def continuity_form(state, test, q, ibp=IntegrateByParts.ONCE, outflow=False):
if outflow and ibp == IntegrateByParts.NEVER:
raise ValueError(
"outflow is True and ibp is None are incompatible options")
Vu = state.spaces("HDiv")
dS_ = (dS_v + dS_h) if Vu.extruded else dS
ubar = Function(Vu)
if ibp == IntegrateByParts.ONCE:
L = -inner(grad(test), outer(q, ubar)) * dx
else:
L = inner(test, div(outer(q, ubar))) * dx
if ibp != IntegrateByParts.NEVER:
n = FacetNormal(state.mesh)
un = 0.5 * (dot(ubar, n) + abs(dot(ubar, n)))
L += dot(jump(test), (un('+') * q('+') - un('-') * q('-'))) * dS_
if ibp == IntegrateByParts.TWICE:
L -= (inner(test('+'),
dot(ubar('+'), n('+')) * q('+')) +
inner(test('-'),
dot(ubar('-'), n('-')) * q('-'))) * dS_
if outflow:
n = FacetNormal(state.mesh)
un = 0.5 * (dot(ubar, n) + abs(dot(ubar, n)))
L += test * un * q * (ds_v + ds_t + ds_b)
form = transporting_velocity(L, ubar)
return ibp_label(transport(form, TransportEquationType.conservative), ibp)
def kinetic_energy_form(state, test, q):
ubar = Function(state.spaces("HDiv"))
L = 0.5 * div(test) * inner(q, ubar) * dx
form = transporting_velocity(L, ubar)
return transport(form, TransportEquationType.vector_invariant)
def linear_advection_form(state, test, qbar):
ubar = Function(state.spaces("HDiv"))
L = test * dot(ubar, state.k) * dot(state.k, grad(qbar)) * dx
form = transporting_velocity(L, ubar)
return transport(form, TransportEquationType.advective)
def vector_invariant_form(state, test, q, ibp=IntegrateByParts.ONCE):
Vu = state.spaces("HDiv")
dS_ = (dS_v + dS_h) if Vu.extruded else dS
ubar = Function(Vu)
n = FacetNormal(state.mesh)
Upwind = 0.5 * (sign(dot(ubar, n)) + 1)
if state.mesh.topological_dimension() == 3:
if ibp != IntegrateByParts.ONCE:
raise NotImplementedError
# <w,curl(u) cross ubar + grad( u.ubar)>
# =<curl(u),ubar cross w> - <div(w), u.ubar>
# =<u,curl(ubar cross w)> -
# <<u_upwind, [[n cross(ubar cross w)cross]]>>
both = lambda u: 2 * avg(u)
L = (inner(q, curl(cross(ubar, test))) * dx -
inner(both(Upwind * q), both(cross(n, cross(ubar, test)))) * dS_)
else:
perp = state.perp
if state.on_sphere:
outward_normals = CellNormal(state.mesh)
perp_u_upwind = lambda q: Upwind('+') * cross(
outward_normals('+'), q('+')) + Upwind('-') * cross(
outward_normals('-'), q('-'))
else:
perp_u_upwind = lambda q: Upwind('+') * perp(q('+')) + Upwind(
'-') * perp(q('-'))
if ibp == IntegrateByParts.ONCE:
L = (-inner(perp(grad(inner(test, perp(ubar)))), q) * dx -
inner(jump(inner(test, perp(ubar)), n), perp_u_upwind(q)) *
dS_)
else:
L = ((-inner(test,
div(perp(q)) * perp(ubar))) * dx -
inner(jump(inner(test, perp(ubar)), n), perp_u_upwind(q)) *
dS_ + jump(inner(test, perp(ubar)) * perp(q), n) * dS_)
L -= 0.5 * div(test) * inner(q, ubar) * dx
form = transporting_velocity(L, ubar)
return transport(form, TransportEquationType.vector_invariant)
def linear_continuity_form(state, test, qbar, facet_term=False):
Vu = state.spaces("HDiv")
ubar = Function(Vu)
L = qbar * test * div(ubar) * dx
if facet_term:
n = FacetNormal(state.mesh)
Vu = state.spaces("HDiv")
dS_ = (dS_v + dS_h) if Vu.extruded else dS
L += jump(ubar * test, n) * avg(qbar) * dS_
form = transporting_velocity(L, ubar)
return transport(form, TransportEquationType.conservative)
def vector_manifold_continuity_form(state,
test,
q,
ibp=IntegrateByParts.ONCE,
outflow=False):
L = continuity_form(state, test, q, ibp, outflow)
Vu = state.spaces("HDiv")
dS_ = (dS_v + dS_h) if Vu.extruded else dS
ubar = Function(Vu)
n = FacetNormal(state.mesh)
un = 0.5 * (dot(ubar, n) + abs(dot(ubar, n)))
L += un('+') * inner(test('-'),
n('+') + n('-')) * inner(q('+'), n('+')) * dS_
L += un('-') * inner(test('+'),
n('+') + n('-')) * inner(q('-'), n('-')) * dS_
form = transporting_velocity(L, ubar)
return transport(form)