26 Unstructured Grids: Procedures and Applications 26.1 26.2 26.3 26.4 26.5 26.6
Introduction Grids Constructed by Delaunay Triangulation — The General Procedure Unstructured Grid Control Using a Background Grid and Sources Unstructured Grids of Triangles Hybrid Grids of Quadrilaterals and Triangles Unstructured Tetrahedral Grids Dassault Falcon • THRUST Supersonic Car
26.7
Nigel P. Weatherill
26.8 26.9
Non-Isotropic Grid Generation for Viscous Flow Simulation Parallel Unstructured Grid Generation Summary Appendix: Graphics User Interfaces
26.1 Introduction The aims of this chapter are to provide some examples of unstructured grids and, moreover, to illustrate the major steps involved in the generation and use of unstructured grids of triangles and tetrahedra. No theory will be presented, since all the basic theory has been introduced in previous chapters.
26.2 Grids Constructed by Delaunay Triangulation — The General Procedure The Delaunay approach for the construction of unstructured grids is a popular method. It is appropriate, therefore, before discussing real examples, to illustrate the general procedure. Chapters 1 and 16 have discussed the technical aspects of the approach and outlined relevant algorithmic details, so they will not be reproduced here. However, the illustrations presented are based upon the construction of the Delaunay triangulation using the Bowyer [1–6] algorithm. Consider a circle as shown in Figure 26.1a. It is described as a set of discrete points. • The first step is to define a convex hull enclosing all the boundary points that describe the geometry.
This can be automatically performed given the coordinates of the circle. In this case, four points are used to define the convex hull. A Delaunay triangulation of these four points is performed, and the resulting grid with the geometry is shown in Figure 26.1a The convex hull encloses all the geometry points and is triangulated. ©1999 CRC Press LLC
FIGURE 26.1A
The convex hull encloses all the geometry points and is triangulated.
FIGURE 26.1B
A Delaunay triangulation of the boundary points is performed.
• Given an initial construction of four points, together with their Delaunay construction, each one
of the geometry boundary points is inserted sequentially and connected into the triangulation structure. Figure 26.1b shows the resulting grid after all the boundary points have been inserted. • To create the grid inside the circle, it is then necessary to systematically refine the triangles inside the circle. There are several methods for performing this task, as described in Chapter 16. However, for this illustration, the insertion strategy involves the addition of points at the centroid of elements until the required point density is achieved (Chapter 1). The grid point density is controlled by the background mesh and any sources that have been specified (see Section 26.3). Points are created by looping over elements within the domain and inserting a point when element refinement
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FIGURE 26.1C Points within the domain are inserted in an iterative process until the required point density is obtained. Shown is the grid during the point insertion phase.
FIGURE 26.1D
The final grid after point insertion and the deletion of elements outside the domain of interest.
is required. Points are connected into the triangulation using the Delaunay based algorithm. Figure 26.1c shows the grid after the insertion of some points, although the grid point density criterion throughout the grid has not yet been satisfied. point density criterion throughout the grid has not yet been satisfied. • Once the grid point density has been achieved, a post-processing step deletes all triangles that are not within the domain, and if appropriate, the grid can then be smoothed using a Laplacian filter. The final grid for this case is shown in Figure 26.1d. As a further illustration, each stage of the process is illustrated, for a simple geometry, in Figure 26.1e. The process illustrated here for very simple geometries and small grids highlights the sequence of steps that are applied for the generation of Delaunay grids in both two and three dimensions. These simple geometries do not illustrate a very important requirement in the generation of grids by Delaunay triangulation. It is important that in the final grid the edges of the initial boundary are preserved. This is the so-called boundary integrity requirement. Hence, to augment the steps given above, it is necessary to add a final step, • Ensure that the initial boundary edges are included within the final grid [7].
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FIGURE 26.1E
Each stage of the grid generation phase is shown for a simple geometry.
