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mhd3d_timedep.py
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# -*- coding: utf-8 -*-
from firedrake import *
import alfi
from alfi.stabilisation import *
from alfi.transfer import *
from pyop2.datatypes import IntType
import ufl.algorithms
from datetime import datetime
import argparse
import numpy
import sys
import os
from mpi4py import MPI
import petsc4py
petsc4py.PETSc.Sys.popErrorHandler()
# Parallel distribution parameters
distribution_parameters = {"partition": True, "overlap_type": (DistributedMeshOverlapType.VERTEX, 2)}
# Definition of Burman Stabilisation term to avoid oscillations in velocity u
def BurmanStab(B, C, wind, stab_weight, mesh):
n = FacetNormal(mesh)
h = FacetArea(mesh)
beta = avg(facet_avg(sqrt(inner(wind, wind)+1e-10)))
gamma1 = stab_weight # as chosen in doi:10.1016/j.apnum.2007.11.001
stabilisation_form = 0.5 * gamma1 * avg(h)**2 * beta * dot(jump(grad(B), n), jump(grad(C), n))*dS
return stabilisation_form
# Definition of outer Schur complement for the order ((E,B),(u,p)), which is Navier-Stokes block + Lorentz force
class SchurPCup(AuxiliaryOperatorPC):
def form(self, pc, V, U):
mesh = V.function_space().mesh()
[u, p] = split(U)
[v, q] = split(V)
Z = V.function_space()
U_ = Function(Z)
state = self.get_appctx(pc)['state']
[u_n, p_n, B_n, E_n] = split(state)
eps = lambda x: sym(grad(x))
# Factor for Schur complement approximation as chosen in doi.org/10.1137/16M1074084
if linearisation == "mdp":
alpha = 1
elif linearisation == "picard":
alpha = dt/(dt+Rem*(1/(baseN*nref))**2)
elif linearisation == "newton":
norm_un = sqrt(assemble(inner(u_n, u_n)*dx))
hh = 1/(baseN*nref)
alpha = dt/(dt + Rem*hh**2 + Rem*hh*norm_un*dt)
# Weak form of NS + (scaled) Lorentz force
A = (
+ 2/Re * inner(eps(u), eps(v)) * dx
+ gamma * inner(div(u), div(v)) * dx
- inner(p, div(v)) * dx
- inner(div(u), q) * dx
+ alpha*S * inner(cross(B_n, cross(u, B_n)), v) * dx
)
# For H(div)-L2 discretization we have to add a DG-formulation of the advection and diffusion term
if discr in ["rt", "bdm"]:
h = CellVolume(mesh)/FacetArea(mesh)
uflux_int = 0.5*(dot(u, n) + abs(dot(u, n))) * u
uflux_ext_1 = 0.5*(inner(u, n) + abs(inner(u, n))) * u
uflux_ext_2 = 0.5*(inner(u, n) - abs(inner(u, n))) * u_ex
A_DG = (
- 1/Re * inner(avg(2*sym(grad(u))), 2*avg(outer(v, n))) * dS
- 1/Re * inner(avg(2*sym(grad(v))), 2*avg(outer(u, n))) * dS
+ 1/Re * sigma/avg(h) * inner(2*avg(outer(u, n)), 2*avg(outer(v, n))) * dS
- inner(outer(v, n), 2/Re*sym(grad(u))) * ds
- inner(outer(u-u_ex, n), 2/Re*sym(grad(v))) * ds(bcs_ids_apply)
+ 1/Re*(sigma/h)*inner(v, u-u_ex) * ds(bcs_ids_apply)
- advect * dot(u, div(outer(v, u))) * dx
+ advect * dot(v('+')-v('-'), uflux_int('+')-uflux_int('-')) * dS
+ advect * dot(v, uflux_ext_1) * ds
+ advect * dot(v, uflux_ext_2) * ds(bcs_ids_apply)
)
if bcs_ids_dont_apply is not None:
A_DG += (
- inner(outer(u, n), 2/Re*sym(grad(v)))*ds(bcs_ids_dont_apply)
+ 1/Re*(sigma/h)*inner(v, u)*ds(bcs_ids_dont_apply)
)
# Linearize A_DG
A_DG = action(A_DG, U_)
A_DG_linear = derivative(A_DG, U_, U)
A_DG_linear = replace(A_DG_linear, {split(U_)[0]: u_n, split(U_)[1]: p_n})
A = A + A_DG_linear
elif discr in ["cg"]:
A = A + advect * inner(dot(grad(u_n), u), v) * dx + advect * inner(dot(grad(u), u_n), v) * dx
if stab:
stabilisation2 = BurmanStabilisation(V.function_space().sub(0), state=z_last_u, h=FacetArea(mesh), weight=stab_weight)
stabilisation_form_2 = stabilisation2.form(u, v)
A = A + advect * stabilisation_form_2
bcs = [DirichletBC(V.function_space().sub(0), 0, "on_boundary"),
PressureFixBC(V.function_space().sub(1), 0, 1),
]
A = inner(u, v) * dx + dt_factor*dt*A
