KR100568562B1 - A real-time simulation and rendering method for fluid flows using continuum fluid mechanics and volume rendering techniques - Google Patents

Abstract

The disclosed fluid flow simulation and rendering method is based on a flow equation analysis method for a continuum. The fluid motion simulation divides the calculation domain into finite elements in the form of a fixed lattice, calculates the velocity and pressure of the fluid, and calculates the shape of the free surface. And a volume rendering step of performing a rendering for the fluid simulated in the fluid motion simulation step by obtaining visual variables for expressing the three-dimensionality of the fluid using the property of light passing through the fluid or reflected from the surface of the fluid. do. According to this method, the simulation is performed using the exact equation of motion that satisfies the physical law. Therefore, it is possible to produce a video with higher realism than the existing fluid effect video production method, and it is not necessary for the operator to modify it repeatedly. This reduces the overall work time and enables real-time video depicting complex flow phenomena.

Description

A real-time simulation and rendering method for fluid flows using continuum fluid mechanics and volume rendering techniques

1 is a flow chart showing a fluid flow simulation and rendering method according to the present invention,

2A-3B show an example of a method of analyzing a moving fluid interface;

4 shows an example of a fixed coordinate system for analyzing a fluid having a free surface,

5A-5E show an example of an algorithm for reconstructing the surface shape of a fluid,

6 shows a direction vector at each element of a free surface;

7 is a diagram illustrating a process of obtaining a wet-out fraction using the direction vector of FIG. 6.

8 illustrates a process of mapping a grid of simulations to a grid of volume rendering;

9a to 9c is a view showing that the fluid is poured into a predetermined container by the fluid flow simulation and rendering method of the present invention,

10A and 10B are diagrams for explaining each variable of the Phong illumination model.

The present invention relates to a fluid flow simulation and rendering method, and more particularly, to a method for realizing a complex flow phenomenon of a fluid in real time using a continuum simulation technique and volume rendering.

As one of the methods for expressing the free motion of a fluid in the field of computer graphics, a particle dynamic method has been proposed that describes the dynamic interactions and motions between fluids using relatively simple mechanical equations. However, the range that can be expressed only by such particle dynamics is quite limited, and many people agree that there are many limitations in expressing the various movements of the fluid in this way. That is, in the particle method, it is difficult to effectively express various fluid inherent properties such as shear stress, viscosity, and vorticity, and it is difficult to effectively express the flow path by collision between particles or collision with the wall surface. There are many problems, such as the rapid change of, in which particles that lose velocity energy accumulate spatially and do not strictly satisfy the law of conservation of total volume.

Another very important aspect of fluid simulation is the stability of the simulation algorithm. In many cases, if the increment of time applied to an operation is not appropriate, the operation may not be completed and may be inappropriately interrupted depending on the mathematical stability of the algorithm. This mathematical instability is a factor that reduces the reliability of the simulation tool and reduces the visual probability of the result obtained by the calculation.

Recently, in the field of computer graphics, studies are being actively conducted to express fluid effects by using continuum simulation through the Navier-Stokes equation by utilizing the techniques of traditional fluid dynamics to overcome the limitations and visual limitations of particle simulation. It is becoming. In 1996, Nick Foster applied a three-dimensional incompressible Navier-Stokes equation to visualize the flow after simulation. Stationary objects and free surfaces are represented as boundary conditions, as well as buoyancy effects. At SIGGRAPH in 1999, Stam presented a way to solve the Navier-Stokes equation relatively accurately and reliably for any time step value, and to guide future research. The numerical analysis technique used here is different from the conventional particle method, which uses a finite difference method using a mesh of lattice structure. In 1999, Yngve et al. Also studied finite difference methods to analyze shock wave problems and the behavior of flames and fumes caused by explosions.

In collaboration with Nick Foster of PDI studio and Ronald Fedkiw of Stanford University and published in SIGGRAPH in 2001, the study of Practical Animation of Liquids combines particle and continuum methods according to the situation and introduces a level set method. It was. This method was attempted to faithfully reproduce the motion of the fluid by applying physical laws suitable for flow analysis. However, the method used by them takes too much computation time to solve the flow problem consisting of a large number of small particles such as water droplets, fractions, and raindrops. Since the geometry exposes errors due to the numerical diffusion of the flow equations used by them without filtration, it is difficult to apply to problems that require accurate calculations in real time due to limitations in accurate reproduction of physical phenomena and excessive computation time. Follow.

As the rendering technique, ray casting technique, blackbody radiation technique, etc. are applied to make the expression more realistic. Other techniques include particle dynamics using finite-sized particles in the concept of droplets, which calculate the overall motion, and then process the delicate textures with mapping techniques.