26.3 Unstructured Grid Control Using a Background Grid and Sources Figure 26.1a of Section 26.2 shows the points that define the geometry of the circle. However, any mesh generation procedure must provide a suitable mechanism for a user to change the number of points on the boundary of any given geometry — perhaps a coarse discretization is required, or a fine discretization. One of the popular approaches to this problem in the generation of unstructured grids is to use a background grid and sources [Chapters 1 and 17]. Figure 26.2a shows a schematic of a particularly simple background grid. The idea is straightforward. • Define a mesh that covers the domain Ω to be gridded. The mesh should consist of nodes and have a topologically valid connectivity that defines the elements. In Figure 26.2a the background grid consists of four nodes and two triangles. • At each node of the grid, a parameter is defined which specifies the point spacing at that position.
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FIGURE 26.2A
The background mesh used to control grid point spacing.
• During the grid generation procedure, and included in this is the point discretization of the
geometry, the required spacing at any place in the domain is interpolated from the background grid. Delaunay and advancing front methods require such information. Hence, given a position P in the domain, 1. Determine the element, E, of the background grid that contains P. 2. Find the nodes {n1,n2,n3} of E. 3. Find the point spacing {d1,d2,d3} specified at each of the nodes {n1,n2,n3}. 4. Using {d1,d2,d3}, interpolate the spacing at P. This procedure can also be used to generate grids with stretching (Chapter 19 and 20). This is a very effective way of controlling the element and point density within an unstructured grid. However, it involves the user in generating a suitable grid and specifying the grid point density parameters at each node of the grid. In two dimensions, using graphics user interfaces, this is not too time-consuming and is readily achieved for most geometries. However, in three dimensions it is a nontrivial task. Hence control of a grid by a background grid is usually augmented with the use of sources. Figure 26.2b shows three basic types of sources. Although there are many variants of the definition of a source now in the literature, the fundamental features of a point source are defined by • • • •
A position, Q, within the domain. At Q, the required grid point spacing is defined, d. A circle is specified of radius r1, within which the user specified grid point density, d, is defined. A second circle is specified of radius r2, where, r2 > r1. Within the region defined between the circle radius r1 and the circle radius r2, the point spacing will decay from d to that specified by the background grid.
Hence, for a point source with the structure just defined, the user must specify four parameters. However, this does not involve the intricacies of a mesh connectivity as required with a background grid. In fact, the background grid which accompanies sources is effectively redundant, since uniform spacing everywhere can be the default condition. Hence, no interpolation is required. The extension to a line source, a triangle source, or even a volume source is straightforward. Figure 26.2b shows, in schematic form, a line and triangle source. These two allow the user to easily specify the required grid point density over regions of the domain. The concept of a source naturally extends to three dimensions.
26.4 Unstructured Grids of Triangles The example chosen to illustrate the use of unstructured grids of triangles is the outline of San Francisco Bay. The geometry is defined as a set of discrete points. The boundary can be modeled as a set of NURBS
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FIGURE 26.2B
Point, line, and triangular sources.
FIGURE 26.3A
The geometry of San Francisco Bay.
or splines. This then enables an arbitrary point distribution to be generated on the boundary for any given grid density. • Figure 26.3a shows the geometry as defined by a discrete set of points. • A background grid is superimposed over the geometry and spacing is defined at the nodes. In
Figure 26.3b, the background grid consists of two elements and the specified spacing is shown by the circles around the nodes. • From the geometrical data, and the background grid, the points which will define the boundary within the grid can be generated. Figure 26.3c shows the point distribution on the boundary. • Figure 26.3d shows the resulting grid within the domain. • To illustrate the use of sources, Figure 26.3e shows two line sources that have been designed to construct a grid that will resolve an imaginary deep water channel. The sources that form the line source have different regions of influence, as shown by the circles.
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FIGURE 26.3B The geometry of San Francisco Bay, together with the background grid. Note that the circles attached to the nodes of the background grid indicate to the user the spacing specified.
FIGURE 26.3C
The boundary grid generated from the point spacing specified on the background grid.