return (A, bcs)
# Definition of outer Schur complement for the order ((u,p),(B,E))
class SchurPCBE(AuxiliaryOperatorPC):
def form(self, pc, V, U):
[B, E] = split(U)
[C, Ff] = split(V)
state = self.get_appctx(pc)['state']
[u_n, p_n, B_n, E_n] = split(state)
A = (
+ inner(E, Ff) * dx
+ inner(cross(u_n, B), Ff) * dx
- 1/Rem * inner(B, curl(Ff)) * dx
+ inner(curl(E), C) * dx
+ 1/Rem * inner(div(B), div(C)) * dx
+ gamma2 * inner(div(B), div(C)) * dx
)
bcs = [DirichletBC(V.function_space().sub(0), 0, "on_boundary"),
DirichletBC(V.function_space().sub(1), 0, "on_boundary"),
]
A = inner(B, C) * dx + dt_factor*dt*A
return (A, bcs)
# We fix the pressure at one vertex on every level
class PressureFixBC(DirichletBC):
def __init__(self, V, val, subdomain, method="topological"):
super().__init__(V, val, subdomain, method)
sec = V.dm.getDefaultSection()
dm = V.mesh().topology_dm
coordsSection = dm.getCoordinateSection()
dim = dm.getCoordinateDim()
coordsVec = dm.getCoordinatesLocal()
(vStart, vEnd) = dm.getDepthStratum(0)
indices = []
for pt in range(vStart, vEnd):
x = dm.getVecClosure(coordsSection, coordsVec, pt).reshape(-1, dim).mean(axis=0)
if x.dot(x) == 0.0: # fix [0, 0] in original mesh coordinates (bottom left corner)
if dm.getLabelValue("pyop2_ghost", pt) == -1:
indices = [pt]
break
nodes = []
for i in indices:
if sec.getDof(i) > 0:
nodes.append(sec.getOffset(i))
if V.mesh().comm.rank == 0:
nodes = [0]
else:
nodes = []
self.nodes = numpy.asarray(nodes, dtype=IntType)
if len(self.nodes) > 0:
print("Fixing nodes %s" % self.nodes)
# else:
# print("Not fixing any nodes")
import sys
sys.stdout.flush()
# Definition of different solver parameters
# Increase buffer size for MUMPS solver
ICNTL_14 = 5000
tele_reduc_fac = int(MPI.COMM_WORLD.size/4)
if tele_reduc_fac < 1:
tele_reduc_fac = 1
# LU solver
lu = {
"snes_type": "newtonls",
"snes_monitor": None,
"snes_atol": 1.0e-8,
"snes_rtol": 1.0e-13,
"snes_linesearch_type": "l2",
"snes_linesearch_maxstep": 1,
"mat_type": "aij",
"ksp_type": "preonly",
"pc_type": "lu",
"pc_factor_mat_solver_type": "mumps",
"mat_mumps_icntl_14": ICNTL_14,
}
nsfsstar = {
"ksp_type": "fgmres",
"ksp_atol": 1.0e-7,
"ksp_rtol": 1.0e-7,
"ksp_max_it": 2,
"pc_type": "fieldsplit",
"pc_fieldsplit_type": "schur",
"pc_fieldsplit_schur_factorization_type": "upper",
"pc_fieldsplit_schur_precondition": "user",
"fieldsplit_0": {
"ksp_type": "gmres",
"ksp_max_it": 1,
"ksp_norm_type": "unpreconditioned",
"pc_type": "mg",
"pc_mg_cycle_type": "v",
"pc_mg_type": "full",
"mg_levels_ksp_type": "fgmres",
"mg_levels_ksp_convergence_test": "skip",
"mg_levels_ksp_max_it": 6,
"mg_levels_ksp_norm_type": "unpreconditioned",
"mg_levels_pc_type": "python",
"mg_levels_pc_python_type": "firedrake.ASMStarPC",
"mg_levels_pc_star_backend": "tinyasm",
# "mg_levels_pc_star_construct_dim": 0,
# "mg_levels_pc_star_sub_sub_ksp_type": "preonly",
# "mg_levels_pc_star_sub_sub_pc_type": "lu",
# "mg_levels_pc_star_sub_sub_pc_factor_mat_solver_type": "umfpack",
"mg_coarse_ksp_type": "richardson",
"mg_coarse_ksp_max_it": 1,
"mg_coarse_ksp_norm_type": "unpreconditioned",
"mg_coarse_pc_type": "python",
"mg_coarse_pc_python_type": "firedrake.AssembledPC",
"mg_coarse_assembled": {
"mat_type": "aij",
"pc_type": "telescope",
"pc_telescope_reduction_factor": tele_reduc_fac,
"pc_telescope_subcomm_type": "contiguous",
"telescope_pc_type": "lu",
"telescope_pc_factor_mat_solver_type": "mumps",
"telescope_pc_factor_mat_mumps_icntl_14": ICNTL_14,
}
},
"fieldsplit_1": {
"ksp_type": "preonly",
"pc_type": "python",
"pc_python_type": "alfi.solver.DGMassInv"
},
}
# Monolithic macrostar solver for (E,B)-block
nsfsmacrostar = {
"ksp_type": "fgmres",
