Post on 27-Jun-2020
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