def main(viscosity=1.3e-3, density=1e3, pout=223e5, uw=-0.01, nelems=10):
# viscosity = 1.3e-3
# density = 1e3
# pout = 223e5
# nelems = 10
# uw = -0.01
domain, geom = mesh.rectilinear([numpy.linspace(0, 1, nelems), numpy.linspace(1, 2, nelems), [0, 2 * numpy.pi]],
periodic=[2])
domain = domain.withboundary(inner='bottom', outer='top')
ns = function.Namespace()
ns.y, ns.r, ns.θ = geom
ns.x_i = '<r cos(θ), y, r sin(θ)>_i'
ns.uybasis, ns.urbasis, ns.pbasis = function.chain([
domain.basis('std', degree=3, removedofs=((0,-1), None, None)), # remove normal component at y=0 and y=1
domain.basis('std', degree=3, removedofs=((0,-1), None, None)), # remove tangential component at y=0 (no slip)
domain.basis('std', degree=2)])
ns.ubasis_ni = '<urbasis_n cos(θ), uybasis_n, urbasis_n sin(θ)>_i'
ns.viscosity = viscosity
ns.density = density
ns.u_i = 'ubasis_ni ?lhs_n'
ns.p = 'pbasis_n ?lhs_n'
ns.sigma_ij = 'viscosity (u_i,j + u_j,i) - p δ_ij'
ns.pout = pout
ns.uw = uw
ns.uw_i = 'uw <cos(phi), 0, sin(phi)>_i'
ns.tout_i = '-pout n_i'
ns.uw_i = 'uw n_i' # uniform outflow
res = domain.integral('(viscosity ubasis_ni,j u_i,j - p ubasis_ni,i + pbasis_n u_k,k) d:x' @ ns, degree=6)
# res -= domain[1].boundary['inner'].integral('ubasis_ni tout_i d:x' @ ns, degree=6)
sqr = domain.boundary['inner'].integral('(u_i - uw_i) (u_i - uw_i) d:x' @ ns, degree=6)
# sqr = domain.boundary['outer'].integral('(u_i - uin_i) (u_i - uin_i) d:x' @ ns, degree=6)
sqr -= domain.boundary['outer'].integral('(p - pout) (p - pout) d:x' @ ns, degree=6)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
lhs = solver.solve_linear('lhs', res, constrain=cons)
plottopo = domain[:, :, 0:].boundary['back']
bezier = plottopo.sample('bezier', 10)
r, y, p, u = bezier.eval([ns.r, ns.y, ns.p, function.norm2(ns.u)], lhs=lhs)
with export.mplfigure('pressure.png', dpi=800) as fig:
ax = fig.add_subplot(111, title='pressure', aspect=1)
ax.autoscale(enable=True, axis='both', tight=True)
im = ax.tripcolor(r, y, bezier.tri, p, shading='gouraud', cmap='jet')
ax.add_collection(
collections.LineCollection(numpy.array([y, r]).T[bezier.hull], colors='k', linewidths=0.2, alpha=0.2))
fig.colorbar(im)
uniform = plottopo.sample('uniform', 1)
r_, y_, uv = uniform.eval([ns.r, ns.y, ns.u], lhs=lhs)
with export.mplfigure('velocity.png', dpi=800) as fig:
ax = fig.add_subplot(111, title='Velocity', aspect=1)
ax.autoscale(enable=True, axis='both', tight=True)
im = ax.tripcolor(r, y, bezier.tri, u, shading='gouraud', cmap='jet')
ax.quiver(r_, y_, uv[:, 0], uv[:, 1], angles='xy', scale_units='xy')
fig.colorbar(im)
def test_sparsity(self):
topo, geom = mesh.rectilinear([6] * self.ndim)
topo = topo.refined_by(
set(
map(
topo.transforms.index,
itertools.chain(topo[1:3].transforms,
topo[-2:].transforms))))
ns = function.Namespace()
ns.x = geom
ns.tbasis = topo.basis('th-spline',
degree=2,
truncation_tolerance=1e-14)
ns.tnotol = topo.basis('th-spline', degree=2, truncation_tolerance=0)
ns.hbasis = topo.basis('h-spline', degree=2)
tA, tA_tol, hA = topo.sample('gauss', 5).integrate_sparse([
ns.eval_ij('tbasis_i,k tbasis_j,k'),
ns.eval_ij('tnotol_i,k tnotol_j,k'),
ns.eval_ij('hbasis_i,k hbasis_j,k')
])
tA_nnz, tA_tol_nnz, hA_nnz = self.vals[self.ndim]
self.assertEqual(len(sparse.prune(sparse.dedup(tA))), tA_nnz)
self.assertEqual(len(sparse.prune(sparse.dedup(tA_tol))), tA_tol_nnz)
self.assertEqual(len(sparse.prune(sparse.dedup(hA))), hA_nnz)
def setUp(self):
super().setUp()
self.ns = function.Namespace()
self.domain, self.ns.geom = mesh.rectilinear([2,2])
self.ns.ubasis = self.domain.basis('std', degree=1)
self.ns.u = 'ubasis_n ?dofs_n'
self.optimize = solver.optimize if not self.minimize else lambda *args, newtontol=0, **kwargs: solver.minimize(*args, **kwargs).solve(newtontol)
def main(nelems:int, etype:str, btype:str, degree:int, poisson:float, angle:float, restol:float, trim:bool):
'''
Deformed hyperelastic plate.