• Figure 26.3f shows the resulting boundary point distribution. The effect of the line sources is
apparent. (Compare with the boundary point distribution from the background grid only as shown in Figure 26.3c). • The resulting grid, controlled by both the background grid and the line sources, is shown in Figure 26.3g. • Following the generation of a mesh, it is good practice to make an assessment of grid quality [8]. In many cases, such an assessment can be included within a grid generator and only if there is a problem would the user be informed. However, it can also be beneficial to have a stand-alone grid analysis package. The assessment of grid quality in relation to an analysis algorithm is still a topic for much research. However, it is possible to identify geometrical measures of the “goodness” of a grid. Some appropriate measures are shown in Table 26.1. After computation of quality measures they can be presented in the form of histograms, as shown in Figure 26.3h.
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FIGURE 26.3D
A grid for the Bay generated with grid control from the background grid.
FIGURE 26.3E The geometry of San Francisco Bay with the background grid augmented by two line sources to resolve a deep-water channel. • It is important to know the location of elements with particular grid quality measures. Figure 26.3i
shows the generated grid and the elements that have been highlighted. In practice, such a presentation will involve the user of color.
26.5 Hybrid Grids of Quadrilaterals and Triangles As a second example of the use of unstructured grids, an approach is presented whereby a hybrid grid is constructed from quadrilaterals and triangles. Shaw (Chapter 23) discusses at length the philosophy behind the use of hybrid grids and presents results for three-dimensional aerospace configurations. The
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FIGURE 26.3F line sources.
The boundary grid generated from the point spacing specified on the background grid and the two
FIGURE 26.3G
The grid generated with control from the background grid and the line sources.
example shown here is based upon early work [9,10] and is presented to further elaborate and possibly clarify some of the comments made in Chapter 23. Hybrid grids are also covered in Chapter 25. Figure 26.4a shows an outline of a four component airfoil system composed of a main airfoil, one leading edge slat and two trailing edge flaps. In the process of generating a hybrid grid, • The first step is to generate a structured grid around the main component airfoil. Any structured
grid technique can be used, but here a conformal mapping grid based upon a Von-Karman–Trefftz transformation is used. Figure 26.4b shows such a grid. The outer boundary, which is not shown, extends about 15 chord lengths away from the airfoil. ©1999 CRC Press LLC
TABLE 26.1
Grid Element Quality Parameters
Radius of circumscribing sphere b = ----------------------------------------------------------------------------Radius of inscribed sphere
βequilateral = 3.0
Maximum edge length s = --------------------------------------------------------------Radius of inscribed sphere
σequilateral = 4.8989
Radius of circumscribed sphere w = --------------------------------------------------------------------------Maximum edge length
ωequilateral = 0.6125
Maximum edge length z = -----------------------------------------------------Minimum edge length
ζequilateral = 1.0 3
(average element edge length) a = -------------------------------------------------------------------------Volume 3
(R.M.S edge length) g = --------------------------------------------------Volume
γequilateral = 8.479 3
(Volume) K = ----------------------------------------------------------------------------------------------------------------------2 (Summation of all surface area of triangle faces)
FIGURE 26.3H
αequilateral = 8.479
Κequilateral = 4.5 × 10–4
Typical histogram display of grid quality statistics.
• The next step is to choose one of the flaps or slat components and construct a grid that does not
extend too far away from the geometry. In the case illustrated, the second flap is chosen and a structured grid, again generated by a conformal mapping, is produced. Figure 26.4c shows the grid in relation to the main component airfoil.
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FIGURE 26.3I
Elements with particular grid quality measures can be visualized.
FIGURE 26.4A
Geometry of the three component airfoil.
