Natural Neighbors and Voronoi Tessellations in Computational Mechanics

Transcription

1 Unversty of Calforna, Davs Natural Neghbors and Vorono Tessellatons n Computatonal Mechancs N. Sukumar Unversty of Calforna, Davs, USA Jgsaw Tessellatons Workshop Lorentz Center, Leden March 09, 2006

2 Collaborators and Contrbutors Numercal smulatons usng natural element method are courtesy of Professor Elas Cueto (Unversdad de Zaragoza, Span) and Dr. Mke Puso (LLNL) Fracture on Vorono networks (wth Professor John Bolander, UC Davs) Polygonal fnte elements and adaptve computatons on quadtrees (wth Alreza Tabarrae, UC Davs)

3 Vorono Tesellatons n Materals and Mechancs Polycrystallne alloy Fber-matrx composte Osteonal bone (Courtesy of Kumar, LLNL) (Bolander and S, PRB, 2004) (Martn and Burr, 1989)

4 Outlne Meshfree/Grdless Approxmaton Schemes Natural Neghbor (NN) Interpolants Fracture on Vorono Networks Polygonal Fnte Element Methods Closure and Outlook

5 Meshfree Approxmaton Schemes Polynomals and splnes Radal bass functons Convex (NN and MAXENT) Approxmants Least-squares and movng least squares

6 Polynomals and Fnte Elements n 1D and 2D node coordnate: x

7 Arbtrary Nodal Dscretzaton h u ( x) = φ ( x)u n = 1 shape (bass) functon

8 Desrable Propertes of Shape Functons Affne Combnaton: φ ( x) = 1, φx = x Convex combnaton: φ 0 ensures convergence Regularty: φ C (Ω) Pece-wse lnear on the boundary: conformty and for mposng essental boundary condtons C 0

9 Movng Least Squares (MLS) Approxmant ) ( ) ( ) ( x x x u T h a = p 1 u 8 u ) ( w 5 x (Lancaster & Salkauskas, 1982) T m T T n x x u x x x w J ],, [1,, ) ( ] ) ( ) ( )[ ( ] [ mn / 2 1 K = = = = p u a P W a p a a

10 Vorono Neghbors

11 Natural Neghbors and NN-Interpolants Convex hull p les outsde the crcumcrcles n green

12 Sbson Interpolant φ ( p) = A ( p) A( p) (Sbson, 1980)

13 Laplace Interpolant = j j p p p ) ( ) ( ) ( α α φ ) ( ) ( ) ( p h p s p α = (Chrst et al., Nuclear Physcs B, 1982; Belkov et al, 1997; Hyosh and Sughara, 1999)

14 Propertes Non-negatve and PU: Interpolate data: Lnear precson: Smoothness: φ LAP C 0 φ 1, φ (x) = φ (x ) = δ φx = j 0 j x ( Ω), φ S C 1 1 ( Ω \ x j ) Lnear essental boundary condtons can be exactly mposed

15 Lnear Precson (Laplace Interpolant) Gauss s theorem: f dv = f n ds V S Let f =1 : n ds = 0 S (Mnkowsk theorem) x x x x x = 0 s ( ) s ( x) = 0 x x h ( x) φ (x)x = x (Chrst et al., 1982)

16 Bass Functon Plots Sbson Laplace

17 Meshfree Approxmatons n CG/Graphcs Surface Reconstructon: Bossonnat (France) Polygonal Graphcs Models: Warren (Rce Unversty), Floater (Norway), Schroeder and Desbrun (Caltech) Fracture and Falure Anmatons: Turk (Georga Tech.) O Bren (UC Berkeley), Mark Pauly (Stanford), etc. Surface and Volume Vsualzaton at UC Davs: Faculty (Graduate Student)areBernd Hamann (Sung Park), Ken Joy (Chrs Co), and Nna Amenta (Yong Kl)

18 Galerkn Fnte Element and Meshfree Methods FEM: Functon-based method to solve partal dfferental equatons steady-state heat conducton Strong Form: 2 u = f n Ω, u = u 3 DT on x 2 Ω 1 Varatonal (Weak) Form: * u = arg mn π [ u] = 2 u Ω ( u u fu) dω

19 Galerkn Methods (Cont d) Varatonal Form ( u u 2 fu) dω 0 δπ[ u] = δ = Ω δu ud Ω Ω Ω fδud Ω = 0 δu 1 H 0( Ω ) Fnte-dmensonal approxmatons for tral functon and admssble varatons u h h ( x) = φ ( x) u, δu = φ ( x) j j j