.. arguments::
nelems [10]
Number of elements along edge.
etype [square]
Type of elements (square/triangle/mixed).
btype [std]
Type of basis function (std/spline).
degree [1]
Polynomial degree.
poisson [.25]
Poisson's ratio, nonnegative and stricly smaller than 1/2.
angle [20]
Rotation angle for right clamp (degrees).
restol [1e-10]
Newton tolerance.
trim [no]
Create circular-shaped hole.
'''
domain, geom = mesh.unitsquare(nelems, etype)
if trim:
domain = domain.trim(function.norm2(geom-.5)-.2, maxrefine=2)
bezier = domain.sample('bezier', 5)
ns = function.Namespace(fallback_length=domain.ndims)
ns.x = geom
ns.angle = angle * numpy.pi / 180
ns.lmbda = 2 * poisson
ns.mu = 1 - 2 * poisson
ns.ubasis = domain.basis(btype, degree=degree)
ns.u_i = 'ubasis_k ?lhs_ki'
ns.X_i = 'x_i + u_i'
ns.strain_ij = '.5 (d(u_i, x_j) + d(u_j, x_i))'
ns.energy = 'lmbda strain_ii strain_jj + 2 mu strain_ij strain_ij'
sqr = domain.boundary['left'].integral('u_k u_k J(x)' @ ns, degree=degree*2)
sqr += domain.boundary['right'].integral('((u_0 - x_1 sin(2 angle) - cos(angle) + 1)^2 + (u_1 - x_1 (cos(2 angle) - 1) + sin(angle))^2) J(x)' @ ns, degree=degree*2)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
energy = domain.integral('energy J(x)' @ ns, degree=degree*2)
lhs0 = solver.optimize('lhs', energy, constrain=cons)
X, energy = bezier.eval(['X', 'energy'] @ ns, lhs=lhs0)
export.triplot('linear.png', X, energy, tri=bezier.tri, hull=bezier.hull)
ns.strain_ij = '.5 (d(u_i, x_j) + d(u_j, x_i) + d(u_k, x_i) d(u_k, x_j))'
ns.energy = 'lmbda strain_ii strain_jj + 2 mu strain_ij strain_ij'
energy = domain.integral('energy J(x)' @ ns, degree=degree*2)
lhs1 = solver.minimize('lhs', energy, arguments=dict(lhs=lhs0), constrain=cons).solve(restol)
X, energy = bezier.eval(['X', 'energy'] @ ns, lhs=lhs1)
export.triplot('nonlinear.png', X, energy, tri=bezier.tri, hull=bezier.hull)
return lhs0, lhs1
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 setUp(self):
ns = function.Namespace()
domain, ns.x = mesh.rectilinear([10], periodic=(0,))
ns.basis = domain.basis('discont', degree=1)
ns.u = 'basis_n ?dofs_n'
ns.f = '.5 u^2'
self.residual = domain.integral('-basis_n,0 f d:x' @ ns, degree=2)
self.residual += domain.interfaces.integral('-[basis_n] n_0 ({f} - .5 [u] n_0) d:x' @ ns, degree=4)
self.inertia = domain.integral('basis_n u d:x' @ ns, degree=5)
self.lhs0 = numpy.sin(numpy.arange(len(ns.basis))) # "random" initial vector
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: 'number of elements' = 12,
lmbda: 'first lamé constant' = 1.,
mu: 'second lamé constant' = 1.,
degree: 'polynomial degree' = 2,
withplots: 'create plots' = True,
solvetol: 'solver tolerance' = 1e-10,
):
# construct mesh
verts = numpy.linspace(0, 1, nelems + 1)
domain, geom = mesh.rectilinear([verts, verts])
# create namespace
ns = function.Namespace(default_geometry_name='x0')
ns.x0 = geom
ns.basis = domain.basis('spline', degree=degree).vector(2)
ns.u_i = 'basis_ni ?lhs_n'
ns.x_i = 'x0_i + u_i'
ns.lmbda = lmbda
ns.mu = mu
ns.strain_ij = '(u_i,j + u_j,i) / 2'
ns.stress_ij = 'lmbda strain_kk δ_ij + 2 mu strain_ij'
# construct dirichlet boundary constraints
sqr = domain.boundary['left'].integral('u_k u_k' @ ns,
geometry=ns.x0,
degree=2)
sqr += domain.boundary['right'].integral('(u_0 - .5)^2' @ ns,
geometry=ns.x0,
degree=2)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
