Anisotropic h and p refinement for
conforming FEM in 3D
Philipp Frauenfelder
Christian Lage
Christoph Schwab
pfrauenf@math.ethz.ch
Seminar for Applied Mathematics
Federal Institute of Technology, ETH Zürich
Anisotropic h and p refinement for conforming FEM in 3D – p.1/48
Goal for Meshes
• Hierarchy of hanging nodes
• Anisotropic refinements
Anisotropic h and p refinement for conforming FEM in 3D – p.2/48
Overview
• Introduction
• Anisotropic h refinements
• S and T matrices
• Assembly of Supports
Anisotropic h and p refinement for conforming FEM in 3D – p.3/48
Overview
• Introduction
• Anisotropic h refinements
• S and T matrices
• Assembly of Supports
• Anisotropic p refinements
Anisotropic h and p refinement for conforming FEM in 3D – p.3/48
Overview
• Introduction
• Anisotropic h refinements
• S and T matrices
• Assembly of Supports
• Anisotropic p refinements
• hp Meshes
Anisotropic h and p refinement for conforming FEM in 3D – p.3/48
Overview
• Introduction
• Anisotropic h refinements
• S and T matrices
• Assembly of Supports
• Anisotropic p refinements
• hp Meshes
• Perspectives
Anisotropic h and p refinement for conforming FEM in 3D – p.3/48
Previous hp Software
• Szabó 1985: PROBE (p only)
• Demkowicz, Oden, Rachowicz et al. 1989: PHLEX, hp90
• Anderson: STRIPE (p only on a-priori generated meshes)
• Flaherty, Shephard: Tetrahedra only (3D anisotropy?)
• Karniadakis, Sherwin: NEKTAR (regular meshes only, tetrahedra,
hexahedra, prisms, p only)
• Devloo
• Szabó since 1995: STRESSCHECK (p only)
• Heuveline et al.: HiFlow
• In development: deal.II (Kanschat & Bangerth), ngsolve (Schöberl
et al.)
Anisotropic h and p refinement for conforming FEM in 3D – p.4/48
FE Method
• Let Ω ⊂ Rd , d = 1, 2, 3 (dimension independent design)
• Find u ∈ V such that
a(u, v) = l(v)
∀v ∈ V,
V a FE space, a(. , .) a bilinear form and l(.) a linear form.
• Standard FE: V ⊂ H 1 (Ω)
V = S 1,p (Ω, T )
1
= u ∈ H (Ω) : u|K ◦ FK ∈ Qp ∀K ∈ T
⇒ u ∈ V is continuous, ie. C 0 (Ω̄).
• Vector valued problems are possible
Anisotropic h and p refinement for conforming FEM in 3D – p.5/48
FE Space: Generalities
K
• Basis {Φi }N
constructed
from
element
shape
functions
φ
j on
i=1
elements K ∈ T .
• Reference element shape functions: Nj , element map: FK : K̂ → K
⇒ φK
j ◦ F K = Nj .
PSfrag replacements
ξ
x
FK
K
K̂
Φj
Nj
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
1
0.8
0
0
0.6
0.2
0.4
x
0.4
0.6
0.8
0.2
1 0
y
0.2
0.4
0.6
x
0.8
1
1.2
1.4
1.6 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
y
Anisotropic h and p refinement for conforming FEM in 3D – p.6/48
FE Meshes
Local refinements as mean to improve
approximation of exact solution by FE
solution
Anisotropic h and p refinement for conforming FEM in 3D – p.7/48
FE Meshes
Local refinements as mean to improve
approximation of exact solution by FE
solution
Anisotropic h and p refinement for conforming FEM in 3D – p.7/48
FE Meshes
Local refinements as mean to improve
approximation of exact solution by FE
