Visualization techniques for curvilinear grids

Transcription

1 Visualization techniques for curvilinear grids Report I. Ari Sadarjoen Willem C. de Leeuw Frits H. Post Technische Universiteit Delft Delft University of Technology Faculteit der Technische Wiskunde en Informatica Faculty of Technical Mathematics and Informatics

2 ISSN Copyright c 1995 by the Faculty of Technical Mathematics and Informatics, Delft, The Netherlands. No part of this Journal may be reproduced in any form, by print, photoprint, microfilm, or any other means without permission from the Faculty of Technical Mathematics and Informatics, Delft University of Technology, The Netherlands. Copies of these reports may be obtained from the bureau of the Faculty of Technical Mathematics and Informatics, Julianalaan 132, 2628 BL Delft, phone A selection of these reports is available in PostScript form at the Faculty s anonymous ftp-site. They are located in the directory /pub/publications/tech-reports at ftp.twi.tudelft.nl

3 Visualization Techniques for Curvilinear Grids I. Ari Sadarjoen Willem C. de Leeuw Frits H. Post Abstract In this paper we describe techniques for the visualization of data defined on curvilinear grids. This type of grid poses some difficulties for visualization techniques not found in regular grids. First we will describe some basic techniques that are useful for many visualization techniques if they are applied to curvilinear grids: grid transformation, tetrahedral decomp osition, point location, and interpolation. Second, we will describe three visualization techniques and their use on curvilinear grids in more detail. Algorithms are given for are volume ray casting, particle tracing, and spot noise. keywords: computer graphics, scientific visualization, volume rendering, particle tracing, texture synthesis. 1 Introduction Scientific visualization [1] concerns the use of computer graphics for interactive data presentation and exploration for scientific and engineering applications. Large amounts of numerical data are produced by numerical simulations and measurements, using high-performance computing and sensing systems. The unique capabilities of the human visual system are exploited to detect patterns and structure in the data. A large class of numerical simulations is based on discrete methods for solving systems of partial differential equations, such as finite element, finite difference, and finite volume methods. Typical engineering applications of these methods are structural analysis, heat transport, electrodynamics, and computational fluid dynamics. All of these areas require simulation of highly complex physical phenomena. The use of these numerical simulation methods involves subdivision of a physical domain or object of analysis into a computational grid, and solving approximate equations for each node or cell of the grid. To simulate time-dependent phenomena, solutions are computed for a large number of time steps. For practical applications, these grids may consist of thousands to millions of nodes, and thus large quantities of numerical data are generated, consisting of discrete fields of physical quantities. These may be scalar, vector, or tensor quantities, computed for each node or cell of the grid. Visualization techniques must be capable of handling these data, to help the scientist or engineer to interpret them, and to understand the underlying physics. The type of grid used is dependent on the solution method and the application domain. The simplest type is the regular orthogonal grid, of which all cells are unit cubes. We will refer to this type of grid as regular in this paper (see Figure 1a). Although regular grids are not very common in practice, many visualization techniques are only suitable for this type of grid. In this paper, we will discuss visualization techniques developed for 3D structured curvilinear grids, as commonly used in computational fluid dynamics (CFD). Such a structured grid (Figure 1b) has a regular topology, with the nodes defined in a 3D index space I(i,j,k). Adjacent nodes can be found by incrementing index values. The geometry of Delft University of Technology, Faculty of Technical Mathematics and Informatics, Julianalaan 132, 2628 BL Delft. fari.sadarjoen, wim.deleeuw, frits.postg@twi.tudelft.nl

