MATH 676
–
Finite element methods in
scientific computing
Wolfgang Bangerth, Texas A&M University
http://www.dealii.org/
Wolfgang Bangerth
Lecture 21.6:
Boundary conditions
Part 3a: Homogenous Dirichlet
boundary conditions
http://www.dealii.org/
Wolfgang Bangerth
Zero Dirichlet conditions
Consider this simple example:
●
Solve the Laplace equation with zero boundary values:
−Δ u = f
u = 0
●
in Ω
on Γ=∂ Ω
Assume we use the following mesh:
3
2
1
0
9
4
5
6
7
http://www.dealii.org/
{1,5,6}:
“real” degrees of
freedom
{0,2…4,7…9}: Constrained
to zero
8
Wolfgang Bangerth
Zero Dirichlet conditions
There are three approaches:
●
●
●
Consider only “free” degrees of freedom
Consider all degrees of freedom
– do local assembly without boundary conditions
– copy to global system
– eliminate boundary DoFs
Consider all degrees of freedom
– do local assembly without boundary conditions
– eliminate boundary DoFs
– copy to global system
Criteria: All approaches are correct; which one we like
best is determined by considerations of algorithm design!
http://www.dealii.org/
Wolfgang Bangerth
Zero Dirichlet conditions
Approach 1:
●
●
Don't enumerate constrained degrees of freedom
Only enumerate the 3 unconstrained ones:
1→0,
5→1,
6→2
3
2
1
0
9
4
5
6
7
http://www.dealii.org/
0
X
X
8
X
X
X
1
2
X
X
Wolfgang Bangerth
Zero Dirichlet conditions
Approach 1:
●
Then assemble a 3x3 linear system as usual:
(φi , f ) K
∑ j=0 [⏟
∑K =0 (∇ φi , ∇ φ j)K ] U j=∑
K =0
⏟
2
11
11
A ij
3
2
1
0
9
6
7
http://www.dealii.org/
Fi
0
X
X
8
X
X
4
5
∀ i=0 ...2
X
1
2
X
X
Wolfgang Bangerth
Zero Dirichlet conditions
Advantages of approach 1:
●
●
All degrees of freedom are really “free” (i.e., determined
by a linear system)
Linear system is much smaller (3x3 vs 10x10)
3
2
1
0
9
5
6
7
http://www.dealii.org/
0
X
X
8
X
X
4
X
1
2
X
X
Wolfgang Bangerth
Zero Dirichlet conditions
Advantages of approach 1:
●
●
All degrees of freedom are really “free” (i.e., determined
by a linear system)
Linear system is much smaller (3x3 vs 10x10)
But: This size reduction doesn't matter on finer meshes!
1 vs 9 DoFs
http://www.dealii.org/
9 vs 25 DoFs
49 vs 81 DoFs
Wolfgang Bangerth
Zero Dirichlet conditions
Disdvantages of approach 1:
●
The FEM is at its best if you can do the same on every
cell:
– when enumerating degrees of freedom
– when assembling
– when evaluating the solution at individual points
–…
X
X
0
X
X
1
2
X
http://www.dealii.org/
X
X
Wolfgang Bangerth
Zero Dirichlet conditions
Disdvantages of approach 1:
●
If Dirichlet conditions only on parts of the boundary:
Some boundary nodes would have to be treated
differently than others.
6
4
0
X
X
1
2
X
http://www.dealii.org/
5
X
Wolfgang Bangerth
Zero Dirichlet conditions
Disdvantages of approach 1:
●
What to do in cases of vector-valued problems and
conditions of the sort u.n=0?
