86 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES - Construct a quad/octree for the points; - Order the elements according to decreasing volume (e.g in a heap-list); - Construct a linked list for all the elements surrounding each point; - do: Loop over the elements, in descending volume, testing: - if the element, denoted in the following by ielem, has not been marked for deletion before: - Obtain the minimum/maximum extent of the coordinates belonging to this element; - Find from the quad/octree all points falling into this search region, storing them in a list lclop(1:nclop); - Find all the unmarked elements with smaller volume than ielem surrounding the points stored in lclop(1:nclop); this yields a list of close elements lcloe(1:ncloe); - Loop over the elements stored in lcloe(1:ncloe): - if the element crosses the faces of or is inside ielem Mark ielem for deletion endif endif enddo The reason for looping over the elements according to descending volumes is that the search region is obtained in a natural way (the extent of the element) Looping according to ascending volumes would imply guessing search regions As negative elements could lead to a failure of this test, the overlap test is performed after the negative elements have been identified and marked The test for overlapping elements can account for a large portion of the overall CPU requirement Therefore, several filtering and marking strategies have to be implemented to speed up the procedure The most important ones are as follows Column marking The mesh crossing test is carried out after all the negative, badly shaped and large elements have been removed This leaves a series of prismatic columns, which go from the surface triangle to the last element of the original column still kept The idea is to test the overlap of these prismatic columns, and mark all the elements in columns that pass the test as not requiring any further crossing tests In order to perform this test, we subdivide the prismatic columns into three tetrahedra as before, and use the usual element crossing tests for tetrahedra Since a large portion of the elements does not require further testing (e.g convex surfaces that are far enough apart), this is a very effective test that leads to a drastic reduction of CPU requirements Prism removal Given that the elements in the semi-structured region were created from prisms, it is an easy matter to identify for each element the triplet of elements stemming from the same prism If an element happens to be rejected, all three elements of the original prism are rejected This avoids subsequent testing of the elements from the same prism GRID GENERATION 87 Marking of surfaces Typically, the elements of a surface segment or patch will not overlap This is because, in most instances, the number of faces created on each of these patches is relatively large, and/or the patches themselves are relatively smooth surfaces The main areas where overlap can occur are corners or ‘coalescing fronts’ For both cases, in the majority of the cases encountered in practice, the elements will originate from different surface patches Should a patch definition of the surface not be available, an alternative is to compute the surface smoothness and concavity from the surface triangulation Then discrete ‘patches’ can be associated with the discretized surface using, e.g., a neighbour-to-neighbour marching algorithm If the assumption of surface patch smoothness and/or convexity can be made, then it is clear that only the elements (and points) that originated from a surface patch other than the one that gave rise to the element currently being examined need to be tested In this way, a large number of unnecessary tests can be avoided Rejection via close points The idea of storing the patch from which an element emanated can also be applied to points If any given point from another patch that is surrounded by elements that are smaller than the one currently being tested is too close, the element is marked for deletion The proximity test is carried out by computing the smallest distance between the close point and the four vertices of the element being tested If this distance is less than approximately the smallest side length of the element, the element is marked for deletion Evaluating four distances is very inexpensive compared to a full crossing test Rejection if a point lies within an element If one of the close points happens to fall within the element being tested, then obviously a crossing situation occurs Testing whether a point falls inside the element is considerably cheaper (by more than an order of magnitude) than testing whether the elements