geometry modeling & grid generation
TRANSCRIPT
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ME469B/2/GI 1
Geometry Modeling &
Grid Generation
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ME469B/2/GI 2
Geometry Modeling & Grid Generation
• Geometry definition (simple shapes, CAD import)• Grid generation algorithms• GAMBIT• Grid quality and improvement• Automation
Acknowledgements: Fluent Inc. Gambit User ManualS. Owen: Introduction to unstructured mesh generation
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ME469B/2/GI 3
Simulation Process
3
2
1. Build CAD Model 2. Mesh 3. Apply BoundaryConditions
4. Computational Analysis 5. Visualization
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ME469B/2/GI 4
Adaptive Simulation Process
3
2
1. Build CAD Model 2. Mesh 3. Apply Boundary Conditions
4. Computational Analysis
7. Visualization
5. Error Estimation
Error?
6. Remesh/Refine/Improve
Adaptivity Loop
Error < ε
Error > ε
Usersupplies meshing
parameters
Analysis Codesupplies meshing
parameters
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Geometry
Mesh Generation
Geometry Engine
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Grid generation package: GAMBIT
Geometry Grid BC Tools
GeometryTools
VolumeTools
VisualizationTools
GraphicsWindow
Special
TextWindow
File I/O Defaults GridFormat
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GAMBIT
gambit -id <namefile>Interactive execution with GUI
gambit -inp <journalfile>Batch execution without GUI
Geometry & Grid are saved in a database file (*.dbs)The mesh is saved into a solver-dependent file (*.msh)
At the end of each session Gambit automatically saves a journal file (*.jou)
Batch and GUI execution are EXACTLY equivalent!
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ME469B/2/GI 8
Geometry
vertices:x,y,z location
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vertices:x,y,z location
curves: boundedby two vertices
Geometry
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ME469B/2/GI 10
vertices:x,y,z location
surfaces: closedset of curves
curves: boundedby two vertices
Geometry
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ME469B/2/GI 11
vertices:x,y,z location
surfaces: closedset of curves
volumes: closedset of surfaces
curves: boundedby two vertices
Geometry
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ME469B/2/GI 12
group: collectionof volumes
vertices:x,y,z location
surfaces: closedset of curves
volumes: closedset of surfaces
curves: boundedby two vertices
Geometry
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ME469B/2/GI 13
Volume 1
Surface 1 Surface 2 Surface 3 Surface 4 Surface 5 Surface 6
Volume 2
Surface 8 Surface 9 Surface 10 Surface 11
Surface 7
Volume 1
Volume 2
Surface 11
Surface 7
Manifold Geometry:Each volume maintainsits own set of uniquesurfaces
Geometry
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ME469B/2/GI 14
Volume 1
Surface 1 Surface 2 Surface 3 Surface 4 Surface 5 Surface 6
Volume 2
Surface 8 Surface 9 Surface 10
Surface 7
Volume 1
Volume 2
Surface 7
Non-ManifoldGeometry: Volumesshare matching surfaces
Geometry
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Geometry & Topology
Geometry types in Gambit
• Real Geometry: entities characterized by a direct definition of their geometry
example: a vertex defined by its coordinates (0,0,0)
• Virtual Geometry: entities characterized ONLY by an indirect definition, i.e. a reference to another entity.