solution
Anisotropic h and p refinement for conforming FEM in 3D – p.7/48
FE Meshes
Local refinements as mean to improve
approximation of exact solution by FE
solution
Anisotropic h and p refinement for conforming FEM in 3D – p.7/48
But standard FE forbids locally refined grids: discontinuities are possible.
FE Meshes
Local refinements as mean to improve
approximation of exact solution by FE
solution
Anisotropic h and p refinement for conforming FEM in 3D – p.7/48
Mortar vs. Enforcing Continuity
• Topolocigal closure
Anisotropic h and p refinement for conforming FEM in 3D – p.8/48
Mortar vs. Enforcing Continuity
• Topolocigal closure
Anisotropic h and p refinement for conforming FEM in 3D – p.8/48
Mortar vs. Enforcing Continuity
• Topolocigal closure
Drawbacks: more elements, more element types, what about
refining a
?
Anisotropic h and p refinement for conforming FEM in 3D – p.8/48
Mortar vs. Enforcing Continuity
• Topolocigal closure
Drawbacks: more elements, more element types, what about
refining a
?
• Our philosophy: hexahedral meshes only (tensorized interpolants,
spectral quadrature techniques)
• Our solution: Treating the constraints induced by the hanging nodes
Why conforming? a(u, v) = a(v, u) and a(u, u) ≥ αkuk2V ⇒ A SPD,
pccg . . .
Anisotropic h and p refinement for conforming FEM in 3D – p.8/48
Our Software: Concepts
• Started by Christian Lage during his Ph.D. studies (1995).
• Used and improved by Frauenfelder, Matache, Schmidlin, Schmidt
and several students.
• Concept Oriented Design using mathematical principles [1].
• Currently two parts: hp-FEM, BEM (wavelet and multipole methods).
• C++
[1] P. F. and Ch. Lage, “Concepts—An Object Oriented Software Package
for Partial Differential Equations”, Mathematical Modelling and Numerical
Analysis 36 (5), pp. 937–951 (2002).
Anisotropic h and p refinement for conforming FEM in 3D – p.9/48
Overview
• Introduction
• Anisotropic h refinements
S and T matrices
•
• Assembly of Supports
• Anisotropic p refinements
• hp Meshes
• Perspectives
Anisotropic h and p refinement for conforming FEM in 3D – p.10/48
T Matrix
K mK
Definition 1 (T Matrix). Element shape functions φj j=1 on element K,
N
global basis functions {Φi }i=1 .
The T matrix T K ∈ RmK ×N of element K is implicitly defined by
Φi | K =
mK
X
[T K ]ji φK
j
j=1
as vectors:
K
Φ|K = T >
Kφ .
Anisotropic h and p refinement for conforming FEM in 3D – p.11/48
Assembly using T Matrices
Assembling:
l = l(Φ) = l
X
K̃
K̃
T>
φ
K̃
=
X
K̃
K̃
T>
l(φ
)
K̃
=
X
T>
l
K̃ K̃
K̃
Anisotropic h and p refinement for conforming FEM in 3D – p.12/48
Assembly using T Matrices
Assembling:
l = l(Φ) = l
X
K̃
T>
φ
K̃
K̃
A = a(Φ, Φ) =
X
=
X
K̃
T>
l(φ
)
K̃
=
K̃
K̃
K
T>
a(φ
,
φ
)T K
K̃
K,K̃
X
T>
l
K̃ K̃
K̃
=
X
T>
A T
K̃ K̃K K
K,K̃
Note: AK̃K = 0 in standard FEM for K̃ 6= K.
Anisotropic h and p refinement for conforming FEM in 3D – p.12/48
Example: Regular Mesh
Two elements with three local shape functions each and four global basis
functions.
3
3
3
2
4