For realistic rendering, the rendering of natural phenomena used a lot of global illumination methods. Among them, the method using the photon mapping proposed in 1998 by Jensen is widely used in recent fluid simulations. The visual quality of the result obtained through this method is very good, but it is not suitable for real-time application because the execution time takes in minutes from several minutes to one frame.

There are several technical limitations in fluid motion simulations such as gases, liquids, flames and climate phenomena.

The first constraint is the computation time associated with the mechanical simulation of fluid motion. In traditional fluid mechanics, a number of computational techniques using the Navier-Stokes equation have been used for high precision fluid simulation. Among the flow analysis techniques based on the Navier-Stokes equation, the technique of simulating the free surface motion is mainly used the fixed grid method and the moving grid method, each of which has advantages and disadvantages. The moving lattice method has the disadvantage of requiring too much lattice and a long calculation time to express the motion of a fluid moving in a complicated shape region instead of the geometrically clear position of the moving fluid interface. In contrast, the fixed lattice method expresses the motion of a fluid in such a way that the entire computational domain is initially decomposed to a predetermined resolution, and then the volume fraction occupied by the fluid changes with time as the fluid moves. do. In addition to this method, solid fractions can be defined to effectively account for fluid interactions with moving objects, and by varying the volume fraction of solids within a given volume over time, the motion of solids and the resulting fluids. The method of more effectively simulating the disturbance phenomenon has been developed and used, but it has a problem that the operation speed is slow to cope with real time.

The second constraint is the constraints on rendering time. Global illumination methods, such as general photon mapping, require significant computational time for realistic rendering. In addition, rendering simulation data using commercial software such as 3D Max and Maya is also time consuming. These methods produce about 5 seconds of 640 x 480 images, which takes approximately 20 minutes or more. Various techniques have been tried to solve these two constraints, and in recent DirectX technology, the fluid effect in real time in the form of wave propagation, texture mapping, height-field, etc. Methods of expressing are being developed. However, these methods are inadequate to represent any free surface and remain at a level that slightly changes the planar effect. Maya and Max are plugin concepts of standard tools, such as Air and Cloud, which make plugins to create three-dimensional atmospheric and cloud-like environmental effects based on physics. Air is responsible for animate the colors of the sky, using the angle and brightness of the light to create a shady sky during rendering. In addition, simulations can be performed, ranging from light fog to stormy fog to rainbow effects. Water creates a realistic water surface. These plug-ins can create natural atmospheric effects by controlling parameters such as cloud density, aggregation, and color. In these methods, many restrictions on expression can be eliminated, but are not suitable for real-time rendering because the amount of data is quite large for producing realistic images. RealFlow, RealWave, and RealTracer, developed by Nextlimmit, Inc., are fluid-related special effects programs that can be used in software for Max, Maya, Lightwave, etc. as standalone plug-ins. The recently released fluid modules in Maya 4.5 also perform well. However, these programs are excellent in quality and compatible with image output, but basically adopt only particle method, so it is difficult to apply them widely to all physical ranges of flow phenomena and does not support real-time rendering. Various standards such as Java3D, OpenGL, and DirectX are being produced as real-time virtual reality rendering tools, and some functions using fluid simulation are being implemented, but it is possible to effectively simulate irregular flow in real time such as complex fluid flow or flame. VR solutions are still in the development stage, and their quality is very primitive and can't deliver enough realism.

As a result, existing methods cannot be interactive due to their computational constraints during simulation and rendering, so creating various effects using existing methods is time-consuming, even if they interact. It is difficult to obtain realistic images.

SUMMARY OF THE INVENTION The present invention has been made in view of the above problems, and provides an improved fluid flow simulation and rendering method for rapidly realizing fluid complex free motions in real time and strictly satisfying physical laws to produce more realistic image results. The purpose is.

In order to achieve the above object, the fluid flow simulation and rendering method of the present invention divides the calculation region into finite elements in the form of fixed lattice and calculates the velocity and pressure of the fluid based on the flow equation analysis method for the continuum. A fluid motion simulation step that calculates the shape of a surface, and visual variables for expressing the three-dimensional effect of the fluid using the properties of light passing through or reflected from the surface of the fluid to obtain a fluid simulated in the fluid motion simulation step It characterized in that it comprises a volume rendering step of performing a rendering for.

Hereinafter, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings.