# construct residual
res = domain.integral('basis_ni,j stress_ij' @ ns,
geometry=ns.x0,
degree=2)
# solve system and substitute the solution in the namespace
lhs = solver.solve_linear('lhs', res, constrain=cons)
ns |= dict(lhs=lhs)
# plot solution
if withplots:
points, colors = domain.elem_eval([ns.x, ns.stress[0, 1]],
ischeme='bezier3',
separate=True)
with plot.PyPlot('stress', ndigits=0) as plt:
plt.mesh(points, colors, tight=False)
plt.colorbar()
return lhs, cons
def project(self, projection):
ns = fn.Namespace()
for name, func in self._kwargs.items():
setattr(ns, name, func)
for p, (name, func) in zip(projection, self._defaults.items()):
if p is not None:
func = fn.matmat(p, func)
setattr(ns, name, func)
integrand = getattr(ns, self._evaluator)(self._code)
domain, geom, ischeme = self.prop('domain', 'geometry', 'ischeme')
with matrix.Numpy():
retval = domain.integrate(integrand * fn.J(geom), ischeme=ischeme)
return NumpyArrayIntegrand(retval)
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 main(
nelems: 'number of elements' = 20,
degree: 'polynomial degree' = 1,
timescale: 'time scale (timestep=timescale/nelems)' = .5,
tol: 'solver tolerance' = 1e-5,
ndims: 'spatial dimension' = 1,
endtime: 'end time, 0 for no end time' = 0,
withplots: 'create plots' = True,
):
# construct mesh, basis
ns = function.Namespace()
domain, ns.x = mesh.rectilinear([numpy.linspace(0, 1, nelems + 1)] * ndims,
periodic=range(ndims))
ns.basis = domain.basis('discont', degree=degree)
ns.u = 'basis_n ?lhs_n'
# construct initial condition (centered gaussian)
lhs0 = domain.project('exp(-?y_i ?y_i) | ?y_i = 5 (x_i - 0.5_i)' @ ns,
onto=ns.basis,
geometry=ns.x,
degree=5)
# prepare residual
ns.f = '.5 u^2'
ns.C = 1
res = domain.integral('-basis_n,0 f' @ ns, geometry=ns.x, degree=5)
res += domain.interfaces.integral(
'-[basis_n] n_0 ({f} - .5 C [u] n_0)' @ ns, geometry=ns.x, degree=5)
inertia = domain.integral('basis_n u' @ ns, geometry=ns.x, degree=5)
# prepare plotting
makeplots = MakePlots(domain, video=withplots
== 'video') if withplots else lambda ns: None
# start time stepping
timestep = timescale / nelems
for itime, lhs in log.enumerate(
'timestep',
solver.impliciteuler('lhs',
res,
inertia,
timestep,
lhs0,
newtontol=tol)):
makeplots(ns | dict(lhs=lhs))
if endtime and itime * timestep >= endtime:
break
return res.eval(arguments=dict(lhs=lhs)), lhs
def check(self, method, theta):
ns = function.Namespace()
topo, ns.x = mesh.rectilinear([1])
ns.u_n = '?u_n + <0>_n'
inertia = topo.integral('?u_n d:x' @ ns, degree=0)
residual = topo.integral('-<1>_n sin(?t) d:x' @ ns, degree=0)
timestep = 0.1
udesired = numpy.array([0.])
uactualiter = iter(method(target='u', residual=residual, inertia=inertia, timestep=timestep, lhs0=udesired, timetarget='t'))
for i in range(5):
with self.subTest(i=i):
uactual = next(uactualiter)
self.assertAllAlmostEqual(uactual, udesired)
udesired += timestep*(theta*numpy.sin((i+1)*timestep)+(1-theta)*numpy.sin(i*timestep))
def main(nelems: int, etype: str, btype: str, degree: int, poisson: float):
'''
Horizontally loaded linear elastic plate.
.. arguments::
nelems [10]
Number of elements along edge.
etype [square]
Type of elements (square/triangle/mixed).
btype [std]
Type of basis function (std/spline), with availability depending on the
configured element type.
degree [1]
Polynomial degree.
poisson [.25]
Poisson's ratio, nonnegative and strictly smaller than 1/2.
