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mhd3d_stationary_upT_BE.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
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 SchurPC(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))
# Weak form of NS + 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
+ 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) + 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
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),
]
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, T_n, B_n, E_n] = split(state)
A = (
+ inner(E, Ff) * dx
+ inner(cross(u_n, B), Ff) * dx
- 1/Pm * inner(B, curl(Ff)) * dx
+ inner(curl(E), C) * dx
+ 1/Pm * 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"),
]
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)
import sys
sys.stdout.flush()
# Definition of different solver parameters
# Increase buffer size for MUMPS solver
ICNTL_14 = 5000
# 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_0_fields": "0,2",
"pc_fieldsplit_1_fields": "1",
"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": 1,
"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_0_fields": "0,2",
"pc_fieldsplit_1_fields": "1",
"pc_fieldsplit_schur_factorization_type": "full",
"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": 1,
"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": 1,
"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": 1,
"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
fs3by2 = {
"snes_type": "newtonls",
"snes_max_it": 10,
"snes_linesearch_type": "basic",
"snes_linesearch_maxstep": 1.0,
"snes_rtol": 1.0e-9,
"snes_atol": 2.5e-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": "0,1,2",
"pc_fieldsplit_1_fields": "3,4",
}
# Main solver with LU solver for (u,p)-block
fs3by2nslu = {
"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": "0,1,2",
"pc_fieldsplit_1_fields": "3,4",
"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
fs3by2slu = {
"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-6,
"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": "0,1,2",
"pc_fieldsplit_1_fields": "3,4",
"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
fs3by2lu = {
"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": "0,1,2",
"pc_fieldsplit_1_fields": "3,4",
"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, "fs3by2": fs3by2, "fs3by2nslu": fs3by2nslu, "fs3by2slu": fs3by2slu, "fs3by2lu": fs3by2lu}
# Definition of problem parameters
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("--Ra", nargs='+', type=float, default=[1])
parser.add_argument("--Pm", nargs='+', type=float, default=[1])
parser.add_argument("--Pr", 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("--hierarchy", choices=["bary", "uniform"], default="bary")
parser.add_argument("--solver-type", choices=list(solvers.keys()), default="lu")
parser.add_argument("--testproblem", choices=["hc", "3dgenerator", "smooth"], default="smooth")
parser.add_argument("--discr", choices=["rt", "bdm", "cg"], required=True)
parser.add_argument("--linearisation", choices=["picard", "mdp", "newton"], required=True)
parser.add_argument("--stab", default=False, action="store_true")
parser.add_argument("--checkpoint", 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
Ra = Constant(args.Ra[0])
Pm = Constant(args.Pm[0])
Pr = Constant(args.Pr[0])
gamma = Constant(args.gamma)
gamma2 = Constant(args.gamma2)
hierarchy = args.hierarchy