The present invention aims to propose a real-time simulation framework that efficiently analyzes irregular movements of liquids, gases, flames, etc. among a wide range of physical domains in a consistent way and visualizes them efficiently by integrating them into a fast real-time rendering environment. do. This method shows an interactive speed and a quality image suitable for use in general real-time applications, especially in extreme cases such as rapid changes in fluid velocity or local congestion. It helps to produce a stable video.

As shown in FIG. 1, the fluid video implementation method of the present invention is divided into three stages. The continuum simulation step (S1) and the step of calculating volume for representing the stereoscopic sense of the simulated continuum and performing volume rendering. (S2) and the flow effect adjustment step S3 that allows the user to adjust various parameters to give a more realistic visual effect.

First, the simulation step (S1) was simulated based on the fluid dynamics that represent the flow by the finite element. This method can represent a smooth, naturally connected surface without the lack of realism due to the discontinuity of data in a limited number of particles or lattice points, a disadvantage of particle dynamics. In addition, the finite element based fluid analysis grid used for continuum simulation and the three-dimensional grid structure of the voxel concept used for rendering have very similar geometrical structures so that the data obtained through the simulation can be easily used for rendering.

However, the fluid to be expressed using the continuum flow equation generally has the form of agglomerated mass, so that a large number of unit grids are included in a continuum. When the regular lattice structure is rendered three-dimensionally, a unique repetitive pattern appears for each layer according to the viewpoint of an observer, which causes a problem of deteriorating visual realism. In order to overcome this disadvantage, in the second stage of volume rendering (S2), a 3D real-time pre-integrated volume rendering method based on a voxel structure is used. As such, the volume rendering method applied in the present invention starts from the ray casting method proposed in 1990 by Levoy. This method is slow because it computes a large amount of volume data, but it can be accelerated by space hopping and early ray interruption. With the recent development of graphics hardware, acceleration methods using hardware textures for real-time volume rendering have been studied. Engel proposed the pre-integration method in 2001, which supplements the problem of OTF in addition to the acceleration method using hardware texture. The rendering method of the present invention is basically based on this Engel method.

In the flow effect adjustment step (S3), various types of flows can be expressed in near real time through the adjustment of various parameters and the use of multiple OTF (Opacity Transfer Function, Opacity Transfer Function).

Hereinafter, the above-described simulation step (S1) will be described in order.

In the present invention, the simulation is performed based on the fluid dynamics representing the flow by the finite element method as described above. First, the fluid having the free surface has a problem in that the boundary moves because the boundary portion is not predetermined and determined according to the analysis process. Like the Navier-Stokes Equation, this problem is very nonlinear. The continuous change of area and the physical discontinuity of the interface make this problem very difficult to deal with.

As a method for solving such a moving boundary problem, there are a Lagrangian method (see FIG. 2A), an Eulerian method (see FIG. 2B), and an ALE (Arbitrary Lagrangian Eulerian) method combining the two methods. In the Lagrangian method, the coordinate system moves at the same speed as the fluid and the free surface coincides with the mesh. As a result, the interface is obtained accurately. However, fluids with severe distortions and complex shapes are difficult to obtain accurate interfaces. In this case, the computation is time consuming, but rejoining the distorted mesh is inevitable.

In the Eulerian method, the coordinate system is fixed and the fluid moves in the coordinate system. Also, the first mesh created is used for the entire calculation time. Using this method, even fluids with high motion can be handled relatively accurately and effectively. It can also handle multiple boundaries and extend two-dimensional code to three dimensions. On the other hand, since the interface does not coincide with the grid point but moves between the grid points, a special method is needed to accurately obtain the interface. Eulerian methods are divided into three types: fixed grid method, moving grid method, and mapping method.

The fixed lattice method divides the entire computational area into a lattice structure of a suitable size in advance, and then uses the ratio between the volume of the introduced fluid and the intrinsic volume of the lattice when the mass of the fluid is introduced into each lattice. Defined as a function, the value is changed over time to track the overall shape of the area occupied by the fluid. Representative techniques include the Volume of Fluid (VOF) method (FIG. 3A), the Marker And Cell (MAC) method (FIG. 3B), the Simple Line Interface Calculation (SLIC) method, and Young's method.

The moving grid method is a method of accurately positioning the interface by adjusting the mesh to match the interface. As the calculation progresses, the velocity of the fluid mass located on the free surface changes, and this data is used to geometrically calculate the changed shape of the computational area as the flow surface advances and define a new computational area every hour. to be.

In the mapping method, the physical domain is transformed into a simple computation domain by a mapping function. This method can solve the boundary accurately, but it is very difficult to apply this method to complex shapes because the overall shape must be able to be expressed as a suitable mathematical function. Therefore, it is difficult to find an appropriate mapping function for flows with complex interfaces.