• The grid around the flap is then superimposed over the main component grid, as shown in
Figure 26.4d. • The next step is governed by user input, which specifies how much of the grid around the flap
should be preserved. A region of the main component grid is then deleted so that the two overlaid grids do not intersect, as shown in Figure 26.4e. • In order to connect the two grids it is then necessary to fill the void region by constructing an unstructured grid. This is readily achieved since the boundary points, together with the boundary edge connectivities, can be easily extracted and sent to the Delaunay grid generator [1–6]. An unstructured grid is then generated, as shown in Figure 26.4f. • A hybrid grid can then be created by connecting together the three generated grids. This is shown in Figure 26.4g. • To introduce the remaining flap and slat, the process already described is repeated. First, introduce a component grid for the leading edge slat and overlay this over the existing grids, Figure 26.4h.
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FIGURE 26.4B A structured grid generated from a conformal mapping is constructed around the main component airfoil.
FIGURE 26.4C
A structured grid is generated around the flap.
• Preserve a portion of the component grid, determine the empty void region, fill with an unstruc-
tured grid and connect all the grids. Figure 26.4i shows the final grid. • To complete the grid, repeat the process again for the second flap component. The final hybrid grid for the complete four component airfoil is shown in Figure 26.4j. The quadrilaterals in the final grid could be directly triangulated if a grid of triangles is required. However, this would defeat the objective of generating a hybrid grid. For high Reynolds number viscous flow simulation, it is very easy to modify the structured grid generation so that appropriate point
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FIGURE 26.4D
The structured grid for the flap is superimposed on the grid for the main component.
FIGURE 26.4E
Overlapping regions of the two grids are deleted leaving two disconnect grids.
clustering in the vicinity of solid boundaries is suitable for capturing boundary layer phenomena. In this way, hybrid grids of the form shown have an important role to play. The major disadvantage of the approach, as illustrated, is that the method is not automatic for general geometries (as defined in the spirit of automatic unstructured grid generators), since a structured grid is required and, since this involves a mapping procedure, the method is geometry-specific. However, the approach is potentially powerful in the sense presented by Shaw (Chapter 23). Hybrid grids are suitable for use with finite volume solvers — in particular, an edge-based scheme — since then the fact that different element types are present is not relevant to the solver [9].
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FIGURE 26.4F An unstructured grid is generated within the void domain, thus connecting the two component grids.
FIGURE 26.4G
The two grids are connected by a ribbon of unstructured grid.
26.6 Unstructured Tetrahedral Grids This section attempts to describe the typical process by which three-dimensional grids of tetrahedra are generated using a Delaunay based approach. Two examples are presented. The first is an aerospace geometry, the Dassault Falcon, consisting of a wing, fuselage, rear-mounted engine, tail and fin, and the second is the geometry of the THRUST Supersonic Car, which broke the world land speed record in October 1997.
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FIGURE 26.4H
FIGURE 26.4I
The third component with a local structured grid is laid over the main component grid.
The component grid is connected to the main grid by a ribbon of unstructured grid.
26.6.1 Dassault Falcon • Figure 26.5a shows the geometry of the Falcon aircraft. For clarity, only half the aircraft will be
considered. The aircraft consists of 12 individual support surfaces. • It will be assumed that a grid is required around the exterior of the geometry, typical for a flow
computation. Hence, the domain must be closed by the addition of bounding surfaces, in this case, an outer hemispherical boundary and a plane of symmetry, as shown in Figure 26.5b.
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FIGURE 26.4J
FIGURE 26.5A
The second component flap grid is introduced and connected to the main component grid.
Shown is the geometry patches. Twelve support surfaces define the shape of the aircraft.
• Given the closed domain, the generation process involves the construction of a grid on the surfaces
which define the domain, followed by a tetrahedral grid generated to fill the domain. Before either of these tasks can be performed it is necessary to define the required spacing of elements within the domain. As in the case of the generation of grids in two dimensions, this is performed using a background grid with added sources. For three dimensions it is not particularly beneficial to present a figure which outlines the background grid. Hence, in Figure 26.5c the geometry of the aircraft is shown together with the representation of a line source. This line source, as can be seen, is shown as a thick line along the leading edge of the wing. The two spheres at the end of the line provide the user with an indication of the region of influence of the line source. The definition of a line source by a user is easily performed within a graphics user environment, since points of
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FIGURE 26.5B
The region around the aircraft is enclosed by a hemispherical boundary and a plane of symmetry.