'''
domain, geom = mesh.unitsquare(nelems, etype)
ns = function.Namespace()
ns.x = geom
ns.basis = domain.basis(btype, degree=degree).vector(2)
ns.u_i = 'basis_ni ?lhs_n'
ns.X_i = 'x_i + u_i'
ns.lmbda = 2 * poisson
ns.mu = 1 - 2 * poisson
ns.strain_ij = '(d(u_i, x_j) + d(u_j, x_i)) / 2'
ns.stress_ij = 'lmbda strain_kk δ_ij + 2 mu strain_ij'
sqr = domain.boundary['left'].integral('u_k u_k J(x)' @ ns,
degree=degree * 2)
sqr += domain.boundary['right'].integral('(u_0 - .5)^2 J(x)' @ ns,
degree=degree * 2)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
res = domain.integral('d(basis_ni, x_j) stress_ij J(x)' @ ns,
degree=degree * 2)
lhs = solver.solve_linear('lhs', res, constrain=cons)
bezier = domain.sample('bezier', 5)
X, sxy = bezier.eval(['X', 'stress_01'] @ ns, lhs=lhs)
export.triplot('shear.png', X, sxy, tri=bezier.tri, hull=bezier.hull)
return cons, lhs
def main(
nelems: 'number of elements' = 10,
degree: 'polynomial degree' = 1,
basistype: 'basis function' = 'spline',
solvetol: 'solver tolerance' = 1e-10,
figures: 'create figures' = True,
):
# construct mesh
verts = numpy.linspace(0, numpy.pi/2, nelems+1)
domain, geom = mesh.rectilinear([verts, verts])
# create namespace
ns = function.Namespace()
ns.x = geom
ns.basis = domain.basis(basistype, degree=degree)
ns.u = 'basis_n ?lhs_n'
ns.fx = 'sin(x_0)'
ns.fy = 'exp(x_1)'
# construct residual
res = domain.integral('-basis_n,i u_,i' @ ns, geometry=ns.x, degree=degree*2)
res += domain.boundary['top'].integral('basis_n fx fy' @ ns, geometry=ns.x, degree=degree*2)
# construct dirichlet constraints
sqr = domain.boundary['left'].integral('u^2' @ ns, geometry=ns.x, degree=degree*2)
sqr += domain.boundary['bottom'].integral('(u - fx)^2' @ ns, geometry=ns.x, degree=degree*2)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
# find lhs such that res == 0 and substitute this lhs in the namespace
lhs = solver.solve_linear('lhs', res, constrain=cons)
ns = ns(lhs=lhs)
# plot solution
if figures:
points, colors = domain.elem_eval([ns.x, ns.u], ischeme='bezier9', separate=True)
with plot.PyPlot('solution', index=nelems) as plt:
plt.mesh(points, colors, cmap='jet')
plt.colorbar()
# evaluate error against exact solution fx fy
err = domain.integrate('(u - fx fy)^2' @ ns, geometry=ns.x, degree=degree*2)**.5
log.user('L2 error: {:.2e}'.format(err))
return cons, lhs, err
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 main(nelems: int, degree: int, reynolds: float, rotation: float,
timestep: float, maxradius: float, seed: int, endtime: float):
'''
Flow around a cylinder.
.. arguments::
nelems [24]
Element size expressed in number of elements along the cylinder wall.
All elements have similar shape with approximately unit aspect ratio,
with elements away from the cylinder wall growing exponentially.
degree [3]
Polynomial degree for velocity space; the pressure space is one degree
less.
reynolds [1000]
Reynolds number, taking the cylinder radius as characteristic length.
rotation [0]
Cylinder rotation speed.
timestep [.04]
Time step
maxradius [25]
Target exterior radius; the actual domain size is subject to integer
multiples of the configured element size.
seed [0]
Random seed for small velocity noise in the intial condition.
endtime [inf]
Stopping time.
