Geometry modeling and grid generation
Geometry Modeling & Grid GenerationME469B/2/GI1Geometry Modeling & Grid Generation? Geometry definition (simple shapes, CAD import) ? Grid generation algorithms ? GAMBIT ? Grid quality and improvement ? AutomationAcknowledgements: Fluent Inc. Gambit User Manual S. Owen: Introduction to unstructured mesh generationME469B/2/GI 2Simulation Process2 31. Build CAD Model2. Mesh3. Apply Boundary Conditions4. Computational AnalysisME469B/2/GI5. Visualization3Adaptive Simulation ProcessUser supplies meshing parameters2 31. Build CAD M odel2. M esh3. Apply Boundary ConditionsAdaptivity LoopError?Error & εAnalysis Code supplies meshing parameters 6. Remesh/Refine/Improve4. Computational Analysis5. Error EstimationError & εME469B/2/GI7. Visualization4GeometryMesh GenerationGeometry EngineME469B/2/GI5Grid generation package: GAMBITFile I/O Defaults Grid FormatSpecial Geometry Grid BC ToolsGraphics WindowGeometry Tools Volume ToolsText WindowVisualization ToolsME469B/2/GI6GAMBITgambit -id &namefile& Interactive execution with GUI gambit -inp &journalfile& Batch execution without GUI Batch and GUI execution are EXACTLY equivalent! 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)ME469B/2/GI7Geometryvertices: x,y,z locationME469B/2/GI8Geometryvertices: x,y,z location curves: bounded by two verticesME469B/2/GI9Geometryvertices: x,y,z location curves: bounded by two vertices surfaces: closed set of curvesME469B/2/GI10Geometryvertices: x,y,z location curves: bounded by two vertices surfaces: closed set of curvesME469B/2/GIvolumes: closed set of surfaces11Geometryvertices: x,y,z location curves: bounded by two vertices surfaces: closed set of curvesME469B/2/GIvolumes: closed set of surfacesgroup: collection of volumes12GeometryManifold Geometry: Each volume maintains its own set of unique surfacesVolume 2Surface 11 Volume 1 Volume 2 Surface 7Surface 8Surface 9Surface 10Surface 11Volume 1Surface 1Surface 2Surface 3Surface 4Surface 5Surface 6Surface 7ME469B/2/GI13GeometryNon-Manifold Geometry: Volumes share matching surfacesVolume 1 Volume 2 Surface 7Volume 2Surface 8Surface 9Surface 10Volume 1Surface 1Surface 2Surface 3Surface 4Surface 5Surface 6Surface 7ME469B/2/GI14Geometry & TopologyGeometry 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 elementME469B/2/GI 15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 entitiesME469B/2/GI 16Vertex:? Input Coordinates…Edge:? 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)ME469B/2/GI 17Edge by NURBSNon-Uniform Rational B-SplinesGeneralization of Bezier interpolants: each point is computed as the weighted sum 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.htmME469B/2/GI 18Face:? Rectangular ? Circular ?… ? Sweep (translation or rotation of an edge) ? Wireframe (connecting 3 or more edges)ME469B/2/GI19Volume:? Cuboid ? Sphere ? Cone ? Pyramids ?… ? Sweep (translation or rotation of a face) ? Wireframe (connecting 3 or more faces)ME469B/2/GI20Manipulate Geometry - Boolean Operations:Volume 1? Unite ? Substract ? IntersectVolume 2it generates the intersection edgeME469B/2/GI21Manipulate Geometry - Blendsmooth sharp edgesME469B/2/GI 22Create Entities - FacesME469B/2/GI23Create Entities - FacesSome entities generated using “primitives” have fewer lower topological entities Example: Cube volume: 6 faces, 12 edges, 8 vertices Cylinder volume: 3 faces, 2 edges, 2 verticesME469B/2/GI 24Manipulate Geometry C Create Entities Create a vertex on a faceParametric representation of the faceME469B/2/GI25Manipulate Geometry C Create Entities Create two edges by splitting an edgeParametric representation of the edgeME469B/2/GI26Manipulate Geometry C Scaling Geometrical scaling of a volume (isotropic)Scaling is based on a Reference Point Default (0,0,0) - origin of the original Cartesian coordinate system It is