def test_newton_relax0(self):
self.assert_resnorm(
solver.newton('dofs',
residual=self.residual,
lhs0=self.lhs0,
constrain=self.cons,
relax0=.1).solve(tol=self.tol, maxiter=5))
def test_newton_relax0(self):
self.assert_resnorm(
solver.newton(self.dofs,
residual=self.residual,
arguments=self.arguments,
constrain=self.cons,
relax0=.1).solve(tol=self.tol, maxiter=5))
def test_newton_medianbased(self):
self.assert_resnorm(
solver.newton('dofs',
residual=self.residual,
lhs0=self.lhs0,
constrain=self.cons,
linesearch=solver.MedianBased()).solve(tol=self.tol,
maxiter=2))
def test_newton_medianbased(self):
self.assert_resnorm(
solver.newton(self.dofs,
residual=self.residual,
arguments=self.arguments,
constrain=self.cons,
linesearch=solver.MedianBased()).solve(tol=self.tol,
maxiter=2))
def test_newton_iter(self):
_test_recursion_cache(
self, lambda:
((types.frozenarray(lhs), resnorm)
for lhs, resnorm in solver.newton('dofs',
residual=self.residual,
lhs0=self.lhs0,
constrain=self.cons)))
def main(nelems: int, degree: int, reynolds: float):
'''
Driven cavity benchmark problem using compatible spaces.
.. arguments::
nelems [12]
Number of elements along edge.
degree [2]
Polynomial degree for velocity; the pressure space is one degree less.
reynolds [1000]
Reynolds number, taking the domain size as characteristic length.
'''
verts = numpy.linspace(0, 1, nelems + 1)
domain, geom = mesh.rectilinear([verts, verts])
ns = function.Namespace()
ns.x = geom
ns.Re = reynolds
ns.ubasis = function.vectorize([
domain.basis('spline',
degree=(degree, degree - 1),
removedofs=((0, -1), None)),
domain.basis('spline',
degree=(degree - 1, degree),
removedofs=(None, (0, -1)))
])
ns.pbasis = domain.basis('spline', degree=degree - 1)
ns.u_i = 'ubasis_ni ?u_n'
ns.p = 'pbasis_n ?p_n'
ns.stress_ij = '(d(u_i, x_j) + d(u_j, x_i)) / Re - p δ_ij'
ns.uwall = domain.boundary.indicator('top'), 0
ns.N = 5 * degree * nelems # nitsche constant based on element size = 1/nelems
ns.nitsche_ni = '(N ubasis_ni - (d(ubasis_ni, x_j) + d(ubasis_nj, x_i)) n(x_j)) / Re'
ures = domain.integral('d(ubasis_ni, x_j) stress_ij d:x' @ ns,
degree=2 * degree)
ures += domain.boundary.integral(
'(nitsche_ni (u_i - uwall_i) - ubasis_ni stress_ij n(x_j)) d:x' @ ns,
degree=2 * degree)
pres = domain.integral('pbasis_n (d(u_k, x_k) + ?lm) d:x' @ ns,
degree=2 * degree)
lres = domain.integral('p d:x' @ ns, degree=2 * degree)
with treelog.context('stokes'):
state0 = solver.solve_linear(['u', 'p', 'lm'], [ures, pres, lres])
postprocess(domain, ns, **state0)
ures += domain.integral('ubasis_ni d(u_i, x_j) u_j d:x' @ ns,
degree=3 * degree)
with treelog.context('navierstokes'):
state1 = solver.newton(('u', 'p', 'lm'), (ures, pres, lres),
arguments=state0).solve(tol=1e-10)
postprocess(domain, ns, **state1)
return state0, state1
def solve_nutils(g, method='Elliptic', ltol=1e-5, **solveargs):
assert len(g) == 2
res = method_library(g, method=method)
init, cons = g.x, g.cons
lhs = solver.newton('target', res, lhs0=init,
constrain=cons.where).solve(ltol, **solveargs)
g.x = lhs
def test_res(self):
for name in 'direct', 'newton':
with self.subTest(name):
if name == 'direct':
lhs = solver.solve_linear('dofs', residual=self.residual, constrain=self.cons)
else:
lhs = solver.newton('dofs', residual=self.residual, constrain=self.cons).solve(tol=1e-10, maxiter=0)
res = self.residual.eval(arguments=dict(dofs=lhs))
resnorm = numpy.linalg.norm(res[~self.cons.where])
self.assertLess(resnorm, 1e-13)
def test_res(self):
for name in 'direct', 'newton':
with self.subTest(name):
if name == 'direct':
lhs = solver.solve_linear('dofs', residual=self.residual, constrain=self.cons)
else:
lhs = solver.newton('dofs', residual=self.residual, constrain=self.cons).solve(tol=1e-10, maxiter=0)
res = self.residual.eval(arguments=dict(dofs=lhs))
resnorm = numpy.linalg.norm(res[~self.cons.where])
self.assertLess(resnorm, 1e-13)
def test_newton_tolnotreached(self):
with self.assertLogs('nutils', logging.WARNING) as cm:
self.assert_resnorm(
solver.newton('dofs',
residual=self.residual,
lhs0=self.lhs0,
constrain=self.cons,
linrtol=1e-99).solve(tol=self.tol, maxiter=2))
for msg in cm.output:
self.assertIn('solver failed to reach tolerance', msg)
def main(nelems: int, degree: int, reynolds: float):
'''
Driven cavity benchmark problem using compatible spaces.
.. arguments::
nelems [12]
Number of elements along edge.
degree [2]
Polynomial degree for velocity; the pressure space is one degree less.
reynolds [1000]
Reynolds number, taking the domain size as characteristic length.