'''
elemangle = 2 * numpy.pi / nelems
melems = int(numpy.log(2 * maxradius) / elemangle + .5)
treelog.info('creating {}x{} mesh, outer radius {:.2f}'.format(
melems, nelems, .5 * numpy.exp(elemangle * melems)))
domain, geom = mesh.rectilinear([melems, nelems], periodic=(1, ))
domain = domain.withboundary(inner='left', outer='right')
ns = function.Namespace()
ns.uinf = 1, 0
ns.r = .5 * function.exp(elemangle * geom[0])
ns.Re = reynolds
ns.phi = geom[1] * elemangle # add small angle to break element symmetry
ns.x_i = 'r <cos(phi), sin(phi)>_i'
ns.J = ns.x.grad(geom)
ns.unbasis, ns.utbasis, ns.pbasis = function.chain([ # compatible spaces
domain.basis(
'spline', degree=(degree, degree - 1), removedofs=((0, ), None)),
domain.basis('spline', degree=(degree - 1, degree)),
domain.basis('spline', degree=degree - 1),
]) / function.determinant(ns.J)
ns.ubasis_ni = 'unbasis_n J_i0 + utbasis_n J_i1' # piola transformation
ns.u_i = 'ubasis_ni ?lhs_n'
ns.p = 'pbasis_n ?lhs_n'
ns.sigma_ij = '(u_i,j + u_j,i) / Re - p δ_ij'
ns.h = .5 * elemangle
ns.N = 5 * degree / ns.h
ns.rotation = rotation
ns.uwall_i = '0.5 rotation <-sin(phi), cos(phi)>_i'
inflow = domain.boundary['outer'].select(
-ns.uinf.dotnorm(ns.x),
ischeme='gauss1') # upstream half of the exterior boundary
sqr = inflow.integral('(u_i - uinf_i) (u_i - uinf_i)' @ ns,
degree=degree * 2)
cons = solver.optimize(
'lhs', sqr, droptol=1e-15) # constrain inflow semicircle to uinf
sqr = domain.integral('(u_i - uinf_i) (u_i - uinf_i) + p^2' @ ns,
degree=degree * 2)
lhs0 = solver.optimize('lhs', sqr) # set initial condition to u=uinf, p=0
numpy.random.seed(seed)
lhs0 *= numpy.random.normal(1, .1, lhs0.shape) # add small velocity noise
res = domain.integral(
'(ubasis_ni u_i,j u_j + ubasis_ni,j sigma_ij + pbasis_n u_k,k) d:x'
@ ns,
degree=9)
res += domain.boundary['inner'].integral(
'(N ubasis_ni - (ubasis_ni,j + ubasis_nj,i) n_j) (u_i - uwall_i) d:x / Re'
@ ns,
degree=9)
inertia = domain.integral('ubasis_ni u_i d:x' @ ns, degree=9)
bbox = numpy.array(
[[-2, 46 / 9],
[-2, 2]]) # bounding box for figure based on 16x9 aspect ratio
bezier0 = domain.sample('bezier', 5)
bezier = bezier0.subset((bezier0.eval(
(ns.x - bbox[:, 0]) * (bbox[:, 1] - ns.x)) > 0).all(axis=1))
interpolate = util.tri_interpolator(
bezier.tri, bezier.eval(ns.x),
mergetol=1e-5) # interpolator for quivers
spacing = .05 # initial quiver spacing
xgrd = util.regularize(bbox, spacing)
with treelog.iter.plain(
'timestep',
solver.impliciteuler('lhs',
residual=res,
inertia=inertia,
lhs0=lhs0,
timestep=timestep,
constrain=cons,
newtontol=1e-10)) as steps:
for istep, lhs in enumerate(steps):
t = istep * timestep
x, u, normu, p = bezier.eval(
['x_i', 'u_i', 'sqrt(u_k u_k)', 'p'] @ ns, lhs=lhs)
ugrd = interpolate[xgrd](u)
with export.mplfigure('flow.png', figsize=(12.8, 7.2)) as fig:
ax = fig.add_axes([0, 0, 1, 1],
yticks=[],
xticks=[],
frame_on=False,
xlim=bbox[0],
ylim=bbox[1])
im = ax.tripcolor(x[:, 0],
x[:, 1],
bezier.tri,
p,
shading='gouraud',
cmap='jet')
import matplotlib.collections
ax.add_collection(
matplotlib.collections.LineCollection(x[bezier.hull],
colors='k',
linewidths=.1,
alpha=.5))
ax.quiver(xgrd[:, 0],
xgrd[:, 1],
ugrd[:, 0],
ugrd[:, 1],
angles='xy',
width=1e-3,
headwidth=3e3,
headlength=5e3,
headaxislength=2e3,
zorder=9,
alpha=.5)
ax.plot(0,
0,
'k',
marker=(3, 2, t * rotation * 180 / numpy.pi - 90),
markersize=20)
cax = fig.add_axes([0.9, 0.1, 0.01, 0.8])
cax.tick_params(labelsize='large')
fig.colorbar(im, cax=cax)
if t >= endtime:
break
xgrd = util.regularize(bbox, spacing, xgrd + ugrd * timestep)
return lhs0, lhs
def main(nelems: int, ndims: int, degree: int, timescale: float,
newtontol: float, endtime: float):
'''
Burgers equation on a 1D or 2D periodic domain.
.. arguments::
nelems [20]
Number of elements along a single dimension.
ndims [1]
Number of spatial dimensions.
degree [1]
Polynomial degree for discontinuous basis functions.
timescale [.5]
Fraction of timestep and element size: timestep=timescale/nelems.
newtontol [1e-5]
Newton tolerance.
endtime [inf]
Stopping time.