possible to introduce additional coordinate systemsME469B/2/GI 27Manipulate Geometry C Align Modify the geometry of an entity with reference to another oneME469B/2/GI28Connect Geometry Building upper topological entities from the lower ones requires that they are properly connectedInconsistent connections Consistent connectionsME469B/2/GI 29Import Geometries? Realistic geometries are TOO complicated to be generated from “simple” shapes ? Engineering design is based on CAD systemsTranslation between CAD and CFD system is a major bottleneck ? Gambit is based on ACIS geometrical libraries ? ACIS (Andy, Charles & Ian’s System) is the most widely used 3D modeling software technology () ? It can also import: ? STEP (STandard for Exchange of P ISO standard) ? IGES (Initial Graphics Exchange S ANSI standard) ? STL (STereo L Rapid Prototyping Standard) ? ….ME469B/2/GI 30Import GeometriesME469B/2/GI31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 partsME469B/2/GI32Example: Helicopter RotorThe rotor-shaft connection is VERY complicatedME469B/2/GI 33Geometrical entitiesThe geometry consists of 10 components 3 blades, 1 shaft support, 6 connectorsME469B/2/GI 34Example: Import IGES ModelIGES export is available from every CAD systemIGES models are a collection of “untrimmed” edges and facesImported geometry consists of: 0 volumes ~250 faces ~1100 edges ~1000 verticesME469B/2/GI 35Example: Import STEP ModelSTEP export is available from many CAD systemSTEP models are a collection of parts or componentsImported geometry consists of: 10 volumes ~190 faces ~450 edges ~300 verticesME469B/2/GI 36Components “exploded”supportconnector-1bladeconnector-2 For aerodynamic analysis the details of the rotor-shaft connection are not important. The geometry of the blades MUST be preservedME469B/2/GI 37Geometry simplificationConnectors eliminatedSupport generated as a “simple” cylinder with blended sideME469B/2/GI38Geometry simplificationBlade edge cleaned and sealedBlade-support connector is a “simple” cuboidME469B/2/GI39Clean GeometryThis example is available on the class Web siteME469B/2/GI 40Virtual GeometryGAMBIT operates on two different type of entities REAL: with corresponding geometrical and topological characteristics VIRTUAL: defined only with reference to REAL or other VIRTUAL entities REAL entities are what we used
VIRTUAL are used to SIMPLIFY, CLEAN UP, DECOMPOSE real entitiesNote that some geometry tools cannot be applied to virtual entities (boolean operation, volume blending, creation of volumes by sweeping faces, etc.)ME469B/2/GI41Virtual GeometrySuperset: Entity that references two or more real entities Interpolant: Entity represents an average/interpolant ot various real entities Parasite: Entity de?ned completely from a real entity The virtual geometry is typically constructed using a host entityReal entities can be transformed in virtual but NOT ALWAYS viceversaME469B/2/GI42Virtual Geometry Clean-UpExample of edge connecting operation: virtual interpolantREALME469B/2/GIVIRTUAL43Clean-Up CracksA &crack& is de?ned as a geometry consisting of an edge pair that 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.ME469B/2/GI44Clean-Up Hard EdgesHard edges (dangling edges) are those that are included in the list of edges that de?ne a face but which do not constitute necessary parts of the closed edge loop that circumscribes the face. Such edges often result from face-split operations in which the splittool face only partially intersects the target face.ME469B/2/GI45Grid Generation? Geometry definition (simple shapes, CAD import) ? Grid generation algorithms ? GAMBIT ? Grid quality and improvement ? AutomationME469B/2/GI46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 2DME469B/2/GI47Grid generation techniquesFrom S. Owen, 2005ME469B/2/GI48Grid generation techniquesGambit is a “commercial” grid