FIGURE 26.5C Sources are used to provide the required grid control. Shown are two line sources along the leading edge of the wind. The outline of the spheres at the ends of the line sources provide the user with an indication of the regions of influence of the sources.
the geometry can be selected and then point sources/line sources created by the push of a button. The line source shown in Figure 26.5c illustrates the concept. However, for a realistic mesh for the Falcon aircraft it is necessary to define many line and point sources, and in such a case it is not effective to show all these in a figure.
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FIGURE 26.5D A coarse surface grid on the aircraft. (Note: The sources used to generate this grid are not the ones shown in Figure 26.5c.)
FIGURE 26.5E
A close-up view of the surface mesh.
• The surface grid on the aircraft, generated using six point sources and ten line sources, is shown
in Figures 26.5d and 26.5e. Surface grid generation is described in detail in Chapters 17 and 19 and in reference [11]. The mesh shown is, for clarity, a coarse mesh, but it does exhibit the required spacing in that the grid has been clustered in regions around leading edges and trailing edges. • Once the surface grid has been generated on all the boundary surfaces, a volume mesh can be created. It is difficult to view tetrahedra, but Figure 26.5f shows the elements that fall inside a cutting arc. This leads to effective pictures, but arguably these are of little value in assessing grid quality. • It is necessary to resort to analysis of the grid quality measures to assess the quality of the grid and histograms are a suitable way to project this data, Figure 26.5g. • If required, elements or nodes within the grid whose associated quality measures are of concern can be viewed, Figure 26.5h.
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FIGURE 26.5F
Sectional cut through the grid of tetrahedra.
FIGURE 26.5G
Histogram of grid quality measures.
• To complete the sequence of figures, the grid generated is suitable for an inviscid flow simulation.
Figure 26.5i shows the geometry of the aircraft, streamlines, contours, and sections through the unstructured grid.
26.6.2 THRUST Supersonic Car • Figure 26.6a shows the geometry of the Thrust car. The car is enclosed within a bounding box, as
shown in Figure 26.6b.
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FIGURE 26.5H
Elements with specific grid quality measures can be viewed in the mesh.
FIGURE 26.5I
Flow simulation for the Dassault Falcon.
• Figure 26.6c shows the geometry of the car together with sources to control the grid point density. • Figure 26.6d shows a grid on the car, plane of symmetry and the ground. • Figure 26.6e shows a cut through the grid of tetrahedra. • A typical flow simulation is shown in Figure 26.6f.
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FIGURE 26.6A
FIGURE 26.6B
FIGURE 26.6C
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Geometrical definition of the car.
The car inside a bounding box.
The geometry of the car showing sources to control grid density.
FIGURE 26.6D
Surface grid on the car, plane of symmetry and the ground.
FIGURE 26.6E
Cut through the domain of tetrahedra.
26.7 Non-Isotropic Grid Generation for Viscous Flow Simulation For some applications, the use of regular isotropic elements can lead to very large meshes. A good example of such a case is the simulation of high Reynolds number viscous flows where, to capture boundary layer effects, very small elements are required. It is appropriate, therefore, knowing the physics of boundary layers, to consider a form of a priori adaptation to reflect the difference in gradients in flowfield variables across a boundary layer as compared with along a boundary layer in the direction of the flow. If such an approach is followed, then elements with high aspect ratios will be required.
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FIGURE 26.6F
Flow solution over the car.