"ksp_atol": 1.0e-7,
"ksp_rtol": 1.0e-7,
"ksp_max_it": 2,
"pc_type": "fieldsplit",
"pc_fieldsplit_type": "schur",
"pc_fieldsplit_schur_factorization_type": "upper",
"pc_fieldsplit_schur_precondition": "user",
"fieldsplit_0": {
"ksp_type": "gmres",
"ksp_max_it": 1,
"ksp_norm_type": "unpreconditioned",
"pc_type": "mg",
"pc_mg_cycle_type": "v",
"pc_mg_type": "full",
"mg_levels_ksp_type": "fgmres",
"mg_levels_ksp_convergence_test": "skip",
"mg_levels_ksp_max_it": 6,
"mg_levels_ksp_norm_type": "unpreconditioned",
"mg_levels_pc_type": "python",
"mg_levels_pc_python_type": "firedrake.PatchPC",
"mg_levels_patch_pc_patch_save_operators": True,
"mg_levels_patch_pc_patch_partition_of_unity": False,
"mg_levels_patch_pc_patch_sub_mat_type": "seqaij",
"mg_levels_patch_pc_patch_construct_dim": 0,
"mg_levels_patch_pc_patch_construct_type": "python",
"mg_levels_patch_pc_patch_construct_python_type": "alfi.MacroStar",
"mg_levels_patch_sub_ksp_type": "preonly",
"mg_levels_patch_sub_pc_type": "lu",
"mg_levels_patch_sub_pc_factor_mat_solver_type": "umfpack",
"mg_coarse_ksp_type": "richardson",
"mg_coarse_ksp_max_it": 1,
"mg_coarse_ksp_norm_type": "unpreconditioned",
"mg_coarse_pc_type": "python",
"mg_coarse_pc_python_type": "firedrake.AssembledPC",
"mg_coarse_assembled": {
"mat_type": "aij",
"pc_type": "telescope",
"pc_telescope_reduction_factor": tele_reduc_fac,
"pc_telescope_subcomm_type": "contiguous",
"telescope_pc_type": "lu",
"telescope_pc_factor_mat_solver_type": "mumps",
"telescope_pc_factor_mat_mumps_icntl_14": ICNTL_14,
}
},
"fieldsplit_1": {
"ksp_type": "preonly",
"pc_type": "python",
"pc_python_type": "alfi.solver.DGMassInv"
},
}
# Fieldsplit solver for outer Schur complement with monolithic star solver
outerschurstar = {
"ksp_type": "fgmres",
"ksp_atol": 1.0e-7,
"ksp_rtol": 1.0e-7,
"pc_type": "python",
"pc_python_type": "__main__.SchurPCBE",
"aux_mg_transfer_manager": __name__ + ".transfermanager",
"ksp_max_it": 2,
"aux": {
"ksp_type": "gmres",
"ksp_max_it": 1,
"ksp_norm_type": "unpreconditioned",
"pc_type": "mg",
"pc_mg_cycle_type": "v",
"pc_mg_type": "full",
"mg_levels_ksp_type": "fgmres",
"mg_levels_ksp_convergence_test": "skip",
"mg_levels_ksp_max_it": 6,
"mg_levels_ksp_norm_type": "unpreconditioned",
"mg_levels_pc_type": "python",
"mg_levels_pc_python_type": "firedrake.ASMStarPC",
"mg_levels_pc_star_backend": "tinyasm",
# "mg_levels_pc_python_type": "firedrake.PatchPC",
# "mg_levels_patch_pc_patch_save_operators": True,
# "mg_levels_patch_pc_patch_partition_of_unity": False,
# "mg_levels_patch_pc_patch_sub_mat_type": "seqaij",
# "mg_levels_patch_pc_patch_construct_dim": 0,
# "mg_levels_patch_pc_patch_construct_type": "star",
# "mg_levels_patch_sub_ksp_type": "preonly",
# "mg_levels_patch_sub_pc_type": "lu",
# "mg_levels_patch_sub_pc_factor_mat_solver_type": "umfpack",
"mg_coarse_ksp_type": "richardson",
"mg_coarse_ksp_max_it": 1,
"mg_coarse_ksp_norm_type": "unpreconditioned",
"mg_coarse_pc_type": "python",
"mg_coarse_pc_python_type": "firedrake.AssembledPC",
"mg_coarse_assembled": {
"mat_type": "aij",
"pc_type": "telescope",
"pc_telescope_reduction_factor": tele_reduc_fac,
"pc_telescope_subcomm_type": "contiguous",
"telescope_pc_type": "lu",
"telescope_pc_factor_mat_solver_type": "mumps",
"telescope_pc_factor_mat_mumps_icntl_14": ICNTL_14,
}
},
}
# Fieldsplit solver for outer Schur complement with monolithic macrostar solver
outerschurmacrostar = {
"ksp_type": "fgmres",
"ksp_atol": 1.0e-7,
"ksp_rtol": 1.0e-7,
"pc_type": "python",
"pc_python_type": "__main__.SchurPCBE",
"aux_mg_transfer_manager": __name__ + ".transfermanager",
"ksp_max_it": 2,
"aux": {
"ksp_type": "gmres",
"ksp_max_it": 1,
"ksp_norm_type": "unpreconditioned",