1
1 1
TI = 
2 0
3 0
J
2
3
4

0 0 0

1 0 0
0 1 0
I
1
1
2
1
2
Anisotropic h and p refinement for conforming FEM in 3D – p.13/48
Example: Regular Mesh
Two elements with three local shape functions each and four global basis
functions.
3
3
3
2
4

1 1
TI = 
2 0
3 0

J
1
I
1
1
1
2
1
2
2
3
4

0 0 0

1 0 0
0 1 0

2
3
1 0 1 0
TJ = 
2 0 0 0
3 0 0 1
4
0

1
0
Anisotropic h and p refinement for conforming FEM in 3D – p.13/48
Example: Irregular Mesh
Three elements with three local shape functions each and four global
basis functions. The hanging node is marked with ◦.
3
3
3
2
K
4
2
TL
1
3
L

1
1 0
=
2 0
3 0
2
3
1
0
1/2
0
0
1/2
4

0

1
0
I
1
1
2
1
2
Anisotropic h and p refinement for conforming FEM in 3D – p.14/48
Example: Irregular Mesh
Three elements with three local shape functions each and four global
basis functions. The hanging node is marked with ◦.
3
3
3
2
K
4
2
TL
1
3
L
1
1
1 0
=
2 0
3 0

1
I
1

2
TK
1
2
1 0
=
2 0
3 0
2
3
1
0
1/2
0
0
1/2
2
3
1/2
1/2
0
0
0
1
4

0

1
0

4
0

1
0
⇒ continuous basis functions.
Anisotropic h and p refinement for conforming FEM in 3D – p.14/48
Generation of T Matrices
• Regular Mesh: Counting and assigning indices with respect to
topological entities such as vertices, edges and faces.
Explained in detail later.
Anisotropic h and p refinement for conforming FEM in 3D – p.15/48
Generation of T Matrices
• Regular Mesh: Counting and assigning indices with respect to
topological entities such as vertices, edges and faces.
Explained in detail later.
• Irregular Mesh: Irregularity due to a refinement of an initially
regular mesh.
OK:
Explanation follows.
, not OK:
Anisotropic h and p refinement for conforming FEM in 3D – p.15/48
T Matrices for Irregular Meshes
Irregularity due to a refinement of an initially regular mesh.
M
refine
M0
Mesh
Basis fcts. B = Brepl ∪ Bkeep
−→
B 0 = Bins ∪ Bkeep
Anisotropic h and p refinement for conforming FEM in 3D – p.16/48
T Matrices for Irregular Meshes
Irregularity due to a refinement of an initially regular mesh.
M
refine
M0
Mesh
Basis fcts. B = Brepl ∪ Bkeep
−→
B 0 = Bins ∪ Bkeep
Brepl : basis fcts. which can be solely
described by elements of M0 \M
Bins : basis fcts. generated by regular parts
of M0 \M
Anisotropic h and p refinement for conforming FEM in 3D – p.16/48
T Matrices for Irregular Meshes
Irregularity due to a refinement of an initially regular mesh.
M
refine
M0
Mesh
Basis fcts. B = Brepl ∪ Bkeep
−→
B 0 = Bins ∪ Bkeep
Brepl : basis fcts. which can be solely
described by elements of M0 \M
Bins : basis fcts. generated by regular parts
of M0 \M
Anisotropic h and p refinement for conforming FEM in 3D – p.16/48
T Matrices for Irregular Meshes
(' ('
&% &%
$# $#
*) *)
,+ ,+
0/ 0/
21 21 .- .
"! "!
Irregularity due to a refinement of an initially regular mesh.
M
refine
M0
Mesh
Basis fcts. B = Brepl ∪ Bkeep
−→
B 0 = Bins ∪ Bkeep
Brepl : basis fcts. which can be solely
described by elements of M0 \M
Bins : basis fcts. generated by regular parts
of M0 \M
Every element of B has a column in the T matrix. Generation is
• easy for Bins (like regular mesh),
• simple for Bkeep : modify column from M by S matrix.
Anisotropic h and p refinement for conforming FEM in 3D – p.16/48
S Matrix
Definition 2 (S Matrix). Let K 0 ⊂ K be the result of a refinement of
element K. The S matrix S K 0 K ∈ RmK 0 ×mK is defined by
mK 0
K
φj
K0
=
X
[S
K0K
K0
]lj φl
l=1
as vectors:
φK
j K0
φK K0
=
K0
>
SK0K φ
is represented as a linear combination of the shape functions
n 0 om K 0
φK
of K 0 .
l
l=1
Anisotropic h and p refinement for conforming FEM in 3D – p.17/48
Application of S Matrix
Proposition 1. Let K 0 ⊂ K be the result of a refinement of an element K.
Then, the T matrix of K 0 can be computed as
ins
+
T
T K 0 = S K 0 K T keep
K0
K
where T keep
denotes the T matrix of element K (with columns not
K
related to functions in Bkeep set to zero) and T ins
K 0 the T matrix for
functions in Bins with respect to K 0 .
Proposition 2. Let K̂ 0 ⊂ K̂ be the result of a refinement of the reference
element K̂ with H : K̂ → K̂ 0 the subdivision map. The element maps are
FK : K̂ → K and FK 0 : K̂ → K 0
and FK 0 ◦ H −1 = FK holds. Then, S K̂ 0 K̂ = S K 0 K .
Anisotropic h and p refinement for conforming FEM in 3D – p.18/48
Meshes
Anisotropic h and p refinement for conforming FEM in 3D – p.19/48
Meshes
Anisotropic h and p refinement for conforming FEM in 3D – p.19/48
Meshes
Anisotropic h and p refinement for conforming FEM in 3D – p.19/48
S Matrix in Dimension d = 1
Subdividing Jˆ = (0, 1) in Jˆ0 = (0, 1/2) and Jˆ? = (1/2, 1) with the reference
element shape functions

j=1
1 − ξ
j=2
Nj (ξ) = ξ

1,1
ξ(1 − ξ)Pj−3
(2ξ − 1) j = 3, . . . , J
yields (solving a linear system) for J = 4:

