University of California, Davis

Barycentric Finite ElementMethods

N. SukumarUniversity of California at DavisSIAM Conference on Geometric

Design and Computing November 8, 2007

Collaborators and Acknowledgements

• Collaborators

• Research support of the NSF is acknowledged

Alireza Tabarraei (Graduate Student, UC Davis)

Elisabeth Malsch (Post-Doc, Germany)

Outline

Motivation

Introduction to Finite Element Method

Conforming Barycentric Finite Elements

Modeling Crack Discontinuities via Partition of Unity Finite Elements

Summary and Outlook

Motivation: Voronoi Tesellations in Mechanics

Polycrystallinealloy

(Courtesy ofKumar, LLNL)

(Martin and Burr, 1989)(Bolander and

S, PRB, 2004)

Fiber-matrix composite Osteonal bone

Motivation: Crack Modeling

FEM

X-FEM (Moes et al., 1999)Nodes are enriched by a• discontinuous function and• near-tip fields

crack

Mesh generated using ‘Triangle’

Motivation: Crack Modeling on Polygonal Meshes

Convex Mesh Non-Convex Mesh

Motivation: Crack Propagation on Quadtree Meshes

Quadtree mesh Zoom

Galerkin Finite Element Method (FEM)

3

1

2

xFEM: Function-based method to solve partial differential equations

Strong Form:

!

"#2u = f in$, u = u on %$

Variational (Weak) Form:

steady-state heat conductionDT

!

u* = argmin

u"[u] = #u•#u /2 $ fu( )

%

& d%'

( )

*

+ ,

Galerkin FEM (Cont’d)

!

"#[u] = " $u•$u /2 % fu( )&

' d& = 0Variational Form

!

"#u•"ud$$

% & f#ud$$

% = 0 '#u( H0

1($)

!

uh(x) = " j (x)u j

j# , $uh = "i(x)

Finite-dimensional approximations for trial function andadmissible variations

must vanish on the bdry

Galerkin FEM (Cont’d)

Discrete Weak Form and Linear System of Equations

!

"#uh •"uhd$$

% = f#uhd$$

%

!

Ku = f

Kij = "#i •"# j d$$

% , fi = f# i d$$

%!

"#i •"# jd$$

%&

' (

)

* + u j

j=1

N

, = f# i d$$

%

Biharmonic EquationStrong Form

!

"4u = f in#

BCs : u = u and $u/$n = 0 on $#

Variational (Weak) Form

!

Find u" S such that

!

"2u"2

w#

$ d# = fw#

$ d# %w & V

!

S = u : u" H2(#), u = u on $#,$u /$n = 0 on $#{ }

!

V = w :w " H2(#),w = 0 on $#,$w /$n = 0 on $#{ }

Nodal Basis Function and Nodal Shape Function

Basis function

a

Shape function

!

"a(x)

!

Na(x)

a a

• Affine combination:

• Convex combination:

• Regularity:

• Piece-wise linear on the boundary: conformity and for imposing Dirichlet boundary conditions

• Fast computations for and ; efficient numerical quadrature over each element

!

"i(#) =1

i

$ ,

!

x := "i(#)x

i

i

$

FEM: Desired Bases Properties and Implementation

!

"i# 0 ensures convergence

for 2nd order PDEs(isoparametric map)

!

"i# C

$(%)

!

C0

!

"i

!

"#i

Convex hull

p lies outside the circumcircles in green

Voronoi (Natural Neighbor)-Based Interpolants

!

"i(p) =Ai(p)

A(p)

Sibson Interpolant

(Sibson, 1980)

Laplace Interpolant

!

"i(p) =# i(p)

# j (p)j

$!

" i(p) =si(p)

hi(p)

(Christ et al., Nuclear Physics B, 1982; Belikov et al., 1997; Hiyoshi and Sugihara, 1999)

• Wachspress basis functions (Wachspress, 1975; Warren, 1996; Meyer et al, 2002; Malsch, 2003)

• Mean value coordinates (Floater, 2003; Hormann, 2005; Floater and Hormann, 2006)

• Laplace and maximum-entropy basis functions

x

(S, 2004; S and Tabarraei, 2004)

Barycentric Coordinates on Polygons

x

• Convex combination

• Partition of unity

• Reproduces affine functions (linear completeness)

!

"i

i=1

n

# (x) =1

!

"i(x)x

i= x

i=1

n

#

Properties of Barycentric Coordinates

!

"i# 0

Laplace Shape Function (Circumscribable Polygons)

Canonical Elements

Identical to Wachspress and Discrete Harmonic Weight

Laplace Shape Function

Isoparametric Transformation

!

"i(p) =# i(p)

# j (p)j

$,

!

" i(p) =si(p)

hi(p)

(S and Tabarraei, IJNME, 2004)

Polygonal Basis Function

• Affine functions:

• Convex combination:

Pos-def mass matrix, total variation diminishing Convex hull property Optimal conditioning

!

