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# Copyright (C) 2016 Lizao Li
"""Biharmonis HHJ demo.
The bilinear form a(u, v) and linear form L(v) for Biharmonic equation
in Hellan-Herrmann-Johnson (HHJ) formulation.
"""
import basix.ufl
from ufl import (
Coefficient,
FacetNormal,
FunctionSpace,
Mesh,
TestFunctions,
TrialFunctions,
dot,
dS,
ds,
dx,
grad,
inner,
jump,
)
HHJ = basix.ufl.element("HHJ", "triangle", 2)
P = basix.ufl.element("P", "triangle", 3)
mixed_element = basix.ufl.mixed_element([HHJ, P])
domain = Mesh(basix.ufl.element("P", "triangle", 1, shape=(2,)))
mixed_space = FunctionSpace(domain, mixed_element)
p_space = FunctionSpace(domain, P)
(sigma, u) = TrialFunctions(mixed_space)
(tau, v) = TestFunctions(mixed_space)
f = Coefficient(p_space)
def b(sigma, v):
"""The form b."""
n = FacetNormal(domain)
return (
inner(sigma, grad(grad(v))) * dx
- dot(dot(sigma("+"), n("+")), n("+")) * jump(grad(v), n) * dS
- dot(dot(sigma, n), n) * dot(grad(v), n) * ds
)
a = inner(sigma, tau) * dx - b(tau, u) + b(sigma, v)
L = f * v * dx
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