Computer Graphics & Geometry

A.Gudzovsky^{**}, A.Aksenov^{**}, S.Klimenko^{***}

******Institute for Computer Aided Design (ICAD) of the Russian Academy of Sciences,

*E-mail: FlowVision@glas.apc.org*

*******Istitute of System Programming (ISP) of the Russian Academy of Sciences

E-mail: __klimenko@mx.ihep.su__

Contents

- Abstracts
- Introduction
- 1. Basic principals
- 2. Peculiarities of the initial data
- 3. Physical basics of volume visualization
- 4. Method of test grid
- References

**Abstracts: **Approaches for building the interactive system of volume visualization and results data analysis in industrial computational fluid dynamics are discussed. Using of simplified laws governed the image building instead laws of physical radiative transfer in analysed domain is proposed. A new numerical method for visualization of 3D scalar field convenient for application in the computer aided engineering systems is advanced.

**Key words**: scientific visualization, computational fluid dynamics, refraction.

Introduction

Advances in computers and numerical methods have led to the increase of complexity of problems that can be solved by numerical simulation. At present we can say about forming of industrial computational fluid dynamics (ICFD) - the branch of science and technology handling with numerical simulation of flows in real-life technical installations. ICFD takes into account complex shape of installations and aggregate of fundamental physical processes. One of the major goal of ICFD is to test a technical installation ahead its appearance in reality.

There are considerable number of ICFD software been used for computer aided engineering (CAE) of technical installations interacted with gas or fluid flows. The software is generally created for workstation and includes advanced interface, tools for geometry and other input data specifying, and analysis of results. For example we can mention road vehicles, boilers of thermal power stations, sewage treatment, ventilated production areas (including cleanrooms), wind load on constructions as industrial application of CAE-ICFD systems.

The peculiarity of present time is the advance of powerful and comparatively inexpensive personal computers (PC). It allows to implement CAE-ICFD systems for PC platform. The deepest gap between capabilities of PC and workstations is in the area of visualization. This makes the very actual problem of developing efficient methods of 3D data visualisation.

It is obvious that universal method for exhaustive representing all possible flows is absent. Only system of visualization tools with the wide spectrum of possibilities can gives answers to different user's questions about processes in real-life technical installation.

The visualization system for ICFD should include different methods. It must include conventional methods for imaging vector (e.g. velocity, vorticity) and scalar (e.g. temperature, pressure) fields, drawing of vectors at the ordered set of points (e.g. in the centres of the calculation cells or any user defined grid); trajectories of particles (strips) moving over velocity field; imaging the scalar isosurfaces, and other.

But the practice shows the need for new visualization tools suitable for analysis of complex flows in real-life technical installation modeled by ICFD code. The volume visualization is effective, but insufficiently developed analysis technique in ICFD. The objective of this paper is to discuss the approaches for building the interactive system of volume visualization and analysis of results data in ICFD.

1. Basic principals

The visualization system may be produced for a different purpose and respectively based on different principals. Therefore it is important to declare from the beginning the foundation of our approach.

a) __The volume visualization is the method of 3D2D transformation__

A matter of the volume visualization is converting of 3D information field into a 2D image. On the one hand, compression of the information is useful for achieving the possibility to analyse 3D data distribution as a whole. On the other hand, the compression of the information creates integral images suitable for quick qualitative comparison of various design options of the investigated technical installation.

b) __The visualized volume contains gas with optical properties specified by user__

We assume that the considered domain contain a gas with variable optical properties. The 3D2D transformation is governed by laws of radiation transfer in gas. At the volume rendering the domain is projected on the screen in such a way that analyzed function f is transformed to a distribution of intensity of light on view plane. This transformation uses some weight function given by the relation between f and optical properties of medium. User should can analyse the 3D distribution of f by control of the relations between optical properties of a medium and function f.

c) __The volume visualization must be understandable__

ICFD visualization has many common approaches and methods with other areas of scientific visualization. At the same time there are some features of ICFD problems, often not completely understood by the specialists in visualization and computer graphics. Then, it is not a surprise that interesting methods for flow visualization are often developed by ICFD specialists [Levi90].