4 (a) (b) Figure 1: Regular grid (a) and curvilinear grid (b) the grid cells is irregular. The cells are usually hexahedra, with a deformed-brick shape; the cell faces are non-planar quadrangles. The grid is called curvilinear, because the grid lines connecting adjacent nodes in each index direction are curved - although in practice the grid lines often consist of straight line segments. The cell size of a curvilinear grid can be highly variable, and thus the resolution of the simulation can be higher in areas of strong variation. Also, the curvilinear shape can be made to conform to the curved boundary of an object, such as an airplane wing in a flow. On this type of grid, discrete fields of scalar or vector data can be defined. Examples of scalar data are pressure, temperature, or density fields. In CFD, velocity vector fields are most important to characterize the structure of a flow. We will assume here that the data are defined at the grid node positions. The purpose of this paper is to identify some basic techniques for visualization with curvilinear grids, and show their application to a variety of visualization techniques. These basic techniques are : transformation of curvilinear cell geometry and data to a regular grid, decomposition of a hexahedral cell into tetrahedra, locating the position of a point in a grid cell, and interpolation of data values at an arbitrary point from the data at the surrounding grid nodes. These basic techniques will be used to adapt three very different types of visualization techniques for use with curvilinear grids: volume ray casting, particle tracing, and spot noise. Volume ray casting is a direct volume rendering technique [2], used to generate semi-transparent images of scalar fields. Particle tracing is the computation of the motion path of a massless, point-shaped particle in a flow. This is done by a stepwise integration of a velocity field. In ray casting and particle tracing, a common problem is the traversal of the grid, stepping along a ray or along a path curve as it is generated. It will be argued that decomposition of hexahedral cells into tetrahedra is a good way to traverse the grid for both purposes. The third technique involves the use of texture for visualization of a flow near a surface. The texture generation technique used is spot noise [3], which produces a random texture that can be locally adapted to visualize a two-dimensional vector field. We will show how spot noise is generated, and which problems have been solved for its use in curvilinear grids. 2 Curvilinear grids General hexahedral curvilinear cells can be difficult to handle. The cell faces consist of four points that may not be coplanar. As a result, containment tests which typically depend on the orientation of points

5 with respect to planes, become more difficult. Also, determining the relative position of an arbitrary point within a cell is more difficult due to the irregular cell shape. One method to cope with these problems is to transform the curvilinear grid to a regular grid, a method commonly applied in numerical simulations. This will be described in subsection 2.1. As an alternative solution, the hexahedral cells can be decomposed into tetrahedra, which will be described in subsection 2.2. Next, we describe how the above approaches can be applied in two basic techniques: point location can is the problem of finding the cell in which an arbitrary point lies, and the relative position within that cell; interpolation determines data values at an arbitrary point from the data at the surrounding grid nodes. 2.1 Grid transformation A curvilinear grid can be transformed into a regular grid in a different domain, called Computational Space (C-space or C). In contrast, the domain where physical quantities such as positions and velocities are defined, is called physical space (P-space or P). Figure 2 shows the transformation in both directions between the two domains. P C 1 β β D α C 1 α 1 β β T D 1 β β α 1 α β C 1 β A α 1 α B T -1 A α 1 α B Figure 2: Transformation between P and C In curvilinear grids with a highly regular shape, it is possible to define a transformation mapping which applies to the entire grid. Examples include cylindrical and spherical grids. In most cases however, the transformation is done locally, meaning that a different mapping is used for each cell. The way in which this is realized will be described later. The quantities defined on the grid that need to be transformed are the positions and vectors. Scalars need not be transformed. The reason for this is that the grid transformation can be regarded as a deformation. When a spatial domain is scaled, twisted or in any other way deformed, positions and vectors change, but scalar quantities are independent of the spatial domain. In the following we will describe point and vector transformation, respectively. Point transformation The transformation of positions from C to P is straightforward. It can be accomplished by mapping the corner nodes of a cubic cell in C to the corner nodes of a curvilinear cell P, and by interpolating all the points in between. All the points lying between the C-corner nodes will then be automatically transformed to points in P lying between the P-corner nodes. Let C-point = (; ; ) consist of the integer parts (I,J,K) and fractional parts (; ; ), with

