Barycentric Finite Element Methods

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1 University of California, Davis Barycentric Finite Element Methods N. Sukumar University of California at Davis SIAM Conference on Geometric Design and Computing November 8, 2007

2 Collaborators and Acknowledgements Collaborators Alireza Tabarraei (Graduate Student, UC Davis) Elisabeth Malsch (Post-Doc, Germany) Research support of the NSF is acknowledged

3 Outline Motivation Introduction to Finite Element Method Conforming Barycentric Finite Elements Modeling Crack Discontinuities via Partition of Unity Finite Elements Summary and Outlook

4 Motivation: Voronoi Tesellations in Mechanics Polycrystalline alloy Fiber-matrix composite Osteonal bone (Courtesy of Kumar, LLNL) (Bolander and S, PRB, 2004) (Martin and Burr, 1989)

5 Motivation: Crack Modeling crack FEM Mesh generated using Triangle X-FEM (Moes et al., 1999) Nodes are enriched by a discontinuous function and near-tip fields

6 Motivation: Crack Modeling on Polygonal Meshes Convex Mesh Non-Convex Mesh

7 Motivation: Crack Propagation on Quadtree Meshes Quadtree mesh Zoom

8 Galerkin Finite Element Method (FEM) FEM: Function-based method to solve partial differential equations steady-state heat conduction Strong Form: "# 2 u = f in $, u = u on %$ 3 DT x 2 1 Variational (Weak) Form: ' * u * = argmin "[u] = & (#u #u/2 $ fu) d% u ), ( + %

9 Galerkin FEM (Cont d) Variational Form "#[u] = " ' ( $u $u/2 % fu) d& = 0 & % "#u "ud$ & % f#ud$ = 0 '#u ( H 1 0 ($) $ $ must vanish on the bdry Finite-dimensional approximations for trial function and admissible variations u h (x) = # " j (x)u j, $u h = " i (x) j

10 Galerkin FEM (Cont d) Discrete Weak Form and Linear System of Equations % "#u h "u h d$ = f#u h d$ $ N & ),( "# i "# j d$ + u j = f# i d$ ' * j=1 Ku = f K ij = % "# i "# j d$, f i = f# i d$ $ % $ % $ % $ % $

11 Biharmonic Equation Strong Form " 4 u = f in # BCs : u = u and $u/$n = 0 on $# Variational (Weak) Form Find u " S such that $ " 2 u" 2 w d# = $ fw d# %w & V # S = { u : u " H 2 (#), u = u on $#,$u/$n = 0 on $# } V = { w : w " H 2 (#), w = 0 on $#, $w /$n = 0 on $# } #

12 Nodal Basis Function and Nodal Shape Function a a a Basis function " a (x) Shape function N a (x)

13 FEM: Desired Bases Properties and Implementation Affine combination: $ " i (#) =1, x := " i (#)x i i $ i Convex combination: Regularity: " i # 0 ensures convergence for 2 nd order PDEs (isoparametric map) " i # C $ (%) Piece-wise linear on the boundary: conformity and for imposing Dirichlet boundary conditions " i "# i C 0 Fast computations for and ; efficient numerical quadrature over each element

14 Voronoi (Natural Neighbor)-Based Interpolants p lies outside the circumcircles in green Convex hull

15 Sibson Interpolant " i ( p) = A i(p) A(p) (Sibson, 1980)

16 Laplace Interpolant " i ( p) = s i( p) h i ( p) " i ( p) = # i(p) $ j # j (p) (Christ et al., Nuclear Physics B, 1982; Belikov et al., 1997; Hiyoshi and Sugihara, 1999)

17 Barycentric Coordinates on Polygons Wachspress basis functions (Wachspress, 1975; Warren, 1996; Meyer et al, 2002; Malsch, 2003) Mean value coordinates (Floater, 2003; Hormann, 2005; Floater and Hormann, 2006) x x Laplace and maximum-entropy basis functions (S, 2004; S and Tabarraei, 2004)

18 Properties of Barycentric Coordinates Convex combination Partition of unity n #" i (x) =1 i=1 Reproduces affine functions (linear completeness) n # i=1 " i # 0 " i (x)x i = x

19 Laplace Shape Function (Circumscribable Polygons) Canonical Elements Identical to Wachspress and Discrete Harmonic Weight

20 Isoparametric Transformation (S and Tabarraei, IJNME, 2004) Laplace Shape Function " i ( p) = # i (p) # j (p) $ j, " i ( p) = s i ( p) h i ( p)

21 Polygonal Basis Function

22 Maximum-Entropy Basis Functions: Constraints (S, IJNME, 2004; Arroyo and Ortiz, IJNME, 2006) Affine functions: Convex combination: u h n n #" i (x) =1, #" i (x)x i = x i=1 i=1 " i (x) # 0 $i,x : convex approximation scheme (Arroyo and Ortiz, IJNME, 2006) Pos-def mass matrix, total variation diminishing Convex hull property Optimal conditioning (Farouki and Goodman, Math. Comp., 96)

