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Implementing Conforming CR #437
See original GitHub issueHi @kinnala,
I finally got back to implementing conforming CR (see 3.2.3 in here for a concise definition of the spaces for velocity and pressure) using the suggestions in #437 for incompressible stokes (hyperelasticity). Since I plan to use this element in my work, and if you deem it fit then include it in scikit-fem. This is what I have for the definitions so far…
3D
class ElementTetCCR(ElementH1):
nodal_dofs = 1
facet_dofs = 1
edge_dofs = 1
interior_dofs = 1
maxdeg = 4
dofnames = ["u", "u", "u", "u"]
doflocs = np.array(
[
[1.0, 0.0, 0.0],
[0.0, 1.0, 0.0],
[0.0, 0.0, 0.0],
[0.0, 0.0, 1.0],
[0.5, 0.5, 0.0],
[0.0, 0.5, 0.0],
[0.5, 0.0, 0.0],
[0.5, 0.0, 0.5],
[0.0, 0.5, 0.5],
[0.0, 0.0, 0.5],
[1.0 / 3, 1.0 / 3, 0.0],
[1.0 / 3, 1.0 / 3, 1.0 / 3],
[0.0, 1.0 / 3, 1.0 / 3],
[1.0 / 3, 0.0, 1.0 / 3],
[1.0 / 4, 1.0 / 4, 1.0 / 4],
]
)
refdom = RefTet
def lbasis(self, X, i):
x, y, z = X
if i == 0: # at (1,0,0)
phi = 3 * x * (y * z + (y + z) * (-x - y - z + 1)) + x * (
2 * x - 4 * y * z * (-x - y - z + 1) - 1
)
dphi = np.array(
[
-3*x*(y + z) + 2*x*(2*y*z + 1) + 2*x + 4*y*z*(x + y + z - 1) + 3*y*z - 3*(y + z)*(x + y + z - 1) - 1,
x*(4*z - 3)*(x + 2*y + z - 1),
x*(4*y - 3)*(x + y + 2*z - 1),
]
)
elif i == 1: # at (1,0,0)
phi = y*(4*x*z*(x + y + z - 1) + 3*x*z + 2*y - 3*(x + z)*(x + y + z - 1) - 1)
dphi = np.array(
[
y*(4*z - 3)*(2*x + y + z - 1),
4*x*z*(x + y + z - 1) + 3*x*z - 3*y*(x + z) + 2*y*(2*x*z + 1) + 2*y - 3*(x + z)*(x + y + z - 1) - 1,
y*(4*x - 3)*(x + y + 2*z - 1),
]
)
elif i == 2: # at (0,1,0)
phi = (x + y + z - 1)*(4*x*y*z - 3*x*y - 3*x*z + 2*x - 3*y*z + 2*y + 2*z - 1)
dphi = np.array(
[
8*x*y*z - 6*x*y - 6*x*z + 4*x + 4*y**2*z - 3*y**2 + 4*y*z**2 - 13*y*z + 7*y - 3*z**2 + 7*z - 3,
4*x**2*z - 3*x**2 + 8*x*y*z - 6*x*y + 4*x*z**2 - 13*x*z + 7*x - 6*y*z + 4*y - 3*z**2 + 7*z - 3,
4*x**2*y - 3*x**2 + 4*x*y**2 + 8*x*y*z - 13*x*y - 6*x*z + 7*x - 3*y**2 - 6*y*z + 7*y + 4*z - 3,
]
)
elif i == 3: # at (0,0,1)
phi = z*(4*x*y*(x + y + z - 1) + 3*x*y + 2*z - 3*(x + y)*(x + y + z - 1) - 1)
dphi = np.array(
[
z*(4*y - 3)*(2*x + y + z - 1),
z*(4*x - 3)*(x + 2*y + z - 1),
4*x*y*(x + y + z - 1) + 3*x*y - 3*z*(x + y) + 2*z*(2*x*y + 1) + 2*z - 3*(x + y)*(x + y + z - 1) - 1,
]
)
elif i == 4: # between (0,1)