6 (a) (b) Figure 3: Two ways to split a hexahedron into five tetrahedra 0 ; ; 1. And let I be the trilinear interpolation function, defined as I(x; I; J; K; ; ; ) = 1X 1X 1X i=0 j=0 k=0 x I+i;J+j;K+k B i ()B j ()B k () (1) The base functions B are defined as B 0 (x) = 1? x and B 1 (x) = x. Then, the corresponding position in P is given by: x = T () = I(x; I; J; K; ; ; ) (2) The principle is illustrated for a 2D cell in figure 2. Vector transformation A vector v in P can be transformed to u in C as u = J?1 v (3) where the matrix J, called the metric Jacobian, contains the partial derivatives of the transformation: J = 0 B@ x x x y y y z z z 1 CA (4) where x is short The matrix J can be considered as three columns (j1 j j2 j j3) containing the @ These derivatives are usually approximated with finite differences, In [4] it is shown the type of differencing can be very important. 2.2 Tetrahedral decomposition Another method to deal with the difficulties caused by the curvilinear cells, consists of decomposing the hexahedral cells into tetrahedra [5]. There are many possibilities to do this. We split a hexahedron into 5 tetrahedra, i.e. 4 corner tetrahedra and an inner tetrahedron. There are two ways to do this (see figure 3 a); to get a grid in which no cracks occur, decomposed adjacent cells in the grid should have alternating orientations. This is shown in figure 3 b.

7 2.3 Point location We can distinguish between global and incremental point location. In global point location, a given point in a grid must be found with no previous known cell. In a curvilinear grid this is not an easy task. As with all search algorithms it is possible to use a simple brute-force algorithm which searches all grid cells one-by-one. Naturally, this is computationally expensive. Auxiliary data structures can be used to speed up this search [6, 7]. Fortunately, in many visualization techniques there is a known starting position and a known starting cell. Starting from there, a new position is to be found. This will be called incremental point location, and we describe two possible approaches. In the stencil walk method [8], first an initial point in computational space is chosen. This point is transformed to physical space using the transformation x = T (x; ; ; ) as described in the previous subsection. The difference between the transformed and the target point P is calculated as x = x? P. This difference vector in physical space is transformed to computational space using = J?1 x and added to the previous point, resulting in a new guess. If one of the elements of is outside the range [0; 1], the centre of the corresponding neighbouring cell is the new guess. The iterative process continues until the right cell has been found. Once the correct cell has been found, one can iterate until the value of is small enough. Alternatively, the hexahedral cells can be decomposed into tetrahedra, as described in the previous section. Incremental point location can now be performed by drawing a line from the previous known position to the next position [9]. Intersections with the faces of the tetrahedron and containment tests in neighbouring cells are used to locate the new point. Tetrahedrization is only performed in the cells along the path of the line. 2.4 Interpolation The most common type of interpolation in regular grids is trilinear interpolation. In a cell having indices (I,J,K) and data values v at the corner nodes, the interpolated data value for all local coordinates (; ; ) is defined as: I(v; I; J; K; ; ; ) = 1X 1X 1X i=0 j=0 k=0 v I+i;J+j;K+k B i ()B j ()B k () (5) where B denotes base functions defined as B 0 (x) = 1? x and B 1 (x) = x. In curvilinear grids, this method can be used, provided that the local coordinates are available. One way to obtain these is through the Stencil Walk method described before. If tetrahedral decomposition is applied, a different interpolation method can be used, which is analog to the trilinear interpolation in a cubic cell with 8 nodes, but which is based on tetrahedra, thus consisting of 4 points. v(; ; ) = v 0 + (v 1? v 0 ) + (v 2? v 0 ) + (v 3? v 0 ) (6) where ; ; denote the local coordinates within the tetrahedron, satisfying 0 ; ; 1 and Volume ray casting In the previous section a framework for calculations on curvilinear grids was given. This section will concentrate on one application of this theoretical framework: volume ray casting in curvilinear grids. Volume ray casting [2] is a technique for the visualization of 3D scalar fields in which the scalar value is mapped to opacity of the medium. The result is that the data is displayed as semitransparent clouds.