1
1/2
S Jˆ0 Jˆ = 
0
0
0
1/2
0
0
1

0
0
/2

1/4
0 
 and S ˆ? ˆ =  0
J J
0
1/4 −3/4
1/8
0
0
1/2
1/4
1
0
0
0
1/4
0

0
0
.
3/4
1/8
Hierarchic shape functions ⇒ hierarchic S matrices.
Anisotropic h and p refinement for conforming FEM in 3D – p.20/48
S Matrices: Tensor Product in 2D
• d > 1 with hexahedral meshes ⇒ S matrices are built from tensor
products of 1D S matrices.
Anisotropic h and p refinement for conforming FEM in 3D – p.21/48
S Matrices: Tensor Product in 2D
• d > 1 with hexahedral meshes ⇒ S matrices are built from tensor
products of 1D S matrices.
• In 2D: Ni,j = Ni ⊗ Nj , the four bilinear shape functions are:
N1,2 (ξ) = N1 (ξ1 ) · N2 (ξ2 )
N2,2 (ξ) = N2 (ξ1 ) · N2 (ξ2 )
N1,1 (ξ) = N1 (ξ1 ) · N1 (ξ2 )
N2,1 (ξ) = N2 (ξ1 ) · N1 (ξ2 )
Anisotropic h and p refinement for conforming FEM in 3D – p.21/48
S Matrices: Tensor Product in 2D
• d > 1 with hexahedral meshes ⇒ S matrices are built from tensor
products of 1D S matrices.
• In 2D: Ni,j = Ni ⊗ Nj , the four bilinear shape functions are:
N1,2 (ξ) = N1 (ξ1 ) · N2 (ξ2 )
N2,2 (ξ) = N2 (ξ1 ) · N2 (ξ2 )
N1,1 (ξ) = N1 (ξ1 ) · N1 (ξ2 )
N2,1 (ξ) = N2 (ξ1 ) · N1 (ξ2 )
PSfrag replacements
• Consider the subdivisions:
K̂ 0
K̂ ?
K̂ ?
K̂ d
K̂ 0
K̂ a K̂ b
K̂ c
Anisotropic h and p refinement for conforming FEM in 3D – p.21/48
S Matrices: Tensor Product in 2D II
Subdivision map of left variant: H : K̂ → K̂ 0 , ξ 7→
ξ1 /2
. S matrix S K̂ 0 K̂
ξ2
is defined by:
Ni,j |K̂ 0 =
X
k,l
S K̂ 0 K̂
−1
N
◦
H
.
k,l
(k,l),(i,j)
PSfrag replacements
K̂ 0 K̂ ?
Anisotropic h and p refinement for conforming FEM in 3D – p.22/48
S Matrices: Tensor Product in 2D II
Subdivision map of left variant: H : K̂ → K̂ 0 , ξ 7→
ξ1 /2
. S matrix S K̂ 0 K̂
ξ2
is defined by:
Ni,j |K̂ 0 =
X
S K̂ 0 K̂
k,l
−1
N
◦
H
.
k,l
(k,l),(i,j)
PSfrag replacements
K̂ 0 K̂ ?
Tensor product shape functions:
(Ni ⊗ Nj )|K̂ 0 =
X
k,l
S K̂ 0 K̂
−1
(N
⊗
N
)
◦
H
.
k
l
(k,l),(i,j)
(1)
Anisotropic h and p refinement for conforming FEM in 3D – p.22/48
S Matrices: Tensor Product in 2D III
S matrices for 1D reference element shape fcts. used in (1):
X
Ni |Jˆ0 =
S Jˆ0 Jˆ mi Nm ◦ G−1
for the ξ1 part and
m
Nj =
X
[E]nj Nn
for the ξ2 part,
n
where G : ξ 7→ ξ/2.
Anisotropic h and p refinement for conforming FEM in 3D – p.23/48
S Matrices: Tensor Product in 2D III
S matrices for 1D reference element shape fcts. used in (1):
X
Ni |Jˆ0 =
S Jˆ0 Jˆ mi Nm ◦ G−1
for the ξ1 part and
m
Nj =
X
for the ξ2 part,
[E]nj Nn
n
where G : ξ 7→ ξ/2. Plugging into the left hand side of (1) yields:
(Ni ⊗ Nj )|K̂ 0 = Ni |Jˆ0
X ⊗ Nj =
S Jˆ0 Jˆ mi Nm ◦ G−1 ⊗ [E]nj Nn
m,n
=
X
m,n
S Jˆ0 Jˆ
−1
·
[E]
N
◦
G
⊗ Nn .
m
nj
mi
Anisotropic h and p refinement for conforming FEM in 3D – p.23/48
S Matrices: Tensor Product in 2D IV
Comparing with the right hand side of (1):
X
S Jˆ0 Jˆ mi · [E]nj Nm ◦ G−1 ⊗ Nn
m,n
=
X
k,l
S K̂ 0 K̂
−1
N
◦
G
⊗ Nl .
k
(k,l),(i,j)
Anisotropic h and p refinement for conforming FEM in 3D – p.24/48
S Matrices: Tensor Product in 2D IV
Comparing with the right hand side of (1):
X
S Jˆ0 Jˆ mi · [E]nj Nm ◦ G−1 ⊗ Nn
m,n
=
X
k,l
S K̂ 0 K̂
−1
N
◦
G
⊗ Nl .
k
(k,l),(i,j)
Therefore for the vertical subdivision:
S K̂ 0 K̂ = S Jˆ0 Jˆ ⊗ E
S K̂ ? K̂ = S Jˆ? Jˆ ⊗ E
for the left quad K̂ 0 ,
PSfrag
replacements
for the right
quad
K̂ ? .
K̂ 0
K̂ ?
Anisotropic h and p refinement for conforming FEM in 3D – p.24/48
S Matrices: Tensor Product in 2D V
Horizontal subdivision:
S K̂ 0 K̂ = E ⊗ S Jˆ0 Jˆ
S K̂ ? K̂ = E ⊗ S Jˆ? Jˆ
for the bottom quad K̂ 0 ,
PSfrag replacements
for the top quad K̂ ? .
K̂ ?
K̂ 0
Anisotropic h and p refinement for conforming FEM in 3D – p.25/48
S Matrices: Tensor Product in 2D V
Horizontal subdivision:
S K̂ 0 K̂ = E ⊗ S Jˆ0 Jˆ
S K̂ ? K̂ = E ⊗ S Jˆ? Jˆ