surrounding this point cross the element Therefore, all the close points are subjected to this test before proceeding Top element in prism test The most likely candidate for element crossing of any given triplet of elements that form a prism is the top one This is because, in the case of ‘coalescing fronts’, the top elements will be the first ones to collide It is therefore prudent to subject only this element to the full (and expensive) element crossing test In fact, only the top face of this element needs to be tested This avoids a very large number of unnecessary tests, and has been found to work very well in practice Avoidance of low layer testing Grids suitable for RANS calculations are characterized by having extremely small grid spacings close to wetted surfaces It is highly unlikely – although of course not impossible – that the layers closest to the body should cross or overlap Therefore, one can, in most 88 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES instances, avoid the element crossing test for the elements closest to wetted surfaces For typical grids, between four and ten layers in the grids are avoided This number is, however, problem dependent, but fairly easy to estimate if a graphical workstation is at hand For most applications, such a device is nowadays commonly used to prepare the data Rejection via faces of an element After all the filtering operations described above, a considerable number of elements will still have to be subjected to the full crossing test The crossing test is carried out face-by-face, using the face-crossing check described in Section 3.6.1 If all the points of a close element lie completely on the outward side of any of the faces of the element being tested, this pair of elements cannot possibly cross each other This filter only involves a few scalar products, and is very effective in pruning the list of close elements tested 3.12.5.4 Elements crossing boundary faces In regions where the distance between surfaces is very small, elements from the semistructured region are likely to cross boundary faces As this test is performed after the element crossing tests are conducted, the only boundaries which need to be treated are those that have no semi-structured grid attached to them In order to detect whether overlapping occurs, a loop is performed over the surface faces, seeing if any element crosses it As before, straightforward testing would result in an expensive O(Nel · Nf ) procedure, where Nf denotes the number of boundary faces By using quad/octrees, this complexity can be reduced to O(Nf log Nel ) The face-crossing check looks essentially the same as the check for overlapping elements, and its explicit description is therefore omitted Needless to say, some of the filtering techniques described before can also be applied here 3.12.5.5 Distribution of points normal to wetted walls/wakes To gain some insight into the distribution of points required for typical high-Reynoldsnumber flows, let us consider the case of the flat plate The friction coefficient cf , given by 2τw cf = , (3.41) ρv∞ is a well-documented quantity Here τw denotes the shear at the wall, ρ a reference density and v∞ a reference velocity The shear in the laminar sublayer is given by τw = µu,y , (3.42) where u denotes the velocity parallel to the wall and y the direction normal to it Assuming that cf = αReβ , (3.43) x where x denotes the streamwise location, we have τw = µu,y = cf ρv∞ , (3.44) 89 GRID GENERATION implying α v∞ 1+β Rex (3.45) x In order to obtain a meaningful gradient of the velocity at the wall, the velocity of the first point away from the wall, u1 , should only be a fraction of the reference velocity v∞ Therefore, v∞ α v∞ 1+β u − u0 = = Rex , u,y ≈ (3.46) y y x which yields the required height of the first point above the wall: u,y = y= x 1+β (3.47) αRex For laminar flow, we have (Schlichting (1979)) α = 0.664, β = −1/2, implying that −1/2 y = xRex (3.48) , while for turbulent flow α = 0.045, β = −1/4, which yields −3/4 y = 44 xRex (3.49) Typical values for range from = 0.05 for laminar flow to = 0.01 for turbulent flows For turbulent flows, an alternative way of obtaining the location of the first point away from the wall can be obtained via the non-dimensional distance parameter y + , defined as y√ y+ = ρτw (3.50) µ Given y + for the first point away from the wall, one can determine y +µ y +x y +µ =√ =√ y=√ 1+β/2 ρτw ρµu,y α/2Rex y from (3.45): (3.51) For α = 0.045, β = −1/4, this results in −7/8 y = 6.7y + xRex (3.52) If the distance between points increases geometrically as yi+1 = c yi , (3.53) one can estimate the number of layers required to assure a