'''
domain, geom = mesh.rectilinear([numpy.linspace(0, 1, nelems + 1)] * ndims,
periodic=range(ndims))
ns = function.Namespace()
ns.x = geom
ns.basis = domain.basis('discont', degree=degree)
ns.u = 'basis_n ?lhs_n'
ns.f = '.5 u^2'
ns.C = 1
res = domain.integral('-basis_n,0 f d:x' @ ns, degree=5)
res += domain.interfaces.integral(
'-[basis_n] n_0 ({f} - .5 C [u] n_0) d:x' @ ns, degree=degree * 2)
inertia = domain.integral('basis_n u d:x' @ ns, degree=5)
sqr = domain.integral(
'(u - exp(-?y_i ?y_i)(y_i = 5 (x_i - 0.5_i)))^2 d:x' @ ns, degree=5)
lhs0 = solver.optimize('lhs', sqr)
timestep = timescale / nelems
bezier = domain.sample('bezier', 7)
with treelog.iter.plain(
'timestep',
solver.impliciteuler('lhs',
res,
inertia,
timestep=timestep,
lhs0=lhs0,
newtontol=newtontol)) as steps:
for itime, lhs in enumerate(steps):
x, u = bezier.eval(['x_i', 'u'] @ ns, lhs=lhs)
export.triplot('solution.png',
x,
u,
tri=bezier.tri,
hull=bezier.hull,
clim=(0, 1))
if itime * timestep >= endtime:
break
return lhs
def setUp(self):
super().setUp()
self.ns = function.Namespace()
self.domain, self.ns.geom = mesh.rectilinear([2, 2])
self.ns.ubasis = self.domain.basis('std', degree=1)
self.ns.u = 'ubasis_n ?dofs_n'
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(
nelems: 'number of elements' = 20,
epsilon: 'epsilon, 0 for automatic (based on nelems)' = 0,
timestep: 'time step' = .01,
maxtime: 'end time' = 1.,
theta: 'contact angle (degrees)' = 90,
init: 'initial condition (random/bubbles)' = 'random',
figures: 'create figures' = True,
):
mineps = 1. / nelems
if not epsilon:
log.info('setting epsilon={}'.format(mineps))
epsilon = mineps
elif epsilon < mineps:
log.warning('epsilon under crititical threshold: {} < {}'.format(
epsilon, mineps))
# create namespace
ns = function.Namespace()
ns.epsilon = epsilon
ns.ewall = .5 * numpy.cos(theta * numpy.pi / 180)
# construct mesh
xnodes = ynodes = numpy.linspace(0, 1, nelems + 1)
domain, ns.x = mesh.rectilinear([xnodes, ynodes])
# prepare bases
ns.cbasis, ns.mubasis = function.chain(
[domain.basis('spline', degree=2),
domain.basis('spline', degree=2)])
# polulate namespace
ns.c = 'cbasis_n ?lhs_n'
ns.c0 = 'cbasis_n ?lhs0_n'
ns.mu = 'mubasis_n ?lhs_n'
ns.f = '(6 c0 - 2 c0^3 - 4 c) / epsilon^2'
# construct initial condition
if init == 'random':
numpy.random.seed(0)
lhs0 = numpy.random.normal(0, .5, ns.cbasis.shape)
elif init == 'bubbles':
R1 = .25
R2 = numpy.sqrt(.5) * R1 # area2 = .5 * area1
ns.cbubble1 = function.tanh(
(R1 - function.norm2(ns.x -
(.5 + R2 / numpy.sqrt(2) + .8 * ns.epsilon)))
/ ns.epsilon)
ns.cbubble2 = function.tanh(
(R2 - function.norm2(ns.x -
(.5 - R1 / numpy.sqrt(2) - .8 * ns.epsilon)))
/ ns.epsilon)
sqr = domain.integral('(c - cbubble1 - cbubble2 - 1)^2 + mu^2' @ ns,
geometry=ns.x,
degree=4)
lhs0 = solver.optimize('lhs', sqr)
else:
raise Exception('unknown init %r' % init)
# construct residual
res = domain.integral(
'epsilon^2 cbasis_n,k mu_,k + mubasis_n (mu + f) - mubasis_n,k c_,k'
@ ns,
geometry=ns.x,
degree=4)
res -= domain.boundary.integral('mubasis_n ewall' @ ns,
geometry=ns.x,
degree=4)
inertia = domain.integral('cbasis_n c' @ ns, geometry=ns.x, degree=4)
# solve time dependent problem
nsteps = numeric.round(maxtime / timestep)
makeplots = MakePlots(domain, nsteps,
ns) if figures else lambda *args: None
for istep, lhs in log.enumerate(
'timestep',
solver.impliciteuler('lhs',
target0='lhs0',
residual=res,
inertia=inertia,
timestep=timestep,
lhs0=lhs0)):
makeplots(lhs)
if istep == nsteps:
break
return lhs0, lhs
def main(
nelems: 'number of elements' = 12,
viscosity: 'fluid viscosity' = 1e-2,
density: 'fluid density' = 1,
tol: 'solver tolerance' = 1e-12,
rotation: 'cylinder rotation speed' = 0,
timestep: 'time step' = 1/24,
maxradius: 'approximate domain size' = 25,
tmax: 'end time' = numpy.inf,