Underline, that the tools have to provide the presentation of analysed data in form obvious for user. User should use visualization tools for analysing of the technical installation functioning instead of deciphering the images. It means that the user must can to explain any peculiarity of the image in terms of peculiarities of fluid flow.

In many cases the difficulties in analysis of the numerical results are associated not with poor computer graphics, but with the lack of appropriate visual images. For instance, the difficulties of visualization of tensor fields are caused by the absence of intuitively clear visual images for tensor characteristics of fluid flow.

d) __The volume visualization must be interactive__

We will be interested in the visualization methods used at the analytical stage of user interaction with the CAE system. This stage assumes using of fast methods of drawing image on a display. At this stage of work the visualization and data analysis system have to minimise the time required for user to understand the structure and major characteristic features of the flow. The system must be interactive to provide the continuous process of analysis. The practice shows that the time for the image building should be no longer than 5 - 10 seconds.

We will not discuss here the problems of the final stage of work - report visualization. At this stage of work the clear understanding of the flow structure exists and the most beautiful presentation of a number of "show" illustrations is only needed. It is evident that in this case time required for making an image is not critical but the attractiveness of the picture and its ability to make strong impression are the most important.

2. Peculiarities of the initial data

At the beginning lets formulate the features of ICFD numerical results that we have to represent on the display.

We already mentioned the first feature, namely, complex geometry of the industrial objects. Besides, the geometry of the calculation domain for the majority of ICFD problems is characterised by the scales differed greatly in sizes. The ratio of the typical size of the whole domain to that of small element defined the flow structure or characteristics may reach hundreds.

Scientific visualization for fundamental research in contrast to the visualization systems for applied research deals with purified phenomena taking place in the domain with simplest geometry. Therefore various techniques including exotic ones are acceptable in visualization of such phenomena. The most of specialists in the area of visualization [Wijk93, Bril94] use quite trivial flows (compared with flows typical for ICFD) for examples of application of the developed methods. The question about applicability of the offered methods for solving real-life applied problems is still open. The practice shows that as a rule the most valuable thing in any idea is its realisation.

We are coming from the fact that analysed scalar and vector fields are three dimensional and linked to the complex geometry of a technical installation. Therefore visualization system should include advanced tools for presentation of complex geometry of technical installation.

The second specific for ICFD feature is associated with the origin of the visualized data from numerical simulation. The data quantity and structure are determined by the used ICFD method instead of the requirements of visualization method. For instance, the simulation may be produced with the use of calculation grid of different type: orthogonal Cartesian grid, multiblock body fitted grids, nonorthogonal grid, local refined grid. The analysis of published ICFD papers shows that the number of cells in the typical used grid is of the order 10^{4} 210^{5}. The use of grids with the number of cells more than 10^{6} requires today's supercomputers and therefore is rare. The typically used size of a calculation grid grows quite slowly - roughly one order of magnitude each 10 years. So the development of visualization methods used grid with a number of cells more than 10^{7} [Wood93] is seen as not today problem of ICFD.

It is important not to be overdiligent in trying to obtain the high quality image with the resolution higher than given by ICFD numerical simulations. Certain threat is associated with the use of operations of filtration, averaging and other actions improving the image. The major purpose of scientific visualization is seen in accurate representation of obtained information instead of drawing a beautiful picture around results data.

It is quite natural to treat a grid cells as a voxel [Elv92] on the visualization of 3D data coupled with a grid. But in the finite volume methods the values at cell are averaged over cell volume. So to restore the value of a parameter at arbitrary point of cell it is necessary to reconstruct the function using it's mean values within the cells. This procedure is not always could be reduced to the simple trilinear interpolation. Ignoring of this fact can cause distortion of the simulation results.