The Arbitrary Lagrangian-Eulerain (ALE) method, which combines the Lagrangian and Eulerain methods, combines the advantages and avoids the disadvantages of the two methods. In this method, the velocity of the mesh is independent of the velocity of the fluid, but the mesh at the interface moves at the same speed as the fluid to easily position the interface.

In the present invention, the irregular grid was traced using the VOF-based method, and the baby-cell method was used to reconstruct the free surface. This method was proposed by Kim and will be described later in Equation 14.

Hereinafter, an analysis algorithm of an incompressible fluid will be described.

First, a method of using the governing equations will be described. The governing equations for incompressible fluids are continuum equations such as Equation 1 and Navier-Stokes Equation of Equation 2.

는 속도 벡터, p는 압력,

는 중력 또는 부력,

는 viscous stress tensor,

는 밀도이다. here,

Is the velocity vector, p is the pressure,

Is gravity or buoyancy,

Viscous stress tensor,

Is the density.

The constitutive equation is expressed as

는 연신율 텐서(strain rate tensor)이고

는 전체 응력 텐서(stress tensor)이다.

는 점성이다.

Is the strain rate tensor

Is the overall stress tensor.

Is viscosity.

의 필수경계조건(essential boundary condition)과

의 정지마찰의 경계조건(traction boundary condition)으로 구성된다. 수학식 5는 경계 조건을 표현한 식이다. The boundary condition for speed is

Essential boundary conditions and

It consists of the traction boundary condition of static friction. Equation 5 is an expression representing a boundary condition.

는

의 속도 벡터이고

는

의 트랙션(traction) 벡터이다.

는

의 단위 normal 벡터이다.The two conditions of Equation 5 are mutually exclusive.

Is

Is the velocity vector of

Is

Is the traction vector of.

Is

The unit of normal vector.

에 대한 초기 조건은 다음과 같다.

에서

와

에서

의 조건을 만족하면

가 해답이 존재하기 위한 초기 조건 이 된다. 만약

가 null이라면, 속도를 위한 경계 조건은 반드시 다음의 수학식 6 조건을 만족해야 한다.Whole area

The initial conditions for are as follows.

in

Wow

in

If you meet the conditions of

Is the initial condition for the solution to exist. if

If is null, the boundary condition for velocity must satisfy the following Equation 6.

는 속도의 초기 값이다.here

Is the initial value of velocity.

Finite Element Method is a numerical analysis technique that divides the entire area of complex shape into finite simple finite elements and treats it as a set of simple geometric elements. It also transforms and interprets boundary conditions for governing equations. FEM is widely used in many areas, including engineering and mathematical physics. In the present invention, the governing equations were analyzed using FEM.

Among the FEMs for the incompressible Navier-Stokes equation, there are a mixed method, a penalty method, and a fractional step method. In the mixed method, velocity and pressure are interpreted simultaneously. That is, mass and momentum conservation are satisfied at the same time. Using the penalty method involves introducing appropriate numerical assumptions between the continuum equation and the momentum equation and converging both within a reasonable margin of error. Using this method, the overall governing equation is simplified to the equation for speed only, and the pressure is further calculated if necessary. The fractional step method satisfies both the momentum conservation equation and the mass conservation equation sequentially.

Next, there is a method called explicit element-by-element fractional step method.

This method has been proposed by Nakayama and Mori and has been applied by Kim et al. Semi-discretized momentum equations can be expressed as

And, the continuum equation can be discretized as follows.

는 발산 행렬을 의미하며 N은 요소의 개수이다. Euler 방법이 반이산화된 운동량 방정식에 다음과 같이 적용될 수 있다.Where u and p are the velocity and pressure vectors of the nodes, respectively. M, C (u), K, and H are mass, convection, divergence, and pressure gradient matrices. F is a vector of forces including external forces and boundary conditions.

Is the divergence matrix and N is the number of elements. Euler's method can be applied to the semi- discretized momentum equation as follows:

는 매개 속도를 나타낸다. 상기 수학식 9에서 삼선형 모양 함수가 속도 변수에 적용되고 단위요소 모양 함수는 압력 변수에 적용된다. 매개 속도

는 발산하지 않는다고 볼 수는 없으나 압력

을 가지고 질량 보존 법칙을 만족하도 록 수정될 수 있다. 압력은 요소의 발산을 없애도록 반복적으로 요소요소를 일소함으로서 조정된다. 속도는 압력에 의해 수정된다. 속도와 압력 수정 단계는 다음과 같이 요약될 수 있다.