generator and includes only few (relatively standard) algorithms. New methods are slow to gain robustness and generality and therefore are not directly available Cubit is a “research” grid generator and the latest approaches are typically included (several of them have been actually invented by the Cubit team)http://cubit.sandia.gov/ME469B/2/GI 49Grid generation techniquesFrom S. Owen, 2005ME469B/2/GI50Structured Grids: Mapping Transfinite InterpolationSide Bj=1=NJ Side A1 Side A2Side A2Side Bj=NJ=1 k=NK Side C k=1Side DSide A1Physical DomainS id eCkjSide DComputational DomainME469B/2/GI51Structured Grids: Sub-mapping Regions are automatically subdivided in “mappable areas”Number of grid elements have to be chosen consistentlyThe grid type is controlled by a vertex attributeME469B/2/GI52Vertex-face type User can specify the behavior of the grid at a certain nodeME469B/2/GI53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 integrityME469B/2/GI54DelaunaycircumcircleME469B/2/GIEmpty Circle (Sphere) Property: No other vertex is contained within the circumcircle (circumsphere) of any triangle (tetrahedron)55DelaunayME469B/2/GIDelaunay Triangulation: Obeys empty-circle (sphere) property56DelaunayNon-Delaunay TriangulationME469B/2/GI 57DelaunayGiven a Delaunay Triangulation of n nodes, How do I insert node n+1 ?XLawson Algorithm ?Locate triangle containing X ?Subdivide triangle ?Recursively check adjoining triangles to ensure emptycircle property. Swap diagonal if needed ?(Lawson,77)ME469B/2/GI 58DelaunayXLawson Algorithm ?Locate triangle containing X ?Subdivide triangle ?Recursively check adjoining triangles to ensure emptycircle property. Swap diagonal if needed ?(Lawson,77)ME469B/2/GI 59DelaunayBowyer-Watson Algorithm ?Locate triangle that contains the point ?Search for all triangles whose circumcircle contain the point (d&r) ?Delete the triangles (creating a void in the mesh) ?Form new triangles from the new point and the void boundary ?(Watson,81)rX d cGiven a Delaunay Triangulation of n nodes, How do I insert node n+1 ?ME469B/2/GI 60DelaunayBowyer-Watson Algorithm ?Locate triangle that contains the point ?Search for all triangles whose circumcircle contain the point (d&r) ?Delete the triangles (creating a void in the mesh) ?Form new triangles from the new point and the void boundary ?(Watson,81)XGiven a Delaunay Triangulation of n nodes, How do I insert node n+1 ?ME469B/2/GI 61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
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 qualityME469B/2/GI62Advancing FrontCAB?Begin with boundary mesh - define as initial front ?For each edge (face) on front, locate ideal node C based on front ABME469B/2/GI 63Advancing FrontrCDA?Determine if any other nodes on current front are within search radius r of ideal location C (Choose D instead of C)ME469B/2/GIB64Advancing FrontD?Book-Keeping: New front edges added and deleted from front as triangles are formed ?Continue until no front edges remain on frontME469B/2/GI 65Advancing Front?Book-Keeping: New front edges added and deleted from front as triangles are formed ?Continue until no front edges remain on frontME469B/2/GI 66Advancing Front?Book-Keeping: New front edges added and deleted from front as triangles are formed ?Continue until no front edges remain on frontME469B/2/GI 67Advancing Front?Book-Keeping: New front edges added and deleted from front as triangles are formed ?Continue until no front edges remain on frontME469B/2/GI 68Advancing FrontrCA?Where multiple choices are available, use best quality (closest shape to equilateral) ?Reject any that would intersect existing front ?Reject any inverted triangles (|AB X AC| & 0) ?