def _assemble(self):
"""
Helper method that assembles the mesh dependent system matrices and
caches them once their are calculated.
"""
if self._assemble_cache: return self._assemble_cache
VS = FunctionSpace(self.mesh.with_shell, "Lagrange", self.order)
VV = VectorFunctionSpace(self.mesh.with_shell, "Lagrange", 1)
v = TestFunction(VS)
u = TrialFunction(VS)
sample_size = self.mesh.data['sample_size']
transformation_order = 1 if (self.order == 1) else 2
gx = MetricMatrix.create_for_cube(sample_size, 0, transformation_order)
gy = MetricMatrix.create_for_cube(sample_size, 1, transformation_order)
gz = MetricMatrix.create_for_cube(sample_size, 2, transformation_order)
# Dummy magnetization for linear form
m = interpolate(Constant((0.0,0.0,0.0)), VV)
# set up forms and boundary conditions
dx = Measure("dx")[self.mesh.cell_domains()]
a = Dx(v, i) * Dx(u, i) * dx(0) + \
Dx(v, i) * Dx(u, i) * dx(1) + \
Dx(v, i) * gx[i,j] * Dx(u, j) * dx(2) + \
Dx(v, i) * gy[i,j] * Dx(u, j) * dx(3) + \
Dx(v, i) * gz[i,j] * Dx(u, j) * dx(4)
L = inner(grad(v), m) * dx(0)
bc = DirichletBC(VS, Constant(0.0), DomainBoundary())
u = Function(VS)
# set up assembler and solver objects
A, b = Matrix(), Vector()
assembler = SystemAssembler(a, L, bc)
assembler.assemble(A)
solver = KrylovSolver(A, "cg", "amg")
#solver.parameters["report"] = False
#solver.parameters["absolute_tolerance"] = 1e-4
#solver.parameters["relative_tolerance"] = 1e-4
solver.parameters["nonzero_initial_guess"] = True
#solver.parameters["preconditioner"]["reuse"] = True # TODO seems to do nothing
self._assemble_cache = (solver, assembler, A, b, bc, m, u)
return self._assemble_cache
def _assemble(self):
"""
Helper method that assembles the mesh dependent system matrices and
caches them once their are calculated.
"""
if self._assemble_cache: return self._assemble_cache
VS = FunctionSpace(self.mesh.with_shell, "Lagrange", self.order)
VV = VectorFunctionSpace(self.mesh.with_shell, "Lagrange", 1)
v = TestFunction(VS)
u = TrialFunction(VS)
sample_size = self.mesh.data['sample_size']
transformation_order = 1 if (self.order == 1) else 2
gx = MetricMatrix.create_for_cube(sample_size, 0, transformation_order)
gy = MetricMatrix.create_for_cube(sample_size, 1, transformation_order)
gz = MetricMatrix.create_for_cube(sample_size, 2, transformation_order)
# Dummy magnetization for linear form
m = interpolate(Constant((0.0, 0.0, 0.0)), VV)
# set up forms and boundary conditions
dx = Measure("dx")[self.mesh.cell_domains()]
a = Dx(v, i) * Dx(u, i) * dx(0) + \
Dx(v, i) * Dx(u, i) * dx(1) + \
Dx(v, i) * gx[i,j] * Dx(u, j) * dx(2) + \
Dx(v, i) * gy[i,j] * Dx(u, j) * dx(3) + \
Dx(v, i) * gz[i,j] * Dx(u, j) * dx(4)
L = inner(grad(v), m) * dx(0)
bc = DirichletBC(VS, Constant(0.0), DomainBoundary())
u = Function(VS)
# set up assembler and solver objects
A, b = Matrix(), Vector()
assembler = SystemAssembler(a, L, bc)
assembler.assemble(A)
solver = KrylovSolver(A, "cg", "amg")
#solver.parameters["report"] = False
#solver.parameters["absolute_tolerance"] = 1e-4
#solver.parameters["relative_tolerance"] = 1e-4
solver.parameters["nonzero_initial_guess"] = True
#solver.parameters["preconditioner"]["reuse"] = True # TODO seems to do nothing
self._assemble_cache = (solver, assembler, A, b, bc, m, u)
return self._assemble_cache
def _assemble(self):
"""
Helper method that assembles the mesh dependent system matrices and
caches them once their are calculated.
"""
if self._assemble_cache: return self._assemble_cache
VS = FunctionSpace(self.mesh.with_shell, "Lagrange", self.order)
v = TestFunction(VS)
u = TrialFunction(VS)
sample_size = self.mesh.data['sample_size']
transformation_order = 1 if (self.order == 1) else 2
gx = MetricMatrix.create_for_cube(sample_size, 0, transformation_order)
gy = MetricMatrix.create_for_cube(sample_size, 1, transformation_order)
gz = MetricMatrix.create_for_cube(sample_size, 2, transformation_order)
# setup system
a = Dx(v, i) * Dx(u, i) * dx(0) + \
Dx(v, i) * Dx(u, i) * dx(1) + \
Dx(v, i) * gx[i,j] * Dx(u, j) * dx(2) + \
Dx(v, i) * gy[i,j] * Dx(u, j) * dx(3) + \
Dx(v, i) * gz[i,j] * Dx(u, j) * dx(4)
bc = DirichletBC(VS, Constant(0.0), DomainBoundary())
A, An = symmetric_assemble(a, bc)
u = Function(VS)
solver = KrylovSolver(A, "cg", "amg")
#solver.parameters["report"] = False
#solver.parameters["absolute_tolerance"] = 1e-4
#solver.parameters["relative_tolerance"] = 1e-4
solver.parameters["nonzero_initial_guess"] = True
#solver.parameters["preconditioner"]["reuse"] = True # TODO seems to do nothing
self._assemble_cache = (solver, An, bc, v, u, A)
return self._assemble_cache
def calculate_v(self, state, dt):
"""
Calculate :math:`\\vec{v}` according to the algorithm introduced above.
