Barycentric Finite Element Methods - USI InformaticsBarycentric Finite Element Methods N. Sukumar...

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