phi = 4*x*y*(3*x + 3*y - 8*z*(x + y + z - 1) - 2)
dphi = np.array(
[
4*y*(-x*(8*z - 3) + 3*x + 3*y - 8*z*(x + y + z - 1) - 2),
4*x*(3*x - y*(8*z - 3) + 3*y - 8*z*(x + y + z - 1) - 2),
32*x*y*(-x - y - 2*z + 1),
]
)
elif i == 5: # between (1,2)
phi = -4*y*(x + y + z - 1)*(8*x*z - 3*x - 3*z + 1)
dphi = np.array(
[
4*y*(-8*x*z + 3*x + 3*z - (8*z - 3)*(x + y + z - 1) - 1),
4*(-x - 2*y - z + 1)*(8*x*z - 3*x - 3*z + 1),
4*y*(-8*x*z + 3*x + 3*z - (8*x - 3)*(x + y + z - 1) - 1),
]
)
elif i == 6: # between (0,2)
phi = -4*x*(x + y + z - 1)*(8*y*z - 3*y - 3*z + 1)
dphi = np.array(
[
4*(-2*x - y - z + 1)*(8*y*z - 3*y - 3*z + 1),
4*x*(-8*y*z + 3*y + 3*z - (8*z - 3)*(x + y + z - 1) - 1),
4*x*(-8*y*z + 3*y + 3*z - (8*y - 3)*(x + y + z - 1) - 1),
]
)
elif i == 7: # between (0,3)
phi = 4*x*z*(3*x - 8*y*(x + y + z - 1) + 3*z - 2)
dphi = np.array(
[
4*z*(-x*(8*y - 3) + 3*x - 8*y*(x + y + z - 1) + 3*z - 2),
32*x*z*(-x - 2*y - z + 1),
4*x*(3*x - 8*y*(x + y + z - 1) - z*(8*y - 3) + 3*z - 2),
]
)
elif i == 8:
phi = -4*y*z*(8*x*(x + y + z - 1) - 3*y - 3*z + 2)
dphi = np.array(
[
32*y*z*(-2*x - y - z + 1),
4*z*(-8*x*(x + y + z - 1) - y*(8*x - 3) + 3*y + 3*z - 2),
4*y*(-8*x*(x + y + z - 1) + 3*y - z*(8*x - 3) + 3*z - 2),
]
)
elif i == 9:
phi = -4*z*(x + y + z - 1)*(8*x*y - 3*x - 3*y + 1)
dphi = np.array(
[
4*z*(-8*x*y + 3*x + 3*y - (8*y - 3)*(x + y + z - 1) - 1),
4*z*(-8*x*y + 3*x + 3*y - (8*x - 3)*(x + y + z - 1) - 1),
4*(-x - y - 2*z + 1)*(8*x*y - 3*x - 3*y + 1),
]
)
elif i == 10:
phi = 27*x*y*(4*z - 1)*(x + y + z - 1)
dphi = np.array(
[
27*y*(4*z - 1)*(2*x + y + z - 1),
27*x*(4*z - 1)*(x + 2*y + z - 1),
27*x*y*(4*x + 4*y + 8*z - 5)
]
)
elif i == 11:
phi = 27*x*y*z*(4*x + 4*y + 4*z - 3)
dphi = np.array(
[
27*y*z*(8*x + 4*y + 4*z - 3),
27*x*z*(4*x + 8*y + 4*z - 3),
27*x*y*(4*x + 4*y + 8*z - 3)
]
)
elif i == 12:
phi = 27*y*z*(4*x - 1)*(x + y + z - 1)
dphi = np.array(
[
27*y*z*(8*x + 4*y + 4*z - 5),
27*z*(4*x - 1)*(x + 2*y + z - 1),
27*y*(4*x - 1)*(x + y + 2*z - 1)
]
)
elif i == 13:
phi = 27*x*z*(4*y - 1)*(x + y + z - 1)
dphi = np.array(
[
27*z*(4*y - 1)*(2*x + y + z - 1),
27*x*z*(4*x + 8*y + 4*z - 5),
27*x*(4*y - 1)*(x + y + 2*z - 1)
]
)
elif i == 14:
phi = 256*x*y*z*(-x - y - z + 1)
dphi = np.array(
[
256*y*z*(-2*x - y - z + 1),
256*x*z*(-x - 2*y - z + 1),
256*x*y*(-x - y - 2*z + 1)
]
)
else:
self._index_error()
return phi, dphi
2D