example: a vertex is defined as the mid-point of an edge
• Faceted Geometry : entities characterized ONLY by an indirect definition with respect to an underlying grid
example: a vertex is defined as the corner of a mesh element
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Geometrical types - Topology
• Vertex
• Edge (2 or more vertices) • Face (3 or more edges)
• Volume (4 or more faces)
Bottom-up approach: generate low dimensional entities and build on top of them higher dimensional entities
Top-bottom approach: generate upper dimensional entities and use boolean operation to define the other entities
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Vertex:
Edge:
• Input Coordinates…
• Line segment (connect 2 vertices)• Circular arc• Quadratic functions• NURBS: Non-Uniform Rational B-Splines (connect N vertices)
Topologically any edge is ALWAYS a connection between 2 vertices(additional vertices used to build the geometry are NOT part of the edge)
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Edge by NURBS
Non-Uniform Rational B-Splines
Generalization of Bezier interpolants: each point is computed as the weightedsum of all the knot points
NURBS can use various blending/control functions (for the weights)Can achieve high degree of continuity
http://www.ibiblio.org/e-notes/Splines/NURBS.htm
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Face:
• Rectangular• Circular• …• Sweep (translation or rotation of an edge)• Wireframe (connecting 3 or more edges)
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Volume:
• Cuboid• Sphere• Cone• Pyramids• …• Sweep (translation or rotation of a face)• Wireframe (connecting 3 or more faces)
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ME469B/2/GI 21
Manipulate Geometry - Boolean Operations:
• Unite• Substract• Intersect
Volume 1
Volume 2
it generates theintersection edge
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ME469B/2/GI 22
Manipulate Geometry - Blend
smooth sharp edges
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ME469B/2/GI 23
Create Entities - Faces
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Create Entities - Faces
Some entities generated using “primitives” have fewer lower topological entitiesExample:
Cube volume: 6 faces, 12 edges, 8 verticesCylinder volume: 3 faces, 2 edges, 2 vertices
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Manipulate Geometry – Create Entities
Parametric representationof the face
Create a vertex on a face
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Manipulate Geometry – Create Entities
Create two edges by splitting an edge
Parametric representationof the edge
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ME469B/2/GI 27
Manipulate Geometry – Scaling
Geometrical scaling of a volume (isotropic)
Scaling is based on a Reference PointDefault (0,0,0) - origin of the original Cartesian coordinate systemIt is possible to introduce additional coordinate systems
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Manipulate Geometry – Align
Modify the geometry of an entity with referenceto another one
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Connect Geometry
Building upper topological entities from the lower onesrequires that they are properly connected
Inconsistentconnections
Consistentconnections
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ME469B/2/GI 30
Import Geometries
• Realistic geometries are TOO complicated to be generated from “simple” shapes• Engineering design is based on CAD systems
• Gambit is based on ACIS geometrical libraries• ACIS (Andy, Charles & Ian’s System) is the most widely used 3D modeling software technology (http://www.spatial.com)
• It can also import:• STEP (STandard for Exchange of Product model data; ISO standard)• IGES (Initial Graphics Exchange Specification; ANSI standard)• STL (STereo Lithography; Rapid Prototyping Standard)• ….
Translation between CAD and CFD system is a major bottleneck
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Import Geometries
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Clean-Up a CAD Model
• Eliminate components not exposed to the flow
• Eliminate duplicated entities
• Eliminate small details
• Water-proofing the surfaces
• Rebuild geometrical connectivity between parts
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Example: Helicopter Rotor
The rotor-shaft connectionis VERY complicated
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Geometrical entities
The geometry consists of 10 components3 blades, 1 shaft support, 6 connectors
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Example: Import IGES Model
IGES export is available from everyCAD system
IGES models are a collection of“untrimmed” edges and faces
Imported geometry consists of:0 volumes~250 faces~1100 edges~1000 vertices
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Example: Import STEP Model
STEP export is available from manyCAD system
STEP models are a collection of partsor components
Imported geometry consists of:10 volumes~190 faces~450 edges~300 vertices
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Components “exploded”
For aerodynamic analysis the details of the rotor-shaftconnection are not important. The geometry of theblades MUST be preserved
blade
supportconnector-1
connector-2
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Geometry simplification
Connectors eliminated
Support generated as a “simple” cylinder with blended side
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Geometry simplification
Blade edge cleaned andsealed
Blade-support connectoris a “simple” cuboid
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ME469B/2/GI 40
Clean Geometry
This example is available on the class Web site
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ME469B/2/GI 41
Virtual Geometry
GAMBIT operates on two different type of entities
REAL: with corresponding geometrical and topological characteristicsVIRTUAL: defined only with reference to REAL or other VIRTUAL entities
REAL entities are what we used and described so far;VIRTUAL are used to SIMPLIFY, CLEAN UP, DECOMPOSE real entities
Note that some geometry tools cannot be applied to virtual entities(boolean operation, volume blending, creation of volumes by sweeping faces, etc.)