Represents the median speed. In Equation 9, the trilinear shape function is applied to the speed variable and the unit element shape function is applied to the pressure variable. Medium speed

Does not dissipate, but pressure

Can be modified to satisfy the law of conservation of mass. The pressure is adjusted by repeatedly sweeping out the element to eliminate divergence of the element. Speed is corrected by pressure. Speed and pressure correction steps can be summarized as follows.

와

의 값은 각각 매개 속도

와

이다. 단위 볼륨

당 발산은 다음과 같이 계산된다.In the beginning

Wow

The value of each parameter

Wow

to be. Unit volume

Sugar divergence is calculated as follows.

는 요소의 볼륨을 말하고,

는 전체 압력 변화량 행렬 H에 상응하는 요소의 값이다. 만약

의 크기가 수렴 범위

이내라면, 유체는 지역적으로 비압축성이고 더 이상 조정이 필요하지 않다. 만약

을 넘는다면 압력은 발산하지 않는다는 조건을 만족하도록 조정될 수 있다. 속도는 수학식 11과 같이 발산하지 않도록 압력에 의해 수정될 수 있다. n + 1 and (k) are the time step and the number of repetitions.

Tells you the volume of the element,

Is the value of the element corresponding to the overall pressure change matrix H. if

The size of the convergence range

If so, the fluid is locally incompressible and no further adjustment is necessary. if

If the pressure is exceeded, the pressure may be adjusted to satisfy the condition of no divergence. The speed may be modified by pressure so as not to diverge as in Equation 11.

은 5×10^-4에서 1×10^-5까지다. 이 방 법은 수렴을 위해 여러 번 반복연산이 필요하지만 전체적인 성능이 우수하다. 움직이는 자유 표면을 가진 유체는 일반적으로 복잡하고 매우 계산시간이 많이 소요된다. explicit element-by-element step방법은 비압축성 유체를 계산하는데 매우 효율적인 알고리즘이다. 본 발명에서는 효율성을 위해 이 방법을 채택하였다. This method uses the row-sum mass lumping method in both the momentum conservation equation and the rate-correction step. In the present invention

Is from 5 × 10 ^-4 to 1 × 10 ^-5 . This method requires multiple iterations for convergence but good overall performance. Fluids with moving free surfaces are generally complex and very time consuming. The explicit element-by-element step method is a very efficient algorithm for calculating incompressible fluids. In the present invention, this method is adopted for efficiency.

Next, an analysis algorithm for a fluid having a free surface will be described.

In general, free surfaces have very complex functions and fluids move very fast. In the present invention, the VOF method is used among many methods based on the Lagrangian, Eulerian, and ALE methods for the complex fluid motion as described above. In the present invention, the following is assumed. First, the system consists of two unmixed fluids. Second, the physical properties of the two unmixed fluids are internally homogeneous. Third, the fluid is incompressible, viscous, and the inertial region covers the range just before turbulence occurs. Fourth, the effect of surface tension is ignored.

The VOF method was first developed by Hirt and Nichols and is a representative algorithm based on a fixed coordinate system for analyzing fluids with moving free surfaces. The volume of the fluid f (x, y, z, t) is defined as the ratio of the fluid in one unit element as shown in equation (12).

When a unit element is completely filled with fluid, the volume of the fluid is f = 1 and the unit element is regarded as the main fluid region. If the unit element is empty (f = 0), it belongs to the empty area and is excluded from the calculation of the fluid. The unit element is considered to be on the free surface when the volume of the fluid is between 0 and 1, i.e. 0 <f <1. As shown in Figure 4, the computational area of the free surface can be easily represented using the variable f.

The motion of the interface is expressed by the following equation (13) for f.

This equation consists of only abnormal and convective terms. Therefore, when discretizing this equation directly and performing arithmetic operations, the analysis can cause significant divergence as the boundary of the fluid surface is blurred. The VOF method can effectively handle the fluid flow of complex movements, but in some cases, the problem arises how to reduce the numerical divergence. To achieve stable numerical results, numerical divergence must be avoided and at the same time the boundary of the fluid surface must move without blurring. Algorithms designed to suppress divergence include the SLIC (Simple Line Interface Calculation) method (see FIG. 5A), SOLA-VOF method (see FIG. 5B), Young's method (see FIG. 5C), and FLAIR (Flux Line-Segment Model for Advection and Interface Reconstruction (FIG. 5D), PLIC (Piecewise-Linear Interface Calculation) method (FIG. 5E), and the like. These algorithms divide and reconstruct the shape of the fluid surface appropriately and move the free surface appropriately in each finite element situation to increase the accuracy of calculation and prevent divergence.