(Lohner,88;96)(Lo,91)BME469B/2/GI69Advancing FrontRemarkable high-quality gridME469B/2/GI 70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 surfacesME469B/2/GI71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) ME469B/2/GI72Unstructured Grids: Paving? Advancing front technique based on quads (instead of triangles) ? Only in 2DTriangulationME469B/2/GIPaving73Unstructured-QuadPaving?Advancing Front: Begins with front at boundary ?Forms rows of elements based on front angles ?Must have even number of intervals for all-quad meshME469B/2/GI(Blacker,92)(Cass,96)74Unstructured-QuadPaving?Advancing Front: Begins with front at boundary ?Forms rows of elements based on front angles ?Must have even number of intervals for all-quad meshME469B/2/GI(Blacker,92)(Cass,96)75Unstructured-QuadPaving?Advancing Front: Begins with front at boundary ?Forms rows of elements based on front angles ?Must have even number of intervals for all-quad meshME469B/2/GI(Blacker,92)(Cass,96)76Unstructured-QuadForm new row and check for overlapPaving?Advancing Front: Begins with front at boundary ?Forms rows of elements based on front angles ?Must have even number of intervals for all-quad meshME469B/2/GI(Blacker,92)(Cass,96)77Unstructured-QuadInsert “Wedge ”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 meshME469B/2/GI(Blacker,92)(Cass,96)78Unstructured-QuadSeamsPaving?Advancing Front: Begins with front at boundary ?Forms rows of elements based on front angles ?Must have even number of intervals for all-quad meshME469B/2/GI(Blacker,92)(Cass,96)79Unstructured-QuadClose Loops and smoothPaving?Advancing Front: Begins with front at boundary ?Forms rows of elements based on front angles ?Must have even number of intervals for all-quad meshME469B/2/GI(Blacker,92)(Cass,96)80Unstructured-QuadReproduces an uniform meshME469B/2/GI 81Unstructured-QuadReproduces an uniform mesh…almost. But it allows flexibility in the edge meshingME469B/2/GI 82Unstructured Quad-to-TriME469B/2/GI83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 eshME469B/2/GI84Coopering/SweepingME469B/2/GISweepingGeometry Requirements ?source and target surfaces topologicaly similar ?linking surfaces mapable or submapable85Coopering/Sweepinglinking surfacestargetsourceGeometry Requirements ?source and target surfaces topologicaly similar ?linking surfaces mapable or submapable86ME469B/2/GISweepingCoopering/SweepingME469B/2/GISweepingGeometry Requirements ?source and target surfaces topologicaly similar ?linking surfaces mapable or submapable87Coopering/SweepingME469B/2/GISweepingGeometry Requirements ?source and target surfaces topologicaly similar ?linking surfaces mapable or submapable88Coopering/SweepingME469B/2/GISweepingGeometry Requirements ?source and target surfaces topologicaly similar ?linking surfaces mapable or submapable89Coopering/SweepingME469B/2/GISweepingGeometry Requirements ?source and target surfaces topologicaly similar ?linking surfaces mapable or submapable90Coopering/SweepingME469B/2/GISweepingGeometry Requirements ?source and target surfaces topologicaly similar ?linking surfaces mapable or submapable91Coopering/SweepingME469B/2/GISweepingGeometry Requirements ?source and target surfaces topologicaly similar ?linking surfaces mapable or submapable92Unstructured Grids: 3D elements? Standard Elements:HexTetPyramidWedge? ? ? ? ?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 & HexME469B/2/GI93Unstructured Grids: Hex or Tets?