smooth transition to the isotropic mesh outside the boundary layer region Denoting by hi the element size in the isotropic mesh region, the number of layers n required is given by n= log hi − log y0 , log c (3.54) and the total thickness of the semi-structured grid region is δ= Typical values for c are 1.10 < c < 1.30 cn+1 − y0 c−1 (3.55) 90 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES 3.13 Filling space with points/arbitrary objects Many simulation techniques in computational mechanics require a space-filling cloud of arbitrary objects For the case of ‘gridless’ or ‘mesh free’ PDE solvers (see Nay and Utku (1972), Gingold and Monahghan (1983), Batina (1993), Belytschko et al (1994), Duarte and Oden (1995), Oñate et al (1996a,b), Liu et al (1996), Löhner et al (2002)) these are simply points For discrete element methods (see Cundall and Stack (1979), Cundall (1988), Cleary (1998), Sakaguchi and Murakami (2002)), these could be spheres, ellipsoids, polyhedra or any other arbitrary shape The task is therefore to fill a prescribed volume with these objects so that they are close but not overlap in an automatic way Several techniques have been used to place these objects in space The so-called ‘fill and expand’ or ‘popcorn’ technique (Sakaguchi and Murakami (2002)) starts by first generating a coarse mesh for the volume to be filled This subdivision of the volume into large, simple polyhedra (elements) is, in most cases, performed with hexahedra The objects required (points, spheres, ellipsoids, polyhedra, etc.) are then placed randomly in each of these elements These are then expanded in size until contact occurs or the desired fill ratio has been achieved An obvious drawback of this technique is the requirement of a mesh generator to initiate the process A second class of techniques are the ‘advancing front’ or ‘depositional’ methods (Löhner and Oñate (1998), Feng et al (2002)) Starting from the surface, objects are added where empty space still exists In contrast to the ‘fill and expand’ procedures, the objects are packed as close as required during introduction Depending on how the objects are introduced, one can mimic gravitational or magnetic deposition, layer growing or size-based growth Furthermore, so-called radius growing can be achieved by first generating a coarse cloud of objects, and then growing more objects around each of these In this way, one can simulate granules or stone In the present section, we consider the direct generation of clouds of arbitrary objects with the same degree of flexibility as advanced unstructured mesh generators The mean distance between objects (or, equivalently, the material density) is specified by means of background grids, sources and density attached to CAD entities In order not to generate objects outside the computational domain, an initial triangulation of the surface that is compatible with the desired mean distance between objects specified by the user is generated Starting from this initial ‘front’ of objects, new objects are added until no further objects can be introduced Whereas the AFT for the generation of volume grids removes one face at a time to generate elements, the present scheme removes one object at a time, attempting to introduce as many objects as possible in its immediate neighbourhood 3.13.1 THE ADVANCING FRONT SPACE-FILLING ALGORITHM Assume as given: - A specification of the desired mean distance between objects in space, as well as the mean size of these objects This can be done through a combination of background grids, sources and mean distance to neighbours attached to CAD-data (see section 3.2 above) - An initial triangulation of the surface, with the face normals pointing towards the interior of the domain to be filled with points GRID GENERATION 91 With reference to Figure 3.48, which shows the filling of a simple 2-D domain with ellipsoids, the complete advancing front space-filling algorithm may be summarized as follows: - Determine the required mean point distance for the points of the triangulation; - while: there are active objects in the front: - Remove the object ioout with the smallest specified mean distance to neighbours from the front; - With the specified mean object distance: determine the coordinates of nposs possible new neighbours This is done using a stencil, examples of which are shown in Figure 3.49; - Find all existing objects in the neighbourhood of ioout; - Find all boundary faces in the neighbourhood of ioout; - do: For each one of the possible new neighbour objects ionew: - If there exists an object closer than a minimum distance dminp from ionew, or if the objects are penetrating each other: ⇒ skip ionew; - If the object ionew crosses existing faces: ⇒ skip ionew; - If the line connecting ioout and ionew crosses existing faces: ⇒ skip ionew; - Determine the required mean point distance for ionew; - Increment the number of objects by one; - Introduce ionew to the list of coordinates and store its attributes; - Introduce ionew to the list of active front objects; enddo endwhile The main search operations required are: - finding the active object with the smallest mean distance to neighbours; - finding the existing objects in the neighbourhood of ioout; - finding the boundary faces in the neighbourhood of ioout These three search operations are performed efficiently using heap lists, octrees and linked lists, respectively 3.13.2 POINT/OBJECT PLACEMENT STENCILS A number of different stencils may be contemplated Each one of these corresponds to a particular space-filling point/object configuration The simplest possible stencil is the one that only considers the six nearest-neighbours on a Cartesian grid (see Figure 3.49(a)) It is easy to see that this stencil, when applied recursively with the present advancing front algorithm, will fill a 3-D volume completely Other Cartesian stencils, which include nearest√ √ neighbours with distances and from ioout, are shown in Figures 3.49(b) and (c) The ‘tetrahedral’ stencil shown in Figure 3.49(d) represents another possibility Furthermore, one can contemplate the use of random stencils, i.e the use of n randomly selected directions 92 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES Active Front Object/Point Surrounded/Inactive Object/Point Domain Boundary Initial Discretization of Boundary Figure 3.48 Advancing front space-filling with ellipses b) c) ) a) ) ) ) d) ) e) Symmetric Figure 3.49 Point stencils: (a) Cartesian [6]; (b) Cartesian [18]; (c) Cartesian [26]; (d) tetrahedral [17]; (e) random GRID GENERATION 93 to place new objects close to an existing one For the generation of points and spheres, it was found that the six-point stencil leads to the smallest amount of rejections and unnecessary testing (Lưhner and ate (1998)) In many instances, it is advisable to generate ‘body conforming’ clouds of points in the vicinity of surfaces In particular, Finite Point Methods (FPMs) or Smoothed Particle Hydrodynamics (SPH) techniques may require these ‘boundary layer point distributions’ Such point distributions can be achieved by evaluating the average point-normals for the initial triangulation When creating new points, the stencil used is rotated in the direction of the normal The newly created points inherit the normal from the point ioout they originated from 3.13.3 BOUNDARY CONSISTENCY CHECKS A crucial requirement for a general space-filling object generator is the ability to only generate objects in the computational domain desired Assuming that the object to be removed from the list of active objects ionew is inside the domain, a new object ionew will cross the boundary triangulation if it lies on the other side of the plane formed by any of the faces that are in the proximity of ionew and can see ionew This criterion is made more stringent by introducing a tolerated closeness or tolerated distance dt of new objects to the exterior faces of the domain Testing for boundary consistency is then carried out using the following algorithm (see Figure 3.50): - Obtain all the faces close to ioout; - Filter, from this list, the faces that are pointing away from ioout; - do: For each of the close faces: - Obtain the normal distance dn from ionew to this face; if: dn < 0: ionew lies outside the domain ⇒ reject ionew and exit; elseif: dn > dt : ionew if far enough from the faces ⇒ proceed to the next close face; elseif: ≤ dn ≤ dt : obtain the closest distance dmin of ionew to this face; if: dmin < dt : ionew is too close to the boundary ⇒ reject ionew and exit; endif - enddo Typical values for dt are 0.707d0 ≤ dt ≤ 0.9d0, where d0 denotes the desired mean average distance between points 3.13.4 MAXIMUM COMPACTION TECHNIQUES For SPH and finite point applications, the use of stencils is sufficient to ensure a proper space filling (discretization) of the computational domain However, many applications that consider not points but volume-occupying objects, such as spheres, ellipsoids and polyhedra, require a preset volume fraction occupied by these objects and, if possible, a minimum 94 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES b) ) a) c) ) ) n n : ioout n dt n : ionew Figure 3.50 Boundary consistency checks: (a) obtain stencil for ioout and close faces; (b) filter faces pointing away from ioout; (c) retain points inside computational domain