8 eyepoint screen data volume samples Figure 4: Ray casting To produce a picture, for each pixel a ray is cast through the volume. The scalar values of the field, sampled along the ray, are integrated resulting in a value which is mapped to a colour. Mostly the colour is chosen to simulate a light ray which is attenuated when it penetrates a semi-opaque medium. This is accomplished by a recurrent relation [10] which is evaluated for all samples beginning at the one closest to the eye position. For the n-th sample the new colour C n is calculated using the colour accumulated so far C n?1, the colour of the sample C 0, the accumulated opacity n?1 and the opacity of the sample 0 : C n = C n?1 + C 0 (1? n?1 ) n = n?1 + 0 (1? n?1 ) (7) In curvilinear grids the problem of ray casting has two important components which are different compared to regular grids: the traversal of the ray through the volume and the calculation of the data value inside a cell (interpolation). We will briefly describe possible solutions to these problems and present the method we used. A possible solution to the problem is resampling of the data to a regular grid. Mostly however curvilinear grids contain a wide range of cell sizes, a ratio of 1000 between the volume of the largest and smallest cell is not uncommon. To get an accurate representation of the data the cell size of the regular grid can not be larger than the smallest cell in the original data. This mostly results in prohibitively large data sets. Therefore this method is not considered as a good approach. Another possibility to attack this problem was proposed by Frühauf [11] who assigned a viewing direction vector to each grid node, and then transformed the grid including the viewing direction vectors to computational space using the transformation described in section 2. The traversal problem in this case is reduced to the integration of field lines in the vector field on a regular grid. A drawback to this method is that it requires considerable storage and also particle tracing (even on a regular grid) is an expensive operation. The problem can also be solved using the interpolation and point location procedures described in section 2. First we determine the sample locations on the ray. Then we locate the cells in which the sample points are located and interpolate the scalar value at these positions. Finally, we compute the colour of the pixel using the values found. In pseudo-code for a single ray this becomes:

9 Figure 5: Ray casting applied to the pressure field of the backward facing step data set initialize colour and opacity find first sample (point location) repeat determine field value (interpolation) update colour and opacity find cell containing next sample (point location) until ray leaves volume or maximum opacity reached In figure 5 we used the ray casting technique to visualize the pressure field in the backward facing step data set. The picture is generated using the H-buffer application [12]. 4 Particle tracing While raycasting is a method for visualizing scalar fields, particle tracing is a method for visualizing vector fields. This technique involves the simulation of releasing particles into a flow and calculating their positions at specific times. Once this is done, the particles can be rendered in an animation. The computation of a particle path is based on a numerical integration of the ordinary differential equation dx = v(x) (8) dt where t denotes time, x the position of the particle and v(x) the velocity field. The starting position x 0 of the particle provides the initial condition: x(t 0 ) = x 0 (9) The solution is a sequence of particle positions (x(t 0 ); x(t 1 ); : : :). A particle tracing algorithm consists of several steps. First, a search is performed for the cell which contains the initial position of the particle. To determine the velocity in this point, the velocities in the cell corners are interpolated. Next, an integration step calculates the next position of the particle.