"pc_type": "mg",
"pc_mg_cycle_type": "v",
"pc_mg_type": "full",
"mg_levels_ksp_type": "fgmres",
"mg_levels_ksp_convergence_test": "skip",
"mg_levels_ksp_max_it": 6,
"mg_levels_ksp_norm_type": "unpreconditioned",
"mg_levels_pc_type": "python",
"mg_levels_pc_python_type": "firedrake.PatchPC",
"mg_levels_patch_pc_patch_save_operators": True,
"mg_levels_patch_pc_patch_partition_of_unity": False,
"mg_levels_patch_pc_patch_sub_mat_type": "seqaij",
"mg_levels_patch_pc_patch_construct_dim": 0,
"mg_levels_patch_pc_patch_construct_type": "python",
"mg_levels_patch_pc_patch_construct_python_type": "alfi.MacroStar",
"mg_levels_patch_sub_ksp_type": "preonly",
"mg_levels_patch_sub_pc_type": "lu",
"mg_levels_patch_sub_pc_factor_mat_solver_type": "umfpack",
"mg_coarse_ksp_type": "richardson",
"mg_coarse_ksp_max_it": 1,
"mg_coarse_ksp_norm_type": "unpreconditioned",
"mg_coarse_pc_type": "python",
"mg_coarse_pc_python_type": "firedrake.AssembledPC",
"mg_coarse_assembled": {
"mat_type": "aij",
"pc_type": "telescope",
"pc_telescope_reduction_factor": tele_reduc_fac,
"pc_telescope_subcomm_type": "contiguous",
"telescope_pc_type": "lu",
"telescope_pc_factor_mat_solver_type": "mumps",
"telescope_pc_factor_mat_mumps_icntl_14": ICNTL_14,
}
},
}
# LU solver for outer Schur complement
outerschurlu = {
"ksp_type": "gmres",
"ksp_max_it": 2,
"pc_type": "python",
"pc_python_type": "__main__.SchurPCBE",
"aux_pc_type": "lu",
"aux_pc_factor_mat_solver_type": "mumps",
"aux_mat_mumps_icntl_14": ICNTL_14,
}
# Main solver
fs2by2 = {
"snes_type": "newtonls",
"snes_max_it": 25,
"snes_linesearch_type": "basic",
"snes_linesearch_maxstep": 1.0,
"snes_rtol": 1.0e-10,
"snes_atol": 1.0e-6,
"snes_monitor": None,
"ksp_type": "fgmres",
"ksp_max_it": 75,
"ksp_atol": 1.0e-7,
"ksp_rtol": 1.0e-7,
"ksp_monitor_true_residual": None,
"ksp_converged_reason": None,
"mat_type": "aij",
"pc_type": "fieldsplit",
"pc_fieldsplit_type": "schur",
"pc_fieldsplit_schur_factorization_type": "upper",
"pc_fieldsplit_0_fields": "0,1",
"pc_fieldsplit_1_fields": "2,3",
}
# Main solver with LU solver for (u,p)-block
fs2by2nslu = {
"snes_type": "newtonls",
"snes_max_it": 30,
"snes_linesearch_type": "l2",
"snes_linesearch_maxstep": 1.0,
"snes_rtol": 1.0e-10,
"snes_atol": 1.0e-7,
"snes_monitor": None,
"ksp_type": "fgmres",
"ksp_max_it": 50,
"ksp_atol": 1.0e-8,
"ksp_rtol": 1.0e-9,
"ksp_monitor_true_residual": None,
"ksp_converged_reason": None,
"mat_type": "aij",
"pc_type": "fieldsplit",
"pc_fieldsplit_type": "schur",
"pc_fieldsplit_0_fields": "0,1",
"pc_fieldsplit_1_fields": "2,3",
"fieldsplit_0_ksp_type": "preonly",
"fieldsplit_0_pc_type": "lu",
"fieldsplit_0_pc_factor_mat_solver_type": "mumps",
"fieldsplit_0_mat_mumps_icntl_14": ICNTL_14,
}
# Main solver with LU solver for Schur complement
fs2by2slu = {
"snes_type": "newtonls",
"snes_max_it": 30,
"snes_linesearch_type": "l2",
"snes_linesearch_maxstep": 1.0,
"snes_rtol": 1.0e-15,
"snes_atol": 1.0e-7,
"snes_monitor": None,
"ksp_type": "fgmres",
"ksp_max_it": 40,
"ksp_atol": 1.0e-7,
"ksp_rtol": 1.0e-7,
"ksp_monitor_true_residual": None,
"ksp_converged_reason": None,
"mat_type": "aij",
"pc_type": "fieldsplit",
"pc_fieldsplit_type": "schur",
"pc_fieldsplit_0_fields": "0,1",
"pc_fieldsplit_1_fields": "2,3",
"fieldsplit_1": outerschurlu,
}
# Main solver with LU solver for (E,B)-block and Schur complement
# Can be used to determine how well the outer Schur complement is approximated by the different linerizations
fs2by2lu = {
"snes_type": "newtonls",
"snes_max_it": 30,
"snes_linesearch_type": "l2",
"snes_linesearch_maxstep": 1.0,
"snes_rtol": 1.0e-10,
"snes_atol": 1.0e-6,
"snes_monitor": None,
"ksp_type": "fgmres",