●
Example: Fluid flow with velocity components (ux,uy)
●
Boundary condition here is nxux=-nyuy
2
3
0
1
8
9
10
11
12
13
14
15
18
19
http://www.dealii.org/
2
X
6
7
4
5
X
1
8
9
10
11
X
13
16
17
X
7
4
X
14
15
18
X
16
X
Wolfgang Bangerth
Zero Dirichlet conditions
Approach 2:
●
Enumerate all degrees of freedom regardless of boundary
conditions
Variation 2a:
●
●
●
●
Compute local contributions regardless
of boundary DoFs
Assemble into one big linear system
Zero out rows and columns
for boundary DoFs {0,2,3,4,7,8,9}
Fix up zero rows
3
2
1
0
9
5
6
7
http://www.dealii.org/
4
8
Wolfgang Bangerth
Zero Dirichlet conditions
Example for approach 2a:
●
●
Compute local contributions regardless of boundary DoFs
Assemble into one big linear system
(forget about sparsity for a moment)
(
a00
a10
a20
a30
a40
a50
a60
a70
a80
a90
a01
a11
a21
a31
a41
a51
a61
a71
a81
a91
a02
a12
a22
a32
a42
a52
a62
a72
a82
a92
a03
a13
a23
a33
a43
a53
a63
a73
a83
a93
a04
a14
a24
a34
a44
a54
a64
a74
a84
a94
http://www.dealii.org/
a05
a15
a25
a35
a 45
a55
a65
a75
a85
a 95
a 06
a16
a 26
a 36
a 46
a56
a 66
a76
a 86
a 96
a07
a17
a27
a37
a 47
a57
a67
a77
a87
a97
a08
a18
a28
a38
a48
a58
a68
a78
a88
a98
a09
a19
a29
a39
a49
a59
a69
a79
a89
a99
u0
f0
u1
f1
u2
f2
u3
f3
u4
f
= 4
u5
f5
u6
f6
u7
f7
u8
f8
u9
f9
)( ) ( )
3
2
1
0
9
4
5
6
7
8
Wolfgang Bangerth
Zero Dirichlet conditions
Example for approach 2a:
●
Compute local contributions regardless of boundary DoFs
●
Assemble into one big linear system
●
Zero out rows and columns for DoFs {0,2,3,4,7,8,9}
(
a00
a10
a20
a30
a40
a50
a60
a70
a80
a90
a01
a11
a21
a31
a41
a51
a61
a71
a81
a91
a02
a12
a22
a32
a42
a52
a62
a72
a82
a92
a03
a13
a23
a33
a43
a53
a63
a73
a83
a93
a04
a14
a24
a34
a44
a54
a64
a74
a84
a94
http://www.dealii.org/
a05
a15
a25
a35
a 45
a55
a65
a75
a85
a 95
a 06
a16
a 26
a 36
a 46
a56
a 66
a76
a 86
a 96
a07
a17
a27
a37
a 47
a57
a67
a77
a87
a97
a08
a18
a28
a38
a48
a58
a68
a78
a88
a98
a09
a19
a29
a39
a49
a59
a69
a79
a89
a99
u0
f0
u1
f1
u2
f2
u3
f3
u4
f
= 4
u5
f5
u6
f6
u7
f7
u8
f8
u9
f9
)( ) ( )
3
2
1
0
9
4
5
6
7
8
Wolfgang Bangerth
Zero Dirichlet conditions
Example for approach 2a:
●
Compute local contributions regardless of boundary DoFs
●
Assemble into one big linear system
●
Zero out rows and columns for DoFs {0,2,3,4,7,8,9}
The resulting system has a 3x3 sub-block for “free” DoFs
(
0 0
0 a11
0 0
0 0
0 0
0 a51
0 a61
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0 a15 a16
0 0
0
0 0
0
0 0
0
0 a55 a56
0 a65 a66
0 0
0
0 0
0
0 0
0
http://www.dealii.org/
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
u0
0
u1
f1
u2
0
u3
0
u4
0
=
f5
u5
f6
u6
0
u7
0
u8
0
u9
)( ) ( )
3
2
1
0
9
4
5
6
7
8
Wolfgang Bangerth
Zero Dirichlet conditions
Example for approach 2a:
●
Compute local contributions regardless of boundary DoFs
●
Assemble into one big linear system
●
Zero out rows and columns for DoFs {0,2,3,4,7,8,9}
●
Fix under-determination:
(
1 0
0 a11
0 0
0 0
0 0
0 a51
0 a61
0 0
0 0
0 0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0 0
0
0 a15 a16
0 0
0
0 0
0
1 0
0
0 a55 a56
0 a65 a66
0 0
0
0 0
0
0 0
0
http://www.dealii.org/
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
u0
0
u1
f1
u2
0
u3
0
u4
0
=
f5
u5
f6
u6
0
u7
0
u8
0
u9
)( ) ( )
3
2
1
0
9
4
5
6
7
8
Wolfgang Bangerth
Zero Dirichlet conditions
Result for approach 2a:
We get a linear system of the form
(
1
a11
a15 a16
1
1
1
a51
a61
a55 a56
a65 a66
u0
0
u1
f1
u2
0
u3
0
u4
0
=
f5
u5
f6
u6
1
0
u7
1
0
u8
1 u
0
9
)( ) ( )
The solution satisfies
U 0 =U 2=U 3 =U 4 =U 7 =U 8 =U 9=0
3
2
1
0
9
5
6
7
http://www.dealii.org/
4
8
Wolfgang Bangerth
Zero Dirichlet conditions
Result for approach 2a:
We get a linear system of the form
(
1
a11
a15 a 16
1
1
1
a 51
a 61
a 55 a 56
a 65 a 66
u0
0
u1
f1
u2
0
u3
0
u4
0
=
f5
u5
f6
u6
1
0
u7
1
0
u8
1 u
0
9
The solution satisfies
(
)( ) ( )
a 11 a15 a16 u1
f1
a51 a55 a56 u 5 = f 5
a61 a65 a66 u 6
f6
http://www.dealii.org/
)( ) ( )
3