number of contacts The modelling of discontinua via discrete element methods represents a typical class of such applications Experience indicates that the use of point stencils does not yield the desired volume fractions and contact neighbours Two complementary techniques have proven useful to achieve these goals: closest object placement and move/enlarge postprocessing 3.13.4.1 Closest object placement Closest object placement attempts to position new objects as close as possible to existing ones (see Figure 3.51) The initial position for new objects is taken from a stencil as before The closest three objects to the new object position and the object being removed from the active front are then obtained This does not represent any substantial increase in effort, as the existing objects in the vicinity are required anyhow for closeness/penetration testing Starting from the three closest objects, an attempt is made to place the new object in such a way that it contacts all of them, does not penetrate any other objects and resides inside the computational domain If this is not possible, an attempt is made with the two closest objects Should this also prove impossible, an attempt is made to move the new objects towards the object being removed from the active front If this attempt is also unsuccessful, the usual stencil position is chosen b) ) ) a) c) ) : Object Being Removed from Active Front : New Object : Close Existing Object Figure 3.51 Closest object placement: (a) move to three neighbours; (b) move to two neighbours; (c) move to one neighbour 95 GRID GENERATION 3.13.4.2 Move/enlarge post-processing Whereas closest object placement is performed while space is being filled with objects, postprocessing attempts to enlarge and/or move the objects in such a way that a higher volume ratio of objects is obtained, and more contacts with nearest-neighbours are established The procedure, shown schematically in Figure 3.52, may be summarized as follows: - while: objects can be moved/enlarged: - do: loop over the objects ioout: - Find the closest existing objects of ioout; - Find all boundary faces in the neighbourhood of ioout; - Move the object away from the closest existing objects so that: - The minimum distance to the closest existing objects increases; - ioout does not penetrate the boundary; - Enlarge object ioout by a small percentage - If the enlarged object ionew penetrates other objects or the boundary: revert to original size; ⇒ skip ioout; enddo endwhile : Object Being Moved/Enlarged Initial Object Size/Distribution Final Object Size/Distribution Figure 3.52 Movement and enlargement of objects The increase factors are progressively decreased for each new loop over the objects Typical initial increase ratios are 5% As the movement of objects is moderate (e.g less than the radius for spherical objects), the spatial search data structures (bins, octrees) required during space filling can be reused without modification This move/enlarge post-processing is very fast and effective, and is used routinely for the generation of discrete element method/discrete particle method datasets 96 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES 3.13.5 ARBITRARY OBJECTS The most time-consuming part of the present technique is given by the penetration/closeness checks These checks are particularly simple for spheres However, for arbitrary objects, the CPU burden can be significant Specialized penetration/closeness checks are available for ellipsoids (Feng et al (2002)), but for general polyhedra the faces have to be triangulated and detailed face/face checks are unavoidable A recourse that has proven effective is to approximate arbitrary objects by a collection of spheres (see Figure 3.53) When adding a new object in space, the penetration/closeness checks are carried out for all spheres comprising the object The new object is only added if all spheres pass the required tests (Löhner and Oñate (2004)) ) ) ) ) d) b) e) ) a) c) etc Figure 3.53 Arbitrary objects as a collection of spheres: (a) sphere; (b) tube; (c) ellipsoid; (d) tetrahedron; (e) cube b) c) ) ) a) ) g Figure 3.54 Deposition patterns: (a) force field; (b) layered growth; (c) seed-point (radius) growing 3.13.6 DEPOSITION PATTERNS Depending on how the objects are removed from the active front, different deposition patterns can be achieved The advancing front space-filling technique always removes the object with the smallest average distance (size) to new neighbours from the active front Different size distributions will therefore lead to different deposition patterns (see Figure 3.54) Gravitational deposition can be achieved by specifying a size