"ksp_max_it": 50,
"ksp_atol": 1.0e-7,
"ksp_rtol": 1.0e-7,
"ksp_monitor_true_residual": None,
"ksp_converged_reason": None,
"mat_type": "aij",
"pc_type": "fieldsplit",
"pc_fieldsplit_type": "schur",
"pc_fieldsplit_0_fields": "0,1",
"pc_fieldsplit_1_fields": "2,3",
"fieldsplit_0_ksp_type": "preonly",
"fieldsplit_0_pc_type": "lu",
"fieldsplit_0_pc_factor_mat_solver_type": "mumps",
"fieldsplit_0_mat_mumps_icntl_14": ICNTL_14,
"fieldsplit_1": outerschurlu,
}
solvers = {"lu": lu, "fs2by2": fs2by2, "fs2by2nslu": fs2by2nslu, "fs2by2slu": fs2by2slu, "fs2by2lu": fs2by2lu}
# Definition of problem paramters
parser = argparse.ArgumentParser(add_help=False)
parser.add_argument("--baseN", type=int, default=10)
parser.add_argument("--k", type=int, default=3)
parser.add_argument("--nref", type=int, default=1)
parser.add_argument("--Re", nargs='+', type=float, default=[1])
parser.add_argument("--Rem", nargs='+', type=float, default=[1])
parser.add_argument("--gamma", type=float, default=10000)
parser.add_argument("--gamma2", type=float, default=0)
parser.add_argument("--advect", type=float, default=1)
parser.add_argument("--dt", type=float, required=True)
parser.add_argument("--Tf", type=float, default=1)
parser.add_argument("--S", nargs='+', type=float, default=[1])
parser.add_argument("--hierarchy", choices=["bary", "uniform"], default="bary")
parser.add_argument("--discr", choices=["rt", "bdm", "cg"], required=True)
parser.add_argument("--solver-type", choices=list(solvers.keys()), default="lu")
parser.add_argument("--testproblem", choices=["ldc"], default="ldc")
parser.add_argument("--linearisation", choices=["picard", "mdp", "newton"], required=True)
parser.add_argument("--stab", default=False, action="store_true")
parser.add_argument("--output", default=False, action="store_true")
args, _ = parser.parse_known_args()
baseN = args.baseN
k = args.k
nref = args.nref
Re = Constant(args.Re[0])
Rem = Constant(args.Rem[0])
gamma = Constant(args.gamma)
gamma2 = Constant(args.gamma2)
S = Constant(args.S[0])
hierarchy = args.hierarchy
discr = args.discr
solver_type = args.solver_type
testproblem = args.testproblem
gamma2 = Constant(args.gamma2)
advect = Constant(args.advect)
dt = Constant(args.dt)
Tf = Constant(args.Tf)
linearisation = args.linearisation
stab = args.stab
output = args.output
# Stabilisation weight for BurmanStabilisation
stab_weight = Constant(3e-3)
if len(args.Re) != 1 and len(args.Rem) != 1 and len(args.S) != 1:
raise ValueError("Re, Rem and S cannot all contain more than one element at the same time")
if k < 3 and hierarchy == "bary":
raise ValueError("Scott Vogelius is not stable for k<3")
if discr == "cg" and hierarchy != "bary":
raise ValueError("SV is only stable on barycentric refined grids")
base = UnitCubeMesh(baseN, baseN, baseN, distribution_parameters=distribution_parameters)
# Callbacks called before and after mesh refinement
def before(dm, i):
for p in range(*dm.getHeightStratum(1)):
dm.setLabelValue("prolongation", p, i+1)
def after(dm, i):
for p in range(*dm.getHeightStratum(1)):
dm.setLabelValue("prolongation", p, i+2)
# Create Mesh hierarchy
if hierarchy == "bary":
mh = alfi.BaryMeshHierarchy(base, nref, callbacks=(before, after))
elif hierarchy == "uniformbary":
bmesh = Mesh(bary(base._plex), distribution_parameters={"partition": False})
mh = MeshHierarchy(bmesh, nref, reorder=True, callbacks=(before, after),
distribution_parameters=distribution_parameters)
elif hierarchy == "uniform":
mh = MeshHierarchy(base, nref, reorder=True, callbacks=(before, after),
distribution_parameters=distribution_parameters)
else:
raise NotImplementedError("Only know bary, uniformbary and uniform for the hierarchy.")
