Smoothing
All degrees of freedom in the mesh must be in one, and only one, of the following smoothing categories:
- Prescribed
- Homogeneous
- Inhomogeneous
- Free
- Exterior
- Interface
- Interior
Figure: Two element test problem.
Table: Nodal coordinates 1-12, with x, y, z, degrees of freedom.
|node|x|y|z|->|||dof|
|:--:|:-:|:-:|:-:|:--:|::|::|:-:|
|1|0.0|0.0|0.0||1|2|3|
|2|1.0|0.0|0.0||4|5|6|
|3|2.0|0.0|0.0||7|8|9|
|4|0.0|1.0|0.0||10|11|12|
|5|1.0|1.0|0.0||13|14|15|
|6|2.0|1.0|0.0||16|17|18|
|7|0.0|0.0|1.0||19|20|21|
|8|1.0|0.0|1.0||22|23|24|
|9|2.0|0.0|1.0||25|26|27|
|10|0.0|1.0|1.0||28|29|30|
|11|1.0|1.0|1.0||31|32|33|
|12|2.0|1.0|1.0||34|35|36|
Table. The node neighbors.
| node | neighbor node(s) |
|---|---|
| 1 | 2, 4, 7 |
| 2 | 1, 3, 5, 8 |
| 3 | 2, 6, 9 |
| 4 | 1, 5, 10 |
| 5 | 2, 4, 6, 11 |
| 6 | 3, 5, 12 |
| 7 | 1, 8, 10 |
| 8 | 2, 7, 9, 11 |
| 9 | 3, 8, 12 |
| 10 | 4, 7, 11 |
| 11 | 5, 8, 10, 12 |
| 12 | 6, 9, 11 |
All Free
Following is a test where all degrees of freedom are and
hierarchical smoothing is OFF.
class DofType(Enum):
"""All degrees of freedom must belong to one, and only one, of the
following smoothing categories.
"""
PRESCRIBED_HOMOGENEOUS = 0
PRESCRIBED_INHOMOGENEOUS = 1
FREE_EXTERIOR = 2
FREE_INTERFACE = 3
FREE_INTERIOR = 4
dofset: DofSet = (
(4, 4, 4),
(4, 4, 4),
(4, 4, 4),
(4, 4, 4),
(4, 4, 4),
(4, 4, 4),
(4, 4, 4),
(4, 4, 4),
(4, 4, 4),
(4, 4, 4),
(4, 4, 4),
(4, 4, 4),
)
Table: Smoothed configuration (x, y, z).
| node | x | y | z |
|---|---|---|---|
| 1 | 0.1 | 0.1 | 0.1 |
| 2 | 1.0 | 0.075 | 0.075 |
| 3 | 1.9 | 0.1 | 0.1 |
| 4 | 0.1 | 0.9 | 0.1 |
| 5 | 1.0 | 0.925 | 0.075 |
| 6 | 1.9 | 0.9 | 0.1 |
| 7 | 0.1 | 0.1 | 0.9 |
| 8 | 1.0 | 0.075 | 0.925 |
| 9 | 1.9 | 0.1 | 0.9 |
| 10 | 0.1 | 0.9 | 0.9 |
| 11 | 1.0 | 0.925 | 0.925 |
| 12 | 1.9 | 0.9 | 0.9 |
Figure: Two element test problem (left) original configuration, (right) subject to one iteration of Laplace smoothing.