The generation of grids that incorporate elements with arbitrary stretching has been the focus of interest for some time. Chapter 20 discusses this issue in some length and presents in detail one approach. An alternative approach is highlighted in Figure 26.7. • Figure 26.7a shows a grid of quadrilaterals which has been generated using an algebraic approach
[14,15]. The approach amounts to growing layers of elements by advancing along lines that are approximately normal to the boundary. These layers of elements are grown until either they selfintersect or reach an aspect ratio of unity. • After this the domain is filled with regular isotropic elements using the standard Delaunay approach, Figure 26.7c. • Figures 26.7e show some of the details of this approach within a concave corner. • The quadrilaterals can be triangulated to provide a grid consisting of triangles, Figures 26.7b, 26.7d, and 26.7f. This approach is equally applicable in three dimensions where grids of tetrahedra or tetrahedra/prisms can be created. This method of advancing layers (or advancing normals) is a pragmatic approach and is clearly applicable for solid boundaries. However, it does not take into account other features of viscous flow phenomena, such as wakes. However, it is relatively easy to modify the approach to include a suitable treatment for wakes. The approach adopted is as follows: • Use an initial mesh to obtain a flow solution. • From the flow solution, determine the wake lines. Figure 26.8a shows a four-component high lift
airfoil system with the computed wake lines. • Attach the wake lines to the existing geometry and then use the advancing layer approach to
construct stretched elements along the geometry boundaries and along the computed wake lines, Figures 26.8b and 26.8c. • Determine the outer points and edges of the grid generated from the advancing layer stage. These define a boundary which is input data for the Delaunay triangulation phase, Figures 26.8d and 26.8e.
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FIGURE 26.7
The advancing layer approach to the generation of stretched elements close to solid boundaries.
• The advancing layers grid and the Delaunay grid are then combined to form the final grid,
Figure 26.8f. • The final grid is suitable for a high Reynolds number viscous flow simulation, Figure 26.8g.
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FIGURE 26.8A
FIGURE 26.8B
Geometry of the high-lift airfoils, together with wake lines.
Highly stretched elements close to the geometrical boundary and the wake lines.
FIGURE 26.8C
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Close-up view of the stretched elements.
FIGURE 26.8D
FIGURE 26.8E
The isotropic grid generated by Delaunay triangulation.
Close-up view of the unstructured isotropic elements.
26.8 Parallel Unstructured Grid Generation The introduction of scalable parallel computers is enabling larger problems to be solved in many areas of computational engineering. In computational electromagnetics (CEM), typical simulations employ meshes of five million triangles in two dimensions and many tens of millions of elements in three dimensions. In computational fluid dynamics (CFD), a mesh of at least ten million elements can be required for a high Reynolds number viscous flow simulation over a complete aircraft. As mesh sizes become as large as this, the process of mesh generation on a serial computer can become problematic both in terms of time for generation and memory requirements of computers. Parallel computers afford the potential to relieve this problem. Chapter 24 discussed in detail many aspects of parallel mesh generation; therefore, here only examples will be given. The approach that will be demonstrated, is based upon geometrical partitioning of the domain [16]. To generate a grid in parallel, the complete domain is divided into a set of smaller
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FIGURE 26.8F
FIGURE 26.8G
FIGURE 26.9A
The final grid.
Flow solution for a high-lift airfoil system.
The inner geometry and the outer boundary is point discretized.
subdomains, and a grid generated in every subdomain independently. A combination of the subdomain grids forms the final grid of the total domain. A manager/worker model is employed, in which the initial work is performed by the manager who then distributes the grid generation tasks to the workers. The manager can recombine all the subdomain grids or, if the grid is particularly large, leave the partitioned grid on disc. Figure 26.9 shows the general procedure.
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FIGURE 26.9B
The initial triangulation formed by triangulating the boundary points.
FIGURE 26.9C
Domain decomposition.
• The geometry is point discretized, Figure 26.9a. • The boundary points are connected using a Delaunay algorithm to produce an initial triangulation,
as shown in Figure 26.9b. • A greedy algorithm, with an area criterion, is employed to give a number of equally sized subdo-
mains, Figure 26.9c. • The interdomain boundaries are discretized leading to a set of independent grid generation tasks, Figure 26.9d. • The data for each subdomain grid is distributed to the processors and the grids generated. The distribution of data is performed using the message passing library MPI. If the number of domains is N, and the number of processors is M, then static load balances results if N = M, and dynamic load balancing if N > M. The parallel procedure is more efficient if dynamic load balancing is employed. ©1999 CRC Press LLC
FIGURE 26.9D
In this example, six independent grid generation problems are created.