distribution that decreases 97 GRID GENERATION (a) (b) Figure 3.55 Space shuttle: (a) outline of domain; (b) close-up and sources; (c) surface mesh slightly in the direction of the gravity vector The objects that are at the ‘bottom’ will then be removed first from the active front and surrounded by new objects The same technique can be applied if magnetic fields are present Layered growth can be obtained be assigning a 98 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (c) Figure 3.55 Continued slight increase in size based on the object number: δ = (1 + nobj )δ0 , (3.56) where δ, δ0 , nobj and denote the size used, original size, object number and a very small number (e.g = 10−10 ), respectively So-called radius growing, used to simulate granules or stone (Sakaguchi and Murakami (2002)), can be achieved by first generating a coarse cloud of objects, and then using layered growth around each of these Another interesting deposition pattern is the so-called bin or sieve placement, whereby spheres or objects of different average diameter are placed in space in such a way that the average number of spheres of a certain diameter in a given region of space corresponds to a given distribution This deposition pattern can be implemented by placing the required distribution of sphere sizes into bins of decreasing sphere size One then performs several passes over the domain, filling it with spheres or objects of a diameter and with a volume packing according to the bin specifications As an example, for concrete one starts with spheres of the aggregate size The next sphere size would be of diameter one-half to one-quarter of the original one, the next again one-half to one-quarter of the previous one, etc Typically, no more than three to four bins are required For each subsequent pass, all points/objects generated so far are considered part of the initial front As the new spheres/objects have a much smaller size than all previous ones, each one of the spheres/objects generated in previous bin passes allows for sufficient ‘space’ for the generation of new spheres/objects 3.14 Applications The following section shows several grids, as well as clouds of points and arbitrary objects, which were constructed using the techniques described in this chapter 99 GRID GENERATION 3.14.1 SPACE SHUTTLE ASCEND CONFIGURATION This example is typical of many aerodynamic data sets The outline of the domain, which was defined via 1271 surface patches, is shown in Figure 3.55(a) A close-up of the shuttle, together with some of the approximately 200 sources used to specify the desired element size in space, can be seen in Figure 3.55(b) The surface triangulation of the final mesh, which had approximately million tetrahedra, is shown in Figure 3.55(c) The smallest and largest specified element side lengths were 5.08 cm and 467.00 cm, respectively, i.e an edge–length ratio of approximately 1:102 and a volume ratio of 1:106 (a) (b) Figure 3.56 F18 pilot ejection: (a) outline of domain; (b) surface mesh (close-up) 100 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (a) (b) (c) Figure 3.57 Circle of Willis: (a) medical images; (b) discrete surface definition; (c) surface mesh 3.14.2 PILOT EJECTING FROM F18 Problems with moving bodies often require many re-meshings during the course of a simulation The case shown here was taken from a pilot ejection simulation recently conducted (Sharov et al (2000)) The outline of the domain, which consisted of 1345 patches, is shown in Figure 3.56(a) The surface triangulation of the final mesh, which had approximately 14 million tetrahedra, is shown in Figure 3.56(b) The smallest and largest specified element side lengths were 0.65 cm and 250.00 cm, respectively, i.e an edge–length ratio of approximately 1:4 × 102 and a volume ratio of 1:5.6 × 107 The spatial variation of element size was specified via approximately 110 sources 101 GRID GENERATION (a) (b) (c) Figure 3.58 DARPA-2 model: (a), (b) surface triangulation; (c) cross-sections in mesh 102 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (a) (b) (c) (d) (e) (f) Figure 3.59 Ahmed Body: (a)–(d) mesh in different cuts; (e), (f) detail of mesh 103 GRID GENERATION (a) (b) (c) (d) (e) Figure 3.60 Truck: (a) CAD definition; (b) triangulation and (c) mixed quad/triangular mesh; (d), (e) final all-quad surface mesh 3.14.3 CIRCLE OF WILLIS This example is typical of patient-specific biomedical CFD simulations (Cebral and Löhner (1999)) The surface of the domain is not specified via analytical patches, but via a 104 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (a) (b) (c) (d) (e) (f) Figure 3.61 F117: (a) CAD definition; (b) global cloud