Applied Computational Fluid Dynamics Techniques - Wiley Episode 1 Part 5 ppsx

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 follo

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 thepoints 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

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 beenidentified and marked

The test for overlapping elements can account for a large portion of the overall CPUrequirement Therefore, several filtering and marking strategies have to be implemented tospeed 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 elementshave been removed This leaves a series of prismatic columns, which go from the surfacetriangle 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 asnot requiring any further crossing tests In order to perform this test, we subdivide theprismatic columns into three tetrahedra as before, and use the usual element crossing testsfor 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 drasticreduction of CPU requirements

Marking of surfaces

Typically, the elements of a surface segment or patch will not overlap This is because, in mostinstances, the number of faces created on each of these patches is relatively large, and/or thepatches themselves are relatively smooth surfaces The main areas where overlap can occurare corners or ‘coalescing fronts’ For both cases, in the majority of the cases encountered inpractice, 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 andconcavity from the surface triangulation Then discrete ‘patches’ can be associated withthe discretized surface using, e.g., a neighbour-to-neighbour marching algorithm If theassumption of surface patch smoothness and/or convexity can be made, then it is clear thatonly the elements (and points) that originated from a surface patch other than the one thatgave rise to the element currently being examined need to be tested In this way, a largenumber 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 thanthe one currently being tested is too close, the element is marked for deletion The proximitytest is carried out by computing the smallest distance between the close point and the fourvertices of the element being tested If this distance is less than approximately the smallestside length of the element, the element is marked for deletion Evaluating four distances isvery 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 considerablycheaper (by more than an order of magnitude) than testing whether the elements surroundingthis point cross the element Therefore, all the close points are subjected to this test beforeproceeding

Top element in prism test

The most likely candidate for element crossing of any given triplet of elements that form aprism 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 (andexpensive) 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 gridspacings 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 TECHNIQUESinstances, avoid the element crossing test for the elements closest to wetted surfaces Fortypical 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 mostapplications, 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 stillhave 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 elementlie 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 structured region are likely to cross boundary faces As this test is performed after theelement crossing tests are conducted, the only boundaries which need to be treated are thosethat have no semi-structured grid attached to them In order to detect whether overlappingoccurs, a loop is performed over the surface faces, seeing if any element crosses it As

semi-before, straightforward testing would result in an expensive O(N el · N f )procedure, where

N f denotes the number of boundary faces By using quad/octrees, this complexity can be

reduced to O(N f log N el ) 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-Reynolds-number flows, let us consider the case of the flat plate The friction coefficient c

f, givenby

u ,y=α2

v∞

x Re

1+β

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,

u ,y≈u1− u0

y =v∞

y =α2

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

one can estimate the number of layers required to assure a smooth transition to the isotropic

mesh outside the boundary layer region Denoting by h i the element size in the isotropic

mesh region, the number of layers n required is given by

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 ofarbitrary 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 withthese objects so that they are close but do not overlap in an automatic way

Several techniques have been used to place these objects in space The so-called ‘fill andexpand’ or ‘popcorn’ technique (Sakaguchi and Murakami (2002)) starts by first generating acoarse mesh for the volume to be filled This subdivision of the volume into large, simplepolyhedra (elements) is, in most cases, performed with hexahedra The objects required(points, spheres, ellipsoids, polyhedra, etc.) are then placed randomly in each of theseelements These are then expanded in size until contact occurs or the desired fill ratio has beenachieved An obvious drawback of this technique is the requirement of a mesh generator toinitiate 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, theobjects are packed as close as required during introduction Depending on how the objects areintroduced, one can mimic gravitational or magnetic deposition, layer growing or size-basedgrowth Furthermore, so-called radius growing can be achieved by first generating a coarsecloud of objects, and then growing more objects around each of these In this way, one cansimulate granules or stone

In the present section, we consider the direct generation of clouds of arbitrary objects withthe same degree of flexibility as advanced unstructured mesh generators The mean distancebetween objects (or, equivalently, the material density) is specified by means of backgroundgrids, sources and density attached to CAD entities In order not to generate objects outsidethe computational domain, an initial triangulation of the surface that is compatible with thedesired mean distance between objects specified by the user is generated Starting from thisinitial ‘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 generateelements, the present scheme removes one object at a time, attempting to introduce as manyobjects as possible in its immediate neighbourhood

3.13.1 THE ADVANCING FRONT SPACE-FILLING ALGORITHM

- An initial triangulation of the surface, with the face normals pointing towards the

interior of the domain to be filled with points

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:

- 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 ofioout;

- finding the boundary faces in the neighbourhood ofioout

These three search operations are performed efficiently using heap lists, octrees and linkedlists, respectively