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)35K Total Elements410K Total ElementsME469B/2/GICross-Section of the Swirler94Quality & ControlWhat is available in GAMBITRobustnessComplex Geometry + + -Speed? Structured gridding (mapping)+++? Unstructured triangulation (2D/3D)-++/-? Unstructured paving (2D)--+/-+-+/-? Unstructured coopering (3D)All GAMBIT meshes are exported as unstructured collection of (mixed) elementsME469B/2/GI 95Mesh sizes + + +Grid generation C 1D - EdgesStraightforward Select number of points Select distribution of points Edge direction is defined from 1st to 2nd vertex Clustering toward one side is defined accordinglyME469B/2/GI96Grid generation C 2D - FacesEasy Select number of points ? use predefined edge meshes ? use uniform spacing Select meshing scheme ? constraints on the edge meshing for mapping and paving schemesME469B/2/GI97Grid generation C 2D - FacesIt is possible to force the cell element type at face-verticeselements forced to be trianglesMixing element-type is one of the main advantages of unstructured mesh technologyME469B/2/GI98Grid generation C 2D - Mesh-patching optionsMatching interfaceNon-conformal interfaceOverlapping interfaceME469B/2/GIMixed-element interface99Grid generation C 3D - VolumesNot 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 schemesME469B/2/GI1003D Grid generation C Advanced Cooper technique“Creative” way of coopering: multisurface to multisurface sweepME469B/2/GI 101Grid generation C Sizing functionsInstead of the bottom-up approach (1D to 3D) grid generation Sizing functions can be specified to mesh volumes directlyME469B/2/GI102Grid generation C Clustering pointsSizing functions can be used effectively to define the size of the cells BUT they cannot provide directional control (anisotropy) One option is to build (grow) elements from the boundaries and to form “viscous” layersME469B/2/GI103Grid generation C Boundary LayersME469B/2/GI104Example C meshing a circleMapping Triangulation Paving Multiblock mappingBoundary Layer PavingBoundary Layer PavingBoundary Layer Multiblock PavingBoundary Layer Transition Multiblock PavingCircle defined as segments ME469B/2/GI 105Mesh linkingEdges, faces and volume meshes can be linked Define corresponding entities and ALSO reference entities Needed to “enforce” coincident grids on different entities (i.e. for periodicity bc)ME469B/2/GI106Grid qualityQuality measures are NOT absolute but should be considered in connection with solution schemes The final accuracy of a procedure is ALWAYS a function of the grid qualitySeveral 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 elementsME469B/2/GI107Quality measures available in GAMBITME469B/2/GI108Examine meshes? Define mesh element to examine ? Define a cutting plane ? Define the quality measureAspect ratioME469B/2/GIEquiangular skew109Mesh improvement? Smoothing operators are applied to redistribute nodesAspect ratioEquiangular skewTypically, improvement in 3D meshes are based on improved 2D meshesME469B/2/GI 110Mesh improvement/smoothingP4 P P3Averaging Methods P5P2 P1(Field, 1988) ME469B/2/GILaplacian111Mesh improvement/smoothingP4 P3 PCentroid of attached nodes Can create inverted elementsAveraging Methods P5P2 P1(Field, 1988) ME469B/2/GILaplacianP = i =1 n! Pi112nMesh improvement/smoothingP4Averaging Methods P P5P1Centroid of attached nodes Can create inverted elementsP3P6ME469B/2/GILaplacianP = i =1 n P2! Pi113nMesh improvement/smoothingP4Averaging Methods P5P1 P P6Centroid of attached nodes Can create inverted elementsP3ME469B/2/GILaplacianP = i =1 n P2! Pi114nMesh improvement/smoothingP4Averaging Methods P P5Weighted Laplacian: Do not move node unless minimum distortion metric is improved Centroid of attached nodes Can create inverted elementsP1P3P6ME469B/2/GILaplacianP = i =1 n P2! Pi115nMesh improvement/smoothingAveraging Methods C4 P C1C4 C3Weighted average of triangle centroidsP = i =1 n! AiCii =1nA1C2 A2! AiAi=area of triangle i Ci=centroid of triangle i116ME469B/2/GIArea Centroid Weighted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 ParametersME469B/2/GI117GAMBIT Journal fileThe command:Volume create width 1 depth 1 height 1 offset 0 0 0 brickGenerates a cube of size 1 centered at 0 0 0 On the other hand the sequence$W = 2.3 $D = 1.5 $H = 4 Volume create width $W depth $D height $H offset 0 0 0 brickGenerates a cuboid of User-Specified