을 어떻게 찾을 것인가 이다. wet-out fraction

과

을 계산하는 알고리즘은 김(Kim)의 baby-cell 방법을 기초로 한다. 유체 표면의 형상을 재구성하는 알고리즘 중에는 상기한 PLIC-type(도 4e)이 있는데 이 방법은 경계면을 normal vector가 f의 기울기(gradient)인 선이나 면으로 가정한다. baby-cell 방법은 PLIC-type을 사용하여 자유 표면을 표현한다. 이 방법에서 방향벡터 r은 수학식 14와 같이 정의된다.The first consideration in the technique of advancing the free surface is the wet-out fraction associated with the geometric reconstruction of the free surface.

How to find wet-out fraction

and

Algorithm for calculating is based on Kim's baby-cell method. Among the algorithms for reconstructing the shape of the fluid surface is the PLIC-type (FIG. 4E) described above, which assumes the interface as a line or face whose normal vector is a gradient of f. The baby-cell method uses a PLIC-type to represent free surfaces. In this method, the direction vector r is defined as in Equation 14.

와

는 각각 i 번째 요소의 j 번째 면의 유체의 볼륨과 단위요소의 볼륨을 의미한다. 또한,

는 요소의 경계

에 대한 단위 normal vector이고

는 i 번째 요소의 전체 영역 중에서 j 번째 면의 비율이다. 수학식 3과 9에서 볼 수 있는 바와 같이, 방향 벡터는 이웃한 유한요소 내의 유체의 체적평균을 구하여 결정된다. 그리고, 도 6에서 보듯이 방향벡터는 유체의 부피가 큰 방향을 나타내는 단위 벡터이다. 방향 벡터 r은 자유 표면의 normal 벡터 대신 사용될 수 있다. 이 벡터는 대수적인(algebraic) 방법으로 정의되므로 비구조적 메쉬(unstructured mesh)에서도 이 벡터를 계산하는 일은 매우 용이하다. 방향 벡터는 도 7에 나타낸 바와 같이 다음의 방법으로 wet-out fraction을 결정하는데 사용된다. 첫째, 방향 벡터 r에 수직되고 유한요소의 중심점을 통과하는 평면을 만든다. 둘째, 각 유한요소는 같은 체적을 가지는 baby-cell들로 분할된다. 셋째, baby-cell은 방향벡터 r의 양의 방향의 평면에서 가장 먼 cell부터 시작해서 채워진다. 그리고 나서, wet-out fraction

가 계산될 수 있다. FEM에서 좌표계 변형 속성을 사용하여 위의 계산이 수행된다. 이러한 시뮬레이션 과정을 통해 유체의 유동이 해석되고 각 셀에서의 속도와 압력 등이 구해지게 된다.

Wow

Denotes the volume of the fluid on the j-th surface of the i-th element and the volume of the unit element, respectively. Also,

Is the bounds of the element

The unit for is normal vector

Is the ratio of the j th face of the entire area of the i th element. As can be seen in equations (3) and (9), the direction vector is determined by obtaining the volume average of the fluid in the adjacent finite element. As shown in FIG. 6, the direction vector is a unit vector representing a direction in which the volume of the fluid is large. The direction vector r can be used instead of the normal vector of the free surface. Since this vector is defined in an algebraic way, it is very easy to calculate this vector even on an unstructured mesh. The direction vector is used to determine the wet-out fraction in the following manner as shown in FIG. First, create a plane perpendicular to the direction vector r and passing through the center point of the finite element. Second, each finite element is divided into baby cells with the same volume. Third, the baby-cell is filled starting from the cell furthest from the plane of the positive direction of the direction vector r. Then, wet-out fraction

Can be calculated. The above calculation is performed using the coordinate system transformation properties in the FEM. Through this simulation process, the flow of fluid is analyzed and the velocity and pressure in each cell are calculated.

Next, step S2 is performed to perform volume rendering. Volume rendering is a process of imparting a three-dimensional effect by calculating brightness, opacity, color, etc., assuming that a fluid particle can transmit light, and first calculating volume data from the continuum simulation. In order to use the simulation result for the volume rendering, a process of mapping the grid of the simulation to the grid of the volume rendering is required as shown in FIG. 8. In the VOF method, which is an analysis algorithm of moving fluid, the value of f, which is the ratio of the volume of the fluid to the volume of one unit element, is a value of how much fluid is filled in the unit element. When this value is mapped to data of a voxel representing a unit volume as shown in FIG. 8, data for volume rendering is generated. Here, the voxel means a point of each cell which is a unit of volume.