"i

i=1

n

# (x) =1,

!

"i(x)x

i= x

i=1

n

#

Maximum-Entropy Basis Functions: Constraints

!

"i(x) # 0 $i,x

(Arroyo and Ortiz, IJNME, 2006)

(Farouki and Goodman, Math. Comp., 96)

: convex approximation scheme

!

uh

(S, IJNME, 2004; Arroyo and Ortiz, IJNME, 2006)

!

max" #R+

n

$ "i

i=1

n

% (x)ln"i(x)

mi(x)

!

"(x) = # $ R+

n: #

i

i=1

n

% =1, #i

i=1

n

% xi= x

& ' (

) * +

(S and Wright, IJNME, 2007; S and Wets, SIOPT, 2007) MAXENT/Minimum Relative-Entropy Formulation

: Prior (weight function)

!

mi(x)

MAXENT Solution

• Numerical solution based on the dual (logsumexp func)• Convex minimization (Agmon et al., JCP, 1979)

!

"i(x) =Zi(x)

Z(x), Zi(x) = mi(x)exp(#x i • $),

Z = Z j (x)j

% (partition function)

Wachspress MVC MAXENT

3

1 a 2

Quadtree

2 3

A

(Tabarraei and S, FEAD, 2005)

Non-Convex Polygons

!

wi(x) =

tan("i#1

2) + tan(

"i

2)

ri

,

!

tan("i

2) =

sin"i

1+ cos"i

=ri# r

i+1

riri+1 + r

i$ r

i+1

Mean Value Coordinates

(Hormann and Floater, ACM Transaction on Graphics, 2006)

!

"i(x) =wi(x)

w j (x)j

#

!

ri= x

i" x, r

i= x

i" x

Introduction of a function within a FE spacesuch that conformity and sparsity of thestiffness matrix are retained

Classical Finite Element Approximation

!

uh(x) = "

i(x)u

i

i

# ,

!

"i(x) =1,

i

# "i(x)x

i

i

# = x

Partition of Unity Finite Element Method (PUFEM)

)(x!

(Melenk and Babuska, CMAME, 1996)

!

C0

!

"

!

uh(x) = "i(x)ui

i#I

$ + " j (x)%(x)j#J

$ a j

classical enrichment

PUFEM/X-FEM (Moes et al., IJNME, 1999)

!

{"i}i#I ${" j%} j#J

Bases

• Index set consists of all nodes in the mesh•, Index set consists of nodes that are enriched

!

I

!

J•

FE space

!

"

FE and Enriched Basis Functions

FE basis function Enriched basis function

!

"a(x)

a a

crack

!

"a(x)H(x)

X-FEM Approximation (Polygonal Mesh)

!

uh(x) = "i(

i#I

$ x)ui + " j (j#J

$ x)H(x)a j + "k (x)k#K

$ %& (x)&=1

4

$ bk&

Heaviside enriched nodes Near-tip enriched nodes

(Tabarraei and S, CMAME, 2008)

Laplace (Polygonal) and Enriched Bases Functions

Crack

MVC (Non-Convex) and Enriched Bases Functions

Crack

Mesh a Mesh b Mesh c

Patch Test

Regularized mesh

Error in the norm = 2L )10( 10!

O

Error in the energy norm = )10( 9!O

Mesh a Mesh b

Error in the norm for meshes a and b areand , respectively

Patch Test (Cont’d)Non-regularized mesh

)10( 7!O )10( 6!

O

2L

Poisson Problem: Localized Potential

( )2221 )1(4

16)(xx

eu+!!

!=x

22 )3,3(in4 !="=#! $%u

!"= on0u

( )2221 )1(4

16xx

e++!

!

Potential

(Tabarraei and S, CMAME, 2007)

Poisson Problem: Mesh Refinements

Mesh a Mesh b Mesh c

Mesh d Mesh e Mesh f

!

KI

= (" 2 sin2 # +"1 cos

2 #) $a

KII

= (" 2 %"1)sin# cos# $a(Aliabadi, IJF, 1987)

Oblique Crack in an Infinite Plate

Oblique Crack (Cont’d)

Quadtree mesh Non-convex mesh292 elements 292 elements

Oblique Crack: Stress Intensity Factors

Inclined Central Crack in Uniaxial Tension

Animation Zoom

!

a

L= 0.01

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

Journal Acronyms• IJNME : International Journal for Numerical Meth. in Engg.

• CMAME : Computer Methods in Applied Mech. and Engg.

• FEAD : Finite Elements in Analysis and Design

• PRB : Physical Review B

• SIOPT : SIAM Journal of Optimization • JCP : Journal of Computational Physics

Links to my journal publications on barycentric FEM areavailable from http://dilbert.engr.ucdavis.edu/~suku