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Superset: Entity that references two or more real entitiesInterpolant: Entity represents an average/interpolant ot various real entitiesParasite: Entity defined completely from a real entity
The virtual geometry is typically constructed using a host entity
Virtual Geometry
Real entities can be transformed in virtual but NOT ALWAYS viceversa
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REAL VIRTUAL
Virtual Geometry Clean-Up
Example of edge connecting operation: virtual interpolant
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Clean-Up CracksA "crack" is defined as a geometry consisting of an edge pairthat meets the following criteria.
・Each edge in the pair serves as a boundary edge for a separate face.・The edges share common endpoint vertices at one or both ends.・The edges are separated along their lengths by a small gap.
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Hard edges (dangling edges) are those that are included in the list ofedges that define a face but which do not constitute necessary parts ofthe closed edge loop that circumscribes the face.
Such edges often result from face-split operations in which the split-tool face only partially intersects the target face.
Clean-Up Hard Edges
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Grid Generation
• Geometry definition (simple shapes, CAD import)• Grid generation algorithms• GAMBIT• Grid quality and improvement• Automation
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ME469B/2/GI 47
Grid generation techniques
• Structured grids• Ordered set of (locally orthogonal) lines• Several Techniques can be used to Map a computational domain into a physical domain: Transfinite Interpolation, Morphing, PDE Based, etc.• The grid lines are curved to fit the shape of the boundaries
• Unstructured grids• Unorganized collection of polygons (polyhedron) • Three main techniques are available to generate automatically triangles (tetrahedra): Delaunay triangulation, Advancing front, OCTREE• Paving for automatic generation of quads in 2D
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Grid generation techniques From S. Owen, 2005
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Grid generation techniques
http://cubit.sandia.gov/
Gambit is a “commercial” grid generator and includes only few (relativelystandard) algorithms. New methods are slow to gain robustness andgenerality and therefore are not directly available
Cubit is a “research” grid generator and the latest approaches are typicallyincluded (several of them have been actually invented by the Cubit team)
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Grid generation techniques From S. Owen, 2005
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k
j
j=NJ=1j=1=NJ
k=1
k=NK
Physical Domain Computational Domain
Side
A1
Sid
e A
2
Side
C
Side B
Side D
Structured Grids: Mapping Si
de A
2
Side A1
Side C
Side B
Side
D
Transfinite Interpolation
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Structured Grids: Sub-mapping
Regions are automatically subdivided in “mappable areas”
Number of grid elementshave to be chosen consistently
The grid type is controlledby a vertex attribute
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Vertex-face type
User can specify the behavior of the grid at a certain node
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Unstructured Grids: Triangulations
• Delaunay• Empty circle principle: any node must not be contained within the circumcircle (circle passing through the vertices of a triangle) on any triangle within the mesh• Automatic triangulation of random set of nodes• Nodes are inserted locally in a triangulation and triangles are redefined locally to satisfy the Delaunay criterion (available mathematical tools)
+ Inherent grid quality+ Elegant mathematical basis - Boundary integrity
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Delaunay
circumcircle
Empty Circle (Sphere) Property:No other vertex is contained within the circumcircle(circumsphere) of any triangle (tetrahedron)
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Delaunay Triangulation:Obeys empty-circle (sphere) property
Delaunay
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ME469B/2/GI 57Non-Delaunay Triangulation
Delaunay
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Lawson Algorithm•Locate triangle containing X•Subdivide triangle•Recursively check adjoiningtriangles to ensure empty-circle property. Swapdiagonal if needed•(Lawson,77)
X
Given a DelaunayTriangulation of n nodes,How do I insert node n+1 ?