*Arguments*
state (:class:`State`)
The simulation state containing the magnetization configuration.
dt (:class:`float`)
The time-step size.
"""
with Timer("Calculate v"):
cache = self._assemble_cache
if cache.requires_update(state):
if isinstance(state.mesh, WrappedMesh):
raise Exception("Use simple mesh with shell instead of WrappedMesh.")
# setup function spaces
cache.VV = VectorFunctionSpace(state.mesh, "CG", 1)
cache.VD = FunctionSpace(state.mesh, "CG", self.demag_order)
cache.VL = FunctionSpace(state.mesh, "CG", 1)
# setup some stuff for the demag field
transformation_order = 2 if (self.demag_order == 1) else 2
shell_width = state.mesh.coordinates().max(axis=0).min() / 2.0
sample_size = state.mesh.coordinates().max(axis=0) - shell_width
cache.gx = MetricMatrix.create_for_cube(sample_size, 0, transformation_order)
cache.gy = MetricMatrix.create_for_cube(sample_size, 1, transformation_order)
cache.gz = MetricMatrix.create_for_cube(sample_size, 2, transformation_order)
# Test and Trial Functions
u = TrialFunction(cache.VD)
v = TrialFunction(cache.VV)
sigma = TrialFunction(cache.VL)
w1 = TestFunction(cache.VD)
w2 = TestFunction(cache.VV)
w3 = TestFunction(cache.VL)
#######################################################
# Define weak forms
#######################################################
f_ex = (- 2.0 * state.material.Aex * Constants.gamma) / \
(Constants.mu0 * state.material.ms * state.scale**2)
# Bilinear form
a12 = - 0.5 * Constant(dt) * inner(v, grad(w1)) * state.dx('magnetic') # Demag Field, Implicit RHS
a11 = inner(grad(w1), grad(u)) * state.dx('all') \
+ inner(grad(w1), grad(u)) * state.dx(1000) \
+ inner(grad(w1), cache.gx * grad(u)) * state.dx(1001) \
+ inner(grad(w1), cache.gy * grad(u)) * state.dx(1002) \
+ inner(grad(w1), cache.gz * grad(u)) * state.dx(1003)
a21 = - Constant(state.material.ms * Constants.gamma) * inner(grad(u), w2) * state.dx('magnetic') # Demag
a22 = Constant(state.material.alpha) * dot(v, w2) * state.dx('magnetic') # LLG
a22 += dot(cross(state.m, v), w2) * state.dx('magnetic')
a22 += - 0.5 * Constant(dt * f_ex) * Dx(v[i],j) * Dx(w2[i],j) * state.dx('magnetic') # Exchange
# Linear form
L1 = inner(state.m, grad(w1)) * state.dx('magnetic') # Demag Field
L2 = Constant(f_ex) * Dx(state.m[i],j) * Dx(w2[i],j) * state.dx('magnetic') # Exchange
for term in self.terms: # Additional Terms
L2 += term.form_rhs(state, w2) * state.dx('magnetic')
#a22 += term.form_lhs(state, w2, Constant(0.5 * dt) * v) * state.dx('magnetic')
#######################################################
# Define boundary conditions
#######################################################
bc1 = DirichletBC(cache.VD, Constant(0.0), DomainBoundary())
#######################################################
# Assemble the system
#######################################################
A11, b1 = assemble_system(a11, L1, bc1)
A12 = assemble(a12)
A21 = assemble(a21)
A22 = assemble(a22, keep_diagonal=True); A22.ident_zeros()
v23 = TestFunction(state.VectorFunctionSpace())
u23 = TrialFunction(state.FunctionSpace())
A23 = assemble(u23 * dot(state.m, v23) * dP)
v32 = TestFunction(state.FunctionSpace())
u32 = TrialFunction(state.VectorFunctionSpace())
A32 = assemble(v32 * dot(state.m, u32) * dP)
#A23 = DofAssembler.assemble(TransScalarProductMatrix(cache.VL, cache.VV, state.m))
#A32 = DofAssembler.assemble(ScalarProductMatrix(cache.VL, cache.VV, state.m))
b2 = assemble(L2)
#######################################################
# Schur Ansatz
#######################################################
A11p = ILU(A11)
A11inv = ConjGrad(A11, precond=A11p)
z1 = A11inv * b1
S11 = A22 - A21 * A11inv * A12
S = block_mat([[S11, A23],
[A32, 0]])
Sp11 = ILU(A22)
Sp22 = InvDiag(collapse(A32 * LumpedInvDiag(A22) * A23))
Sp = block_mat([[Sp11, A23],
[0, -Sp22]]).scheme('sgs')
Sinv = BiCGStab(S, precond=Sp, tolerance=1e-8/dt)
b = block_vec([b2 - A21*z1, 0])
v, _ = Sinv * b
return Function(cache.VV, v)