3. Physical basics of volume visualization

Human beings distinguish the objects in the surrounding space due interaction of these objects with light: objects radiate, absorb, reflect, scatter or refract light. The same physical principles as in reality are used for creating an image on the screen of monitor. There are some examples of the images have been built by using different optical processes in the works, devoted to volume visualization [Kru91, Kauf94, Hav94, Max95].

So, the most common computer presentation of objects as a set of surfaces (surface visualization) is based on the properties of reflection and refraction of incident light. The process of reflection is typical to surfaces of rigid and liquid phase not to gas volume. Therefore we shall not consider the reflection below.

Assess then the role of light scattering in the volume visualization. Lets consider the radiative transfer in a domain contained radiative, absorptive, and scattering medium. From the definition, the spectral intensity of light is a flux of energy at point through a unit of square, a unit solid angle around direction , within unit spectral interval around wavelength (color) . The radiation transfer is described by equation for the spectral intensity of light

where - absorptivity, radiance, scattering power, and indicatrix of gas.

The solving of integro-differential equation (1) in common case of multiple scattering medium may be produced only by iterative procedure. So it requires for a considerable processor's time analogous to rendering of scene with the multiple scattering of light from the surfaces by ray tracing method. Therewith taking into account the scattering can only fog (in a literal sense) the image, make it not ideal as in poor quality experiment. An illustrative example is given in [Max95].

Note the relations between experimental and computer visualization. After obtaining of the numerical data one can simulate the image as it is seen in the experiment. As the computer technology progresses our capabilities in this area will only increase. To an extent that it will be possible to reproduce badly done experiment - with spots of light, scattering, diffraction and other phenomena which hide the information searched in the experiment.

A race after realistic presentation in this direction looks as doubtful. The aim of ICFD visualization is to give a method of cognition but not for hiding the truth by means of reproduction of nonideal experiment. Lets give such an analogy. There is some interest in solving a task of building on the display the realistic image of the screen covered by a thick layer of a dust. But it seems that there will be quite few volunteers willing to work at such a conditions when on the image is automatically placed such "antireflecting coating".

As mentioned above the image building must be quickly enough. So image drawing based on volume scattering at the analytical stage of works with CAE-ICFD system is unnecessary. Therefore, at further discussion we will restrict ourselves with ray casting method for building an image.

Discuss further in more details peculiarities of the building images of volume distribution f by radiation and absorption based method (RABM) and refraction based method (RBM).

The losses of information at 3D field projection on 2D plane inherent for volume visualisation are inevitable. Therewith it is worthwhile to constrict images not in accordance with exact physical laws but being governed by some simplified laws. These simplified laws must reflect the matter of real physics, produce the image clear to the user and lead to more simple mathematics to provide more quickly calculation. The examples for absorption (radiation) and refraction are presented below.

3.1 Absorption and radiation

Consider the case when image is formed by radiative and absorptive medium. Let axis z going along the line of sight, z=0 is image plane. As follow from (1) the intensity of light I(x,y) on image plane (x,y) is

(2)

where is the optical thickness of a layer between s and z.; z=s<0 is position of the most distant from the screen boundary of integration. Subscript s denotes boundary conditions at z=s.

In the method of ray casting brightness of a pixel at point (x,y) on a screen is proportional to I(x,y). In common case the value of I is defined by the relation between f and optical properties of medium and q. Let consider some particular cases.

In case of nonradiative medium (like for X-ray image) q=0 and illumination I(x,y) depends on distribution of absorptivity along z axis. For example, at =Cf a part of the region with higher integral optical thickness

(3)

will look on the screen more dark.

If the integral optical thickness is small t_{l}(0)<<1 (i.e. t_{l}(0)<0.3), then and

(4)

In contrary case of radiative, but nonabsorptive medium absorptivity =0, the illumination I(x,y) depends on distribution of radiance q along z axis. For example, at q=Cf

(5)

a part of the region with higher value of will look lighter on the screen.

