BiharmonicHHJ.py

# 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