3.13.2 POINT/OBJECT PLACEMENT STENCILS

A number of different stencils may be contemplated Each one of these corresponds to aparticular space-filling point/object configuration The simplest possible stencil is the onethat 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 frontalgorithm, will fill a 3-D volume completely Other Cartesian stencils, which include nearest-neighbours with distances√

92 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES

Domain Boundary Initial Discretization of Boundary

Active Front Object/Point Surrounded/Inactive Object/Point

Figure 3.48 Advancing front space-filling with ellipses

)

Figure 3.49 Point stencils: (a) Cartesian [6]; (b) Cartesian [18]; (c) Cartesian [26]; (d) tetrahedral [17];

(e) random

to place new objects close to an existing one For the generation of points and spheres, it wasfound that the six-point stencil leads to the smallest amount of rejections and unnecessarytesting (Löhner and Oñate (1998)).

In many instances, it is advisable to generate ‘body conforming’ clouds of points inthe vicinity of surfaces In particular, Finite Point Methods (FPMs) or Smoothed ParticleHydrodynamics (SPH) techniques may require these ‘boundary layer point distributions’.Such point distributions can be achieved by evaluating the average point-normals for theinitial triangulation When creating new points, the stencil used is rotated in the direction ofthe normal The newly created points inherit the normal from the pointiooutthey originatedfrom

3.13.3 BOUNDARY CONSISTENCY CHECKS

A crucial requirement for a general space-filling object generator is the ability to onlygenerate objects in the computational domain desired Assuming that the object to be removedfrom the list of active objectsionewis inside the domain, a new objectionewwill crossthe boundary triangulation if it lies on the other side of the plane formed by any of the facesthat are in the proximity ofionewand can seeionew This criterion is made more stringent

by introducing a tolerated closeness or tolerated distance d t of new objects to the exteriorfaces of the domain Testing for boundary consistency is then carried out using the followingalgorithm (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 d n from ionew to this face;

if: d n < 0: ionew lies outside the domain

⇒ reject ionew and exit;

elseif: d n > d t: ionew if far enough from the faces

⇒ proceed to the next close face;

elseif: 0 ≤ d n ≤ d t:

obtain the closest distance dmin of ionew to this face;

if: dmin< d t: ionew is too close to the boundary

⇒ reject ionew and exit;

endif

- enddo

Typical values for d t are 0.707d0≤ d t ≤ 0.9d0, where d0denotes the desired mean averagedistance between points

3.13.4 MAXIMUM COMPACTION TECHNIQUES

For SPH and finite point applications, the use of stencils is sufficient to ensure a properspace filling (discretization) of the computational domain However, many applications thatconsider 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

n

: ionew : ioout

3.13.4.1 Closest object placement

Closest object placement attempts to position new objects as close as possible to existingones (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 theactive front are then obtained This does not represent any substantial increase in effort, as theexisting objects in the vicinity are required anyhow for closeness/penetration testing Startingfrom the three closest objects, an attempt is made to place the new object in such a way that itcontacts all of them, does not penetrate any other objects and resides inside the computationaldomain If this is not possible, an attempt is made with the two closest objects Should thisalso prove impossible, an attempt is made to move the new objects towards the object beingremoved from the active front If this attempt is also unsuccessful, the usual stencil position

Figure 3.51 Closest object placement: (a) move to three neighbours; (b) move to two neighbours;

(c) move to one neighbour

3.13.4.2 Move/enlarge post-processing

Whereas closest object placement is performed while space is being filled with objects, processing attempts to enlarge and/or move the objects in such a way that a higher volumeratio of objects is obtained, and more contacts with nearest-neighbours are established Theprocedure, shown schematically in Figure 3.52, may be summarized as follows:

post 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 ionewpenetrates other objects or

the boundary: revert to original size;

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 thanthe radius for spherical objects), the spatial search data structures (bins, octrees) requiredduring 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 elementmethod/discrete particle method datasets

96 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES3.13.5 ARBITRARY OBJECTS

The most time-consuming part of the present technique is given by the penetration/closenesschecks These checks are particularly simple for spheres However, for arbitrary objects, theCPU 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 toapproximate arbitrary objects by a collection of spheres (see Figure 3.53) When adding a newobject in space, the penetration/closeness checks are carried out for all spheres comprisingthe object The new object is only added if all spheres pass the required tests (Löhner andOñate (2004))

etc.

a) b) c)

e) d)

Figure 3.53 Arbitrary objects as a collection of spheres: (a) sphere; (b) tube; (c) ellipsoid; (d)

tetrahe-dron; (e) cube

(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 techniquecan be applied if magnetic fields are present Layered growth can be obtained be assigning a