# Change mesh from [0,1]^3 to [-0.5,0.5]^3
for m in mh:
m.coordinates.dat.data[:, 0] -= 0.5
m.coordinates.dat.data[:, 1] -= 0.5
m.coordinates.dat.data[:, 2] -= 0.5
mesh = mh[-1]
area = assemble(Constant(1, domain=mh[0])*dx)
def message(msg):
if mesh.comm.rank == 0:
warning(msg)
# Define mixed function space
if discr == "rt":
Vel = FiniteElement("N1div", mesh.ufl_cell(), k, variant=f"integral({k+1})")
V = FunctionSpace(mesh, Vel)
elif discr == "bdm":
Vel = FiniteElement("N2div", mesh.ufl_cell(), k, variant=f"integral({k+1})")
V = FunctionSpace(mesh, Vel)
elif discr == "cg":
V = VectorFunctionSpace(mesh, "CG", k)
Q = FunctionSpace(mesh, "DG", k-1) # p
Rel = FiniteElement("N1curl", mesh.ufl_cell(), k, variant=f"integral({k+1})")
R = FunctionSpace(mesh, Rel) # E
Wel = FiniteElement("N1div", mesh.ufl_cell(), k, variant=f"integral({k+1})")
W = FunctionSpace(mesh, Wel)
Z = MixedFunctionSpace([V, Q, W, R])
# used for BurmannStabilisation
z_last_u = Function(V)
# time variable
t = Constant(0)
(x, y, zz) = SpatialCoordinate(Z.mesh())
n = FacetNormal(mesh)
if testproblem == "ldc":
# example taken from https://doi.org/10.1016/j.jcp.2016.04.019
u_ex = Constant((1, 0, 0), domain=mesh)
B_ex = Constant((0, 1, 0), domain=mesh)
B_ex = interpolate(B_ex, W)
E_ex = Constant((0, 0, 0), domain=mesh)
E_ex = interpolate(E_ex, R)
p_ex = Constant(0, domain=mesh)
bcs_ids_apply = 4
bcs_ids_dont_apply = (1, 2, 3, 5, 6)
bcs = [DirichletBC(Z.sub(0), u_ex, bcs_ids_apply), # 4 == upper boundary (y==1)
DirichletBC(Z.sub(0), 0, bcs_ids_dont_apply),
DirichletBC(Z.sub(2), B_ex, "on_boundary"),
DirichletBC(Z.sub(3), 0, "on_boundary"),
PressureFixBC(Z.sub(1), 0, 1)
]
rhs = None
# Do we know what the exact solution of the problem is?
solution_known = False
# Do the boundary conditions depend on parameters
bc_varying = False
# Cell diameter used in DG formulation
h = CellVolume(mesh)/FacetArea(mesh)
# Penalty parameter used in DG formulation
sigma = Constant(1) * Z.sub(0).ufl_element().degree()**2
theta = Constant(1)
# Definition of weak form
# timelevel = 0 is time we solve for; timelevel = 1 means previous timestep
def form(z, test_z, Z, timelevel):
(u, p, B, E) = split(z)
(v, q, C, Ff) = split(test_z)
eps = lambda x: sym(grad(x))
F = (
+ 2/Re * inner(eps(u), eps(v))*dx
# + advect * inner(dot(grad(u), u), v) * dx
+ gamma * inner(div(u), div(v)) * dx
+ S * inner(cross(B, E), v) * dx
+ S * inner(cross(B, cross(u, B)), v) * dx
- inner(p, div(v)) * dx
- inner(div(u), q) * dx
+ inner(E, Ff) * dx
+ inner(cross(u, B), Ff) * dx
- 1/Rem * inner(B, curl(Ff)) * dx
+ inner(curl(E), C) * dx
+ 1/Rem * inner(div(B), div(C)) * dx
+ gamma2 * inner(div(B), div(C)) * dx
)
if discr in ["rt", "bdm"]:
uflux_int = 0.5*(dot(u, n) + theta*abs(dot(u, n))) * u
uflux_ext_1 = 0.5*(inner(u, n) + theta*abs(inner(u, n))) * u
uflux_ext_2 = 0.5*(inner(u, n) - theta*abs(inner(u, n))) * u_ex
F_DG = (
- 1/Re * inner(avg(2*sym(grad(u))), 2*avg(outer(v, n))) * dS
- 1/Re * inner(avg(2*sym(grad(v))), 2*avg(outer(u, n))) * dS
+ 1/Re * sigma/avg(h) * inner(2*avg(outer(u, n)), 2*avg(outer(v, n))) * dS
- inner(outer(v, n), 2/Re*sym(grad(u))) * ds
- advect * dot(u, div(outer(v, u))) * dx
+ advect * dot(v('+')-v('-'), uflux_int('+')-uflux_int('-')) * dS
+ advect * dot(v, uflux_ext_1) * ds
)
# Only add boundary conditions on timelevel we solve for
# Otherwise we run into problems if initial guess doesn't fullfill right boundary conditions
if timelevel == 0:
F_DG += (
- inner(outer(u-u_ex, n), 2/Re*sym(grad(v))) * ds(bcs_ids_apply)
+ 1/Re*(sigma/h)*inner(v, u-u_ex) * ds(bcs_ids_apply)
+ advect * dot(v, uflux_ext_2) * ds(bcs_ids_apply)
)
if bcs_ids_dont_apply is not None:
F_DG += (
- inner(outer(u, n), 2/Re*sym(grad(v))) * ds(bcs_ids_dont_apply)
+ 1/Re*(sigma/h)*inner(v, u) * ds(bcs_ids_dont_apply)
)
F += F_DG
elif discr == "cg":
F += advect * inner(dot(grad(u), u), v) * dx
return F
def J_form(F, z, test_z):
(u, p, B, E) = split(z)
(v, q, C, Ff) = split(test_z)
w = TrialFunction(Z)
[w_u, w_p, w_B, w_E] = split(w)