solver_type = args.solver_type
testproblem = args.testproblem
gamma2 = Constant(args.gamma2)
advect = Constant(args.advect)
linearisation = args.linearisation
discr = args.discr
stab = args.stab
checkpoint = args.checkpoint
output = args.output
#Re = Constant(1)
#Rem = Constant(1)
S = Constant(1)
# Stabilisation weight for BurmanStabilisation
stab_weight = Constant(3e-3)
if len(args.Ra) != 1 and len(args.Pm) != 1 and len(args.Pr) != 1:
raise ValueError("Ra, Pm and Pr cannot all contain more than one element at the same time")
if discr == "cg" and hierarchy != "bary":
raise ValueError("SV is only stable on barycentric refined grids")
if k < 3 and hierarchy == "bary":
raise ValueError("Scott Vogelius is not stable for k<3")
# Define the base Mesh for the different problems
if testproblem == "3dgenerator":
base = BoxMesh(5*baseN, baseN, baseN, 5, 1, 1, distribution_parameters=distribution_parameters)
else:
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
if testproblem != "3dgenerator":
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 spaces
if discr == "rt":
Vel = FiniteElement("N1div", mesh.ufl_cell(), k, variant="integral")
V = FunctionSpace(mesh, Vel)
elif discr == "bdm":
Vel = FiniteElement("N2div", mesh.ufl_cell(), k, variant="integral")
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="integral")
R = FunctionSpace(mesh, Rel) # E
Wel = FiniteElement("N1div", mesh.ufl_cell(), k, variant="integral")
W = FunctionSpace(mesh, Wel)
TT = FunctionSpace(mesh, "CG", k)
Z = MixedFunctionSpace([V, Q, TT, W, R])
z = Function(Z)
(u, p, T, B, E) = split(z)
(v, q, s, C, Ff) = split(TestFunction(Z))
# used for BurmanStabilisation
z_last_u = Function(V)
# For continuation we might want to start from checkpoint
if checkpoint:
try:
if len(args.Pr) == 1 or len(args.Ra) == 1:
chk = DumbCheckpoint("dump/"+str(float(S*Pm))+str(linearisation)+str(testproblem), mode=FILE_READ)
else:
chk = DumbCheckpoint("dump/"+str(float(Pm))+str(linearisation)+str(testproblem), mode=FILE_READ)
chk.load(z)
except Exception as e:
message(e)
(u_, p_, T_, B_, E_) = z.split()
z_last_u.assign(u_)
(x, y, zz) = SpatialCoordinate(Z.mesh())
n = FacetNormal(mesh)
eps = lambda x: sym(grad(x))
g = as_vector([0,1,0])
# Base weak form of problem
F = (
2*Pr * 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
- Pr/Pm * inner(B, curl(Ff)) * dx
+ inner(curl(E), C) * dx
+ Pr/Pm * inner(div(B), div(C)) * dx
- Ra*Pr * inner(g*T , v) * dx
+ inner(grad(T), grad(s)) * dx
+ inner(dot(u, grad(T)), s) * dx
)
# Compute RHS for Method of Manufactured Solution (MMS)
def compute_rhs(u_ex, B_ex, T_ex, p_ex, E_ex):
E_ex_ = interpolate(E_ex, R)
f1 = (-2*Pr * div(eps(u_ex)) + advect * dot(grad(u_ex), u_ex) - gamma * grad(div(u_ex))
+ grad(p_ex) + S * cross(B_ex, (E_ex + cross(u_ex, B_ex))) - Ra*Pr * g*T_ex)
f2 = + curl(E_ex_) - Pr/Pm * grad(div(B_ex))
f3 = -Pr/Pm * curl(B_ex) + E_ex + cross(u_ex, B_ex)
f4 = -div(grad(T_ex)) + dot(u_ex, grad(T_ex))
return (f1, f2, f3, f4)
if testproblem == "smooth":
u_ex = as_vector([cos(zz), sin(x), sin(y)])
B_ex = as_vector([sin(y), cos(zz), sin(x)])
T_ex = cos(x)
p_ex = sin(x)
E_ex = as_vector([sin(x)*cos(x), sin(y), cos(x)]) # 1.0e-13*x
(f1, f2, f3, f4) = compute_rhs(u_ex, B_ex, T_ex, p_ex, E_ex)
rhs = True # because RHS is zero
bcs = [DirichletBC(Z.sub(0), u_ex, "on_boundary"),
DirichletBC(Z.sub(2), T_ex, "on_boundary"),
DirichletBC(Z.sub(3), B_ex, "on_boundary"),
DirichletBC(Z.sub(4), E_ex, "on_boundary"),
PressureFixBC(Z.sub(1), 0, 1)]