'''
verts = numpy.linspace(0, 1, nelems + 1)
domain, geom = mesh.rectilinear([verts, verts])
ns = function.Namespace()
ns.x = geom
ns.Re = reynolds
ns.uxbasis, ns.uybasis, ns.pbasis, ns.lbasis = function.chain([
domain.basis('spline',
degree=(degree, degree - 1),
removedofs=((0, -1), None)),
domain.basis('spline',
degree=(degree - 1, degree),
removedofs=(None, (0, -1))),
domain.basis('spline', degree=degree - 1),
[1], # lagrange multiplier
])
ns.ubasis_ni = '<uxbasis_n, uybasis_n>_i'
ns.u_i = 'ubasis_ni ?lhs_n'
ns.p = 'pbasis_n ?lhs_n'
ns.l = 'lbasis_n ?lhs_n'
ns.stress_ij = '(u_i,j + u_j,i) / Re - p δ_ij'
ns.uwall = domain.boundary.indicator('top'), 0
ns.N = 5 * degree * nelems # nitsche constant based on element size = 1/nelems
ns.nitsche_ni = '(N ubasis_ni - (ubasis_ni,j + ubasis_nj,i) n_j) / Re'
res = domain.integral(
'(ubasis_ni,j stress_ij + pbasis_n (u_k,k + l) + lbasis_n p) d:x' @ ns,
degree=2 * degree)
res += domain.boundary.integral(
'(nitsche_ni (u_i - uwall_i) - ubasis_ni stress_ij n_j) d:x' @ ns,
degree=2 * degree)
with treelog.context('stokes'):
lhs0 = solver.solve_linear('lhs', res)
postprocess(domain, ns, lhs=lhs0)
res += domain.integral('ubasis_ni u_i,j u_j d:x' @ ns, degree=3 * degree)
with treelog.context('navierstokes'):
lhs1 = solver.newton('lhs', res, lhs0=lhs0).solve(tol=1e-10)
postprocess(domain, ns, lhs=lhs1)
return lhs0, lhs1
def main(nelems: int, etype: str, degree: int, reynolds: float):
'''
Driven cavity benchmark problem.
.. arguments::
nelems [12]
Number of elements along edge.
etype [square]
Element type (square/triangle/mixed).
degree [2]
Polynomial degree for velocity; the pressure space is one degree less.
reynolds [1000]
Reynolds number, taking the domain size as characteristic length.
'''
domain, geom = mesh.unitsquare(nelems, etype)
ns = function.Namespace()
ns.Re = reynolds
ns.x = geom
ns.ubasis, ns.pbasis = function.chain([
domain.basis('std', degree=degree).vector(2),
domain.basis('std', degree=degree - 1),
])
ns.u_i = 'ubasis_ni ?lhs_n'
ns.p = 'pbasis_n ?lhs_n'
ns.stress_ij = '(u_i,j + u_j,i) / Re - p δ_ij'
sqr = domain.boundary.integral('u_k u_k d:x' @ ns, degree=degree * 2)
wallcons = solver.optimize('lhs', sqr, droptol=1e-15)
sqr = domain.boundary['top'].integral('(u_0 - 1)^2 d:x' @ ns,
degree=degree * 2)
lidcons = solver.optimize('lhs', sqr, droptol=1e-15)
cons = numpy.choose(numpy.isnan(lidcons), [lidcons, wallcons])
cons[-1] = 0 # pressure point constraint
res = domain.integral('(ubasis_ni,j stress_ij + pbasis_n u_k,k) d:x' @ ns,
degree=degree * 2)
with treelog.context('stokes'):
lhs0 = solver.solve_linear('lhs', res, constrain=cons)
postprocess(domain, ns, lhs=lhs0)
res += domain.integral(
'.5 (ubasis_ni u_i,j - ubasis_ni,j u_i) u_j d:x' @ ns,
degree=degree * 3)
with treelog.context('navierstokes'):
lhs1 = solver.newton('lhs', res, lhs0=lhs0,
constrain=cons).solve(tol=1e-10)
postprocess(domain, ns, lhs=lhs1)
return lhs0, lhs1
def main(nelems:int, etype:str, degree:int, reynolds:float):
'''
Driven cavity benchmark problem.
.. arguments::
nelems [12]
Number of elements along edge.
etype [square]
Element type (square/triangle/mixed).
degree [2]
Polynomial degree for velocity; the pressure space is one degree less.
reynolds [1000]
Reynolds number, taking the domain size as characteristic length.