FIGURE 26.10A
The initial geometry.
The approach outlined also applies to the generation of grids in three dimensions. The generation of grids on the interdomain boundaries is significantly more difficult [16]. As an illustration of the procedure, Figure 26.10 shows some of the stages in the generation of a grid for a realistic geometry. • Figure 26.10a shows the initial geometry. • Figure 26.10b shows the surface grid of triangles. • Figure 26.10c shows each of the four partitions, first in the form following the initial decompo-
sition, and then after the surface grid has been suitably modified to provide input data for the volume generation. • Figure 26.10d shows sections cut through the four volume grids. The procedure outlined is capable of generating very large meshes. As an example, Figure 26.11 shows the profile data of a mesh with almost 50 million tetrahedra. The manipulation of such large meshes becomes very difficult, and the user interaction with a graphics user interface described in the different sections of this chapter is not practical. Therefore, it is necessary to use graphics frameworks based upon parallel computer platforms [17]. Figure 26.12 shows an illustration of the parallel visualization of a large mesh generated by the parallel mesh generator. ©1999 CRC Press LLC
FIGURE 26.10B
The surface mesh.
26.9 Summary In this chapter, an attempt has been made to provide examples of unstructured grids and to indicate the procedures followed in the process of grid generation. In this way, it should augment much of the material presented in the other chapters of this part of the handbook. All the grids have been generated using software developed at Swansea and are snapshots taken of results presented with graphics user interfaces [18,19] (see also the Appendix to this chapter). The literature now provides many impressive examples of grids generated for real-world problems, and the interested reader is directed to proceedings of recent grid generation conferences [20–24] and survey papers [25].
Appendix: Graphics User Interfaces With the wide-scale availability of high-resolution computer graphics, the process of user-grid interaction has been revolutionized. It is now common practice for grid generation algorithms to be embedded within easy-to-use graphics user interfaces where users can be shown relevant data in a visually meaningful way [18,19]. This technology has reduced both the time taken to generate grids and the training time required for new users to become proficient at generating grids. Images from some typical windows of two graphics user environments for grid generation [18,19] are presented in Figure 26.A1.
Acknowledgment The author would like to acknowledge Dr. O Hassan, Dr. M. J. Marchant, Mr. R. Said, Mr. E. TurnerSmith, Mr. J. Jones for helping to produce the figures used in the chapter.
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FIGURE 26.10C An example of a domain decomposed into four partitions. Shown is the surface grid, the interdomain surface triangles (faces of the initial tetrahedra that fill the domain), followed by the final surface grids prior to volume meshing.
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FIGURE 26.10D
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Sections through the volume grids of the individual partitions.
FIGURE 26.11
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Details of a mesh of almost 50 million elements generated in parallel.
FIGURE 26.12
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Sections through a grid computed using parallel visualization.
FIGURE 26.A1
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Typical Windows environments for more effective interactive grid generation.