2
1
0
9
4
5
6
7
8
Wolfgang Bangerth
Zero Dirichlet conditions
Summary approach 2a:
●
Compute local contributions regardless of boundary DoFs
●
Assemble into one big linear system
●
Zero out rows and columns for boundary DoFs
●
Fix up zero rows
2
1
This results in a linear system that:
●
Has the desired solution
9
6
●
Is invertible
7
●
Has a few more zeros in it
●
Is symmetric if the original matrix was symmetric
●
Is SPD if the original matrix was SPD
0
http://www.dealii.org/
3
4
5
8
Wolfgang Bangerth
Zero Dirichlet conditions
Implementation approach 2a (step-4):
void Step4::assemble_system {
…;
for (cell=...) {
...assemble cell_matrix, cell_rhs...;
...add cell_matrix, cell_rhs to system_matrix, system_rhs...;
}
std::map<types::global_dof_index,double> boundary_values;
VectorTools::interpolate_boundary_values (dof_handler, 0,
ZeroFunction<dim>(),
boundary_values);
MatrixTools::apply_boundary_values (boundary_values,
system_matrix, solution, system_rhs);
}
http://www.dealii.org/
Wolfgang Bangerth
Zero Dirichlet conditions
Summary approach 2a:
●
Compute local contributions regardless of boundary DoFs
●
Assemble into one big linear system
●
Zero out rows and columns for boundary DoFs
●
Fix up zero rows
2
3
4
1
5
Advantages:
●
Can do the exact same thing on
9
6
every cell during assembly/postprocessing
7
●
Relegate handling b.c. to a separate
8
part of the code, independent of assembly
●
Can use same solver regardless of boundary conditions
0
http://www.dealii.org/
Wolfgang Bangerth
Zero Dirichlet conditions
Summary approach 2a:
●
Compute local contributions regardless of boundary DoFs
●
Assemble into one big linear system
●
Zero out rows and columns for boundary DoFs
●
Fix up zero rows
Disadvantages:
●
We have a few more rows and columns
●
We have to modify the linear system
after global assembly
http://www.dealii.org/
3
2
1
0
9
4
5
6
7
8
Wolfgang Bangerth
Zero Dirichlet conditions
Approach 2:
●
Enumerate all degrees of freedom regardless of boundary
conditions
Variation 2b:
●
●
●
●
Compute local contributions regardless
of boundary DoFs
Zero out rows and columns
for boundary DoFs {0,2,3,4,7,8,9}
Fix up zero rows
Assemble into one big linear system
3
2
1
0
9
5
6
7
http://www.dealii.org/
4
8
Wolfgang Bangerth
Zero Dirichlet conditions
Example for approach 2b:
Recall that the linear system
AU = F
is computed from contributions of all cells:
A =
F =
10
∑K =0
10
AK
∑K =0 F
3
2
K
0
0
7
8
1
4
10
5
1
6
2
9
9
6
5
3
7
http://www.dealii.org/
4
8
Wolfgang Bangerth
Zero Dirichlet conditions
Example for approach 2b:
Recall that the linear system
AU = F
is computed from contributions of all cells. For example:
(
1
1
1
1
1
a00 a 01 0 0 0 0 a06 0 0 0
a10 a11
0
0
0
0
1
0
A= 0
0
0
1
1
a60 a 61
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0 a16
0 0
0 0
0 0
0 0
1
0 a66
0 0
0 0
0 0
http://www.dealii.org/
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
) ()
0
0
0
0 ,
0
0
0
0
0
f0
1
f1
0
0
1
F = 0
0
1
f6
0
0
0
3
2
0
0
7
8
1
4
10
5
1
6
2
9
9
6
5
3
7
4
8
Wolfgang Bangerth
Zero Dirichlet conditions
Example for approach 2b:
Recall that the linear system
AU = F
is computed from contributions of all cells.
In reality, of course, we only store the
dense 3x3 sub-block:
(
1
1
1
0
1
) ()
a00 a01 a06
1, loc
A = a110 a111 a116 ,
1
1
1
a60 a61 a66
f0
1, loc
F = f 11
1
f6
3
2
0
7
8
1
4
10
5
1
6
2
9
9
6
5
3
7
http://www.dealii.org/
4
8
Wolfgang Bangerth
Zero Dirichlet conditions
Example for approach 2b:
Recall that the linear system
AU = F
is computed from contributions of all cells.
Zero out rows and columns DoFs {0,2,3,4,7,8,9}:
3
2
(
1
1
1
0
1
) ()
a00 a01 a06
1, loc
A = a110 a111 a116 ,
1
1
1
a60 a61 a66
f0
1, loc
F = f 11
1
f6
0
7
8
1
4
10
5
1
6
2
9
9
6
5
3
7
http://www.dealii.org/
4
8
Wolfgang Bangerth
Zero Dirichlet conditions
Example for approach 2b:
Recall that the linear system
AU = F
is computed from contributions of all cells.