degree: 'polynomial degree' = 2,
withplots: 'create plots' = True,
):
log.user('reynolds number: {:.1f}'.format(density / viscosity)) # based on unit length and velocity
# create namespace
ns = function.Namespace()
ns.uinf = 1, 0
ns.density = density
ns.viscosity = viscosity
# construct mesh
rscale = numpy.pi / nelems
melems = numpy.ceil(numpy.log(2*maxradius) / rscale).astype(int)
log.info('creating {}x{} mesh, outer radius {:.2f}'.format(melems, 2*nelems, .5*numpy.exp(rscale*melems)))
domain, x0 = mesh.rectilinear([range(melems+1),numpy.linspace(0,2*numpy.pi,2*nelems+1)], periodic=(1,))
rho, phi = x0
phi += 1e-3 # tiny nudge (0.057 deg) to break element symmetry
radius = .5 * function.exp(rscale * rho)
ns.x = radius * function.trigtangent(phi)
domain = domain.withboundary(inner='left', inflow=domain.boundary['right'].select(-ns.uinf.dotnorm(ns.x), ischeme='gauss1'))
# prepare bases (using piola transformation to maintain u/p compatibility)
J = ns.x.grad(x0)
detJ = function.determinant(J)
ns.unbasis, ns.utbasis, ns.pbasis = function.chain([ # compatible spaces using piola transformation
domain.basis('spline', degree=(degree+1,degree), removedofs=((0,),None))[:,_] * J[:,0] / detJ,
domain.basis('spline', degree=(degree,degree+1))[:,_] * J[:,1] / detJ,
domain.basis('spline', degree=degree) / detJ,
])
ns.ubasis_ni = 'unbasis_ni + utbasis_ni'
# populate namespace
ns.u_i = 'ubasis_ni ?lhs_n'
ns.p = 'pbasis_n ?lhs_n'
ns.sigma_ij = 'viscosity (u_i,j + u_j,i) - p δ_ij'
ns.hinner = 2 * numpy.pi / nelems
ns.c = 5 * (degree+1) / ns.hinner
ns.nietzsche_ni = 'viscosity (c ubasis_ni - (ubasis_ni,j + ubasis_nj,i) n_j)'
ns.ucyl = -.5 * rotation * function.trignormal(phi)
# create residual vector components
res = domain.integral('ubasis_ni,j sigma_ij + pbasis_n u_k,k' @ ns, geometry=ns.x, degree=2*(degree+1))
res += domain.boundary['inner'].integral('nietzsche_ni (u_i - ucyl_i)' @ ns, geometry=ns.x, degree=2*(degree+1))
oseen = domain.integral('density ubasis_ni u_i,j uinf_j' @ ns, geometry=ns.x, degree=2*(degree+1))
convec = domain.integral('density ubasis_ni u_i,j u_j' @ ns, geometry=ns.x, degree=3*(degree+1))
inertia = domain.integral('density ubasis_ni u_i' @ ns, geometry=ns.x, degree=2*(degree+1))
# constrain full velocity vector at inflow
sqr = domain.boundary['inflow'].integral('(u_i - uinf_i) (u_i - uinf_i)' @ ns, geometry=ns.x, degree=9)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
# solve unsteady navier-stokes equations, starting from stationary oseen flow
lhs0 = solver.solve_linear('lhs', res+oseen, constrain=cons)
makeplots = MakePlots(domain, ns, timestep=timestep, rotation=rotation) if withplots else lambda *args: None
for istep, lhs in log.enumerate('timestep', solver.impliciteuler('lhs', residual=res+convec, inertia=inertia, lhs0=lhs0, timestep=timestep, constrain=cons, newtontol=1e-10)):
makeplots(lhs)
if istep * timestep >= tmax:
break
return lhs0, lhs
class NutilsDelayedIntegrand(Integrand):
optimized = False
def __init__(self, code, indices, variables, **kwargs):
super().__init__()
self._code = code
self._defaults = OrderedDict([(name, kwargs[name]) for name in variables])
self._kwargs = {name: func for name, func in kwargs.items() if name not in variables}
self._evaluator = 'eval_' + indices
self._kwargs, self._arg_shapes = {}, {}
self.update_kwargs({k: v for k, v in kwargs.items() if k not in variables})
if code is not None:
self.shape = self._integrand().shape
self.ndim = len(self.shape)
def update_kwargs(self, kwargs):
for name, func in kwargs.items():
if isinstance(func, tuple):
func, shape = func
self._arg_shapes[name] = shape
self._kwargs[name] = func
def add(self, code, **kwargs):
self.update_kwargs(kwargs)
if self._code is None:
self._code = f'({code})'
self.shape = self._integrand().shape
