Applications of DEC:
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1 Applcatons of DEC: Flud Mechancs and Meshng Dscrete Dfferental Geometry Overvew Puttng DEC to good use Fluds, fluds, fluds geometrc nterpretaton of classcal models dscrete geometrc nterpretaton new geometry-based ntegraton technque Quadrangle Meshng dscrete noton of harmoncty practcal method to drectly create quads 2
2 Part I Computatonal Fluds wth DEC 3 Flud Models (I) Euler Equatons pressure momentum eq. velocty mass body eq. forces nvscd fluds (not vscous) ncompressble non-lnear PDE, wth lnear constrant 4
3 Flud Models (II) Naver-Stokes Equatons only change: vscosty coeffcent loss of total energy durng moton 5 Geometry of Fluds? Euler equatons seem clear advecton + dv-free projecton ad nfntum Stam s Stable Fluds do ths wonderfully well numerous follow-up work (Fedkw et al.) but what does t mean, geometrcally? total energy s rather unntutve s there a noton of momentum preservaton? Yes but of course, we need to massage the PDE so as to reveal the geometrc structure u t u u p 6
4 u Geometry Revealed t u u p Pressure dsappears when we take the curl: (vortcty) ( t) C ( t) u. dl vortcty measures the spn of a parcel vortcty s advected along the flow the crculaton around any closed loop s constant as t gets advected (by Stokes) known as Kelvn s theorem call t preserv. of angular momentum f you want 7 Geometry Revealed So we know: Integral of vortcty constant on advected sheet Addtonally, defnes u f we gnore complex topology for a moment because u s dvergence free! Vortcty s the only real varable here and Kelvn s s a defnng property (Naver-Stokes: loss along the way) 8
5 Towards a Proper Dscretzaton Doman dscretzaton = smplcal complex fluxes through faces for velocty ntrnsc (coordnate-free) and Euleran» remnscent of staggered grds net flux for dvergence what comes n must come out flux spn for vortcty Torque created on a paddle wheel vald for any grd 9 Enter Dscrete Exteror Calculus Need for proper lnk btw flux, vortcty, dv hopefully matchng dfferental counterparts to create a dscrete dfferental structure.e., structure-preservng dscretzaton Fortunately, that s DEC we know how to do all that, rght?? flux = 2-form dv = exteror dervatve of flux curl = of flux d 10
6 Dvergence Operator Smply d of 2-form summng face values of tets returnng values n tets pont-based scalar feld edge-based face-based cell-based scalar feld cell-based scalar feld face-based edge-based pont-based scalar feld 11 Curl Operator Curl requres gong to the dual from faces to dual edges frst then d (sum of dual edge values) then back onto prmal edges pont-based scalar feld edge-based face-based cell-based scalar feld cell-based scalar feld face-based edge-based pont-based scalar feld 12
7 Gradent Operator (for completeness) Wat, constant per tet, rght? (FEM 101) yes u 0 but can be stored as values on edges, as announced u du u u d e {, j} k ( u u 1 u j u ) e k u2 u 3 13 u Laplacan Operator u For Naver-Stokes, Laplacan needed from faces to faces d d d d t u p u Try t for 0-forms at home: you ll get the cot formula pont-based scalar feld edge-based face-based cell-based scalar feld cell-based scalar feld face-based edge-based pont-based scalar feld 14
8 Integratng Equatons of Moton We have all the computatonal set-up But how do we ntegrate the moton? Through preservng mportant structures? Crculaton/vortcty preservaton Crucal for vsual mpact volutes n smoke vortces n lquds 15 Dscrete Kelvn s Theorem Smple way to ntegrate Euler equatons: For each 1-smplex backtrack local loop n current velocty feld deduce new crculaton.e., new dscrete vortcty Fnd new velocty feld smple Posson equaton 16
9 Dscrete Kelvn s Theorem Guarantees crculaton preservaton for any dscrete loop! bg loop = unon of small ones even on curved spaces Dfference wth Stable Fluds? trace back ntegrals, not pont values 17 Results New method exact dscrete vortcty preservaton arbtrary smplcal meshes see also [Feldman et al. 05, Bargtel et al 06] everythng s ntrnsc basc operators very smple (super parse) great flows for small meshes! computatonally effcent even on coarse mesh no need for mllons of vortex partcles 18
10 Channel 19 Smokng Bunny 7k vertces, 32k tets; 0.45s per frame on PIV (3GHz) 20
11 Mergng Vortces 21 Move 22
12 Part II Quad Meshng wth DEC 23 Quadrangulatons Needed n CAGD, Reverse Engneerng Ubqutous (tensor-product nature) Modelng ansotropy/symmetres FEM, texture atlasng But global topology constrants A Varety of Requrements: Isotropy vs ansotropy Orthogonalty, Algnment Regularty, Szng 24
13 Quad Meshes: Reverse Engneerng For a local patch of quadrangulaton nduce natural (u,v) parametrzaton Edges: nteger-valued socurves of u/v Nce mesh square mesh n certan metrc Cauchy-Remann equatons usng language of dfferental form, ths s one-forms Thus, u and v are both harmonc (Laplacan=0) du and dv too! Cool, DEC seems perfect for that 25 Methods for Quadrangulatons Among many: clusterng/morse [Boer-Martn et al. 03, Carr et al. 06] curvature lnes [Allez et al. 03, Marnov/Kobbelt 05] socontours ldong et al. 04] two contnuous potentals (much) more robust than streamlnes perodc global param (PGP) [Ray et al. 06] Pbs: PGP non lnear + no real control What about usng dscrete forms? global conformal param [Gu/Yau 03] 26
14 Problems, Problems Solvng for two contnuous potentals (u,v) wth gradents felds satsfyng CR eqs Alas, sngulartes unavodable ether poles or lne sngularty T-junctons Can we fnd a better way? Only requrement: contnuty of 1-forms so we can actually use dscontnuous (u,v)! [Yyng Tong et al. 2006] 27 Dscontnuous Potentals Tweaked Laplacan contnuty of 1-form nduces: N - N + Smlarly, generate smooth felds modulo the jump! 28
15 Smple Example of Tweaked Once solnes of u and v are extracted: u (dscontnuous) v (contnuous) 29 Possble Dscontnutes Only three dfferent cases: only way to guarantee pure quads n the 3 cases, just a tweak of the Laplacan stll only a lnear system to solve! 30
16 Examples 31 Example: Pure-Quad Bunny 32
17 More Results 33 Advantages of DEC Foundatons of dscrete forms powerful good grasp on sngulartes control of rregular valences lnk wth cone sngulartes [Kharevych et al. 06] stll just a lnear system to solve no need for well-talored cuts provdes parameterzaton too! 34
18 Take-Home Message Don t Arbtrarly Dscretze! dscretze geometrc structures PDEs often hde these structures uncover the nature of the varables nvolved usually, natural locatons on mesh turn the crank wth some DEC tools. 35
19 Harmonc Form Bass wth DEC Closed manfold Compute homology bass 1 harmonc 1-form per generator Manfold wth m boundares m-1 addtonal normal harmonc 1-forms Tangental form bass dual to hom. gens Take dual of normal forms Compute perod matrx P, multply by P -1 VxV symmetrc lnear systems 37 Gradent Operator (for completeness) Wat, constant per tet, rght? (FEM 101) yes u 0 but can be stored as values on edges, as announced u du u u d e {, j} k ( u u 1 u j u ) e k u 2 u 3 38
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