20 Galerkn Methods (Cont d) Dscrete Weak Form and Lnear System of Equatons Ω Ω Ω Ω = d u f d u u h h h δ δ Ω = Ω Ω = Ω d f u d M j j j φ φ φ 1 Ω Ω Ω = Ω = = d f f d K j j φ φ φ, f Ku

21 Fnte Element Method M h u ( ξ, η) = N j ( ξ, η) u j j= 1 shape functon h δu ( ξ, η) = N ( ξ, η), = 1, 2, K, M 1 3 x 2 Facltates modelng complex-geometres Local nterpolant (polynomals n ξ-space) ``Exact numercal ntegraton Accuracy, robustness, and convergence

22 Meshfree Methods (Revews: Belytschko et al., 1996; L and Lu, 2002) (Atlur and Shen, 2002; Lu, 2003; L and Lu, 2004) SPH, RBFs, and MLS r x Natural neghbors (NEM) (Braun and Sambrdge, 1995) w ( r ) = 0 Maxmum entropy approxmants (S, 2004/2005; Arroyo and Ortz, 2006)

23 Nodal Shape Functon Support FEM MLS Compact support Boundary behavor

24 Support (Cont d) ω I I ω I ω J I ω I ω J J J ω J ω I ω J Natural Neghbor MLS

25 Important and Unresolved Issues Imposng essental boundary condtons Numercal ntegraton of the Galerkn weak form Handlng non-convex boundares (especally pertnent n large deformatons) Stablty and robustness of the method

26 NEM and α-shapes Shape constructors are geometrc structures that transform fnte pont sets nto contnuous shapes Use α-shapes (Edelsbrunner and Mucke, ACT, 1994) Each cloud of ponts defnes a fnte famly of shapes rangng from coarse to fner level of detal α= α=0

27 Shape Functon Constructon Constructon of natural neghbor nterpolants over an approprate α-shape leads to nterpolaton along the essental boundary (Cueto et al., IJNME, 2000) h B x C u A

28 Extruson of Hollow Profles (Alfaro et al., CMAME, n press, 2006)

29 Bomechancs Cloud of ponts α-nem model (Doblare et al., CMAME, 2005)

30 Bubble Burstng at Free Surface (Buoyancy Only) (Gonzalez et al., n revew, 2006)

31 Irregular Lattce Model Dmensonal reducton usng two-node elements

32 Doman Dscretzaton

33 Rgd-Body Sprng Network (RBSN)

34 Elastc Unformty Stran producton n 3D network subjected to unform thermal loadng

35 Crack Intaton and Propagaton Crack band (Bazant, 1984) Softenng Relaton

36 Crack Propagaton straght lne dscretzaton random dscretzaton Energy consumpton along lgament length

37 Plate Structure Dryng From Top Surface

38 Shrnkage Crackng Anmaton (plan vew) Expt (clay-rch mud)

39 Constructon of Polygonal Interpolants Wachspress bass functons (Wachspress, 1975; Meyer et al, 2002; Hormann, 2004) Mean value coordnates (Floater, 2003) x x Laplace shape functons (S et al., 2004, 2005) Maxmum entropy (MAXENT) shape functons (S, 2004)

40 Laplace Shape Functons Canoncal Elements

41 Polygonal Interpolant Usng Isoparametrc Mappng Laplace Shape Functon

42 Polygonal Bass Functon

43 Quadtree Data Structure b a Quadtree s a herarchcal data structure based on the prncple of recursve decomposton

44 Handlng Hangng Nodes 3 1 a 2 2 3

45 Shape Functon (Hangng Node) Support of bass functon for node a

46 Mesh Adaptvty Uthman and Askes, 2005 p- r- h-

47 Numercal Example: Corner Sngularty Model Drchlet Problem: 2 u = 0 n Ω 2 3 u( r,θ) = r sn 2θ 3

48 Closure and Outlook An overvew of meshfree approxmaton schemes was presented wth partcular emphass on natural neghbor nterpolants and NEM A natural neghbor-based scalng on Vorono meshes was used to perform fracture smulatons on rregular lattces and polygonal fnte elements were proposed Development of meshfree methods that are sutable for evolvng (non-convex) domans wth stable nodal numercal ntegraton schemes are needed