J_newton = ufl.algorithms.expand_derivatives(derivative(F, z, w))
if linearisation == "newton":
J = J_newton
elif linearisation == "mdp":
J_mdp = (
J_newton
- dt_factor * dt * inner(cross(w_u, B), Ff) * dx # G
)
J = J_mdp
elif linearisation == "picard":
J_picard = (
J_newton
- dt_factor * dt * S * inner(cross(w_B, E), v) * dx # J_tilde
- dt_factor * dt * S * inner(cross(w_B, cross(u, B)), v) * dx # D_1_tilde
- dt_factor * dt * S * inner(cross(B, cross(u, w_B)), v) * dx # D_2_tilde
- dt_factor * dt * inner(cross(u, w_B), Ff) * dx # G_tilde
)
J = J_picard
else:
raise ValueError("only know newton, mdp and picard as linearisation method")
return J
def initial_condition():
z = Function(Z)
t.assign(0)
z.sub(0).interpolate(u_ex)
z.sub(1).interpolate(p_ex)
z.sub(2).interpolate(B_ex)
z.sub(3).interpolate(E_ex)
return z
# Initial Condition
z0 = initial_condition()
z1 = initial_condition()
z = Function(z0)
z_test = TestFunction(Z)
dt_factor = Constant(1)
# Crank Nicolson form
F_cn = (
inner(split(z)[0], split(z_test)[0])*dx
- inner(split(z0)[0], split(z_test)[0])*dx
+ inner(split(z)[2], split(z_test)[2])*dx
- inner(split(z0)[2], split(z_test)[2])*dx
+ 0.5*dt*(form(z, z_test, Z, 0) + form(z0, z_test, Z, 1))
)
# Implicit Euler form
F_ie = (
inner(split(z)[0], split(z_test)[0])*dx
- inner(split(z0)[0], split(z_test)[0])*dx
+ inner(split(z)[2], split(z_test)[2])*dx
- inner(split(z0)[2], split(z_test)[2])*dx
+ dt*(form(z, z_test, Z, 0))
)
# BDF2 form
F_bdf2 = (
inner(split(z)[0], split(z_test)[0])*dx
- 4.0/3.0*inner(split(z0)[0], split(z_test)[0])*dx
+ 1.0/3.0*inner(split(z1)[0], split(z_test)[0])*dx
+ inner(split(z)[2], split(z_test)[2])*dx
- 4.0/3.0*inner(split(z0)[2], split(z_test)[2])*dx
+ 1.0/3.0*inner(split(z1)[2], split(z_test)[2])*dx
+ 2.0/3.0*dt*form(z, z_test, Z, 0)
)
if stab:
initial = interpolate(as_vector([sin(y), x, x]), V)
z_last_u.assign(initial)
stabilisation = BurmanStabilisation(Z.sub(0), state=z_last_u, h=FacetArea(mesh), weight=stab_weight)
stabilisation_form_u = stabilisation.form(split(z)[0], split(z_test)[0])
F_ie += dt*(advect * stabilisation_form_u)
F_cn += 0.5*dt*(advect * stabilisation_form_u)
F_bdf2 += 2.0/3.0*dt*(advect * stabilisation_form_u)
J_cn = J_form(F_cn, z, z_test)
J_bdf2 = J_form(F_bdf2, z, z_test)
appctx = {"Re": Re, "gamma": gamma, "nu": 1/Re, "Rem": Rem, "gamma2": gamma2}
params = solvers[args.solver_type]
# Depending on the Mesh Hierarchy we have to use star or macrostar solver
if args.solver_type in ["fs2by2", "fs2by2slu"]:
params["fieldsplit_0"] = nsfsstar if hierarchy == "uniform" else nsfsmacrostar
if args.solver_type in ["fs2by2", "fs2by2nslu"]:
params["fieldsplit_1"] = outerschurstar if hierarchy == "uniform" else outerschurmacrostar
# Set up nonlinear solver for first time step
nvproblem_cn = NonlinearVariationalProblem(F_cn, z, bcs=bcs, J=J_cn)
solver_cn = NonlinearVariationalSolver(nvproblem_cn, solver_parameters=params, options_prefix="", appctx=appctx)
# Set up nonlinear solver for later time steps
nvproblem_bdf2 = NonlinearVariationalProblem(F_bdf2, z, bcs=bcs, J=J_bdf2)
solver_bdf2 = NonlinearVariationalSolver(nvproblem_bdf2, solver_parameters=params, options_prefix="", appctx=appctx)
# Definition of solver and transfer operators
qtransfer = NullTransfer()
Etransfer = NullTransfer()
vtransfer = SVSchoeberlTransfer((1/Re, gamma), 2, hierarchy)
Btransfer = SVSchoeberlTransfer((1/Rem, gamma2), 2, hierarchy)
dgtransfer = DGInjection()
transfers = {
Q.ufl_element(): (prolong, restrict, qtransfer.inject),
VectorElement("DG", mesh.ufl_cell(), args.k): (dgtransfer.prolong, restrict, dgtransfer.inject),
VectorElement("DG", mesh.ufl_cell(), args.k-1): (dgtransfer.prolong, restrict, dgtransfer.inject),
}
# On barycentric refined grids we need special prolongation operators
if hierarchy == "bary":
transfers[V.ufl_element()] = (vtransfer.prolong, vtransfer.restrict, inject)
transfermanager = TransferManager(native_transfers=transfers)
solver_cn.set_transfer_manager(transfermanager)
solver_bdf2.set_transfer_manager(transfermanager)