# Do we know what the exact solution of the problem is?
solution_known = True
# Do the boundary conditions depend on parameter thats change during continuation
bc_varying = False
# On what ids of the boundary do we want to apply the boundary conditions
# This is needed for DG-Form of H(div)-L2 formulation
bcs_ids_apply = (1, 2, 3, 4, 5, 6)
bcs_ids_dont_apply = None
elif testproblem == "3dgenerator":
# Hartmann number
Ha = sqrt(S*Rem*Re)
# Problem parameter
B0 = Constant(1.0)
x_on = Constant(2.0)
x_off = Constant(2.5)
delta = Constant(0.1)
B_z_gen = (B0/2)*(tanh((x-x_on)/delta) - tanh((x-x_off)/delta))
# *_ex are just defined for boundary conditions. We don't know exact solution
u_ex = Constant((1, 0, 0), domain=mesh)
B_ex = as_vector([Constant(0, domain=mesh), Constant(0, domain=mesh), B_z_gen])
E_ex = 1/Rem * curl(B_ex)
p_ex = Constant(0, domain=mesh)
T_ex = sin(x)
# Inflow BCs for u at x=0, outflow at x=1, no-slip on 4 sides
# On what ids of the boundary do we want to apply the boundary conditions
# This is needed for DG-Form of H(div)-L2 formulation
bcs_ids_apply = (1)
bcs_ids_dont_apply = (3, 4, 5, 6)
bcs = [DirichletBC(Z.sub(0), u_ex, bcs_ids_apply),
DirichletBC(Z.sub(0), Constant((0., 0., 0.)), bcs_ids_dont_apply),
DirichletBC(Z.sub(2), T_ex, "on_boundary"),
DirichletBC(Z.sub(3), B_ex, "on_boundary"),
DirichletBC(Z.sub(4), 0, "on_boundary"),
PressureFixBC(Z.sub(1), 0, 1)]
rhs = None # because rhs is zero for this problem
# Do we know what the exact solution of the problem is?
solution_known = False
# Do the boundary conditions depend on parameter thats change during continuation
bc_varying = True
elif testproblem == "hc":
# example taken from https://doi.org/10.1016/j.jcp.2016.04.019
u_ex = Constant((0, 0, 0), domain=mesh)
B_ex = Constant((0, 1, 0), domain=mesh)
B_ex = project(B_ex, W)
bcs_ids_apply = (1, 2, 3, 4, 5, 6)
bcs_ids_dont_apply = None
bcs = [DirichletBC(Z.sub(0), u_ex, bcs_ids_apply), # 4 == upper boundary (y==1)
DirichletBC(Z.sub(2), 0, 2),
DirichletBC(Z.sub(2), 1, 1),
DirichletBC(Z.sub(3), B_ex, "on_boundary"),
DirichletBC(Z.sub(4), 0, "on_boundary"),
PressureFixBC(Z.sub(1), 0, 1)]
rhs = None
solution_known = False
bc_varying = False
if solution_known:
u_ex_ = interpolate(u_ex, V)
B_ex_ = interpolate(B_ex, W)
T_ex_ = interpolate(T_ex, TT)
p_ex_ = interpolate(p_ex, Q)
E_ex_ = interpolate(E_ex, R)
# Add Burman Stabilisation
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(u, v)
F += (advect * stabilisation_form_u)
# 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)
sigma = Constant(1) * Z.sub(0).ufl_element().degree()**2 # penalty term
theta = Constant(1) # theta=1 means upwind discretization of nonlinear advection term
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 = (
- Pr * inner(avg(2*sym(grad(u))), 2*avg(outer(v, n))) * dS
- Pr * inner(avg(2*sym(grad(v))), 2*avg(outer(u, n))) * dS
+ Pr * sigma/avg(h) * inner(2*avg(outer(u, n)), 2*avg(outer(v, n))) * dS
- inner(outer(v, n), 2*Pr*sym(grad(u))) * ds
- inner(outer(u-u_ex, n), 2*Pr*sym(grad(v))) * ds(bcs_ids_apply)
+ Pr*(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:
F_DG += (
- inner(outer(u, n), 2*Pr*sym(grad(v))) * ds(bcs_ids_dont_apply)
+ Pr*(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
if rhs is not None:
F -= inner(f1, v) * dx + inner(f3, Ff) * dx + inner(f2, C) * dx + inner(f4, s) * dx
# Definition of the three different linearizations
w = TrialFunction(Z)
[w_u, w_p, w_T, 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
- inner(cross(w_u, B), Ff) * dx # G
)
J = J_mdp
elif linearisation == "picard":
J_picard = (
J_newton