'''
domain, geom = mesh.unitsquare(nelems, etype)
ns = function.Namespace()
ns.Re = reynolds
ns.x = geom
ns.ubasis = domain.basis('std', degree=degree).vector(domain.ndims)
ns.pbasis = domain.basis('std', degree=degree-1)
ns.u_i = 'ubasis_ni ?u_n'
ns.p = 'pbasis_n ?p_n'
ns.stress_ij = '(d(u_i, x_j) + d(u_j, x_i)) / Re - p δ_ij'
usqr = domain.boundary.integral('u_k u_k d:x' @ ns, degree=degree*2)
wallcons = solver.optimize('u', usqr, droptol=1e-15)
usqr = domain.boundary['top'].integral('(u_0 - 1)^2 d:x' @ ns, degree=degree*2)
lidcons = solver.optimize('u', usqr, droptol=1e-15)
ucons = numpy.choose(numpy.isnan(lidcons), [lidcons, wallcons])
pcons = numpy.zeros(len(ns.pbasis), dtype=bool)
pcons[-1] = True # constrain pressure to zero in a point
cons = dict(u=ucons, p=pcons)
ures = domain.integral('d(ubasis_ni, x_j) stress_ij d:x' @ ns, degree=degree*2)
pres = domain.integral('pbasis_n d(u_k, x_k) d:x' @ ns, degree=degree*2)
with treelog.context('stokes'):
state0 = solver.solve_linear(('u', 'p'), (ures, pres), constrain=cons)
postprocess(domain, ns, **state0)
ures += domain.integral('.5 (ubasis_ni d(u_i, x_j) - d(ubasis_ni, x_j) u_i) u_j d:x' @ ns, degree=degree*3)
with treelog.context('navierstokes'):
state1 = solver.newton(('u', 'p'), (ures, pres), arguments=state0, constrain=cons).solve(tol=1e-10)
postprocess(domain, ns, **state1)
return state0, state1
def test_newton_boolcons(self):
self.assert_resnorm(
solver.newton('dofs',
residual=self.residual,
constrain=self.boolcons).solve(tol=self.tol,
maxiter=7))
def finitestrain_patch(bottom, right, top, left):
from nutils import version
if int(version[0]) != 4:
raise ImportError(
'Mismatching nutils version detected, only version 4 supported. Upgrade by \"pip install --upgrade nutils\"'
)
from nutils import mesh, function
from nutils import _, log, solver
# error test input
if not (left.dimension == right.dimension == top.dimension ==
bottom.dimension == 2):
raise RuntimeError(
'finitestrain_patch only supported for planar (2D) geometries')
if left.rational or right.rational or top.rational or bottom.rational:
raise RuntimeError(
'finitestrain_patch not supported for rational splines')
# these are given as a oriented loop, so make all run in positive parametric direction
top.reverse()
left.reverse()
# in order to add spline surfaces, they need identical parametrization
Curve.make_splines_identical(top, bottom)
Curve.make_splines_identical(left, right)
# create an initial mesh (correct corners) which we will morph into the right one
p1 = bottom.order(0)
p2 = left.order(0)
p = max(p1, p2)
linear = BSplineBasis(2)
srf = Surface(linear, linear, [bottom[0], bottom[-1], top[0], top[-1]])
srf.raise_order(p1 - 2, p2 - 2)
for k in bottom.knots(0, True)[p1:-p1]:
srf.insert_knot(k, 0)
for k in left.knots(0, True)[p2:-p2]:
srf.insert_knot(k, 1)
# create computational mesh
n1 = len(bottom)
n2 = len(left)
dim = left.dimension
domain, geom = mesh.rectilinear(srf.knots())
ns = function.Namespace()
ns.basis = domain.basis('spline',
degree(srf),
knotmultiplicities=multiplicities(srf)).vector(2)
ns.phi = domain.basis('spline',
degree(srf),
knotmultiplicities=multiplicities(srf))
ns.eye = np.array([[1, 0], [0, 1]])
ns.cp = controlpoints(srf)
ns.x_i = 'cp_ni phi_n'
ns.lmbda = 1
ns.mu = 1
# add total boundary conditions
# for hard problems these will be taken in steps and multiplied by dt every
# time (quasi-static iterations)
constraints = np.array([[[np.nan] * n2] * n1] * dim)
for d in range(dim):
constraints[d, 0, :] = (left[:, d] - srf[0, :, d])
constraints[d, -1, :] = (right[:, d] - srf[-1, :, d])
constraints[d, :, 0] = (bottom[:, d] - srf[:, 0, d])
constraints[d, :, -1] = (top[:, d] - srf[:, -1, d])
# TODO: Take a close look at the logic below
# in order to iterate, we let t0=0 be current configuration and t1=1 our target configuration
# if solver divergeces (too large deformation), we will try with dt=0.5. If this still
# fails we will resort to dt=0.25 until a suitable small iterations size have been found
# dt = 1
# t0 = 0
# t1 = 1
# while t0 < 1:
# dt = t1-t0
n = 10
dt = 1 / n
for i in range(n):
# print(' ==== Quasi-static '+str(t0*100)+'-'+str(t1*100)+' % ====')
print(' ==== Quasi-static ' + str(i / (n - 1) * 100) + ' % ====')
# define the non-linear finite strain problem formulation
ns.cp = np.reshape(srf[:, :, :].swapaxes(0, 1), (n1 * n2, dim),
order='F')
ns.x_i = 'cp_ni phi_n' # geometric mapping (reference geometry)
ns.u_i = 'basis_ki ?w_k' # displacement (unknown coefficients w_k)
ns.X_i = 'x_i + u_i' # displaced geometry
ns.strain_ij = '.5 (u_i,j + u_j,i + u_k,i u_k,j)'
ns.stress_ij = 'lmbda strain_kk eye_ij + 2 mu strain_ij'
# try:
residual = domain.integral(ns.eval_n('stress_ij basis_ni,j d:X'),