The obvious disadvantage of using (2) is the necessity to operate with exponent procedure which is processor's time consuming. At the same time it should be remembered that the relationship between f and , q is controlled by user and is essentially artificial. In this case it is not clear for what purpose the comprehensive modeling of absorption (radiation) process is required.

From the above reason the next simplification of (1) may be proposed. Define the relation between f and so that t_{l}(0) not exceed 0.3 and the arbitrary relation between f and q. Then combine (4) and (5) and get the simplified form of (2)

(6)

So we get that the volume is actually projected on the screen in such a way that distribution f(x,y,z) is integrated along line of sight (z axis) using some weight function given by the relation between f and optical properties of medium and q. User can view the distribution of f in the view plane (x,y) as a variety of color and brightness of picture. Moreover he can view the distribution of f along line of sight (z axis) too if he changes the relation between optical properties of a medium and function f (or for example the gradient of f). Analogy is pertinent to X-ray imaging, where for selection of the interested objects a special substance increasing absorption is injected into the object.

Additional possibilities for drawing of information saturated pictures arise when color instead of monochrome palette is used. This could be done by calculating distribution I(x,y) for 2-3 wavelengths (colors) for various dependencies of and q from f. By further mixing of colors one can obtain a picture reflecting simultaneously several features of distribution of f in volume. Anyway, due to the uncertain character of this process it looks as suited better for final (report) stage of visualization. At the analytical stage it is better to use monochrome images [Wood93].

3.2. Refraction

Consider now domain contained only refractive (not radiative and absorptive) medium. The processes of radiative transfer in refractive and in radiative and absorptive gases are principally different.

In radiative and absorptive gas the path of light is srtraight line, but the light intensity is changed along the line in accordance with (1). When a set of parallel light beams enter the domain, each beam goes through the domain parallel to another, so they does not intersect. In contrast to this the radiative transfer in refractive medium does not described by equation like (1). In refractive gas the light intensity is not changed along the path, but the path becomes twisted, not straight line. As consequence the situation when any paths pass through one point may take place.

The refractivity of medium is defined by index of refraction n. In common case the equation of the path of light beam is

(7)

where k and are curvature and principal normal of the path. Underline that calculation from (7) the three dimensional twisted path for arbitrary distribution of n requires is considerable processor's time. Besides the mentioned above intersection of light beams leads to difficulty of explanation of resulting picture. Therefore the methods of visualization based on refraction [Hav94] are used less frequently than methods based on other optical processes.

The problem is simplified in case when refraction is small and variation of n along z axis is negligible, i.e. n=n(x,y). Such conditions are realised in a set of aerodynamic experiments, for example, boundary layer on heated plane, flow around 2D wing profile, etc. Methods of visualization based on refraction (Schliren and shadow methods, interferometric [Gold83]) are one of the most powerful instruments for study of high speed flows of gas and density heterogeneous mediums in experimental fluid dynamics. Assume x(z), y(z) is the equations of light beam, denote , then the equations for the path of the light beam are [Gold83]

(8)

As follows from (8) the entity of refraction is the sensibility of the light path to gradient of n in direction perpendicular to the light path. The image of domain contained medium with nonuniform distribution of n deforms due to this reason. Note that the beam of light deviates in the direction of n increase.

Modification of the optic laws of refraction has to be so that outlined feature should remain but the mathematical form should simplify. In particular

- the image deformation should be proportional to gradient of n in the plane perpendicular to view direction;
- the variation of n along z axis and respective image deformation should be ignored.

