Hat–Function
defined on a triangulation of the domain D
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 1
Hat–Function
defined on a triangulation of the domain D
Bi , i ∈ I : basis for piecewise linear functions
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 1
Hat–Function
defined on a triangulation of the domain D
Bi , i ∈ I : basis for piecewise linear functions
Bi (pi ) = 1, 0 at other vertices pj
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 1
Hat–Function
defined on a triangulation of the domain D
Bi , i ∈ I : basis for piecewise linear functions
Bi (pi ) = 1, 0 at other vertices pj
Bi
pi
τ
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 1
Hat–Function
defined on a triangulation of the domain D
Bi , i ∈ I : basis for piecewise linear functions
Bi (pi ) = 1, 0 at other vertices pj
Bi
pi
τ
approximation, determined by Lagrange data
X
uh =
ui Bi , ui = uh (pi )
i∈I
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 1
Ritz-Galerkin System for Hat–Functions
system entries: sum contributions from each triangle τ = [pi , pj , pk ]
XZ
XZ
g`,m =
grad B` grad Bm , f` =
f B`
τ
SIAM FR26:
FEM with B-Splines
τ
τ
τ
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 2
Ritz-Galerkin System for Hat–Functions
system entries: sum contributions from each triangle τ = [pi , pj , pk ]
XZ
XZ
g`,m =
grad B` grad Bm , f` =
f B`
τ
τ
τ
compute gradients via directional derivatives


grad Bi
 grad Bj  pj − pi pk − pj = R,
grad Bk
|
{z
}
τ


−1 0
R =  1 −1 
0
1
Gτ
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 2
G : add submatrix of
area(τ ) Gτ Gτt =
| det P|
RP −1 (P t )−1 R t ,
2
corresponding to inner vertices
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 3
G : add submatrix of
area(τ ) Gτ Gτt =
corresponding to inner vertices
F : add subvector of