for the bottom quad K̂ 0 ,
PSfrag replacements
?
for the topPSfrag
quad K̂replacements
.
K̂ ?
0
K̂ 0
K̂
K̂ ?
Subdivision into four quads:
• subdivide K̂ horizontally into two children
• subdivide upper and lower
child vertically into K̂ d and K̂ c and K̂ a and K̂ b resp.
S K̂ d K̂ = S Jˆ0 Jˆ ⊗ E · E ⊗ S Jˆ? Jˆ
S K̂ a K̂ = S Jˆ0 Jˆ ⊗ E · E ⊗ S Jˆ0 Jˆ
K̂ d
K̂ c
K̂ a
K̂ b
S K̂ c K̂ = S Jˆ? Jˆ ⊗ E · E ⊗ S Jˆ? Jˆ
S K̂ b K̂ = S Jˆ? Jˆ ⊗ E · E ⊗ S Jˆ0 Jˆ
Anisotropic h and p refinement for conforming FEM in 3D – p.25/48
S Matrices: Tensor-Product in 3D
Same idea as in 2D, just of this form:
S K̂ 0 K̂ =
Y
(A ⊗ B ⊗ C)
in each of the factors, one of A, B or C is an 1D S matrix.
Depending on the factors, 7 subdivisions are possible:
Concepts: allow arbitrary number and combination of these 7 subdivisions
in 3D.
Anisotropic h and p refinement for conforming FEM in 3D – p.26/48
Overview
• Introduction
• Anisotropic h refinements
• S and T matrices
•
Assembly of Supports
• Anisotropic p refinements
• hp Meshes
• Perspectives
Anisotropic h and p refinement for conforming FEM in 3D – p.27/48
Anisotropic and Conforming
Main point: find the cells (either coarse or fine) which belong to the
support of a certain basis function.
Anisotropic h and p refinement for conforming FEM in 3D – p.28/48
Anisotropic and Conforming
Main point: find the cells (either coarse or fine) which belong to the
support of a certain basis function.
Anisotropic h and p refinement for conforming FEM in 3D – p.28/48
Anisotropic and Conforming
Main point: find the cells (either coarse or fine) which belong to the
support of a certain basis function.
Anisotropic h and p refinement for conforming FEM in 3D – p.28/48
Anisotropic and Conforming
Main point: find the cells (either coarse or fine) which belong to the
support of a certain basis function.
Anisotropic h and p refinement for conforming FEM in 3D – p.28/48
Anisotropic and Conforming
Main point: find the cells (either coarse or fine) which belong to the
support of a certain basis function.
Anisotropic h and p refinement for conforming FEM in 3D – p.28/48
Anisotropic and Conforming
Main point: find the cells (either coarse or fine) which belong to the
support of a certain basis function.
Anisotropic h and p refinement for conforming FEM in 3D – p.28/48
Anisotropic and Conforming II
acements
• Can easily be treated since all edges are
broken
Anisotropic h and p refinement for conforming FEM in 3D – p.29/48
Anisotropic and Conforming II
acements
• Can easily be treated since all edges are
broken
• “Level of refinement” on each cell is enough to
handle hanging nodes
Anisotropic h and p refinement for conforming FEM in 3D – p.29/48
Anisotropic and Conforming II
acements
• Can easily be treated since all edges are
broken
• “Level of refinement” on each cell is enough to
handle hanging nodes
• More complicated as not all
edges are broken
acements
PSfrag replacements
Anisotropic h and p refinement for conforming FEM in 3D – p.29/48
Anisotropic and Conforming II
acements
acements
PSfrag replacements
• Can easily be treated since all edges are
broken
• “Level of refinement” on each cell is enough to
handle hanging nodes
• More complicated as not all
edges are broken
• “Level of refinement” (also a
vector valued level) is not
enough
Anisotropic h and p refinement for conforming FEM in 3D – p.29/48
Anisotropic and Conforming II
acements
acements
PSfrag replacements
• Can easily be treated since all edges are
broken
• “Level of refinement” on each cell is enough to
handle hanging nodes
• More complicated as not all
edges are broken
• “Level of refinement” (also a
vector valued level) is not
enough
PSfrag replacements
•
conforming
should be seen as