class ElementTriCCR(ElementH1):
nodal_dofs = 1
facet_dofs = 1
interior_dofs = 1
maxdeg = 3
dofnames = ['u', 'u', 'u']
doflocs = np.array([[1., 0.],
[0., 1.],
[0., 0.],
[0.5, 0.5],
[0, 0.5],
[0.5, 0.],
[1./3, 1./3]])
refdom = RefTri
def lbasis(self, X, i):
x, y = X
if i == 0:
phi = -x*(-2*x + 3*y*(x + y - 1) + 1)
dphi = np.array([6*x*y + 4*x + 3*y**2 - 3*y - 1,
3*x*(x + 2*y - 1)])
elif i == 1:
phi = -y*(3*x*(x + y - 1) - 2*y + 1)
dphi = np.array([3*y*(-2*x - y + 1),
-3*x**2 - 6*x*y + 3*x + 4*y - 1])
elif i == 2:
phi = -(-2*x + y*(3*x - 2) + 1)*(x + y - 1)
dphi = np.array([-6*x*y + 4*x - 3*y**2 + 7*y - 3,
-3*x**2 - 6*x*y + 7*x + 4*y - 3])
elif i == 3: # 0->1
phi = 4*x*y*(3*x + 3*y - 2)
dphi = np.array([4*y*(6*x + 3*y - 2),
4*x*(3*x + 6*y - 2)])
elif i == 4: # 1->2
phi = 4*y*(3*x - 1)*(x + y - 1)
dphi = np.array([4*y*(6*x + 3*y - 4),
4*(3*x - 1)*(x + 2*y - 1)])
elif i == 5: # 0->2
phi = 4*x*(3*y - 1)*(x + y - 1)
dphi = np.array([4*(3*y - 1)*(2*x + y - 1),
4*x*(3*x + 6*y - 4)])
elif i == 6:
phi = 27*x*y*(-x - y + 1)
dphi = np.array(
[
27*y*(-2*x - y + 1),
27*x*(-x - 2*y + 1)
]
)
else:
self._index_error()
return phi, dphi
I am looking to first test this in example 36 (in doing so I realized that example 36 was written in a non-standard fashion – I plan to clean that up soon as well). Since the pressure space is discontinuous P1, and now that you have ElementTetDG and ElementTriDG already in scikit-fem, it seems that example 36 should be reproducible by simply changing the relevant spaces to
uelem = ElementVectorH1(ElementTetCCR())
pelem = ElementTetDG(ElementTetP1())
Please let me know what you think. In the meanwhile, I will check if I have implemented everything correctly (basis functions and their gradients)
Issue Analytics
- State:
- Created 2 years ago
- Comments:9 (9 by maintainers)
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Here are some tests that are run against the elements (if applicable):
We also test the convergence rates, e.g.: https://github.com/kinnala/scikit-fem/blob/master/tests/test_convergence.py#L213
A new file is created:
skfem/element/element_tri/element_tri_ccr.pyand imports added toskfem/element/element_tri/__init__.pyandskfem/element/__init__.pyso that wildcard works properly. Then the above tests are modified accordingly.If you want you can open a pull request with these changes.
Merged