Zero out rows and columns DoFs {0,2,3,4,7,8,9}:
3
2
0 0
0
1, loc
1
A = 0 a11 a116 ,
1
1
0 a61 a66
(
)
0
1, loc
F = f 11
1
f6
()
0
0
7
8
1
4
10
5
1
6
2
9
9
6
5
3
7
http://www.dealii.org/
4
8
Wolfgang Bangerth
Zero Dirichlet conditions
Example for approach 2b:
Recall that the linear system
AU = F
is computed from contributions of all cells.
Fix up under-determinedness:
3
2
1 0
0
1, loc
1
A = 0 a11 a116 ,
1
1
0 a61 a66
(
)
0
1, loc
F = f 11
1
f6
()
0
0
7
8
1
4
10
5
1
6
2
9
9
6
5
3
7
http://www.dealii.org/
4
8
Wolfgang Bangerth
Zero Dirichlet conditions
Example for approach 2b:
Recall that the linear system
AU = F
is computed from contributions of all cells.
Then add to global linear system:
A := A +expand ( A
1, loc
0
F := F +expand ( F
1, loc
)
3
2
)
0
7
8
1
4
10
5
1
6
2
9
9
6
5
3
7
http://www.dealii.org/
4
8
Wolfgang Bangerth
Zero Dirichlet conditions
Summary approach 2b:
●
Compute local contributions regardless of boundary DoFs
●
Zero out rows and columns for boundary DoFs
●
Fix up zero rows
●
Assemble into one big linear system
Actual implementation:
●
The last three operations are all done
at the same time
●
The local contributions are not modified
http://www.dealii.org/
3
2
1
0
9
4
5
6
7
8
Wolfgang Bangerth
Zero Dirichlet conditions
Implementation approach 2b (step-6):
void Step6::assemble_system {
ConstraintMatrix constraints;
VectorTools::interpolate_boundary_values (dof_handler, 0,
ZeroFunction<dim>(),
constraints);
…;
for (cell=...) {
...assemble cell_matrix, cell_rhs...;
cell­>get_dof_indices (local_dof_indices);
constraints.distribute_local_to_global (cell_matrix, cell_rhs,
local_dof_indices,
system_matrix, system_rhs);
}
}
http://www.dealii.org/
Wolfgang Bangerth
Zero Dirichlet conditions
Summary approach 2b:
●
Compute local contributions regardless of boundary DoFs
●
Zero out rows and columns for boundary DoFs
●
Fix up zero rows
●
Assemble into one big linear system
2
3
4
1
5
Advantages:
●
Can do the exact same thing on
9
6
every cell during assembly/postprocessing
7
●
Relegate handling b.c. to a separate
8
part of the code, independent of assembly
●
Can use same solver regardless of boundary conditions
0
http://www.dealii.org/
Wolfgang Bangerth
Zero Dirichlet conditions
Summary approach 2b:
●
Compute local contributions regardless of boundary DoFs
●
Zero out rows and columns for boundary DoFs
●
Fix up zero rows
●
Assemble into one big linear system
Disdvantages:
●
We have a few more rows and columns
●
We have to modify the linear system
after local integration or during
copy-local-to-global
(both options are easy and cheap)
http://www.dealii.org/
3
2
1
0
9
4
5
6
7
8
Wolfgang Bangerth
Summary
We are driven by algorithm design considerations:
●
We want to do the exact same thing on every cell
●
We want to isolate handling of boundary values to as few
code pieces as possible
●
Dealing with small, dense, local
contributions is simpler than with
global linear systems
3
2
1
0
9
5
6
Consequently:
7
●
Enumerate all degrees of freedom
●
Integrate locally regardless of boundary conditions
●
Deal with boundary values during assembly or after
http://www.dealii.org/
4
8
Wolfgang Bangerth
A final implementation detail
As a postscript:
Consider a linear system of the form
(
1
a11
a15 a 16
1
1
1
a 51
a 61
a 55 a 56
a 65 a 66
This matrix is ill-conditioned if
u0
0
u1
f1
u2
0
u3
0
u4
0
=
f5
u5
f6
u6
1
0
u7
1
0
u8
1 u
0
9
)( ) ( )
|aij|≫1 or |aij|≪1
Consequence:
●
We should not place 1.0 on the diagonal
●
Algorithm instead use values “similar to other entries”
http://www.dealii.org/
Wolfgang Bangerth
MATH 676
–
Finite element methods in
scientific computing
Wolfgang Bangerth, Texas A&M University
http://www.dealii.org/
Wolfgang Bangerth