Delaunay
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X
Lawson Algorithm•Locate triangle containing X•Subdivide triangle•Recursively check adjoiningtriangles to ensure empty-circle property. Swapdiagonal if needed•(Lawson,77)
Delaunay
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Bowyer-Watson Algorithm•Locate triangle that containsthe point•Search for all triangleswhose circumcircle containthe point (d<r)•Delete the triangles (creatinga void in the mesh)•Form new triangles from thenew point and the voidboundary•(Watson,81)
X
r cd
Given a DelaunayTriangulation of n nodes,How do I insert node n+1 ?
Delaunay
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X
Bowyer-Watson Algorithm•Locate triangle that containsthe point•Search for all triangleswhose circumcircle containthe point (d<r)•Delete the triangles (creatinga void in the mesh)•Form new triangles from thenew point and the voidboundary•(Watson,81)
Given a DelaunayTriangulation of n nodes,How do I insert node n+1 ?
Delaunay
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Unstructured Grids: Triangulations
• Advancing front• Triangles are built inward from the boundary surfaces• The last layer of elements constitutes the active front • An optimal location for a new nodes is generated for each segment on the front; the new node is generated by checking all existing nodes and this new optimal location • Intersection checks are required to avoid front overlap
+ Surface grid preserved+ Specialized layers near surfaces - Computationally complex - Low quality
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Advancing Front
A B
C
•Begin with boundary mesh - define as initial front•For each edge (face) on front, locate ideal node C based on front AB
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Advancing Front
A B
Cr
•Determine if any other nodes on current front are within searchradius r of ideal location C (Choose D instead of C)
D
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Advancing Front
•Book-Keeping: New front edges added and deleted from front astriangles are formed•Continue until no front edges remain on front
D
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Advancing Front
•Book-Keeping: New front edges added and deleted from front astriangles are formed•Continue until no front edges remain on front
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ME469B/2/GI 67
Advancing Front
•Book-Keeping: New front edges added and deleted from front astriangles are formed•Continue until no front edges remain on front
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ME469B/2/GI 68
Advancing Front
•Book-Keeping: New front edges added and deleted from front astriangles are formed•Continue until no front edges remain on front
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ME469B/2/GI 69
Advancing Front
A
B
C
•Where multiple choices are available, use best quality (closestshape to equilateral)•Reject any that would intersect existing front•Reject any inverted triangles (|AB X AC| > 0)•(Lohner,88;96)(Lo,91)
r
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ME469B/2/GI 70
Advancing Front
Remarkable high-quality grid
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ME469B/2/GI 71
Unstructured Grids: Triangulations
• OCTREE• Squares containing the boundaries are recursively subdivided until desired resolution is obtained• Irregular cells (or triangulation) are generated near the surface where square intersect the boundary
+ Requires least of surface representation+ Highly automated - Cannot match surface grid - Low quality near surfaces
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ME469B/2/GI 72
Octree/Quadtree
•Define intial bounding box (root of quadtree)•Recursively break into 4 leaves per root to resolve geometry•Find intersections of leaves with geometry boundary•Mesh each leaf using corners, side nodes and intersections with geometry•Delete Outside•(Yerry and Shephard, 84), (Shepherd and Georges, 91)
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ME469B/2/GI 73
Unstructured Grids: Paving
• Advancing front technique based on quads (instead of triangles)• Only in 2D
Triangulation Paving
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ME469B/2/GI 74
Unstructured-Quad
Paving•Advancing Front: Begins with front at boundary•Forms rows of elements based on front angles•Must have even number of intervals for all-quad mesh
(Blacker,92)(Cass,96)
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ME469B/2/GI 75
Unstructured-Quad
Paving•Advancing Front: Begins with front at boundary•Forms rows of elements based on front angles•Must have even number of intervals for all-quad mesh
(Blacker,92)(Cass,96)