of points; (c) close-up of surface mesh; (d) cloud of points; (e), (f) cuts at x = 0, 120; (g), (h) cuts at x = 120 (detail) and x = 190 triangulation Figure 3.57(a) shows the maximum intensity projection of the medical images used as a starting point to generate the arterial surfaces The (discrete) surface defining the outline of the CFD domain is shown in Figure 3.57(b) On this surface, the surface 105 GRID GENERATION (g) (h) Figure 3.61 Continued triangulation for a subsequent CFD run is generated (Figure 3.57(c)) The final mesh had approximately 3.1 million tetrahedra 3.14.4 GENERIC SUBMARINE BODY This example demonstrates the use of boundary-layer gridding techniques A RANS grid for the generic DARPA-2 appended submarine model is considered The triangulation of the surfaces on which RANS grids are required had 27 389 faces The semi-structured grid generated from the surface triangulation consisted of 150 333 elements Application of the removal criteria reduced this number to 682 401 elements The fully unstructured grid generator then introduced another 441 040 elements The surface of the final grid is given in Figures 3.58(a) and (b) A close-up of the semi-structured region is shown in Figure 3.58(c), where one can again see the smooth transition between semi-structured and unstructured grid regions 3.14.5 AHMED CAR BODY This example demonstrates the use of boundary layer gridding techniques, as well as tetrahedra obtained from Cartesian cores for the field The geometry and discretization used are shown in Figure 3.59 One can clearly see the RANS grid, the adaptively refined Cartesian cores of the flow domain, as well as the transition to an unstructured grid between these two zones The number of elements in the RANS region is nelns=1,302K, in the Cartesian core region necrt=820K and in the remaining unstructured grid region nelun=415K 3.14.6 TRUCK This example shows the generation of quad-shells on a truck configuration The surface definition is shown in Figure 3.60(a) Based on this surface definition, a triangulation with elements of approximately twice the length as that desired for the final mesh is generated (Figure 3.60(b)) As many triangles as possible are merged into acceptable quads, and the 106 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (a) (b) (c) (d) Figure 3.62 (a), (b) Hopper filled with beans; (c), (d) hopper filled with ellipsoids mesh is smoothed (Figure 3.60(c)) One level of refinement yields the desired all-quad mesh, which is displayed in Figures 3.60(d) and (e) As one can see, the majority of quads are of very good quality 3.14.7 POINT CLOUD FOR F117 A cloud of points for the aerodynamic simulation of inviscid, transonic flow past (half) an F117 fighter via FPM (Löhner et al (2002)) is considered next The point density was specified through the combination of a background grid and background sources The surface triangulation consisted of approximately 50 000 points and 100 000 triangles The advancing front point generator added another 200 000 points using simple stencil placement Figure 3.61, taken from Lưhner and ate (2004), shows the CAD definition of the computational domain, the global cloud of points, a close-up of the surface mesh, the cloud of points close to the plane, as well as some slices through the volume The spatial variation of point density is clearly visible The complete generation process (boundary 107 GRID GENERATION triangulation, volume fill, output of files, etc.) took 44 s on a PC with Intel P4 chip running at 3.2 GHz, Gbyte Ram, a Linux OS and an Intel Compiler 3.14.8 HOPPER FILLED WITH BEANS/ELLIPSOIDS Granular materials that require simulation include grains, ground stone, woodchips and many other materials (Cundall and Stack (1979), Cleary (1998)) Bridging in silos and hoppers can cause severe structural damage and has always been a concern Figures 3.62(a) and (b) show a hopper configuration with bean-like objects composed of four spheres each The total number of beans is 2124, i.e 8496 spheres, and took s to generate Figures 3.62(c) and (d) show the same configuration filled with ellipsoidal objects composed of five spheres each The total number of ellipsoids is 2794, i.e 13 970 spheres, and took 10 s to generate Figure 3.63 Cube filled with spheres of different sizes 3.14.9 CUBE FILLED WITH SPHERES OF DIFFERENT SIZES Concrete and other aggregate materials often exhibit spherical or ellipsoidal objects of different sizes The number of objects can be recorded in a statistical distribution of size versus the number of particles Figure 3.63 shows a case for concrete: three different average sphere sizes, each on average half the size of