Anisotropic h and p refinement for conforming FEM in 3D – p.29/48
Condition for Continuity
• In order to have continuous global basis functions Φi , the unisolvent
sets on the interfaces in the support of Φi have to match.
Anisotropic h and p refinement for conforming FEM in 3D – p.30/48
Condition for Continuity
• In order to have continuous global basis functions Φi , the unisolvent
sets on the interfaces in the support of Φi have to match.
• Unisolvent set for Q1 in a quad / hex are the corners:
⇒ matching edges which coincide in a vertex is sufficient
Anisotropic h and p refinement for conforming FEM in 3D – p.30/48
Condition for Continuity
• In order to have continuous global basis functions Φi , the unisolvent
sets on the interfaces in the support of Φi have to match.
• Unisolvent set for Q1 in a quad / hex are the corners:
⇒ matching edges which coincide in a vertex is sufficient
• Unisolvent set for Qp are
• additional p − 1 points on every edge
• additional (p − 1)2 points on every face
• additional (p − 1)3 points in the interior
⇒ matching faces which coincide in an edge is sufficient for
continuous edge modes
Anisotropic h and p refinement for conforming FEM in 3D – p.30/48
Algorithm for Continuity (Vertex)
• In every cell of the finest mesh, register all edges and cells in their
vertices.
Anisotropic h and p refinement for conforming FEM in 3D – p.31/48
Algorithm for Continuity (Vertex)
• In every cell of the finest mesh, register all edges and cells in their
vertices.
• For every vertex, while something changed in the last loop:
• Check if some of the edges of the vertex have a relationship
(ancestor / descendant).
Anisotropic h and p refinement for conforming FEM in 3D – p.31/48
Algorithm for Continuity (Vertex)
• In every cell of the finest mesh, register all edges and cells in their
vertices.
• For every vertex, while something changed in the last loop:
• Check if some of the edges of the vertex have a relationship
(ancestor / descendant).
• If two edges are related, exchange the smaller cell in the list of
the vertex by the cell matching the larger cell.
• Delete the list of edges and rebuild it from the list of cells.
Anisotropic h and p refinement for conforming FEM in 3D – p.31/48
Overview
• Introduction
• Anisotropic h refinements
• S and T matrices
• Assembly of Supports
•
Anisotropic p refinements
• hp Meshes
• Perspectives
Anisotropic h and p refinement for conforming FEM in 3D – p.32/48
Anisotropic p
Why anisotropic p? Necessary for thin plates, shells, films.
• Every edge has p, every face has p = (p0 , p1 ), every cell has
p = (p0 , p1 , p2 ). They can differ in a cell!
• The minimum rule for edges and faces is enforced.
PSfrag replacements
K
K0
The common edge of K and K 0 has p = min {pK , pK 0 }.
• Higher p on an edge than neighbouring elements prescribe is
possible!
Anisotropic h and p refinement for conforming FEM in 3D – p.33/48
p Enrichment on Edges
• p? ≥ 2 for the basis functions Φi on the marked edge must be
possible to achieve exponential convergence.
3
PSfrag replacements
2
1
3
2
2
3
Analogly for edge and faces in 3D.
Anisotropic h and p refinement for conforming FEM in 3D – p.34/48
p Enrichment on Edges
• p? ≥ 2 for the basis functions Φi on the marked edge must be
possible to achieve exponential convergence.
• The basis functions on the red edges contribute to Φi
⇒ p ≥ p? must be possible and enforced.
3
PSfrag replacements
2
1
3
2
2
3
Analogly for edge and faces in 3D.
Anisotropic h and p refinement for conforming FEM in 3D – p.34/48
Trunk Spaces
• Tensor Product Space: p3 shape functions, internal shape functions
have indices
i = 2, . . . , pξ ,
j = 2, . . . , pη and
k = 2, . . . , pζ