degree=2 * p)
cons = np.ndarray.flatten(constraints * dt, order='C')
lhs = solver.newton('w', residual, constrain=cons).solve(
tol=state.controlpoint_absolute_tolerance, maxiter=8)
# store the results on a splipy object and continue
geom = lhs.reshape((n2, n1, dim), order='F')
srf[:, :, :] += geom.swapaxes(0, 1)
# t0 += dt
# t1 = 1
# except solver.SolverError: # newton method fail to converge, try a smaller step length 'dt'
# t1 = (t1+t0)/2
return srf
def main(inflow: 'inflow velocity' = 10,
viscosity: 'kinematic viscosity' = 1.0,
density: 'density' = 1.0,
theta=0.5,
timestepsize=0.01):
# mesh and geometry definition
grid_x_1 = numpy.linspace(-3, -1, 7)
grid_x_1 = grid_x_1[:-1]
grid_x_2 = numpy.linspace(-1, -0.3, 8)
grid_x_2 = grid_x_2[:-1]
grid_x_3 = numpy.linspace(-0.3, 0.3, 13)
grid_x_3 = grid_x_3[:-1]
grid_x_4 = numpy.linspace(0.3, 1, 8)
grid_x_4 = grid_x_4[:-1]
grid_x_5 = numpy.linspace(1, 3, 7)
grid_x = numpy.concatenate(
(grid_x_1, grid_x_2, grid_x_3, grid_x_4, grid_x_5), axis=None)
grid_y_1 = numpy.linspace(0, 1.5, 16)
grid_y_1 = grid_y_1[:-1]
grid_y_2 = numpy.linspace(1.5, 2, 4)
grid_y_2 = grid_y_2[:-1]
grid_y_3 = numpy.linspace(2, 4, 7)
grid_y = numpy.concatenate((grid_y_1, grid_y_2, grid_y_3), axis=None)
grid = [grid_x, grid_y]
topo, geom = mesh.rectilinear(grid)
domain = topo.withboundary(inflow='left', wall='top,bottom', outflow='right') - \
topo[18:20, :10].withboundary(flap='left,right,top')
# Nutils namespace
ns = function.Namespace()
# time approximations
# TR interpolation
ns._functions['t'] = lambda f: theta * f + (1 - theta) * subs0(f)
ns._functions_nargs['t'] = 1
# 1st order FD
ns._functions['δt'] = lambda f: (f - subs0(f)) / dt
ns._functions_nargs['δt'] = 1
# 2nd order FD
ns._functions['tt'] = lambda f: (1.5 * f - 2 * subs0(f) + 0.5 * subs00(f)
) / dt
ns._functions_nargs['tt'] = 1
# extrapolation for pressure
ns._functions['tp'] = lambda f: (1.5 * f - 0.5 * subs0(f))
ns._functions_nargs['tp'] = 1
ns.nu = viscosity
ns.rho = density
ns.uin = inflow
ns.x0 = geom # reference geometry
ns.dbasis = domain.basis('std', degree=1).vector(2)
ns.d_i = 'dbasis_ni ?meshdofs_n'
ns.umesh_i = 'dbasis_ni (1.5 ?meshdofs_n - 2 ?oldmeshdofs_n + 0.5 ?oldoldmeshdofs_n ) / ?dt'
ns.x_i = 'x0_i + d_i' # moving geometry
ns.ubasis, ns.pbasis = function.chain([
domain.basis('std', degree=2).vector(2),
domain.basis('std', degree=1),
])
ns.F_i = 'ubasis_ni ?F_n' # stress field
ns.urel_i = 'ubasis_ni ?lhs_n' # relative velocity
ns.u_i = 'umesh_i + urel_i' # total velocity
ns.p = 'pbasis_n ?lhs_n' # pressure
# initialization of dofs
meshdofs = numpy.zeros(len(ns.dbasis))
oldmeshdofs = meshdofs
oldoldmeshdofs = meshdofs
oldoldoldmeshdofs = meshdofs
lhs0 = numpy.zeros(len(ns.ubasis))
# for visualization
bezier = domain.sample('bezier', 2)
# preCICE setup
configFileName = "../precice-config.xml"
participantName = "Fluid"
solverProcessIndex = 0
solverProcessSize = 1
interface = precice.Interface(participantName, configFileName,
solverProcessIndex, solverProcessSize)
# define coupling meshes
meshName = "Fluid-Mesh"
meshID = interface.get_mesh_id(meshName)
couplinginterface = domain.boundary['flap']
couplingsample = couplinginterface.sample(
'gauss', degree=2) # mesh located at Gauss points
dataIndices = interface.set_mesh_vertices(meshID,
couplingsample.eval(ns.x0))
# coupling data
writeData = "Force"
readData = "Displacement"
writedataID = interface.get_data_id(writeData, meshID)
readdataID = interface.get_data_id(readData, meshID)
# initialize preCICE
precice_dt = interface.initialize()
dt = min(precice_dt, timestepsize)
# boundary conditions for fluid equations
sqr = domain.boundary['wall,flap'].integral('urel_k urel_k d:x0' @ ns,
degree=4)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
sqr = domain.boundary['inflow'].integral(
'((urel_0 - uin)^2 + urel_1^2) d:x0' @ ns, degree=4)
cons = solver.optimize('lhs', sqr, droptol=1e-15, constrain=cons)
# weak form fluid equations
res = domain.integral('t(ubasis_ni,j (u_i,j + u_j,i) rho nu d:x)' @ ns,
degree=4)
res += domain.integral('(-ubasis_ni,j p δ_ij + pbasis_n u_k,k) d:x' @ ns,
degree=4)
res += domain.integral('rho ubasis_ni δt(u_i d:x)' @ ns, degree=4)
res += domain.integral('rho ubasis_ni t(u_i,j urel_j d:x)' @ ns, degree=4)
# weak form for force computation
resF = domain.integral('(ubasis_ni,j (u_i,j + u_j,i) rho nu d:x)' @ ns,
degree=4)
resF += domain.integral('tp(-ubasis_ni,j p δ_ij d:x)' @ ns, degree=4)
resF += domain.integral('pbasis_n u_k,k d:x' @ ns, degree=4)
resF += domain.integral('rho ubasis_ni tt(u_i d:x)' @ ns, degree=4)
resF += domain.integral('rho ubasis_ni (u_i,j urel_j d:x)' @ ns, degree=4)
resF += couplinginterface.sample('gauss',
4).integral('ubasis_ni F_i d:x' @ ns)
consF = numpy.isnan(