Size centered at 0 0 0ME469B/2/GI118GAMBIT Journal fileIt is possible to perform operations on input parameters$SUM = $A + 0.5*$BMath. functions are available:$SUM = SIN($A)In addition, geometrical operations$X = INTERSECTING(volume, “volume.5”, “volume.12”) $X = BBOX(&volume.3&) $X = GETNORMAL(&face.3&, 1, 13, 89) (…)ME469B/2/GI119GAMBIT Journal fileConditional statements:if cond ($A .eq. 5) volume create sphere radius ($A+3) endifLoops:$Z = 0 do para &$Z& init 6 cond ($Z .le. 24) incr ($Tmp*3) volume create sphere radius $Z EnddoRelation and logical operators :.eq. .le. .lt. (…) .and. .or.ME469B/2/GI120Example of Journal fileME469B/2/GI/ -------------------------------------------------------/ 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 / /121Grid generation research? Structured grids: automatic generationof mappable subdomains NLR
Report (PDF available from class web site)? Unstructured tetrahedral grids: anisotropic Delaunay schemesShimada et al. “High quality anisotropic tetrahedral mesh generation via ellipsoidal bubble packing” (PDF available)ME469B/2/GI122Grid generation research? Unstructured hex-dominant grids: OCTREE basedSAMM C Computational Dynamics Ltd. Hexpress C Numeca International Inc.? Unstructured purely hexahedral grids: Whisker-WeavingCUBIT C Sandia National Lab. (PS report available)ME469B/2/GI123Grid generation using Medial AxisMedial Axis?Medial Object - Roll a Maximal circle or sphere through the model. The center traces the medial object ?Medial Object used as a tool to automatically decompose model into simpler mapable or sweepable parts (Price, 95;97)(Tam,91)ME469B/2/GI 124Grid generation using Medial AxisMedial Axis?Medial Object - Roll a Maximal circle or sphere through the model. The center traces the medial object ?Medial Object used as a tool to automatically decompose model into simpler mapable or sweepable parts (Price, 95;97)(Tam,91)ME469B/2/GI 125Grid generation using Medial AxisMedial Axis?Medial Object - Roll a Maximal circle or sphere through the model. The center traces the medial object ?Medial Object used as a tool to automatically decompose model into simpler mapable or sweepable parts (Price, 95;97)(Tam,91)ME469B/2/GI 126Grid generation using Medial AxisMedial Axis?Medial Object - Roll a Maximal circle or sphere through the model. The center traces the medial object ?Medial Object used as a tool to automatically decompose model into simpler mapable or sweepable parts (Price, 95;97)(Tam,91)ME469B/2/GI 127Grid generation using Medial AxisMedial Axis?Medial Object - Roll a Maximal circle or sphere through the model. The center traces the medial object ?Medial Object used as a tool to automatically decompose model into simpler mapable or sweepable parts (Price, 95;97)(Tam,91)ME469B/2/GI 128Grid generation using Medial AxisMedial Axis?Medial Object - Roll a Maximal circle or sphere through the model. The center traces the medial object ?Medial Object used as a tool to automatically decompose model into simpler mapable or sweepable parts (Price, 95;97)(Tam,91)ME469B/2/GI 129Grid generation using Medial AxisMedial Axis?Medial Object - Roll a Maximal circle or sphere through the model. The center traces the medial object ?Medial Object used as a tool to automatically decompose model into simpler mapable or sweepable parts (Price, 95;97)(Tam,91)ME469B/2/GI 130Grid generation using Medial Axis3D Medial Object examples (from FEGS website, URL: http://www.fegs.co.uk/medial.html)ME469B/2/GI 131Grid generation C Links and References? Links ? Mesh generation and grid generation on the Web rmatik.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.? References:? Handbook of grid generation. Thompson, Soni, Weatherill, CRC Press ? Numerical Grid Generation: Foundation & Applications. Thompson, Warsi, Mastin. North Holland PressME469B/2/GI 132