results = {}
res = args.Re
rems = args.Rem
Ss = args.S
pvd = File("output/mhd.pvd")
def run(re, rem, s):
(u, p, B, E) = z.split()
Re.assign(re)
Rem.assign(rem)
S.assign(s)
z0.assign(initial_condition())
z1.assign(initial_condition())
z.assign(z0)
# Indices for output depending on Re-S, Re-Rem or S-Rem table
if len(args.S) == 1 or len(args.Rem) == 1:
ind1 = re
ind2 = s*rem
else:
ind1 = re*s
ind2 = rem
# Set things up for timestepping
T = args.Tf # final time
t.assign(0.0) # current time we are solving for
global dt
global bcs
ntimestep = 0 # number of timesteps solved
total_nonlinear_its = 0 # number of total nonlinear iterations
total_linear_its = 0 # number of total linear iterations
while (float(t) < float(T-dt)+1.0e-10):
t.assign(t+dt)
if mesh.comm.rank == 0:
print(BLUE % ("\nSolving for time: %f" % t), flush=True)
if bc_varying:
if not solution_known:
raise ValueError("Sorry, don't know how to reconstruct the BCs")
else:
u_ex_ = interpolate(u_ex, V)
B_ex_ = interpolate(B_ex, W)
p_ex_ = interpolate(p_ex, Q)
E_ex_ = interpolate(E_ex, R)
bcs[0].function_arg = u_ex_
bcs[1].function_arg = B_ex_
bcs[2].function_arg = E_ex_
if mesh.comm.rank == 0:
print(GREEN % ("Solving for #dofs = %s, Re = %s, Rem = %s, gamma = %s, gamma2 = %s, S = %s, baseN = %s, nref = %s, "
"linearisation = %s, testproblem = %s, discr = %s, k = %s"
% (Z.dim(), float(re), float(rem), float(gamma), float(gamma2), float(S), int(baseN), int(nref),
linearisation, testproblem, discr, int(float(k)))), flush=True)
if not os.path.exists("results/"):
os.mkdir("results/")
# Update z_last_u in Burman Stabilisation
if stab:
stabilisation.update(z.split()[0])
z_last_u.assign(u)
# Do CN in first timestep, after that BDF2
start = datetime.now()
if ntimestep < 1:
dt_factor.assign(0.5)
solver = solver_cn
solver_cn.solve()
else:
dt_factor.assign(2.0/3.0)
solver = solver_bdf2
solver_bdf2.solve()
end = datetime.now()
# Iteration numbers for this time step
linear_its = solver.snes.getLinearSolveIterations()
nonlinear_its = solver.snes.getIterationNumber()
time = (end-start).total_seconds() / 60
if nonlinear_its == 0:
nonlinear_its = 1
if linear_its == 0:
linear_its = 1
if mesh.comm.rank == 0:
print(GREEN % ("Time taken: %.2f min in %d nonlinear iterations, %d linear iterations (%.2f Krylov iters per Newton step)"
% (time, nonlinear_its, linear_its, linear_its/float(nonlinear_its))), flush=True)
print("%.2f @ %d @ %d @ %.2f" % (time, nonlinear_its, linear_its, linear_its/float(nonlinear_its)), flush=True)
(u, p, B, E) = z.split()
B.rename("MagneticField")
u.rename("VelocityField")
p.rename("Pressure")
E.rename("ElectricFieldf")
norm_div_u = sqrt(assemble(inner(div(u), div(u))*dx))
norm_div_B = sqrt(assemble(inner(div(B), div(B))*dx))
if mesh.comm.rank == 0:
print("||div(u)||_L^2 = %s" % norm_div_u, flush=True)
print("||div(B)||_L^2 = %s" % norm_div_B, flush=True)
if solution_known:
B_ex_ = interpolate(B_ex, B.function_space())
u_ex_ = interpolate(u_ex, u.function_space())
p_ex_ = interpolate(p_ex, p.function_space())
E_ex_ = interpolate(E_ex, E.function_space())
B_ex_.rename("ExactSolutionB")
u_ex_.rename("ExactSolutionu")
p_ex_.rename("ExactSolutionp")
E_ex_.rename("ExactSolutionE")
# Compute errors for MMS
error_u = errornorm(u_ex, u, 'L2')
error_B = errornorm(B_ex, B, 'L2')
error_E = errornorm(E_ex, E, 'L2')
error_p = errornorm(p_ex, p, 'L2')
if mesh.comm.rank == 0:
print("Error ||u_ex - u||_L^2 = %s" % error_u, flush=True)
print("Error ||p_ex - p||_L^2 = %s" % error_p, flush=True)
print("Error ||B_ex - B||_L^2 = %s" % error_B, flush=True)
print("Error ||E_ex - E||_L^2 = %s" % error_E, flush=True)
f = open("error.txt", 'a+')
f.write("%s,%s,%s,%s\n" % (error_u, error_p, error_B, error_E))
f.close()
if output:
pvd.write(u, u_ex_, p, p_ex_, B, B_ex_, E, E_ex_, time=float(t))
sys.stdout.flush()
info_dict = {
"Re": re,
"Rem": rem,
"S": s,
"krylov/nonlin": linear_its/nonlinear_its,
"nonlinear_iter": nonlinear_its,
"error_u": error_u,
"error_p": error_p,
"error_B": error_B,