- S * inner(cross(w_B, E), v) * dx # J_tilde
- S * inner(cross(w_B, cross(u, B)), v) * dx # D_1_tilde
- S * inner(cross(B, cross(u, w_B)), v) * dx # D_2_tilde
- inner(cross(u, w_B), Ff) * dx # G_tilde
)
J = J_picard
else:
raise ValueError("only know newton, mdp and picard as linearisation method")
problem = NonlinearVariationalProblem(F, z, bcs, J=J)
appctx = {"Ra": Ra, "gamma": gamma, "nu": 1/Ra, "Pm": Pm, "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 ["fs3by2", "fs3by2slu"]:
params["fieldsplit_0"] = nsfsstar if hierarchy == "uniform" else nsfsmacrostar
if args.solver_type in ["fs3by2", "fs3by2mlu"]:
params["fieldsplit_1"] = outerschurstar if hierarchy == "uniform" else outerschurmacrostar
if args.hierarchy == "bary":
params["snes_linesearch_type"] = "l2"
# Definition of solver and transfer operators
solver = NonlinearVariationalSolver(problem, solver_parameters=params, options_prefix="", appctx=appctx)
qtransfer = NullTransfer()
Etransfer = NullTransfer()
vtransfer = SVSchoeberlTransfer((Pr, gamma), 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.set_transfer_manager(transfermanager)
results = {}
Ras = args.Ra
Pms = args.Pm
Prs = args.Pr
pvd = File("output/mhd.pvd")
def get_ind_dict(ra, pm, pr):
# Indices for output depending on Ra-Pr, Ra-Pm or Pr-Pm table
if len(args.Pm) == 1:
ind_dict = {"Ra": ra, "Pr": pr}
elif len(args.Ra) == 1:
ind_dict = {"Pr": pr, "Pm": pm}
elif len(args.Pr) == 1:
ind_dict = {"Ra": ra, "Pm": pm}
else:
raise ValueError("Can only iterate over two elements of Ra, Pm, Pr")
return ind_dict
def run(ra, pm, pr):
(u, p, T, B, E) = z.split()
Ra.assign(ra)
Pm.assign(pm)
Pr.assign(pr)
ind_dict = get_ind_dict(ra, pm, pr)
val1, val2 = list(ind_dict.values())
if bc_varying:
global p_ex
u_ex_ = interpolate(u_ex, V)
B_ex_ = interpolate(B_ex, W)
T_ex_ = interpolate(T_ex, TT)
E_ex_ = interpolate(E_ex, R)
pintegral = assemble(p_ex*dx)
p_ex = p_ex - Constant(pintegral/area)
# bcs[0] is u, bcs[1] is B, bcs[2] is E
bcs[0].function_arg = u_ex_
if checkpoint:
try:
chk = DumbCheckpoint("dump/"+str(float(val2))+str(linearisation)+str(testproblem), mode=FILE_READ)
chk.load(z)
except Exception as e:
message(e)
if mesh.comm.rank == 0:
print(GREEN % ("Solving for #dofs = %s, Ra = %s, Pm = %s, gamma = %s, Pr = %s, baseN = %s, nref = %s, "
"linearisation = %s, testproblem = %s, discr = %s, k = %s"
% (Z.dim(), float(ra), float(pm), float(gamma), float(pr), 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)
# Solve the problem and measure time
start = datetime.now()
solver.solve()
end = datetime.now()
# Iteration numbers
linear_its = solver.snes.getLinearSolveIterations()
nonlinear_its = solver.snes.getIterationNumber()
time = (end-start).total_seconds() / 60
if mesh.comm.rank == 0:
if nonlinear_its == 0:
nonlinear_its = 1
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, T, B, E) = z.split()
# Make sure that average of p is 0
pintegral = assemble(p*dx)
p.assign(p - Constant(pintegral/area))
B.rename("MagneticField")
u.rename("VelocityField")
T.rename("Temperature")
p.rename("Pressure")
E.rename("ElectricFieldf")
# Compute divergence of u and B
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())
T_ex_ = interpolate(T_ex, T.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")
T_ex_.rename("ExactSolutionT")
p_ex_.rename("ExactSolutionp")
E_ex_.rename("ExactSolutionE")
# Compute error for MMS
error_u = errornorm(u_ex, u, 'L2')
error_B = errornorm(B_ex, B, 'L2')
error_T = errornorm(T_ex, T, '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 ||T_ex - T||_L^2 = %s" % error_T, 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)