Generalized barycentric coordinates for degenerate geometry in FEM Andrew Gillette Department of Mathematics University of Arizona joint work with Alexander Rand, CD-adapco Andrew Gillette - U. Arizona GBCs for degenerate geometry MAFELAP June 2016 1 / 23
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Generalized barycentric coordinatesfor degenerate geometry in FEM
Andrew GilletteDepartment of Mathematics
University of Arizona
joint work withAlexander Rand, CD-adapco
Andrew Gillette - U. Arizona ( )GBCs for degenerate geometry MAFELAP June 2016 1 / 23
Table of Contents
1 How are generalized barycentric coordinates used in FEM?
2 Deconstructing the ‘a priori’ estimate
3 Lessons from the triangle case
4 Experiments: GBCs on degenerate geometry
5 Comparison to theoretical results
Andrew Gillette - U. Arizona ( )GBCs for degenerate geometry MAFELAP June 2016 2 / 23
Outline
1 How are generalized barycentric coordinates used in FEM?
2 Deconstructing the ‘a priori’ estimate
3 Lessons from the triangle case
4 Experiments: GBCs on degenerate geometry
5 Comparison to theoretical results
Andrew Gillette - U. Arizona ( )GBCs for degenerate geometry MAFELAP June 2016 3 / 23
The growing world of polytopal meshing
full polyhedral meshing surface-conforming meshingStar-CCM+ (CD-adapco) VoroCrust (Sandia)
tet-hex-pyramid meshing NURBS meshing∗
CADfix (ITI Transcendata) Continuity (UCSD)∗image from Krishnamurthy et al, CAGD 2016
Andrew Gillette - U. Arizona ( )GBCs for degenerate geometry MAFELAP June 2016 4 / 23
Generalized barycentric coordinates applications
From the table of contents of the upcoming book:
“Generalized Barycentric Coordinates in Computer Graphicsand Computational Mechanics”
Discrete Laplacians
Mesh parameterization
Shape deformation
Self-supporting surfaces
Extreme deformations
BEM-based FEM
Virtual element methods
. . . and much more!
Chapters in preparation by various authors; to appear in 2017.
In this talk, we focus on the use of GBCs in finite element methods, although ourresults apply to broader questions about interpolation quality.
Andrew Gillette - U. Arizona ( )GBCs for degenerate geometry MAFELAP June 2016 5 / 23
The generalized barycentric coordinate approach
Let P be a convex polytope with vertex set V . We say that
λv : P → R are generalized barycentric coordinates (GBCs) on P
if they satisfy λv ≥ 0 on P and L =∑v∈V
L(vv)λv, ∀ L : P → R linear.
Familiar properties are implied by this definition:∑v∈V
λv ≡ 1︸ ︷︷ ︸partition of unity
∑v∈V
vλv(x) = x︸ ︷︷ ︸linear precision
λvi (vj) = δij︸ ︷︷ ︸interpolation
traditional FEM family of GBC reference elements
Bilinear Map
Physical
Element
Reference
Element
Affine Map TUnit
Diameter
TΩΩ
Andrew Gillette - U. Arizona ( )GBCs for degenerate geometry MAFELAP June 2016 6 / 23
Some ‘degenerate’ geometry families
1
ε
1
1
ε
1
n vertices1
ε
1
large angle short edge many vertices vertex nearnon-incident edge
Generalized barycentric coordinates help describe the relation between degenerateelement geometry and interpolation error.
Andrew Gillette - U. Arizona ( )GBCs for degenerate geometry MAFELAP June 2016 7 / 23
Outline
1 How are generalized barycentric coordinates used in FEM?
2 Deconstructing the ‘a priori’ estimate
3 Lessons from the triangle case
4 Experiments: GBCs on degenerate geometry
5 Comparison to theoretical results
Andrew Gillette - U. Arizona ( )GBCs for degenerate geometry MAFELAP June 2016 8 / 23
Bounded interpolation estimate pairs
Define:• φ, a type of generalized barycentric coordinates
• P, a polygon with vertices v1, . . . , vn
• h = diam(P)
• The interpolation procedure:
Iu =n∑
i=1
u(vi)φi
Suppose we can find a constant CBI and class of polygons p such that∣∣∣∣∣∣∣∣∣∣
n∑i=1
u(vi)φi
∣∣∣∣∣∣∣∣∣∣H1(P)
≤ CBIh ||u||H2(P) u ∈ H2(P), P ∈ p
The we say the pair (φ, p) has a bounded interpolation estimate.
Andrew Gillette - U. Arizona ( )GBCs for degenerate geometry MAFELAP June 2016 9 / 23
The a priori error estimateFix a second order elliptic PDE.
Suppose (φ, p) has a bounded interpolation estimate:∣∣∣∣∣∣∣∣∣∣
n∑i=1
u(vi)φi
∣∣∣∣∣∣∣∣∣∣H1(P)
≤ CBIh ||u||H2(P)
The a priori error estimate for a Galerkin FEM on a mesh of P ∈ p is:
||u − uh||H1(Ω)︸ ︷︷ ︸finite element error
≤ CC︸︷︷︸Cea’s
Lemma
||u − Iu||H1(Ω)︸ ︷︷ ︸interpolation error
≤ CC Cie h |u|H2(Ω)︸ ︷︷ ︸2nd orderoscillation
,
whereCie :=
(C2
BH
(1 + C2
BI
)+ C2
BI
)1/2
CBH from Bramble-Hilbert Lemma
CBI from bounded interpolation estimate
Our goal: Identify pairs (φ, p) with bounded interpolation estimates.
Andrew Gillette - U. Arizona ( )GBCs for degenerate geometry MAFELAP June 2016 10 / 23
Outline
1 How are generalized barycentric coordinates used in FEM?
2 Deconstructing the ‘a priori’ estimate
3 Lessons from the triangle case
4 Experiments: GBCs on degenerate geometry
5 Comparison to theoretical results
Andrew Gillette - U. Arizona ( )GBCs for degenerate geometry MAFELAP June 2016 11 / 23
Bounded interpolation pairs for triangles
Only choice for φ is regular barycentric coordinates.
Two classes of triangles to consider:
tmina All angles are bounded away from zero: αi > α∗ > 0
tmaxa All angles are bounded away from 180: αi < α∗ < 180