• Trunk Space: O(p2 ) shape functions, internal shape functions have
indices
i = 2, . . . , pξ − 4,
j = 2, . . . , pη − 4 and
k = 2, . . . , pζ − 4 where i + j + k = 6, . . . , max {pξ , pη , pζ }, [2].
[2] Szabó and Babuška, “Finite Element Analysis”, John Wiley & Sons,
1991.
Anisotropic h and p refinement for conforming FEM in 3D – p.35/48
Overview
• Introduction
• Anisotropic h refinements
• S and T matrices
• Assembly of Supports
• Anisotropic p refinements
•
hp Meshes
• Perspectives
Anisotropic h and p refinement for conforming FEM in 3D – p.36/48
Some Basis Functions
Anisotropic h and p refinement for conforming FEM in 3D – p.37/48
Some Basis Functions
Anisotropic h and p refinement for conforming FEM in 3D – p.37/48
Some Basis Functions
Anisotropic h and p refinement for conforming FEM in 3D – p.37/48
Some Basis Functions
Anisotropic h and p refinement for conforming FEM in 3D – p.37/48
Some Basis Functions
Anisotropic h and p refinement for conforming FEM in 3D – p.37/48
Exponential Convergence in Pseudo-3D
Edge type singularity.
relative energy error
0.1
0.01
0.001
0.0001
1e-05
2
4
6
8
10
12
14
16
18
dof^1/3
−∆u + u = f in Ω = (−1, 1) × (0, 1) × (0, 1/2)
√
u(r, φ, z) = r sin(φ/2)z(1 − z)
u=0
in Ω
on {z = 0} ⊂ ∂Ω
and on {y = 0} ∩ {x ≥ 0} ⊂ ∂Ω
Anisotropic h and p refinement for conforming FEM in 3D – p.38/48
Exponential Convergence in 3D
0.1
relative energy error
0.01
0.001
0.0001
1e-05
1e-06
1e-07
2
3
4
5
6
7
dof^1/4
8
9
10
11
Vertex type singularity.
1000
100
time [s]
10
−∆u + u = f in Ω = (0, 1)3
√
u(r, θ, φ) = r sin θ sin φ
1
0.1
mesh
solve
integration
0.01
0.001
10
100
1000
dof
10000
100000
u=0
in Ω
on {y = 0} ⊂ ∂Ω
Anisotropic h and p refinement for conforming FEM in 3D – p.39/48
Exp. Conv. in 3D, Edge Mesh
0.1
relative energy error
0.01
0.001
0.0001
1e-05
1e-06
1
2
3
4
5
6
7
8
dof^1/5
Vertex type singularity.
−∆u + u = f in Ω = (0, 1)3
√
u(r, θ, φ) = r sin θ sin φ
u=0
in Ω
on {y = 0} ⊂ ∂Ω
Anisotropic h and p refinement for conforming FEM in 3D – p.40/48
Maxwell EVP
Find Eigenvalues λ = ω 2 such that ∃(E, H) 6= 0 satisfying
curl E − iωµH = 0
and
curl H + iωεE = 0
in Ω,
with perfect conductor b.c. E × n = 0, H · n = 0 on ∂Ω. E ∈ H0 (curl; Ω).
Anisotropic h and p refinement for conforming FEM in 3D – p.41/48
Maxwell EVP
Find Eigenvalues λ = ω 2 such that ∃(E, H) 6= 0 satisfying
and
curl E − iωµH = 0
curl H + iωεE = 0
in Ω,
with perfect conductor b.c. E × n = 0, H · n = 0 on ∂Ω. E ∈ H0 (curl; Ω).
“Electric” variational form:
Find the frequencies ω > 0 such that ∃E ∈ H0 (curl; Ω) \ {0} with
Z
1/µ curl E
Ω
· curl F = ω 2
Z
Ω
εE · F and div εE = 0 ∀F ∈ H0 (curl; Ω).
Anisotropic h and p refinement for conforming FEM in 3D – p.41/48
Weighted Regularization for Maxwell EVP
Find the frequencies ω > 0 such that ∃u ∈ XN with
Z
Ω
curl u · curl v + hu, viY = ω 2
Z
Ω
u·v
∀v ∈ XN := {u ∈ H0 (curl; Ω) : div u ∈ L2 (Ω)}
Anisotropic h and p refinement for conforming FEM in 3D – p.42/48
Weighted Regularization for Maxwell EVP
Find the frequencies ω > 0 such that ∃u ∈ XN with
Z
Ω
curl u · curl v + hu, viY = ω 2
Z
Ω
u·v
∀v ∈ XN := {u ∈ H0 (curl; Ω) : div u ∈ L2 (Ω)}
Z
hu, viY = s
ρ(x) div u div v
Ω
Properly chosen weight ρ(x) and s ∈ R+ .
Anisotropic h and p refinement for conforming FEM in 3D – p.42/48
Weighted Regularization for Maxwell EVP
Find the frequencies ω > 0 such that ∃u ∈ XN with
Z
Ω
curl u · curl v + hu, viY = ω 2
Z
Ω
u·v
∀v ∈ XN := {u ∈ H0 (curl; Ω) : div u ∈ L2 (Ω)}
Z
hu, viY = s
ρ(x) div u div v
Ω
Properly chosen weight ρ(x) and s ∈ R+ . Good choice: ρ(x) = r α where
r is the distance to a reentrant corner and α ≥ 0 in a range depending on
the angle of the reentrant corner.
[3] Martin Costabel and Monique Dauge, “Weighted regularization of