solver.optimize('F',
couplinginterface.sample('gauss',
4).integral('F_i F_i' @ ns),
droptol=1e-10))
# boundary conditions mesh displacements
sqr = domain.boundary['inflow,outflow,wall'].integral('d_i d_i' @ ns,
degree=2)
meshcons0 = solver.optimize('meshdofs', sqr, droptol=1e-15)
# weak form mesh displacements
meshsqr = domain.integral('d_i,x0_j d_i,x0_j d:x0' @ ns, degree=2)
# better initial guess: start from Stokes solution, comment out for comparison with other solvers
#res_stokes = domain.integral('(ubasis_ni,j ((u_i,j + u_j,i) rho nu - p δ_ij) + pbasis_n u_k,k) d:x' @ ns, degree=4)
#lhs0 = solver.solve_linear('lhs', res_stokes, constrain=cons, arguments=dict(meshdofs=meshdofs, oldmeshdofs=oldmeshdofs, oldoldmeshdofs=oldoldmeshdofs, oldoldoldmeshdofs=oldoldoldmeshdofs, dt=dt))
lhs00 = lhs0
timestep = 0
t = 0
while interface.is_coupling_ongoing():
# read displacements from interface
if interface.is_read_data_available():
readdata = interface.read_block_vector_data(
readdataID, dataIndices)
coupledata = couplingsample.asfunction(readdata)
sqr = couplingsample.integral(((ns.d - coupledata)**2).sum(0))
meshcons = solver.optimize('meshdofs',
sqr,
droptol=1e-15,
constrain=meshcons0)
meshdofs = solver.optimize('meshdofs', meshsqr, constrain=meshcons)
# save checkpoint
if interface.is_action_required(
precice.action_write_iteration_checkpoint()):
lhs_checkpoint = lhs0
lhs00_checkpoint = lhs00
t_checkpoint = t
timestep_checkpoint = timestep
oldmeshdofs_checkpoint = oldmeshdofs
oldoldmeshdofs_checkpoint = oldoldmeshdofs
oldoldoldmeshdofs_checkpoint = oldoldoldmeshdofs
interface.mark_action_fulfilled(
precice.action_write_iteration_checkpoint())
# solve fluid equations
lhs1 = solver.newton(
'lhs',
res,
lhs0=lhs0,
constrain=cons,
arguments=dict(
lhs0=lhs0,
dt=dt,
meshdofs=meshdofs,
oldmeshdofs=oldmeshdofs,
oldoldmeshdofs=oldoldmeshdofs,
oldoldoldmeshdofs=oldoldoldmeshdofs)).solve(tol=1e-6)
# write forces to interface
if interface.is_write_data_required(dt):
F = solver.solve_linear('F',
resF,
constrain=consF,
arguments=dict(
lhs00=lhs00,
lhs0=lhs0,
lhs=lhs1,
dt=dt,
meshdofs=meshdofs,
oldmeshdofs=oldmeshdofs,
oldoldmeshdofs=oldoldmeshdofs,
oldoldoldmeshdofs=oldoldoldmeshdofs))
# writedata = couplingsample.eval(ns.F, F=F) # for stresses
writedata = couplingsample.eval('F_i d:x' @ ns, F=F, meshdofs=meshdofs) * \
numpy.concatenate([p.weights for p in couplingsample.points])[:, numpy.newaxis]
interface.write_block_vector_data(writedataID, dataIndices,
writedata)
# do the coupling
precice_dt = interface.advance(dt)
dt = min(precice_dt, timestepsize)
# advance variables
timestep += 1
t += dt
lhs00 = lhs0
lhs0 = lhs1
oldoldoldmeshdofs = oldoldmeshdofs
oldoldmeshdofs = oldmeshdofs
oldmeshdofs = meshdofs
# read checkpoint if required
if interface.is_action_required(
precice.action_read_iteration_checkpoint()):
lhs0 = lhs_checkpoint
lhs00 = lhs00_checkpoint
t = t_checkpoint
timestep = timestep_checkpoint
oldmeshdofs = oldmeshdofs_checkpoint
oldoldmeshdofs = oldoldmeshdofs_checkpoint
oldoldoldmeshdofs = oldoldoldmeshdofs_checkpoint
interface.mark_action_fulfilled(
precice.action_read_iteration_checkpoint())
if interface.is_time_window_complete():
x, u, p = bezier.eval(['x_i', 'u_i', 'p'] @ ns,
lhs=lhs1,
meshdofs=meshdofs,
oldmeshdofs=oldmeshdofs,
oldoldmeshdofs=oldoldmeshdofs,
oldoldoldmeshdofs=oldoldoldmeshdofs,
dt=dt)
with treelog.add(treelog.DataLog()):
export.vtk('Fluid_' + str(timestep), bezier.tri, x, u=u, p=p)
interface.finalize()
def main(fname: str, degree: int, Δuload: unit['mm'], nsteps: int, E: unit['GPa'], nu: float, σyield: unit['MPa'], hardening: unit['GPa'], referencedata: typing.Optional[str], testing: bool):
'''
Plastic deformation of a perforated strip.
.. arguments::
fname [strip.msh]
Mesh file with units in [mm]
degree [1]
Finite element interpolation order
Δuload [5μm]
Load boundary displacement steps
nsteps [11]
Number of load steps
E [70GPa]
Young's modulus
nu [0.2]
Poisson ratio
σyield [243MPa]
Yield strength
hardening [2.25GPa]
Hardening parameter
referencedata [zienkiewicz.csv]
Reference data file name
testing [False]
Use a 1 element mesh for testing
.. presets::
testing
nsteps=30
referencedata=
testing=True
'''
# We commence with reading the mesh from the specified GMSH file, or, alternatively,
# we continue with a single-element mesh for testing purposes.
domain, geom = mesh.gmsh(pathlib.Path(__file__).parent/fname)
Wdomain, Hdomain = domain.boundary['load'].integrate([function.J(geom),geom[1]*function.J(geom)], degree=1)
Hdomain /= Wdomain
if testing:
domain, geom = mesh.rectilinear([numpy.linspace(0,Wdomain/2,2),numpy.linspace(0,Hdomain,2)])
domain = domain.withboundary(hsymmetry='left', vsymmetry='bottom', load='top')