| det P| 
6
| det P|
RP −1 (P t )−1 R t ,
2
1
2
1
4
1
4
1
4
1
2
1
4
1
4
1
4
1
2


fi
  fj 
fk
corresponding to inner vertices
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 3
Details
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 4
Details
exact Taylor approximation for linear functions
B(q) = B(p) + grad B (q − p)
{z
}
|
directional derivative
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 4
Details
exact Taylor approximation for linear functions
B(q) = B(p) + grad B (q − p)
{z
}
|
directional derivative
quadrature formula for triangle τ = [pi , pj , pk ]
Z
area τ
g=
[g ((pi + pj )/2) + g ((pj + pk )/2) + g ((pk + pi )/2)]
3
τ
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 4
Details
exact Taylor approximation for linear functions
B(q) = B(p) + grad B (q − p)
{z
}
|
directional derivative
quadrature formula for triangle τ = [pi , pj , pk ]
Z
area τ
g=
[g ((pi + pj )/2) + g ((pj + pk )/2) + g ((pk + pi )/2)]
3
τ
exact for quadratic polynomials
SIAM FR26:
FEM with B-Splines
error O(h5 ), h = diam τ
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 4
Details
exact Taylor approximation for linear functions
B(q) = B(p) + grad B (q − p)
{z
}
|
directional derivative
quadrature formula for triangle τ = [pi , pj , pk ]
Z
area τ
g=
[g ((pi + pj )/2) + g ((pj + pk )/2) + g ((pk + pi )/2)]
3
τ
exact for quadratic polynomials
error O(h5 ), h = diam τ
apply with g = fB` and linear interpolation of f and B`
[. . .] = f` /2 + fm /4 + fm0 /4,
m, m0 6= ` ,
since g ((p` + pm )/2) = ((f` + fm )/2) ((1 + 0)/2)
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 4
Bivariate Finite Elements
Lagrange
Argyris
Clough–Tocher
degree 4, ∈ C 0
dimension 15
degree 5, ∈ C 1
dimension 21
degree 3, ∈ C 1
dimension 12
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 5
Lagrange Elements
R
nodes, labeled (i, j, k) with i + j + k = n
Q
j
k
i
P+ Q+ R
n
n
n
P
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 6
Lagrange Elements
R
nodes, labeled (i, j, k) with i + j + k = n
Q
j
k
i
P+ Q+ R
n
n
n
P
basis functions
Bi,j,k
SIAM FR26:
n 0
i−1 n
0
j−1 n
=
u ···u
v ···v
w 0 · · · w k−1
i
j
k
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 6
Lagrange Elements
R
nodes, labeled (i, j, k) with i + j + k = n
Q
j
k
i
P+ Q+ R
n
n
n
P
basis functions
Bi,j,k
n 0
i−1 n
0
j−1 n
=
u ···u
v ···v
w 0 · · · w k−1
i
j
k
u ` linear, with u ` (P) = 1 and u ` = 0 on nodes (`, α, β)
(v ` , w ` defined similarly)
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 6
Lagrange Elements
R
nodes, labeled (i, j, k) with i + j + k = n
Q
j
k
i
P+ Q+ R
n
n
n
P
basis functions
Bi,j,k
n 0
i−1 n
0
j−1 n
=
u ···u
v ···v
w 0 · · · w k−1
i
j
k
u ` linear, with u ` (P) = 1 and u ` = 0 on nodes (`, α, β)
(v ` , w ` defined similarly)
interpolation of Lagrange data
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 6
Details
node
X =
SIAM FR26:
FEM with B-Splines
i
j
k
P + Q + R
n
n
n
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 7
Details
node
X =
i
j
k
P + Q + R
n
n
n
u ` ((n/n)P) = 1, u ` ((`/n)P + · · · ) = 0 =⇒
i
i −`
`
u
P + ··· =
n
n−`
(view i as variable, ranging from ` to n)
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 7
Details
node
X =
i
j
k
P + Q + R
n
n
n
u ` ((n/n)P) = 1, u ` ((`/n)P + · · · ) = 0 =⇒
i
i −`
`
u
P + ··· =
n
n−`
(view i as variable, ranging from ` to n)
value of u 0 u 1 · · · u i−1 at X
1
i i −1
···
=
n n−1
n−i +1
−1
n
i
=⇒ Bi,j,k (X ) = 1
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 7
Argyris Triangle
degree 5, ∈ C 1
dimension 21
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 8
Argyris Triangle
degree 5, ∈ C 1
dimension 21
defining data
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 8
Argyris Triangle
degree 5, ∈ C 1
dimension 21
defining data
partial derivatives of order ≤ 2 at vertices
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 8
Argyris Triangle
degree 5, ∈ C 1
dimension 21
defining data
partial derivatives of order ≤ 2 at vertices
normal derivatives at edge mid-points
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 8
Argyris Triangle
degree 5, ∈ C 1
dimension 21
defining data
partial derivatives of order ≤ 2 at vertices
normal derivatives at edge mid-points
=⇒ values and derivatives prescribed at triangle boundaries
(quintic and quartic polynomials, respectively)
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 8
Clough-Tocher Macro-Triangle
degree 3, ∈ C 1
dimension 12
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 9
Clough-Tocher Macro-Triangle
degree 3, ∈ C 1
dimension 12
defining data
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 9
Clough-Tocher Macro-Triangle
degree 3, ∈ C 1
dimension 12
defining data
values and gradients at vertices
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 9
Clough-Tocher Macro-Triangle
degree 3, ∈ C 1
dimension 12
defining data
values and gradients at vertices
normal derivatives at edge mid-points
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 9
Clough-Tocher Macro-Triangle
degree 3, ∈ C 1
dimension 12
defining data
values and gradients at vertices
normal derivatives at edge mid-points
=⇒ values and derivatives prescribed along the outer boundaries
(cubic and quadratic polynomials, respectively)
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 9
Properties of Finite Elements
The basis functions Bi of standard mesh-based finite element subspaces
are piecewise polynomials of degree ≤ n with support on few neighboring
mesh cells. They are at least continuous and compatible with
homogeneous boundary conditions on piecewise linear boundaries.
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 10
Disadvantages of Standard Mesh-Based Finite Elements
principal drawbacks
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 11
Disadvantages of Standard Mesh-Based Finite Elements
principal drawbacks
Generating good meshes can be difficult, time-consuming, and might
require user interaction.
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 11
Disadvantages of Standard Mesh-Based Finite Elements
principal drawbacks
Generating good meshes can be difficult, time-consuming, and might
require user interaction.
Using quadratic or higher degree leads to excessively large systems.
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 11
Disadvantages of Standard Mesh-Based Finite Elements
principal drawbacks
Generating good meshes can be difficult, time-consuming, and might
require user interaction.
Using quadratic or higher degree leads to excessively large systems.
Only moderately accurate approximations are possible.
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 11
Disadvantages of Standard Mesh-Based Finite Elements
principal drawbacks
Generating good meshes can be difficult, time-consuming, and might
require user interaction.
Using quadratic or higher degree leads to excessively large systems.
Only moderately accurate approximations are possible.
Boundary conditions are merely approximated for general free-form
domains.
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 11
Disadvantages of Standard Mesh-Based Finite Elements
principal drawbacks
Generating good meshes can be difficult, time-consuming, and might
require user interaction.
Using quadratic or higher degree leads to excessively large systems.
Only moderately accurate approximations are possible.
Boundary conditions are merely approximated for general free-form
domains.
Weighted spline-based finite elements overcome these difficulties:
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 11
Disadvantages of Standard Mesh-Based Finite Elements
principal drawbacks
Generating good meshes can be difficult, time-consuming, and might
require user interaction.
Using quadratic or higher degree leads to excessively large systems.
Only moderately accurate approximations are possible.
Boundary conditions are merely approximated for general free-form
domains.
Weighted spline-based finite elements overcome these difficulties:
No mesh generation is required, accurate smooth approximations are
possible with relatively low-dimensional finite element subspaces, and
boundary conditions are satisfied exactly.
SIAM FR26:
FEM with B-Splines
Basic Finite Element Concepts – Mesh Based Elements
2-2, page 11