10 Again, a search is performed, now for the cell containing the new position. The process of point location, interpolation, and integration is repeated until the particle leaves the grid. This process can be translated into pseudo-code representing the general structure of a particle tracing algorithm: find cell containing initial position while particle in grid determine velocity at current position calculate new position find cell containing new position endwhile (point location) (interpolation) (integration) (point location) Note that the above pseudo-code is merely meant to show what the main algorithm components are; in real implementations many refinements and optimizations could be made. As to the integration, one of the well-known integration methods described in the literature can be used, for example second-order or fourth-order Runge-Kutta. First-order methods, such as Euler were found to be inadequate for these purposes. Methods for point location and interpolation in physical space have been discussed in section 2. For particle tracing, these have turned out to be more robust and efficient than their computational space alternatives [4]. An example of particle tracing in the Backward-Facing Step is shown in figure 6, which was produced using the Plankton program developed by Andrea Hin [13]. Figure 6: Particle tracing in the backward facing step 5 Spot noise In this section we will describe the generation of spot noise on curvilinear surfaces in 3D. First a brief introduction into spot noise is given then problems specific to the use of spot noise for the visualization of vector fields on curvilinear surfaces are described Spot noise [3] is a technique for texture synthesis. A texture can be characterized by a scalar function f of position x. A spot noise texture is defined as f (x) = X ai h(x? x i ); (10) in which h(x) is called the spot function. It is a function everywhere zero except for an area that is small compared to the texture size. a i is a random scaling factor with a zero mean, x i is a random position. In

11 non-mathematical terms: spots of random intensity are drawn and blended together on random positions on a plane (figure 7a,b). Vector fields can be effectively visualized, if the shape of the spot is adapted to the data at the position of the spot (figure 7c). (a) (b) (c) Figure 7: Principle of spot noise: (a) single spot (b) spot noise texture (c) spot noise used to visualize a vector field A spot noise texture is a planar image, generated on a uniform grid (texture space), whereas data is defined on a curvilinear grid (an arbitrary surface in 3D). For the final image the spot noise textures generated are mapped to the curvilinear surface. This mapping is equivalent to the Jacobian transformation described in section 2. To prevent distortion of the texture, and thus wrong visualization of the data, the spots have to be predistorted in texture space [14]. By applying the inverse Jacobian matrix transformation to the spots before they are rendered, the spots will have the desired shape in physical space. For texture-mapping triangular mapping is used, i.e. triangles in texture space are mapped linearly onto triangles of the surface in physical space. As the result of this triangular mapping from physical space to computational space, the velocity field transformed to texture space becomes discontinuous. This can be seen in figure 8, which shows a constant field on a single trapezium-shaped cell with spot noise rendered onto it. In texture space there is a discontinuity along the diagonal of the square. The velocity in computational space at a grid point depends on the triangle bordering the point chosen. For data interpolation we also used linear interpolation. Many curvilinear grids are highly irregular with respect to cell sizes. In regions in which high velocity gradients are expected, such as boundary layers, the simulation is carried out at a higher resolution. If a uniform grid is used for texture synthesis, the density of the generated texture is equal for each cell. If the texture is transformed back to physical space, in small cells many texels (texture elements) are squeezed into a small area while texel-resolution is low in large cells. Hence, it is more efficient to adapt the size of the texture cells to the size of the physical cells. We realized this by using a rectilinear texture space instead of a uniform texture space. (A rectilinear grid is a regular grid with the exception that the distance between grid lines is variable.) The distance between two lines in the rectilinear texture space is chosen such that the area of the strip between the two lines has the same proportion to the total area as the equivalent strip in physical space (figure 9). Although this method does not guarantee improvements, in most cases it will give less variation in texel size in the final image. In figure 10 we used spot noise to visualize the velocity on the symmetry plane of the backward facing step data set. Note the recirculation zone behind the step and the velocity magnitude differences between this zone and the main stream (line like texture in regions of high velocity magnitude)

12 (a) (b) Figure 8: Texture space (a) and physical space (b) for spot noise texture in a single cell. The field is constant and vertical x i+1 x i+1 x i+1 B x i B x i B x i A A A Figure 9: By using rectilinear texture space (middle) instead of regular texture space (left) to represent the curvilinear physical space (right) less irregular scaling results. Figure 10: Spot noise on a slice of the backward facing step.