References 1. Weatherill, N.P., The generation of unstructured grids using Dirichlet tessellations, Department of Mechanical and Aerospace Engineering, Report No. 1715, Princeton University, 1985. 2. Jameson, A., Baker, T.J., and Weatherill, N.P., Calculation of inviscid transonic flow over a complete aircraft, 24th Aerospace Sciences Meeting, AIAA Paper 86-0103. Reno, NV, 1986. 3. Weatherill, N.P., A method for generating irregular computational grids in multiply connected planar domains, Int. J. for Numerical Methods in Fluids. 1988, Vol. 8, pp. 181–197. 4. Weatherill, N.P., Delaunay triangulation in computational fluid dynamics, Computers and Mathematics with Applications. 1992, Vol. 24, No. 5/6, pp. 129–150. 5. Weatherill, N.P. and Hassan, O., Efficient three-dimensional grid generation using the Delaunay triangulation, Proc. of the 1st European Computational Fluid Dynamics Conf. Brussels, Belgium, Hirsch, C., Periaux, J., Kodulla, W., (Eds.), Elsevier, Amsterdam, 1992. 6. Weatherill, N.P. and Hassan, O., Efficient three-dimensional delaunay triangulation with automatic point creation and imposed boundary constraints, Int. J. for Numerical Methods in Engineering. 1994, Vol. 37, pp. 2005–2039. 7. Weatherill, N.P., The reconstruction of boundary contours and surfaces in arbitrary unstructured triangular and tetrahedral grids, Engineering Computations. 1996, Vol. 3, No. 8, pp. 66–81. 8. Parmley, K.L., Dannenhoffer J.F. III, and Weatherill, N.P., Techniques for the visual evaluation of computational grids, AIAA Paper 93-3353. AIAA CFD Meeting Orlando, FL, July 6-9, 1993. 9. Weatherill, N.P., Mixed structured and unstructured meshes for aerodynamic flow simulation, Aeronautical Journal. 1990, 94, pp. 111–123. 10. Weatherill, N.P. and Natakusumah, D., The simulation of potential flow around multiple bodies using overlapping connected meshes, Appl. Math. Comput., 1991, 46, pp. 1–21. 11. Peraire, J., Peiro, J., Formaggia, L., Morgan,K., and Zienkiewicz, O.C., Finite element Euler computations in three dimensions, 1988, Vol. 26, pp. 2135–2159. 12. Weatherill, N.P., Mixed structured and unstructured meshes for aerodynamic flow simulation, Aeronautical Journal. 1990, 94, pp. 111–123. 13. Weatherill, N.P. and Natakusumah, D., The simulation of potential flow around multiple bodies using overlapping connected meshes, Appl. Math. Comput., 1991, 46, pp. 1–21. 14. Marchant, M.J., Weatherill, N.P., and Hassan, O., FEA. 15. Hassan, O., AIAA. 16. Said, R., Weatherill, N.P., Morgan, K., and Verhoeven, N.A., Distributed Delaunay mesh generation for very large meshes, submitted for publication, January 1998. 17. Jones, J. and Weatherill, N.P., Parallel visualisation, submitted for publication. 18. Marchant, M.J. and Weatherill, N.P., The design of a software tutorial for computational aerodynamics, Proc. of the Eng. Education Conf., Professional Standards and Quality. Sheffield, UK, Bramhall, M.D. and Robinson, I.M., (Eds.), SHU Press, 1997. 19. Marchant, M.J., Weatherill, N.P., Turner–Smith, E., Zheng, Y., and Sotirakos, M., A parallel simulation user environment for computational engineering, Proceedings of the 5th International Conference on Numerical Grid Generation in Computational Field Simulation. April 1996, Soni, B., Hauser, J., Eiseman, P., Thompson, J.F., (Eds.), MSU Press, 1996. 20. Proc. of the 1st Int.Conf. on Grid Generation. Landshut, West Germany, Pineridge Press, UK, 1986. 21. Proc. of the 2nd Int. Conf. on Grid Generation. Miami, FL, Pineridge Press, UK, 1988. 22. Proc. of the 5th Int. Conf. on Grid Generation in Computational Fluid Dynamics and Related Fields. Starkville, MS, North-Holland, 1991. 23. Proc. of the 4th Int. Conf. on Grid Generation in Computational Fluid Dynamics and Related Fields. Swansea, UK., North-Holland, 1994. 24. Proc. of the 3rd Int. Conf. on Grid Generation in Computational Fluid Dynamics and Related Fields Barcelona, Spain. North-Holland, 1991.
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25. Thompson J.R. and Weatherill, N.P., Aspects of numerical grid generation, AIAA Applied Aerodynamics Meeting, Monterey, CA, August 1993.
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