![Page 76: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/76.jpg)
ME469B/2/GI 76
Unstructured-Quad
Paving•Advancing Front: Begins with front at boundary•Forms rows of elements based on front angles•Must have even number of intervals for all-quad mesh
(Blacker,92)(Cass,96)
![Page 77: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/77.jpg)
ME469B/2/GI 77
Unstructured-Quad
Paving•Advancing Front: Begins with front at boundary•Forms rows of elements based on front angles•Must have even number of intervals for all-quad mesh
Form new rowand check for
overlap
(Blacker,92)(Cass,96)
![Page 78: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/78.jpg)
ME469B/2/GI 78
Unstructured-Quad
Paving•Advancing Front: Begins with front at boundary•Forms rows of elements based on front angles•Must have even number of intervals for all-quad mesh
Insert“Wedge
”
(Blacker,92)(Cass,96)
![Page 79: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/79.jpg)
ME469B/2/GI 79
Unstructured-Quad
Paving•Advancing Front: Begins with front at boundary•Forms rows of elements based on front angles•Must have even number of intervals for all-quad mesh
Seams
(Blacker,92)(Cass,96)
![Page 80: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/80.jpg)
ME469B/2/GI 80
Unstructured-Quad
Paving•Advancing Front: Begins with front at boundary•Forms rows of elements based on front angles•Must have even number of intervals for all-quad mesh
CloseLoops and
smooth
(Blacker,92)(Cass,96)
![Page 81: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/81.jpg)
ME469B/2/GI 81
Unstructured-Quad
Reproduces an uniform mesh
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ME469B/2/GI 82
Unstructured-Quad
Reproduces an uniform mesh…almost. But itallows flexibility in the edge meshing
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ME469B/2/GI 83
Unstructured Quad-to-Tri
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ME469B/2/GI 84
Unstructured Grids: Coopering
• 2D mesh sweeping• Only for cylindrical volumes• unstructured surface mesh is generated on surface A (source face)
• structured grids are generated on cylindrical surfaces C & D
• mesh on surface A is sweeped in the volume to generate the full 3D m esh
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ME469B/2/GI 85Sweeping
Geometry Requirements•source and target surfacestopologicaly similar•linking surfaces mapableor submapable
Coopering/Sweeping
![Page 86: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/86.jpg)
ME469B/2/GI 86
source
target
Sweeping
Geometry Requirements•source and target surfacestopologicaly similar•linking surfaces mapableor submapable
linkingsurfaces
Coopering/Sweeping
![Page 87: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/87.jpg)
ME469B/2/GI 87Sweeping
Geometry Requirements•source and target surfacestopologicaly similar•linking surfaces mapableor submapable
Coopering/Sweeping
![Page 88: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/88.jpg)
ME469B/2/GI 88Sweeping
Geometry Requirements•source and target surfacestopologicaly similar•linking surfaces mapableor submapable
Coopering/Sweeping
![Page 89: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/89.jpg)
ME469B/2/GI 89Sweeping
Geometry Requirements•source and target surfacestopologicaly similar•linking surfaces mapableor submapable
Coopering/Sweeping
![Page 90: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/90.jpg)
ME469B/2/GI 90Sweeping
Geometry Requirements•source and target surfacestopologicaly similar•linking surfaces mapableor submapable
Coopering/Sweeping
![Page 91: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/91.jpg)
ME469B/2/GI 91Sweeping
Geometry Requirements•source and target surfacestopologicaly similar•linking surfaces mapableor submapable
Coopering/Sweeping
![Page 92: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/92.jpg)
ME469B/2/GI 92Sweeping
Geometry Requirements•source and target surfacestopologicaly similar•linking surfaces mapableor submapable
Coopering/Sweeping
![Page 93: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/93.jpg)
ME469B/2/GI 93
Standard Elements:
Hex Tet Pyramid Wedge
Hex: Maximum Volume Covered per Edge Size Hex: Maximum Ratio Nodes/Elements Hex/Wedges: Clustering at Solid Wall with High Quality Elements Tets: Automatic Meshing of Extremely Complicated Regions Pyramids/Wedges: Transition Between Tets & Hex
Unstructured Grids: 3D elements
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ME469B/2/GI 94
35K Total Elements 410K Total Elements
We NEED Hex-Based Meshing because: Equiangular Tets are NOT Good for Thin Volumes Too Many Elements for Reasonable Resolutions (estimated >2M grid Points in conical-annular Swirler)
Cross-Section of the Swirler
Unstructured Grids: Hex or Tets?