# We next initiate the point set in which the constitutive bahvior will be evaluated.
# Note that this is also the point set in which the history variable will be stored.
gauss = domain.sample('gauss', 2)
# Elasto-plastic formulation
# ==========================
# The weak formulation is constructed using a `Namespace`, which is initialized and
# populated with the necessary problem parameters and coordinate system. Note that the
# coefficients `mu` and `λ` have been defined such that 'C_{ijkl}' is the fourth-order
# **plane stress** elasticity tensor.
ns = function.Namespace(fallback_length=domain.ndims)
ns.x = geom
ns.Δuload = Δuload
ns.mu = E/(1-nu**2)*((1-nu)/2)
ns.λ = E/(1-nu**2)*nu
ns.delta = function.eye(domain.ndims)
ns.C_ijkl = 'mu ( delta_ik delta_jl + delta_il delta_jk ) + λ delta_ij delta_kl'
# We make use of a Lagrange finite element basis of arbitrary `degree`. Since we
# approximate the displacement field, the basis functions are vector-valued. Both
# the displacement field at the current load step `u` and displacement field of the
# previous load step `u0` are approximated using this basis:
ns.basis = domain.basis('std',degree=degree).vector(domain.ndims)
ns.u0_i = 'basis_ni ?lhs0_n'
ns.u_i = 'basis_ni ?lhs_n'
# This simulation is based on a standard elasto-plasticity model. In this
# model the *total strain*, ε_kl = ½ (∂u_k/∂x_l + ∂u_l/∂x_k)', is comprised of an
# elastic and a plastic part:
#
# ε_kl = εe_kl + εp_kl
#
# The stress is related to the *elastic strain*, εe_kl, through Hooke's law:
#
# σ_ij = C_ijkl εe_kl = C_ijkl (ε_kl - εp_kl)
ns.ε_kl = '(u_k,l + u_l,k) / 2'
ns.ε0_kl = '(u0_k,l + u0_l,k) / 2'
ns.gbasis = gauss.basis()
ns.εp0_ij = 'gbasis_n ?εp0_nij'
ns.κ0 = 'gbasis_n ?κ0_n'
ns.εp = PlasticStrain(ns.ε, ns.ε0, ns.εp0, ns.κ0, E, nu, σyield, hardening)
ns.εe_ij = 'ε_ij - εp_ij'
ns.σ_ij = 'C_ijkl εe_kl'
# Note that the plasticity model is implemented through the user-defined function `PlasticStrain`,
# which implements the actual yielding model including a standard return mapping algorithm. This
# function is discussed in detail below.
#
# The components of the residual vector are then defined as:
#
# r_n = ∫_Ω (∂N_ni/∂x_j) σ_ij dΩ
res = domain.integral('basis_ni,j σ_ij d:x' @ ns, degree=2)
# The problem formulation is completed by supplementing prescribed displacement boundary conditions
# (for a load step), which are computed in the standard manner:
sqr = domain.boundary['hsymmetry,vsymmetry'].integral('(u_k n_k)^2 d:x' @ ns, degree=2)
sqr += domain.boundary['load'].integral('(u_k n_k - Δuload)^2 d:x' @ ns, degree=2)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
# Incremental-iterative solution procedure
# ========================================
# We initialize the solution vector for the first load step `lhs` and solution vector
# of the previous load step `lhs0`. Note that, in order to construct a predictor step
# for the first load step, we define the previous load step state as the solution
# vector corresponding to a negative elastic loading step.
lhs0 = -solver.solve_linear('lhs', domain.integral('basis_ni,j C_ijkl ε_kl d:x' @ ns, degree=2), constrain=cons)
lhs = numpy.zeros_like(lhs0)
εp0 = numpy.zeros((gauss.npoints,)+ns.εp0.shape)
κ0 = numpy.zeros((gauss.npoints,)+ns.κ0.shape)
# To store the force-dispalcement data we initialize an empty data array with the
# inital state solution substituted in the first row.
fddata = numpy.empty(shape=(nsteps+1,2))
fddata[:] = numpy.nan
fddata[0,:] = 0
# Load step incrementation
# ------------------------
with treelog.iter.fraction('step', range(nsteps)) as counter:
for step in counter:
# The solution of the previous load step is set to `lhs0`, and the Newton solution
# procedure is initialized by extrapolation of the state vector:
lhs_init = lhs + (lhs-lhs0)
lhs0 = lhs
# The non-linear system of equations is solved for `lhs` using Newton iterations,
# where the `step` variable is used to scale the incremental constraints.
lhs = solver.newton(target='lhs', residual=res, constrain=cons*step, lhs0=lhs_init, arguments={'lhs0':lhs0,'εp0':εp0,'κ0':κ0}).solve(tol=1e-6)