13 6 Conclusions We presented the extension of visualization techniques from regular grids to curvilinear grids. A set of basic operations on curvilinear grids was described. Three very different visualization techniques were adapted to curvilinear grids using these operations, illustrating their generic use. In general, there are two global approaches can be discerned, the first is to transform the curvilinear grid to a regular grid and solve the problem on this regular grid. The second is to do the operations directly on the curvilinear grid. An advantage of the first approach is that simple and fast operations for point location and interpolation on regular grids can be used. With the second approach is better in that the original data can be used directly and therefore no additional errors are introduced due to the transformation. Which approach is the best depends on the visualization technique used. For volume ray casting and particle tracing, we have avoided the grid transformation, and all operations are performed directly in physical space, using tetrahedral decomposition. This leads to robust and efficient algorithms [4]. The tetrahedral techniques can also be extended for use in unstructured tetrahedral grids with irregular topology (such as finite element grids), provided that cell adjacency information is available. For spot noise, the transformation to a regular grid is needed to generate textures which have to be planar while the data surface may be curved in the third dimension. Curvilinear grids are only a first step in generalizing visualization algorithms. In practice, even more complex situations occur, such as unstructured grids, multiple and mixed-type grids, and time-dependent simulations, with moving and deforming grids. Adaptation of visualization algorithms for these cases is necessary for practical applicability, and will pose challenging problems in visualization research. Acknowledgements The invaluable contributions of Theo van Walsum and Andrea Hin to this work are highly appreciated. Also we would like to thank Remko Vaatstra for his implementation work on H-buffer and spot noise. The backward facing step data set is courtesy of the Department of Technical Mathematics of Delft University. Wim de Leeuw s work is supported by the Netherlands Computer Science Research Foundation(SION), with financial support of the Netherlands Organization for Scientific Research (NWO). References [1] L.J. Rosenblum et al. (editors). Scientific Visualization: Advances and Challenges. Academic press, [2] M. Levoy. Efficient ray tracing of volume data. ACM Transactions on Graphics, 9(3): , [3] J.J. van Wijk. Spot noise texture synthesis for data visualization. In Thomas W. Sederberg, editor, Computer Graphics (SIGGRAPH 91 Proceedings), volume 25, pages , July [4] I.A. Sadarjoen, T. van Walsum, A.J.S. Hin, and F.H. Post. Particle tracing algorithms for 3D curvilinear grids. In Proc. 5th Eurographics Workshop on Visualization in Scientific Computing, [5] T. van Walsum. Selective Visualization Techniques for Curvilinear Grids. PhD thesis, Delft University of Technology, [6] H. Neeman. A decomposition algorithm for visualizing irregular grids. Computer Graphics, 24(5):49 62, November 1990.

14 [7] P. Williams. Interactive Direct Volume Rendering of Curvilinear and Unstructured Data. PhD thesis, University of Illinois, [8] P.G. Buning. Numerical algorithms in CFD post-processing. In Computer Graphics and Flow Visualization in Computational Fluid Dynamics. Von Karman Institute Lecture Series , [9] M. Garrity. Raytracing irregular volume data. Computer Graphics, 24(5):35 40, November [10] J. Wilhelms and A. Van Gelder. A coherent projection approach for direct volume rendering. In Thomas W. Sederberg, editor, Computer Graphics (SIGGRAPH 91 Proceedings), volume 25, pages , July [11] T. Frühauf. Raycasting of nonregularly structured volume data. In M. Dæhlen and L. Kjelldahl, editors, Computer Graphics Forum (Conference issue), pages Blackwell Publishers, [12] T. van Walsum and A.J.S Hin. Hybrid rendering of scientific data using an H-buffer. In J.L.G. Dietz, editor, Proceedings CSN 92, pages , [13] A.J.S. Hin. Visualization of Turbulent Flow. PhD thesis, Delft University of Technology, [14] W.C. de Leeuw and J.J. van Wijk. Enhanced spot noise for vector field visualization. In D. Silver and G.M. Nielson, editors, Proceedings Visualization 95. IEEE Computer Society Press, 1995.