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ME469B/2/GI 95
What is available in GAMBIT
• Structured gridding (mapping)
• Unstructured triangulation (2D/3D)
• Unstructured paving (2D)
• Unstructured coopering (3D)
All GAMBIT meshes are exported as unstructured collection of (mixed) elements
+-+/--+
+++/---
-++/-+-
+-+++
Spee
d
Rob
ustn
ess
Qua
lity
& C
ontro
l
Com
plex
Geo
met
ry
Mes
h si
zes
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ME469B/2/GI 96
Grid generation – 1D - Edges
Straightforward
Select number of pointsSelect distribution of points
Edge direction is defined from 1st to 2nd vertexClustering toward one side is defined accordingly
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ME469B/2/GI 97
Grid generation – 2D - Faces
Easy
Select number of points• use predefined edge meshes• use uniform spacing
Select meshing scheme• constraints on the edge meshing for mapping and paving schemes
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ME469B/2/GI 98
Grid generation – 2D - Faces
It is possible to force the cell element typeat face-vertices
Mixing element-type is one of the main advantages of unstructured meshtechnology
elementsforced tobe triangles
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ME469B/2/GI 99
Grid generation – 2D - Mesh-patching options
Matching interface Non-conformal interface
Overlapping interface Mixed-element interface
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ME469B/2/GI 100
Grid generation – 3D - Volumes
Not so easy
Select number of points• use predefined face meshes• use uniform spacing
Select meshing scheme• constraints on the face meshing for mapping and cooper schemes
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ME469B/2/GI 101
3D Grid generation – Advanced Cooper technique
“Creative” way of coopering: multisurface to multisurface sweep
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ME469B/2/GI 102
Grid generation – Sizing functions
Instead of the bottom-up approach (1D to 3D) grid generationSizing functions can be specified to mesh volumes directly
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ME469B/2/GI 103
Grid generation – Clustering points
Sizing functions can be used effectively to define the size of thecells BUT they cannot provide directional control (anisotropy)
One option is to build (grow) elements from the boundaries andto form “viscous” layers
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ME469B/2/GI 104
Grid generation – Boundary Layers
![Page 105: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/105.jpg)
ME469B/2/GI 105
Example – meshing a circle
Mapping Triangulation Paving Multiblock mapping
Boundary Layer Boundary Layer Boundary Layer Boundary Layer Transition Paving Paving Multiblock Paving Multiblock Paving
Circle defined as segments
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ME469B/2/GI 106
Mesh linking
Edges, faces and volume meshescan be linked
Define corresponding entitiesand ALSO reference entities
Needed to “enforce” coincidentgrids on different entities (i.e.for periodicity bc)
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ME469B/2/GI 107
Grid quality
Quality measures are NOT absolute but should be considered inconnection with solution schemes
The final accuracy of a procedure is ALWAYS a function of the gridquality
Several geometrical measures can be defined:
• Depending on the size of the elements• Depending on the shape of the elements• Depending on relative dimensions of neighboring elements