# The computed solution is post-processed in the form of a loading curve - which
# plots the normalized mean stress versus the maximum 'ε_{yy}' strain
# component - and a contour plot showing the 'σ_{yy}' stress component on a
# deformed mesh. Note that since the stresses are defined in the integration points
# only, a post-processing step is involved that transfers the stress information to
# the nodal points.
εyymax = gauss.eval(ns.ε[1,1], arguments=dict(lhs=lhs)).max()
basis = domain.basis('std', degree=1)
bw, b = domain.integrate([basis * ns.σ[1,1] * function.J(geom), basis * function.J(geom)], degree=2, arguments=dict(lhs=lhs,lhs0=lhs0,εp0=εp0,κ0=κ0))
σyy = basis.dot(bw / b)
uyload, σyyload = domain.boundary['load'].integrate(['u_1 d:x'@ns,σyy * function.J(geom)], degree=2, arguments=dict(lhs=lhs,εp0=εp0,κ0=κ0))
uyload /= Wdomain
σyyload /= Wdomain
fddata[step,0] = (E*εyymax)/σyield
fddata[step,1] = (σyyload*2)/σyield
with export.mplfigure('forcedisp.png') as fig:
ax = fig.add_subplot(111, xlabel=r'${E \cdot {\rm max}(\varepsilon_{yy})}/{\sigma_{\rm yield}}$', ylabel=r'${\sigma_{\rm mean}}/{\sigma_{\rm yield}}$')
if referencedata:
data = numpy.genfromtxt(pathlib.Path(__file__).parent/referencedata, delimiter=',', skip_header=1)
ax.plot(data[:,0], data[:,1], 'r:', label='Reference')
ax.legend()
ax.plot(fddata[:,0], fddata[:,1], 'o-', label='Nutils')
ax.grid()
bezier = domain.sample('bezier', 3)
points, uvals, σyyvals = bezier.eval(['(x_i + 25 u_i)' @ ns, ns.u, σyy], arguments=dict(lhs=lhs))
with export.mplfigure('stress.png') as fig:
ax = fig.add_subplot(111, aspect='equal', xlabel=r'$x$ [mm]', ylabel=r'$y$ [mm]')
im = ax.tripcolor(points[:,0]/unit('mm'), points[:,1]/unit('mm'), bezier.tri, σyyvals/unit('MPa'), shading='gouraud', cmap='jet')
ax.add_collection(collections.LineCollection(points.take(bezier.hull, axis=0), colors='k', linewidths=.1))
ax.autoscale(enable=True, axis='both', tight=True)
cb = fig.colorbar(im)
im.set_clim(0, 1.2*σyield)
cb.set_label(r'$σ_{yy}$ [MPa]')
# Load step convergence
# ---------------------
# At the end of the loading step, the plastic strain state and history parameter are updated,
# where use if made of the strain hardening relation for the history variable:
#
# Δκ = √(Δεp_ij Δεp_ij)
Δεp = ns.εp-ns.εp0
Δκ = function.sqrt((Δεp*Δεp).sum((0,1)))
κ0 = gauss.eval(ns.κ0+Δκ, arguments=dict(lhs0=lhs0,lhs=lhs,εp0=εp0,κ0=κ0))
εp0 = gauss.eval(ns.εp, arguments=dict(lhs0=lhs0,lhs=lhs,εp0=εp0,κ0=κ0))
def test_newton_iter(self): _test_recursion_cache(self, lambda: ((self.frozen(lhs), info.resnorm) for lhs, info in solver.newton(self.dofs, residual=self.residual, constrain=self.cons)))
def mixed_fem(g, ltol=1e-5, coordinate_directions=None, **solveargs):
assert len(g) == 2
n = len(g.x)
basis = g.basis
def c0(g):
return tuple(
[i for i in range(2) if g.degree[i] in g.knotmultiplicities[i]])
if coordinate_directions is None:
coordinate_directions = c0(g)
assert len(coordinate_directions) in (1, 2) and all(
[i in (0, 1) for i in coordinate_directions])
s = function.stack
veclength = {1: 4, 2: 6}[len(coordinate_directions)]
target = function.Argument('target', [veclength * len(basis)])
U = g.basis.vector(veclength).dot(target)
G = metric_tensor(g, target[-n:])
(g11, g12), (g21, g22) = G
scale = g11 + g22
if len(coordinate_directions) == 2:
u, v, x_ = U[:2], U[2:4], U[4:]
expr = function.concatenate(x_.grad(g.geom) - s([u, v], axis=1))
res1 = (basis.vector(4) * expr).sum(-1)
res2 = g22 * u.grad(g.geom)[:, 0] - g12 * u.grad(
g.geom)[:, 1] - g12 * v.grad(g.geom)[:, 0] + g11 * v.grad(
g.geom)[:, 1]
if len(coordinate_directions) == 1:
index = coordinate_directions[0]
u, x_ = U[:2], U[2:4]
expr = x_.grad(g.geom)[:, index] - u
res1 = (basis.vector(2) * expr).sum(-1)
if index == 0:
x_eta = x_.grad(g.geom)[:, 1]
res2 = g22 * u.grad(g.geom)[:, 0] - g12 * u.grad(
g.geom)[:, 1] - g12 * x_eta.grad(
g.geom)[:, 0] + g11 * x_eta.grad(g.geom)[:, 1]
if index == 1:
x_xi = x_.grad(g.geom)[:, 0]
res2 = g22 * x_xi.grad(g.geom)[:, 0] - g12 * u.grad(
g.geom)[:, 0] - g12 * x_xi.grad(g.geom)[:, 1] + g11 * u.grad(
g.geom)[:, 1]
res = function.concatenate(
[res1, (basis.vector(2) * res2).sum(-1) / scale])
res = g.domain.integral(res, geometry=g.geom, degree=g.ischeme * 4)
mapping0 = g.basis.vector(2).dot(g.x)
cons = util.NanVec(n * (len(coordinate_directions) + 1))
cons[-n:] = g.cons
init = np.concatenate([
g.domain.project(mapping0.grad(g.geom)[:, i],
geometry=g.geom,
onto=g.basis.vector(2),
ischeme='gauss12') for i in coordinate_directions
] + [g.x])
lhs = solver.newton('target', res, lhs0=init,
constrain=cons.where).solve(ltol, **solveargs)
g.x = lhs[-n:]
def elliptic_control_mapping(g,
f,
eps=1e-4,
ltol=1e-5,
degree=None,
**solveargs):
'''
Nutils implementation of the corresponding method from the ``fsol`` module.
It's slower but accepts any control mapping function. Unline in the ``fsol``
case, all derivatives are computed automatically.