The lattice structure used in the continuum finite element analysis and the voxel structure of volume rendering have very similar hexahedral arrangements, so the shape is very similar. When the cell spacing between the finite element grid and the voxel is the same, the fill fraction (f) defined in Equation 12 of each finite element may be directly corresponded to the voxel data as it is. At this time, the mass of the microvolume surrounding the point is treated as the concentrated mass at the nodes of each finite element, and the data can be assigned to each node of the voxel structure and used as it is for rendering.

If the lattice structure of the finite element does not coincide with the lattice structure of the voxel data, the concentrated mass assigned to the position of each finite element node occupies a position within a particular hexahedron in the voxel data lattice. The volume fraction is calculated by calculating how much the effective volume occupied by the concentrated mass of the finite element node occupies within a cube in the voxel data grid and uses this volume fraction as the volume fraction of the voxel node. In some cases, if the radius corresponding to the effective volume of the concentrated mass exceeds the size of one voxel, the surplus volume is transferred to the adjacent voxels and added. In this way, the fill fraction occupied by the fluid for the entire finite element node is matched one-to-one with the voxel data, and then rendered to visualize the distribution of the fluid.

The basic volume rendering formula is given by the following form using pre-integration.

I: final brightness value

λ: distance from the viewpoint,

x: three-dimensional space coordinates,

s: subsequent scalar sheet,

: 기본 칼라,

Base color

: 감쇄도,

: Decay degree,

D: Maximum distance from the viewpoint.

Herein, when d = D / n, the inside of the exponential function of Equation 15 may be approximated as in Equation 16 below.

는 i번째 광선 부분의 불투명도를 나타낸다.here

Denotes the opacity of the i-th ray portion.

는 다음의 수학식 17과 같이 근사될 수 있다.Finally, the opacity of the i th ray segment

Can be approximated as Equation 17 below.

는 수학식 18로 근사될 수 있다.And, the color emitted from the i th light portion in the equation (15)

Can be approximated by Equation 18.

As a result, the integration of Equation 15 may be arranged as in Equation 19 below.

That is, the final brightness value of the i th light portion is obtained by multiplying the opacity from 0 to i-1 by the color value of the i th light portion. Then, the sum of all the final brightness values of the 0th to nth light beam parts becomes I.

Therefore, the back-to-front compositing algorithm is calculated as shown in Equation 20 below.

은 n번째 광선 부분에서부터 i번째 광선 부분까지의 밝기의 총합을 나타낸다.

Denotes the sum of the brightnesses from the n th light portion to the i th light portion.

는

로 교체될 수 있으므로, i번째 광선 부분에서 방출되는 색깔

는 수학식 21과 같이 근사될 수 있다. Also,

Is

Color emitted from the i th light portion

Can be approximated as Equation 21.

Using the above equation, the integral of Equation 15 may be expressed in a more generalized form as shown in Equation 22 below.

는 색깔과 불투명도가 곱해진 값이므로 풀어서

로 쓴 것이다. That is, in equation (19)

Is the product of color and opacity, so solve it

It is written as.

The first scalar value of the segment (density value) s _f : = s (x (id)) for the i th segment, and the scalar value s _b : = s (x ((i + 1) after the segment. d)) Using this value, the opacity of the i-th segment can be approximated by Equation 23.

w means integral parameter.

는 s_f, s_b, d의 함수가 된다. 따라서, i번째 광선 부분에서 방출되는 색깔

는 다음의 수학식 24와 같이 계산될 수 있다. finally

Becomes a function of s _f , s _b , and d. Thus, the color emitted from the i th light portion

May be calculated as in Equation 24 below.

를 정의하면,

를 수학식 25과 같이 T를 사용해서 나타낼 수 있다. here

If you define

Can be expressed using T as in Equation 25.

를 정의하면,

를 K를 사용해서 수학식 26처럼 나타낼 수 있다. In a similar way

If you define

Can be expressed as Equation 26 using K.

By using these formulas, data for expressing volume such as final brightness, color, and opacity is calculated, and rendering is performed in real time based on the data. 9A-9C show rendering results depicting scenes where fluid is poured into a container.

Next, it enters the flow effect adjustment step (S3) that provides an input environment for the user to adjust the parameter, and reflects the input value to impart various flow effects to the rendering result. The simulation parameters used here include fluid density, air viscosity, water viscosity, and the like. As rendering parameters, the ambient lighting constant k _a and the dispersion lighting constant k _{d of the} phong lighting model equation shown in Equation 18 below. , Reflection lighting constant k _s, exponent term n, and so on. For example, by adjusting the n value, you can change the brightness value of the reflected light to express the degree of sparkling.

