를 T를 사용해서 수학식 23과 같이 나타낼 수 있다.From here

If you define

Can be expressed by Equation 23 using T.

를 정의하면,

를 K를 사용해서 수학식 24로 나타낼 수 있다.In the same way,

If you define

Can be represented by Equation 24 using K.

를 이용하여 유체의 색상을 표현할 수 있다.Obtained color information like this

You can express the color of the fluid using.

As described above, the three-dimensional rendering is performed by obtaining variables (brightness, color, opacity, etc.) of the fluid to express the three-dimensional effect of the fluid.

Next, a detailed image quality adjustment step (S806) by adjusting rendering parameters will be described.

Phong illumination model method as shown in Equation 25 gives the effect of shading and reflection on the surface of the rendered fluid.

는 유체의 한 지점의 밝기(intensity),

는 주변반사(ambient-reflection) 상수,

는 주변 광원의 밝기,

는 광원의 밝기,

는 확산반사(diffuse-reflection) 상수,

는 거울반사(specular reflection) 상수,

은 해당 지점의 법선벡터,

은 빛의 방향 벡터,

은 빛의 반사벡터,

이다(

는 시점과 물체가 이루는 단위벡터이다).here

Is the intensity of a point in the fluid,

Is the ambient-reflection constant,

Is the brightness of the ambient light,

Is the brightness of the light source,

Is the diffuse-reflection constant,

Is the specular reflection constant,

Is the normal vector at that point,

Silver light direction vector,

Silver Reflection,

to be(

Is the unit vector of the viewpoint and the object).

6A and 6B show this Phong illumination vector.

을 실 시간으로 조절할 수 있게 되어 세밀한 이미지 질(quality)의 실시간 조작이 가능해진다.Equation (25) using pixel shader technology supported by DirectX, a multimedia application program interface,

Can be adjusted in real time, enabling real-time manipulation of fine image quality.

By the method described above, the volume rendering results as shown in FIGS. 7A to 7D can be obtained.

As described above, according to the present invention, the motion of the fluid is simulated in real time by combining the characteristics of the non-lattice method capable of performing numerical analysis without using a lattice structure and a volume rendering technique suitable for real-time rendering of irregular shapes. Can render.

Claims (6)

Simulating and analyzing the momentum of a fluid including a free surface by a non-lattice method that expresses the mathematical correlation between each randomly distributed matter in the calculation domain as a function of distance; A non-lattice analysis and volume rendering technique including volume rendering step of generating volume data using coordinates and velocity components of a material point obtained in the simulation step, and obtaining and rendering variables for expressing a three-dimensional effect of a fluid composed of the material points. Real-time simulation and rendering method of fluid effect using The method according to claim 1, The simulating includes using a smooth particle hydrodynamics (SPH) lattice technique to calculate velocity, acceleration, position, and density, which are fluid variables. The volume rendering step includes a non-lattice analysis technique and a volume rendering technique comprising storing and rendering a flow variable of the fluid, which is obtained through the use of the SPH grating technique, as a voxel data structure in real time. Real-time simulation and rendering method using fluid effects. The method according to claim 2, wherein the field variable at any position for each material point is represented by a three-dimensional position function, and the field variable is approximated in the area by the SPH method, and then the momentum with respect to the speed by the SPH method from the approximated value. A fluid effect real-time simulation and rendering method using a non-lattice analysis method and a volume rendering method characterized in that the equation is calculated and calculated and then rendered using a volume rendering method. The method according to claim 1, Variables for expressing the three-dimensional effect of the volume rendering step are the brightness, color, opacity of the fluid, and the non-lattice analysis method using the non-lattice method, characterized in that these variables are calculated by using a pre-integration method Real-time Simulation and Rendering Method of Fluid Effects Using Intensity and Volume Rendering Techniques. The method according to claim 1 or 4, wherein the volume data, Subdividing the space in which quality points move freely into a unitary grid; Obtaining a density of a quality point contained per unit volume for a space corresponding to each lattice; By looking at the density value as the fluid content data of the voxel representing the unit volume by the step of proportionally matching the real value from 0 to 1 to the number of quality points, A fluid effect real-time simulation and rendering method using a non-lattice method characterized in that the generated. The method according to claim 1, After the volume rendering step, Using the Phong illumination model, represented by Equation 26, further performs adjustment steps that effect shading and reflection on the surface of the volume rendered fluid. Effect real time simulation and rendering method.

(here

Is the intensity of a point in the fluid,

Is the ambient-reflection constant,

Is the brightness of the ambient light,

Is the brightness of the light source,

Is the diffuse-reflection constant,

Is the specular reflection constant,

Is the normal vector at that point,

Silver light direction vector)

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