'''
assert len(g) == len(f) == 2
assert eps >= 0
if degree is None:
degree = 4 * g.ischeme
if isinstance(f, go.TensorGridObject):
if not (f.knotvector <= g.knotvector).all():
raise AssertionError('''
Error, the control-mapping knotvector needs
to be a subset of the target GridObject knotvector
''')
# refine, just in case
f = go.refine_GridObject(f, g.knotvector)
G = metric_tensor(g, f.x)
else:
# f is a nutils function
jacT = function.transpose(f.grad(g.geom))
G = function.outer(jacT, jacT).sum(-1)
target = function.Argument('target', [len(g.x)])
basis = g.basis.vector(2)
x = basis.dot(target)
(g11, g12), (g21, g22) = metric_tensor(g, target)
lapl = g22 * x.grad(g.geom)[:, 0].grad(g.geom)[:, 0] - \
2 * g12 * x.grad(g.geom)[:, 0].grad(g.geom)[:, 1] + \
g11 * x.grad(g.geom)[:, 1].grad(g.geom)[:, 1]
G_tilde = G / function.sqrt(function.determinant(G))
(G11, G12), (G12, G22) = G_tilde
P = g11 * G12.grad(g.geom)[1] - g12 * G11.grad(g.geom)[1]
Q = 0.5 * (g11 * G22.grad(g.geom)[0] - g22 * G11.grad(g.geom)[0])
S = g12 * G22.grad(g.geom)[0] - g22 * G12.grad(g.geom)[0]
R = 0.5 * (g11 * G22.grad(g.geom)[1] - g22 * G11.grad(g.geom)[1])
control_term = -x.grad( g.geom )[:, 0] * ( G22*(P-Q) + G12*(S-R) ) \
-x.grad( g.geom )[:, 1] * ( G11*(R-S) + G12*(Q-P) )
scale = g11 + g22 + eps
res = g.domain.integral((basis * (control_term + lapl)).sum(-1) / scale,
geometry=g.geom,
degree=degree)
init, cons = g.x, g.cons
lhs = solver.newton('target', res, lhs0=init,
constrain=cons.where).solve(ltol, **solveargs)
g.x = lhs
def test_newton(self):
self.assert_resnorm(solver.newton('dofs', residual=self.residual, lhs0=self.lhs0, constrain=self.cons).solve(tol=self.tol, maxiter=2))
def test_newton_boolcons(self):
self.assert_resnorm(solver.newton('dofs', residual=self.residual, constrain=self.boolcons).solve(tol=self.tol, maxiter=7))
def main(
nelems: 'number of elements' = 12,
viscosity: 'fluid viscosity' = 1e-3,
density: 'fluid density' = 1,
degree: 'polynomial degree' = 2,
warp: 'warp domain (downward bend)' = False,
figures: 'create figures' = True,
):
log.user('reynolds number: {:.1f}'.format(
density / viscosity)) # based on unit length and velocity
# create namespace
ns = function.Namespace()
ns.viscosity = viscosity
ns.density = density
# construct mesh
verts = numpy.linspace(0, 1, nelems + 1)
domain, ns.x0 = mesh.rectilinear([verts, verts])
# construct bases
ns.uxbasis, ns.uybasis, ns.pbasis, ns.lbasis = function.chain([
domain.basis('spline',
degree=(degree + 1, degree),
removedofs=((0, -1), None)),
domain.basis('spline',
degree=(degree, degree + 1),
removedofs=(None, (0, -1))),
domain.basis('spline', degree=degree),
[1], # lagrange multiplier
])
ns.ubasis_ni = '<uxbasis_n, uybasis_n>_i'
# construct geometry
if not warp:
ns.x = ns.x0
else:
xi, eta = ns.x0
ns.x = (eta + 2) * function.rotmat(xi * .4)[:, 1] - (
0, 2) # slight downward bend
ns.J_ij = 'x_i,x0_j'
ns.detJ = function.determinant(ns.J)
ns.ubasis_ni = 'ubasis_nj J_ij / detJ' # piola transform
ns.pbasis_n = 'pbasis_n / detJ'
# populate namespace
ns.u_i = 'ubasis_ni ?lhs_n'
ns.p = 'pbasis_n ?lhs_n'
ns.l = 'lbasis_n ?lhs_n'
ns.sigma_ij = 'viscosity (u_i,j + u_j,i) - p δ_ij'
ns.c = 5 * (degree + 1) / domain.boundary.elem_eval(
1, geometry=ns.x, ischeme='gauss2', asfunction=True)
ns.nietzsche_ni = 'viscosity (c ubasis_ni - (ubasis_ni,j + ubasis_nj,i) n_j)'
ns.top = domain.boundary.indicator('top')
ns.utop_i = 'top <n_1, -n_0>_i'
# solve stokes flow
res = domain.integral(
'ubasis_ni,j sigma_ij + pbasis_n (u_k,k + l) + lbasis_n p' @ ns,
geometry=ns.x,
degree=2 * (degree + 1))
res += domain.boundary.integral('nietzsche_ni (u_i - utop_i)' @ ns,
geometry=ns.x,
degree=2 * (degree + 1))
lhs0 = solver.solve_linear('lhs', res)
if figures:
postprocess(domain, ns(lhs=lhs0))
# solve navier-stokes flow
res += domain.integral('density ubasis_ni u_i,j u_j' @ ns,
geometry=ns.x,
degree=3 * (degree + 1))
lhs1 = solver.newton('lhs', res, lhs0=lhs0).solve(tol=1e-10)
if figures:
postprocess(domain, ns(lhs=lhs1))
return lhs0, lhs1
def test_newton_iter(self):
_test_recursion_cache(self, lambda: ((types.frozenarray(lhs), info.resnorm) for lhs, info in solver.newton('dofs', residual=self.residual, constrain=self.cons)))