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ME469B/2/GI 108
Quality measures available in GAMBIT
![Page 109: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/109.jpg)
ME469B/2/GI 109
Examine meshes
Equiangular skewAspect ratio
• Define mesh element to examine• Define a cutting plane • Define the quality measure
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ME469B/2/GI 110
Mesh improvement
• Smoothing operators are applied to redistribute nodes
Equiangular skewAspect ratio
Typically, improvement in 3D meshes are based on improved 2D meshes
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ME469B/2/GI 111
P1
P2
P3
P4
P5
P
Laplacian
AveragingMethods
(Field, 1988)
Mesh improvement/smoothing
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ME469B/2/GI 112
P1
P2
P3
P4
P5P
Laplaciann
n
i
i!== 1
P
P
AveragingMethods
Centroid ofattached nodes
Can createinverted elements
(Field, 1988)
Mesh improvement/smoothing
![Page 113: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/113.jpg)
ME469B/2/GI 113
P1
P2
P3
P4
P5
P
Laplaciann
n
i
i!== 1
P
P
AveragingMethods
Centroid ofattached nodes
Can createinverted elements
P6
Mesh improvement/smoothing
![Page 114: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/114.jpg)
ME469B/2/GI 114
P6 P2
P3
P4
P5
P
Laplaciann
n
i
i!== 1
P
P
AveragingMethods
Centroid ofattached nodes
Can createinverted elements
P1
Mesh improvement/smoothing
![Page 115: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/115.jpg)
ME469B/2/GI 115
P1
P2
P3
P4
P5
P
Laplaciann
n
i
i!== 1
P
P
AveragingMethods
Centroid ofattached nodes
Can createinverted elements
P6
Weighted Laplacian: Do notmove node unless minimumdistortion metric is improved
Mesh improvement/smoothing
![Page 116: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/116.jpg)
ME469B/2/GI 116
C1
P
Area Centroid Weighted
AveragingMethods
Weightedaverage oftriangle centroids
C2
C3
C4C4
A1
A2
!
!
=
==n
i
i
n
i
ii
A
A
1
1
C
P
Ai=area of triangle i
Ci=centroid of triangle i
Mesh improvement/smoothing
![Page 117: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/117.jpg)
ME469B/2/GI 117
Grid Generation Automation
• GAMBIT saves a “journal” file with the commands issues during a session
• Journal file are ASCII editable files
• Commands are quasi-English and easy-to-use
• They are useful to trace-back sessions to find errors
• They can be made general by introducing User Defined Parameters
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ME469B/2/GI 118
GAMBIT Journal file
The command:
Volume create width 1 depth 1 height 1 offset 0 0 0 brick
Generates a cube of size 1 centered at 0 0 0
On the other hand the sequence
$W = 2.3$D = 1.5$H = 4Volume create width $W depth $D height $H offset 0 0 0 brick
Generates a cuboid of User-Specified Size centered at 0 0 0
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ME469B/2/GI 119
GAMBIT Journal file
It is possible to perform operations on input parameters
$SUM = $A + 0.5*$B
Math. functions are available:
$SUM = SIN($A)
In addition, geometrical operations
$X = INTERSECTING(volume, “volume.5”, “volume.12”)$X = BBOX("volume.3")$X = GETNORMAL("face.3", 1, 13, 89)(…)
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ME469B/2/GI 120
GAMBIT Journal file
Conditional statements:
if cond ($A .eq. 5) volume create sphere radius ($A+3)endif
Loops:
$Z = 0do para "$Z" init 6 cond ($Z .le. 24) incr ($Tmp*3)
volume create sphere radius $ZEnddo
Relation and logical operators :
.eq. .le. .lt. (…)
.and. .or.
