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

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