Maxwell equations in polyhedral domains”, Numer. Math. 93 (2),
pp. 239–277 (2002).
Anisotropic h and p refinement for conforming FEM in 3D – p.42/48
EVP in the Thick L Shaped Domain
0.1
0.01
relativ error in eigenvalue
0.001
0.0001
1e-05
1e-06
1e-07
1e-08
1e-09
1e-10
ev 1
ev 2
ev 3
ev 4
ev 5
ev 6
ev 7
ev 8
ev 9
5
10
15
20
25
dof^1/3
24
σ = 0.15
22
eigenvalue
20
α=2
18
16
14
12
10
8
0
20
40
60
s
80
100
Anisotropic h and p refinement for conforming FEM in 3D – p.43/48
EVP in the Thick L Shaped Domain
0.1
0.01
relativ error in eigenvalue
0.001
0.0001
1e-05
1e-06
1e-07
1e-08
1e-09
1e-10
ev 1
ev 2
ev 3
ev 4
ev 5
ev 6
ev 7
ev 8
ev 9
5
10
15
20
25
30
dof^1/3
24
σ = 0.5
22
eigenvalue
20
α=2
18
16
14
12
10
8
30
35
40
45
s
50
55
60
Anisotropic h and p refinement for conforming FEM in 3D – p.44/48
Perspectives
• Maxwell EVP in the Fichera corner
• Anisotropic error estimation,
anistropic regularity estimation
• Improved mesh handling
• Iterative multilevel domain decompositioning solvers:
Toselli (Zürich), Schöberl (Linz)
• Stochastic Eigenvalue Problems (e.g. stochastic ε and µ for
Maxwell)
Anisotropic h and p refinement for conforming FEM in 3D – p.45/48
Perspectives
• Maxwell EVP in the Fichera corner
• Anisotropic error estimation,
anistropic regularity estimation
• Improved mesh handling
• Iterative multilevel domain decompositioning solvers:
Toselli (Zürich), Schöberl (Linz)
• Stochastic Eigenvalue Problems (e.g. stochastic ε and µ for
Maxwell)
Anisotropic h and p refinement for conforming FEM in 3D – p.45/48
Hanging Nodes in Isotropic Meshes
• Traverse all cells on locally finest level: mark every vertex / edge /
face being used.
• On next (hierarchical) traversal of the mesh:
• Add dofs which are marked to be on the current level to the list
L of local dofs. Mark dof as registered.
• If cell is on finest level L → T matrix
• Otherwise S · L is added to L of child (next deeper level)
Anisotropic h and p refinement for conforming FEM in 3D – p.46/48
Mortar
• Give up C 0 , introduce Lagrange multiplier (the mortar)
• −∆u = f in Ω with hom. Dirichlet bc. using mortar method leads to
A Λ
f
u
·
=
, ie.
λ
0
Λ> 0
SPD PDE 6⇒ SPD matrix
⇒ conjugate gradients not applicable
⇒ no standard domain decompositioning solvers
⇒ inf-sup condition needed
• The inf-sup cond. is OK in 2D, 3D for shape regular meshes.
Not OK for hp FEM, existing proofs only for uniform meshes.
• Analogly for Discontinous Galerkin in 3D: Stability of hp DG on
geometric meshes is not clear. First results by Schwab, Toselli,
Schötzau for Stokes (not Mortar).
Anisotropic h and p refinement for conforming FEM in 3D – p.47/48
Shape Functions
The reference element shape functions on (−1, 1) of order p [4]:
 1−ξ
 2
i=0
1+ξ 1,1
Ni (ξ) = 1−ξ
2
2 Pi−1 (ξ) 1 ≤ i ≤ p − 1
 1+ξ
i=p
2
1,1
(ξ) are integrated Legendre Polynomials: Li (ξ) = Pi0,0 (ξ) and
Pi−1
Z
ξ
−1
α
(1 − x) (1 + x)
⇒
Z
β
ξ
−1
Piα,β (x) dx
−1
α+1,β+1
(1 − ξ)α+1 (1 + ξ)β+1 Pi−1
=
(ξ)
2i
Pi0,0 (x) dx =
−1
1,1
(ξ)
(1 − ξ)(1 + ξ)Pi−1
2i
[4] Karniadakis and Sherwin, “Spectral/hp Element Methods for CFD”, Oxford University Press, 1999.
Anisotropic h and p refinement for conforming FEM in 3D – p.48/48
Shape Functions
The reference element shape functions on (−1, 1) of order p [4]:
 1−ξ
 2
i=0
1+ξ 1,1
Ni (ξ) = 1−ξ
2
2 Pi−1 (ξ) 1 ≤ i ≤ p − 1
 1+ξ
i=p
2
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0
-0.2
-0.2
-0.2
-0.4
-0.4
-1
-0.5
0
0.5
1
-1
-0.5
x
0
0.5
1
-0.4
1
1
1
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0
-0.2
-0.2
-0.2
-1
-0.5
0
x
-0.5
0.5
1
-0.4
-1
-0.5
0
x
0
0.5
1
0.5
1
x
0.8
-0.4
-1
x
0.5
1
-0.4
-1
-0.5
0
x
Anisotropic h and p refinement for conforming FEM in 3D – p.48/48