We propose the next form of modified optic laws of refraction satisfied to these conditions. The equations for the path of the light beam for arbitrary distribution of n in domain are

(9)

4. Method of test grid

Both RABM and a computer version of popular experimental fluid dynamics RBM (Schliren and shadow methods, interferometric) produce a 'dense' image. It means that all pixel of image are informatively significant. So if the user want to combine two image simultaneously (for example, distributions of temperature and absolute value of velocity) he have to make one of them transparent. Therewith the studied technical installation must be shown too and the best form of its presentation is the semitransparent technique. As result the picture becomes considerably overloaded and hard for understanding.

As a rule one of two been combined images is more significant, and another is needed as a background. So it is important to construct not 'dense' but 'slight' image for using as a background for another more important image.

In this connection we propose a computer version of experimental method of test grid. This method is proposed in [Boya88] for study of surface waves caused by perturbation of a fluid in a tank.

In the method of test grid the flux of light entered the studied domain with refractive gas has a form of a rectangular grid consisted of set lines. For instance, grid may be formed as a black lines on white background. We will see undistorted grid with rectangular cells if index of refraction n in the domain does not change in plane (x,y). The image of a grid will be deformed if a gradient of n in plane (x,y) exists in the volume. It is similar to a deformation of a tile on the bottom of a swimming pool as seen through the surface of water covered with waves.

The building of image may be produced in the next way. A set of points laying at intersection of grid lines are the source of light beams passed through the domain. The trajectory of light is described by (9). Points of crossing of these beams with the view plane connected in the appropriate order, that gives the transformed image of the grid. Giving user a possibility to define relation between analyzed function f and index of refraction n, size of the grid, as well as limits of the region in which the medium refracts light, we get an instrument for analysis of distribution of f in a direction perpendicular to line of sight.

Among advantages of test grid method are the following:

- high speed of image drawing because of we calculate the path of light emitted from several points at the vertexes of initial grid, not from all pixels;
- compatibility with RABM;
- monochrome image of a test grid easily separated from other objects presented on the picture.

Below, we give two examples of images of a test grid for various distributions of scalar function f. Because of need for use of black-white pictures in this publication we present examples with simple geometry of calculation domain and simple shape of objects in it. The first example presented in Figure 1 is the visualization of model distribution that consists of two "dense" sphere with "atmosphere" with density decreasing to zero at some radius. Part a) of Figure 1 is rendered by RABM, part b) -by the method of tested grid, and part c) is the application of parts a) and b).

The second example fall into a cleanroom aerodynamics. The cleanroom studied is presented in Figure 2a). A clean air flows in the room in direction from ceiling to grating and then is removed through outlet on side under grating. There is a table with heat and contamination source on it. This source model the fire of specific type. The jet of hot gas produces circulation torus-looking flow in the room. The general idea of velocity field structure is shown in Figure 2b). Both RABM and RBM (method of test grid) are used for simultaneous presentation of contamination and temperature field in cleanroom in Figure 3.

Given examples only illustrate applications of the proposed method and not aimed at making an impression that this method is a mean for solving all visualization problems arising in ICFD. To our mind offering to user a wide spectrum of possibilities for forming the images in every particular case is more important than search for panacea - evidently non existent universal method for representing the entity of all possible flows, giving answers to all user's questions.

a) |
b) |
c) |

Figure 1. Model distribution of scalar function in form of two "dense" sphere with "atmosphere" with density decreasing to zero at some radius. a) rendering by RABM, b) rendering by the method of tested grid, and part c) the application of a) and b)

a) |
b) |

Figure 2. Fire and air flow in a cleanroom: a) general view of a cleanroom; b) velocity vectors and flow structure in two perpendicular planes (central and along the wall) passed through source; arrows show the direction of air flow

a) |

b) |

Figure 3. Fire and air flow in cleanroom: a) volume visualization of contamination field by RABM; b) the same and presentation of temperature field with the method of test grid

Acknowledgements

This work performed under support of RFBR: grant # 95-01-00815 (AVG and AAA) and #96-01-01273 (SVK).

Refereces

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Computer Graphics & Geometry