: 물체의 수직 벡터 (도 10a 참조),

: 빛의 방향 벡터,

: 빛의 반사벡터,

: 시점과 물체가 이루는 방향벡터, I : 밝기, n : 지수항,

(도 10b 참조)를 각각 의미한다. here,

: The vertical vector of the object (see Fig. 10a),

: Direction of light vector,

: Light reflection vector,

: Direction vector between the viewpoint and the object, I: brightness, n: exponential term,

(See FIG. 10B), respectively.

Therefore, when the user adjusts these parameters, corresponding simulation results are obtained in real time.

Such a fluid flow simulation and rendering method of the present invention can be effectively used in various virtual reality visualization techniques, since it is possible to quickly obtain detailed image quality of the fluid while adjusting various parameters in real time as described above.

For example, virtual reality visualization technology for complex natural phenomena such as fluids and flames can be applied to 3D stereoscopic contents, digital cultural property restoration, virtual reality travel and tourism, nature and experience, education and learning, etc. It can be effectively applied to various fields of In addition, the virtual reality visualization technology can be effectively applied not only to the cultural contents field but also to specialized fields such as education, medical, and engineering. It can also be effectively applied to virtual reality content for special educational purposes, such as manipulating the aircraft in heavy rain or dense fog. In particular, while medical and engineering data require a large amount of precision and precision, the visualization technology for realistic visualization of the data is insufficient. Cooling technology and flow control technology used for cooling and circulation of high-speed server-class computers including various home appliances and electronic products, analysis of air flow outside the body of automobiles and aircraft, and fluid flow around ships, etc. It is a key item in the optimization design. In order to effectively visualize the results in such an engineering field, realistic visualization techniques using virtual reality techniques are essential. In addition, various fields including medical and biotechnology, including visualization of fluid flow inside each organ in the body, ciliary movement of each organ, inhalation of air or tobacco smoke in the lungs, flow of blood or urine, digestion, etc. It can be widely used in. In addition, it can predict various applications in web 3D rendering, online broadcasting, and interactive contents for high-definition television (HDTV).

As an example to confirm the effect of the present invention, a scene of about 10 seconds was poured using a method of the present invention and the existing software Nextlimit realflow. As a result, when realflow is used to derive a video output of similar precision, it takes several tens of hours for various iterations, calculations, and rendering until the rendered result is obtained. If entered correctly, the result could be obtained in near tens of seconds in real time. Particularly, if the flow is complicated in shape, if the software is operated with existing software, the physical constraints are not satisfied in the part where the flow direction is greatly changed or sudden congestion occurs. Therefore, the operator must manually correct it and produce satisfactory results. Impossible cases also occurred. However, when using the method of the present invention, if only the appropriate initial condition and boundary condition are inputted, the subsequent processes are automatically calculated by the strict physical laws based on the geometrical shape, the physical properties of the fluid, the external force, etc. The time required for video production can be dramatically shortened.

As described above, the fluid flow simulation and rendering method of the present invention provides the following effects.

First, by adopting continuum simulation and volume rendering, the video can be more faithful to the laws of physics, and the fluid video can be realized in real time by reducing the computation time by improving the numerical method.

Second, it is possible to change various video images immediately by receiving user's parameter input, so that the user can implement the desired video more quickly and accurately.

It is to be understood that the invention is not limited to that described above and illustrated in the drawings and that many more modifications and variations are possible within the scope of the following claims.

Claims (4)

A fluid motion simulation step of calculating a free surface shape by dividing the calculation region into finite elements in the form of fixed lattice and calculating the velocity and pressure of the fluid, based on the flow equation analysis method for the continuum; A volume rendering step of performing a rendering for the fluid simulated in the fluid motion simulation step by obtaining visual variables for expressing a three-dimensional effect of the fluid using a property of light passing through or reflected from the surface of the fluid; And a flow effect adjustment step of providing a user for inputting parameters for calculation of the fluid flow simulation and volume rendering step, and reflecting the input value to a fluid video in real time. Render method. The method of claim 1, Variables for expressing the three-dimensional effect of the volume rendering step are the brightness, color, opacity, etc. of the fluid, these variables are fluid flow simulation and rendering method characterized in that it is calculated using a pre-integration method. delete The method of claim 1, Wherein said parameters include at least one of a density and a viscosity of the fluid and an illumination constant.

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