the previous one and each with a standard (Gaussian) variation of 10%, are used to fill the cube shown The total number of spheres is 3958 4 APPROXIMATION THEORY The ideas of approximation theory are at the root of any discretization scheme, be it finite difference, finite volume, finite element, spectral element or boundary element Whenever a PDE is solved numerically, the exact solution is being approximated by some known set of functions How this is accomplished is described in the next two chapters The present chapter is devoted to the basic notions in approximation theory Throughout the book the Einstein summation convention will be used Every time an index appears twice in an equation, the summation over all possible values of the index is assumed 4.1 The basic problem Simply stated, the basic problem is the following Given a function u(x) in the domain , approximate u(x) by known functions N i (x) and a set of free parameters : u(x) ≈ uh (x) = N i (x)ai (4.1) The situation has been sketched in Figure 4.1 u uh u x Figure 4.1 Approximation of functions Examples of approximations that are often employed are: - truncated Taylor series u(x) ≈ uh (x) = a0 + a1 x + a2 x + · · · + am x m ⇒ aj = dj u j ! dxj ; (4.2a) x=0 - Fourier expansions (e.g truncated sine series) u(x) ≈ uh (x) = aj sin j πx ⇒ aj = L L L u(x) sin j πx dx; L Applied Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods, Second Edition Rainald Lưhner © 2008 John Wiley & Sons, Ltd ISBN: 978-0-470-51907-3 (4.2b) 110 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES - Legendre polynomials; - Hermite polynomials, etc In general, a complete set of trial functions N j is chosen: u(x) ≈ uh (x) = N j aj ; j = 1, 2, , m (4.3) An open question that remains to be answered is how best to choose the unknown constants in (4.1) and (4.3) Several possibilities are considered in what follows 4.1.1 POINT FITTING Here uh = u is enforced at m selected points in space: uh |xk = u|xk , k = 1, 2, , m ⇒ N i (xk )aj = u|xk (4.4) This is by far the simplest choice, and has been sketched in Figure 4.2 The set of m conditions will lead to a linear system of coupled equations which may be represented in matrix notation as K · a = f (4.5) u eh uh u x xk Figure 4.2 Point fitting 4.1.2 WEIGHTED RESIDUAL METHODS Weighted residual methods (WRMs) are a more general class of methods, and we define the error h = u − uh , and require h (x) → 0, x∈ (4.6) Obviously h does not vanish in most of the domain In order for h to be as small as possible, it must be penalized, or weighted, in some rational way The easiest way to accomplish this is by introducing a set of weighting functions W i , i = 1, 2, , m The requirement given by (4.6) is restated as Wi h d = 0, i = 1, 2, , m (4.7) 111 APPROXIMATION THEORY Then, given a complete set of trial and weighting functions, as m → ∞ the error satisfies h → for all points in Inserting the expression for uh yields W i (u − N j aj ) d = (4.8) As before, the m integrals obtained for the different W i -functions will lead to a linear system of coupled equations of the form K · a = r, (4.9a) where K ij = WiNj d , ri = Wiu d (4.9b,c) An important remark is that the choice of W i defines the method Two examples are included here to illustrate this (a) Point collocation: obtained by choosing W i = δ(x − xi ), xi ∈ (4.10) The WR statement becomes δ(x − xi ) h d = h (xi ) = 0, i = 1, 2, , m, (4.11) or N j (xi )aj = u(xi ) (4.12) Observe that this is the same as point fitting (b) Galerkin method: obtained by choosing Wi = Ni (4.13) The WR statement becomes Ni NiNj d h d = aj = N i (u − N j aj ) d Niu d , = 0, i = 1, 2, , m, i = 1, 2, , m, (4.14a) (4.14b) or Mc · a = r (4.14c) The matrix Mc is the so-called consistent mass matrix Observe that simply summing up the rows into the diagonal or ‘lumping’ of Mc does not lead to point fitting, as r contains the spatial variation in the integrals The Galerkin choice of W i = N i not only yields a symmetric matrix for Mc , but also has some important properties which will be discussed in what follows ... (3 .54 ) and the total thickness of the semi-structured grid region is δ= Typical values for c are 1. 10 < c < 1. 30 cn +1 − y0 c? ?1 (3 .55 ) 90 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES 3 .13 ... Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods, Second Edition Rainald Lưhner © 2008 John Wiley & Sons, Ltd ISBN: 97 8-0 -4 7 0 -5 19 0 7-3 (4.2b) 11 0 APPLIED COMPUTATIONAL. .. approximately 11 0 sources 10 1 GRID GENERATION (a) (b) (c) Figure 3 .58 DARPA-2 model: (a), (b) surface triangulation; (c) cross-sections in mesh 10 2 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES