Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
Generatzione di grigrila in ambito
industriale
Grid generation using Background
grid based Size Functions
Presented By
Erling Eklund <[email protected]>
Fluent France SA
Prepared By
Jin Zhu <[email protected]>
Fluent, Inc.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
Outline
1. Introduction
2. Geometry based size functions
3. Initialisation and generation
4. Examples
5. Meshed based size function
6. Related development
7. Conclusion
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• The traditional key aspects of a High quality
Industrial CFD Mesh
– Good cell quality
• Low skew
• Low aspect ratio (non-uniform flow)
– Sufficient mesh density to capture physics to
the prescribed accuracy
– Sufficient mesh resolution to capture geometry
curvature and proximity
– Mesh smoothness
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Size control methods:
– Size functions using background mesh.
– Premesh boundaries of the domain.
• Example: Using faceting
– Size control by spatial decomposition.
• Often structural meshing
– Direct size transfer from source entities.
• Example: adaptive meshing
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Example:
Circle pre-meshed with size 0.05,
Face meshed with size 1.0
Total of 5656 elements
5
Meshed with fixed size on circle
and controlled mesh growth of 1.2
from the circle
Total of 1950 elements
Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Size Function requirements
– General and versatile in size control for all
kinds of geometry, all meshing algorithms
and all types of element.
– Fast.
– Local geometric effects have to be able to
radiate, or influence size on a more nonlocal area.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
•
Terminology
– Source entities
Different size function requires different entities types:
• Fixed – vertex, edge, face, volume;
• Curvature – edges (2D), faces (3D);
• Proximity – faces, volumes (i.e. all faces of the given
volumes);
– Attached entities
• Can be any meshable entities (edges, faces, or volumes);
– Growth rate
• The pace to progress from initial size on source;
– Size limit
• Mesh size will stay unchanged once reach size limit.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
•
Fixed SF definitions:
–
•
constant start size on source + common parameters;
Curvature SF definitions
–
–
normal angle + common parameters;
independent of geometry size, purely based on
curvature.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
•
Proximity SF definitions:
–
Including:
• face to face proximity (3D proximity, left)
• edge to edge proximity of same face (2D proximity, right).
–
cells/gap + common parameters,
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
•
Size function initialization
– Initialization establishes the desired sizes
everywhere on the sources.
– Initialize fixed SF:
•
Mesh-size on source = start size.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
•
Initialize 3D curvature SF
–
–
–
Create facets for source faces so that the angle
between the normal vectors of any two adjacent
facets does not exceed given angle.
Global Control of the shape of curvature facets
might be needed
Mesh-size is computed for and stored to each
node of triangle facets:
mesh-size = 2*sin(def-α /2)/max-curvature
which is dependent only upon pre-defined angle
and local curvature.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
– A min-size tolerance is provided to prevent too small
size from very large curvature due to abnormal
geometry features.
– Later the size at each node is smoothed amongst all
nodes of facets on the same face.
n2 ’
n+1
n2
α
n1
n2 ”
n
α’
α
f1
n-1
f2
Splitting big facet if local
angle α>defined angle
local circle
90
90
o
α/2
Mesh size at a node=standard distance
between (n, n+1) (i.e. using def-α, instead of
locally varying angle).
Size = 2Rsin(α/2), R=1/curvature
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Initialize 2D curvature SF:
– Create coarse facets of source edges;
– Refine edge facets (i.e. bisect facet and then project
mid-point to edge) until the offset distance from
chord to arc segment is smaller than a distance
tolerance, or the angle change across the arc
segment is less than an angle tolerance.
mesh-size on source facet = 2*sin(angle/2)/mid-curvature.
– Mesh-size is computed for and stored to each edge
facet.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Example:
Meshed curvature edges with similar size
distributions to initialized edge segments
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Initialize proximity SF:
– Create coarse facets for source faces according to
some specified normal angle criteria
– Refine coarse facets to localize the influence of
proximity SF:
• Checking the distance between each facet and all other
opposing (i.e. visible) facets, and choose the closest one.
– Compute mesh size at each facet:
Mesh size = gap-distance/cells-per-gap.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Example:
influence area
influence area
gap-dist
f1
f2
(size-1)
max-edge
Unrefined facet in gap: identical
mesh size on a big facet
gap-dist
Split a facet from
its longest edge
max-edge (size-2)
(size-f)
split facet
Refined facets: gap
influence is localized:
size-1>size-f, size-2>size-f
Gap influence w.r.t. facets refinement at a gap
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
•
Size calculations from fixed SF
– Get closest projection from a given point to a source
–
–
–
entity;
Compute distance from given point to the projected
point;
Grow mesh size from source entity along distance to
given point;
Compare mesh sizes obtained from all sources and
take smallest one.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
•
Size from curvature SF:
– Get closest projection from a point to a closest facet on
source entity (an edge or a face);
– For 3D: interpolate the size at projected point from the 3
nodes of the closest facet;
– Compute distance from given point to projected point on
source entity;
– Grow mesh size along the computed distance;
– Compare mesh sizes obtained from all source entities
and take smallest one.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Size from 3D proximity SF
– Get closest projection from a given point to facets (sizeby-closeness);
– If the given point is within the gap, directly take the
mesh size from the chosen facet (i.e. uniform mesh
size inside and around gap);
– If the given point is outside the gap, compute distance
from given point to projection point, minus gapdistance, and grow mesh size by this distance (i.e.
radiated mesh size outside gap);
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
P
uniform size domain
growth distance
gap-distance
proximity facets
gap-distance
gap-distance
• Size from 3D proximity SF-cont.
– Grow the min-size of face from the min-size-facet to
given point (size-from-min);
– Compare two obtained sizes, take the smaller one;
– Compare mesh sizes obtained from all sources and
take the smallest one.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Size from 3D proximity SF-cont.
– Compare results using only size-by-closeness and
including size-from-min;
size-by-closeness only
size-by-closeness &
size-from-min
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Size from 2D proximity SF.
– Re-project the surface projection obtained in 3D case
to the closest edge segment of the same face;
– Grow mesh size of the closest edge segment from
surface projection to given point (size-by-closeness);
– Grow mesh size of the min-size-segment to given
point (size-from-min);
R2
P
d1
– Compare two obtained sizes,
R1
take the smaller one;
d2
– Compare with 3D proximity result,
and take the smaller one.
min-size-seg
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
•
Background grid initialization
–
–
Generate a united bounding box from all entities
having same size functions attachment.
Establish values at the B-grid corner nodes.
•
•
Assuming r is the distance from a source entity (i.e.
projection point) to a given point (here the grid node);
Find the distance region between two subsequent grown
distances in which the grid node of radius r locates:
rn-1 <= r <= rn
•
•
Get associated mesh sizes, sn-1 and sn, at rn-1 and rn;
Compute parameter α at the point between rn-1 and rn:
α = (r - rn-1)/(rn - rn-1)
•
Compute the mesh size at given node:
s=(1 – α )sn-1 + αsn
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
Mesh size on
source entity
s0
0
s = s × g n−1
n−1
1
2
step 1
step 2
s =s ×gn
0
n-1
n
p
a
1-a
rn-1
r
rn
Growing mesh size from source entity and
interpolating size for a given point
g = growth rate
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0
n
Mesh size
Progressive growth
Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
•
B-grid refinement and control
–
–
–
If a B-grid’s central mesh size difference between
average value and the value computed from SF is
larger than a user specified error percentage, the Bgrid will be refined;
If a B-grid contains source entities whose smallest
size is less than the minimum size at the 8 corner
points of the cell, the cell is refined;
The number of B-grid refinement levels could also be
user controlled, to avoid over refinement
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Use Interpolate mesh size at given location
•
The mesh size at any given point during meshing
process is interpolated through the chain of B-grid to
its leaf grids by tri-linear shape functions:
∑ NiSi
•
•
Si – mesh size at 8 corner points of the grid;
Ni – interpolation function for each corner point
which is defined as the functions of local coordinates
of the given point inside the leaf grid.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Example:
– Meshing the clown using a single curvature size
function
(a) Whole head
(b) Eyeball
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Example:
– Meshing the clown using a single curvature size
function
(c) Hat-tail
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Example:
– Use of 3D proximity SF in volume meshing
Meshing results
Geometry
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Example:
– Use of combined size functions
(a) Elliptical holes
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Example:
– Use of combined size functions
(b) Airfoil shape
geometry
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(c) Local mesh
Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
Why meshed size function?
• Problems encountered during the meshing processes
in using previously described size function
capabilities:
– Can not handle clustered source mesh sizes defined by
user’s manual manipulation;
– Alternatively, use of multiple fixed size functions for the
following application is both inconvenient and inaccurate.
l
k
j
a
i
b
c
h
d
e
f
g
6 different fixed SFs are needed to specify
size distributions for the airfoil face
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Inconsistency of size functions when attached
to connected geometries
Volume.1
Volume.2
Curvature and proximity SFs are attached to connected volumes, and
mesh size conflict occurs at their common face because proximity sf
dominates exterior volume.1 whereas curvature sf dominates interior
volume.2. The common face follows volume.1 and mismatches volume.2
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Two connected faces have mesh conflict on
their common edge, especially when heavy
interval count adjustment is required by some
mesh schemes, like map.
a
Source
edge
b
Noticeable size jump across the common edge
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Another similar case
Source
vertex
a
b
Mapped face mesh
Paved face mesh
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Unable to handle mesh creation from lower
topologies with imported mesh
A
Volume with imported
face mesh
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Solution
– Therefore, creating a mesh that is radiated in a
controlled manner from some pre-meshed boundaries
of the domain can be an efficient and supplemental
way of obtaining desired mesh transition and
gradation.
• Definition
– The definition of meshed size function is similar to
existing size functions,
• Source edges/faces and attachment entities;
• growth rate and size limit are required;
• does not need the parameter for initialization.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Flow chart, including Meshed size function
Geometry
Entities
•Geometry
Entities
Source
Attachment
SF Definition
SF•Definition
•Fixed
Fixed
Curvature
•Curvature
Proximity
•Proximity
•Initializations
Initializations
•B-Grid
BG GridGeneration
Generation
Evaluator
•MeshingTools
Tools
Meshing
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Meshed
•Proximity
Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
Initialization
• Initialization establishes the desired sizes everywhere
on the sources.
• The mesh SF directly uses the meshes on source
edge or source face as start size.
– For edge source, each element is converted to a segment
and the length of the element is stored to the segment and
represents the local mesh size on the source edge.
– For face source, each triangle element is converted to a
facet, and the average length of the 3 sides of the triangle
element is stored to the facet and represents the local mesh
size on the source face.
– If the source face has quad elements, each quad element is
split into two triangle elements each of which is converted as
above.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
Use of meshed size function
• When evaluating the mesh size at a point:
– For edge source: projecting the point to all edge segments
– For face source: projecting the point to the best facet of the
face.
– If the projection is valid, the mesh size stored in that edge
segment or face facet is taken as the start size and then
grown to the given point, according to specified growth rate.
– Choose the smallest size grown from all source entities of
the meshed SF.
– Choose the smallest size from all size functions.
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
Example 1
• Meshing results from connected elliptical cylinder:
– A mesh SF is created using the common face as source and
is attached to the interior volume;
– Prevented the size jump in the interior volume;
– The meshes of the interior volume are radiated nicely from
the common face, in spite of the varying mesh sizes on
common face.
Without mesh SF
With mesh SF
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
Example 2
• Updated meshing results from connected faces
– Defined a second mesh SF using the common edge ab as
source and attach it to the right face;
– The mesh SF dominates the whole domain of the right face;
– The tri/pave algorithm created smooth mesh transitions from
the common edge and across the whole right face, and the
adjustment to the mesh distributions on the common edge
no longer deteriorates the mesh quality on the right face.
a
Source
edge
b
Without mesh SF
With mesh SF
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
Example 3
• Similar updated results from introduction 5 ?
– No matter what scheme is used, the meshes on the right
face are grown in such a way that no sudden changes of
mesh size can be noticed;
– The whole meshes are smoothly radiated from the same
Source
vertex upper-left vertex without size jumping.
a
b
With mesh SF
Without mesh SF
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
Example 4
• Generating a volume mesh from imported face mesh
– Specify the face having imported mesh as source, and
volume to be meshed as attachment;
– The volume is meshed with desired radiation.
Existing mesh
Volume mesh
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Internal mesh patterns
Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
• Related development
– Meshed Size Function on Boundary layer cap
No SF.
9097 tets.
Max-skew 0.76
Meshed SF from bottom face
2943 tets.
Max-skew 0.92
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Meshed SF from BL cap
11868 tets.
Max-skew 0.75
Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
– An Example illustrating conflict:
• At the intersection of the pipe with the box, there is
a fillet with a radius of 0.1. After structural meshing
of the pipe, assign curvature and meshed size
function to the brick and tet mesh it.
• This produces a bad mesh – why?
Skewed elements without
blending. 56622 elements.
max skewness: 0.98
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
– How can size Function be used to improve
Quad paving :
• Quad paver technology does not work well in
situation where you have internal edge loops with
significantly different mesh density compared to
external edge loops
With Size Function
worst skew 0.6
Without size function
worst skew 0.8
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
– Faster Background grid generation
• Direct solution of n :
– Analytically summing the terms of the geometrical series
instead of an iterative method
• Define sources Sn = S0*gn
– Distances to source entities at a given point then is
• R0 = 0, R1 = S0*g, R2 = S0*g2 ….
– Knowing the Euclidean distance R from the source, we can
directly solve for the exponent
• Rn = S0 (gn-1)/(g-1)-S0
• n = ln( R(g-1)/S0+g ) / ln(g)
• Taking the integer part of n
– The Source at Sp can the be evaluated using linear
interpolation using R, Rn, Rn+1, Sn and Sn+1
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Tecniche di Generazione di Griglie per il Calcolo Scientifico 2005
Conclusions
• Automatic and smooth mesh distribution is essential
•
•
in producing high quality CFD results for all industry
sectors
Background grid based Size Functions (fixed,
curvature, proximity and meshed) and their
combinations handle virtually all geometric features,
and provide rapid evaluators for all element types in
different meshing tools.
The algorithms has been successfully tested on a
wide variety of models with excellent results.
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