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ME469B/2/GI 121
/ --------------------------------------------------------/ CYCLONE GRID GENERATION/ ME269B - Spring 2002/ --------------------------------------------------------// R1 = External radius of Cyclone/ R2 = Gas Outlet Pipe (External)/ R3 = Gas Outlet Pipe (Internal)/ RB = Particles Outlet (Bottom)/ H1 = Height of the Cylindrical Part of Cyclone/ H2 = Height of the Conical Part/ HE = Depth of the Outlet Channel into the Cyclone// inletl = Length (x) of the Gas Inlet Channel/ inleta = Height (z) of the Gas Inlet Channel/ inletb = Span (y) of the Gas Inlet Channel/ outletl = Length (z) of the outlet (gas) pipe// cellsize = Average size of the cells// Remark: z-axis is the Cyclone Axis/ --------------------------------------------------------// Input Quantities/$R1 = 1.555$R2 = 0.45$R3 = 0.4$RB = 0.75$H1 = 4.5$H2 = 5$HE = 3.6$inletl = 2$inleta = 1.03$inletb = 0.7$outletl = 7.2$cellsize = 0.18//
Example of Journal file
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ME469B/2/GI 122
Grid generation research
Structured grids: automatic generation of mappable subdomains
NLR 2000-366 Report(PDF available from class web site)
Unstructured tetrahedral grids: anisotropic Delaunay schemesShimada et al. “High quality anisotropic tetrahedral mesh generationvia ellipsoidal bubble packing” (PDF available)
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ME469B/2/GI 123
Grid generation research
Unstructured hex-dominant grids: OCTREE basedSAMM – Computational Dynamics Ltd.Hexpress – Numeca International Inc.
Unstructured purely hexahedral grids: Whisker-WeavingCUBIT – Sandia National Lab. (PS report available)
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ME469B/2/GI 124
Medial Axis
(Price, 95;97)(Tam,91)
•Medial Object - Roll a Maximal circle or sphere through the model. Thecenter traces the medial object•Medial Object used as a tool to automatically decompose model intosimpler mapable or sweepable parts
Grid generation using Medial Axis
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ME469B/2/GI 125
Medial Axis•Medial Object - Roll a Maximal circle or sphere through the model. Thecenter traces the medial object•Medial Object used as a tool to automatically decompose model intosimpler mapable or sweepable parts (Price, 95;97)(Tam,91)
Grid generation using Medial Axis
![Page 126: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/126.jpg)
ME469B/2/GI 126
Medial Axis•Medial Object - Roll a Maximal circle or sphere through the model. Thecenter traces the medial object•Medial Object used as a tool to automatically decompose model intosimpler mapable or sweepable parts (Price, 95;97)(Tam,91)
Grid generation using Medial Axis
![Page 127: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/127.jpg)
ME469B/2/GI 127
Medial Axis•Medial Object - Roll a Maximal circle or sphere through the model. Thecenter traces the medial object•Medial Object used as a tool to automatically decompose model intosimpler mapable or sweepable parts (Price, 95;97)(Tam,91)
Grid generation using Medial Axis
![Page 128: Geometry Modeling & Grid Generation](https://reader033.vdocument.in/reader033/viewer/2022052313/584ab87b1a28ab85738bd08e/html5/thumbnails/128.jpg)
ME469B/2/GI 128
Medial Axis•Medial Object - Roll a Maximal circle or sphere through the model. Thecenter traces the medial object•Medial Object used as a tool to automatically decompose model intosimpler mapable or sweepable parts (Price, 95;97)(Tam,91)
Grid generation using Medial Axis
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ME469B/2/GI 129
Medial Axis•Medial Object - Roll a Maximal circle or sphere through the model. Thecenter traces the medial object•Medial Object used as a tool to automatically decompose model intosimpler mapable or sweepable parts (Price, 95;97)(Tam,91)
Grid generation using Medial Axis
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Medial Axis•Medial Object - Roll a Maximal circle or sphere through the model. Thecenter traces the medial object•Medial Object used as a tool to automatically decompose model intosimpler mapable or sweepable parts (Price, 95;97)(Tam,91)
Grid generation using Medial Axis
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3D Medial Object examples(from FEGS website, URL: http://www.fegs.co.uk/medial.html)
Grid generation using Medial Axis
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Grid generation – Links and References
• Links• Mesh generation and grid generation on the Web http://www-users.informatik.rwth-aachen.de/~roberts/meshgeneration.html• Meshing research corner http://www.andrew.cmu.edu/user/sowen/mesh.html• General CFD: Topic mesh generation http://www.cfd-online.com
• References:• Handbook of grid generation. Thompson, Soni, Weatherill, CRC Press• Numerical Grid Generation: Foundation & Applications. Thompson, Warsi, Mastin. North Holland Press