23 MAXENT/Minimum Relative-Entropy Formulation (S and Wright, IJNME, 2007; S and Wets, SIOPT, 2007) n max $ %" (x)ln " i (x) n i " #R m (x) + i m i (x) i=1 : Prior (weight function) & "(x) = '# $ R n : + ( n n %# =1, # i i i=1 i=1 % x i = x ) * +

24 MAXENT Solution " i (x) = Z i(x) Z(x), Z i (x) = m i (x)exp(#x i $), Z = % Z j (x) (partition function) j Numerical solution based on the dual (logsumexp func) Convex minimization (Agmon et al., JCP, 1979) Wachspress MVC MAXENT

25 Quadtree 3 1 a 2 A (Tabarraei and S, FEAD, 2005) 2 3

26 Non-Convex Polygons (Hormann and Floater, ACM Transaction on Graphics, 2006) Mean Value Coordinates w i (x) = tan( " i#1 2 ) + tan(" i 2 ) r i " i (x) = w i (x) w j (x), tan(" i 2 ) = sin" i 1+ cos" i = r i = x i " x, r i = x i " x # j r i # r i+1 r i r i+1 + r i $ r i+1

27 Partition of Unity Finite Element Method (PUFEM) (Melenk and Babuska, CMAME, 1996) Introduction of a function! (x) within a FE space such that C 0 conformity and sparsity of the stiffness matrix are retained Classical Finite Element Approximation u h (x) = #" i (x)u i, #" i (x) =1, #" i (x)x i = x " i i i

28 PUFEM/X-FEM (Moes et al., IJNME, 1999) u h (x) = $ " i (x)u i + $ " j (x)%(x) a j Bases i#i j #J classical enrichment I {" i } i#i ${" j %} j #J Index set consists of all nodes in the mesh, Index set consists of nodes that are enriched J " FE space

29 FE and Enriched Basis Functions FE basis function a " a (x) Enriched basis function a crack " a (x)h(x)

30 X-FEM Approximation (Polygonal Mesh) Heaviside enriched nodes u h (x) = $ " i ( x)u i + $ " j ( x)h(x)a j + $ " k (x) $ % & (x) b k& i#i j #J k #K &=1 (Tabarraei and S, CMAME, 2008) Near-tip enriched nodes 4

31 Laplace (Polygonal) and Enriched Bases Functions Crack

32 MVC (Non-Convex) and Enriched Bases Functions Crack

33 Patch Test Regularized mesh Mesh a Mesh b Mesh c L O(10 ) 2!10 Error in the norm =!9 Error in the energy norm = O(10 )

34 Patch Test (Cont d) Non-regularized mesh Mesh a Mesh b 2 Error in the L norm for meshes a and b are!7!6 O (10 ) and O(10 ), respectively

35 Poisson Problem: Localized Potential 2! # u = 4 $% in " = (! 3,3) u = 0 on "! 2! 4 ( x1! 1) + x2 u( x) =! 16e 2 ) 2! 4 ( x1 + 1) + x2! 16e 2 ) 2 Potential (Tabarraei and S, CMAME, 2007)

36 Poisson Problem: Mesh Refinements Mesh a Mesh b Mesh c Mesh d Mesh e Mesh f

37 Oblique Crack in an Infinite Plate K I = (" 2 sin 2 # + " 1 cos 2 #) $a K II = (" 2 %" 1 )sin# cos# $a (Aliabadi, IJF, 1987)

38 Oblique Crack (Cont d) Quadtree mesh Non-convex mesh 292 elements 292 elements

39 Oblique Crack: Stress Intensity Factors

40 Inclined Central Crack in Uniaxial Tension Animation a L = 0.01 Zoom

41 Summary and Outlook Barycentric coordinates on irregular polygons were used to develop finite element methods Mesh-independent modeling of cracks on polygons and quadtree meshes was presented Potential use of barycentric coordinates in FE: smooth interpolants for higher-order PDEs; construction of convex approximants; meshing microstructures in 3D; polyhedral/octree FE

42 Journal Acronyms IJNME : International Journal for Numerical Meth. in Engg. CMAME : Computer Methods in Applied Mech. and Engg. FEAD PRB SIOPT : Finite Elements in Analysis and Design : Physical Review B : SIAM Journal of Optimization JCP : Journal of Computational Physics Links to my journal publications on barycentric FEM are available from

Barycentric Finite Element Methods

University of California, Davis Barycentric Finite Element Methods N. Sukumar UC Davis Workshop on Generalized Barycentric Coordinates, Columbia University July 26, 2012 Collaborators and Acknowledgements