self.ndim = len(self.shape)
else:
self._code = f'{self._code} + ({code})'
def write(self, group, name):
sub = group.require_group(name)
data = {key: getattr(self, key) for key in ['_code', '_defaults', '_kwargs', '_evaluator', 'shape']}
util.to_dataset(data, sub, 'data')
sub.attrs['type'] = 'NutilsDelayedIntegrand'
self.write_props(sub)
return sub
@staticmethod
def read(group):
retval = NutilsDelayedIntegrand.__new__(NutilsDelayedIntegrand)
data = util.from_dataset(group['data'])
retval.__dict__.update(data)
retval.ndim = len(retval.shape)
retval.read_props(group)
return retval
def _integrand(self, contraction=None, mu=None, case=None):
if contraction is None:
contraction = (None,) * len(self._defaults)
ns = fn.Namespace()
for name, func in self._kwargs.items():
if isinstance(func, fn.Array):
setattr(ns, name, func)
elif mu is not None and case is not None:
assert callable(getattr(case, func))
setattr(ns, name, getattr(case, func)(mu))
else:
# This code path should ONLY be used for inferring the shape of the integrand
assert not hasattr(self, 'shape')
setattr(ns, name, fn.zeros(self._arg_shapes[name]))
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 neutrondiffusion(mat_flag='scatterer',
BC_flag='vacuum',
bigsquare=1,
smallsquare=.1,
degree=1,
basis='lagrange'):
W = bigsquare #width of outer box
Wi = smallsquare #width of inner box
nelems = int((2 * 1 * W / Wi))
if mat_flag == 'scatterer':
sigt = 2
sigs = 1.99
elif mat_flag == 'reflector':
sigt = 2
sigs = 1.8
elif mat_flag == 'absorber':
sigt = 10
sigs = 2
elif mat_flag == 'air':
sigt = .01
sigs = .006
topo, geom = mesh.unitsquare(nelems,
'square') #unit square centred at (0.5, 0.5)
ns = function.Namespace()
ns.basis = topo.basis(basis, degree=degree)
ns.x = W * geom #scales the unit square to our physical square
ns.f = function.max(
function.abs(ns.x[0] - W / 2),
function.abs(ns.x[1] - W / 2)) #level set function for inner square
inner, outer = function.partition(
ns.f, Wi / 2) #indicator function for inner square and outer square
ns.phi = 'basis_A ?dofs_A'
ns.SIGs = sigs * outer #scattering cross-section
ns.SIGt = 0.1 * inner + sigt * outer #total cross-section
ns.SIGa = 'SIGt - SIGs' #absorption cross-section
ns.D = '1 / (3 SIGt)' #diffusion co-efficient
ns.Q = inner #source term
if BC_flag == 'vacuum':
sqr = topo.boundary.integral('(phi - 0)^2 J(x)' @ ns,
degree=degree * 2)
cons = solver.optimize(
'dofs', sqr,
droptol=1e-14) #this applies the boundary condition to u
#residual
res = topo.integral(
'(D basis_i,j phi_,j + SIGa basis_i phi - basis_i Q) J(x)' @ ns,
degree=degree * 2)
#solve for degrees of freedom
dofs = solver.solve_linear('dofs', res, constrain=cons)
elif BC_flag == 'reflecting':
#residual
res = topo.integral(
'(D basis_i,j phi_,j + SIGa basis_i phi - basis_i Q) J(x)' @ ns,
degree=degree * 2)
#solve for degrees of freedom
dofs = solver.solve_linear('dofs', res)
#select lower triangular half of square domain. Diagonal is one of its boundaries
triang = topo.trim('x_0 - x_1' @ ns, maxrefine=5)
triangbnd = triang.boundary #select boundary of lower triangle
# eval the vertices of the boundary elements of lower triangle:
verts = triangbnd.sample(*element.parse_legacy_ischeme("vertex")).eval(
'x_i' @ ns)
# now select the verts of the boundary
diag = topology.SubsetTopology(triangbnd, [
triangbnd.references[i] if
(verts[2 * i][0] == verts[2 * i][1] and verts[2 * i + 1][0]
== verts[2 * i + 1][1]) else triangbnd.references[i].empty
for i in range(len(triangbnd.references))
])
bezier_diag = diag.sample('bezier', degree + 1)
x_diag = bezier_diag.eval('(x_0^2 + x_1^2)^(1 / 2)' @ ns)
phi_diag = bezier_diag.eval('phi' @ ns, dofs=dofs)
bezier_bottom = topo.boundary['bottom'].sample('bezier', degree + 1)
x_bottom = bezier_bottom.eval('x_0' @ ns)
phi_bottom = bezier_bottom.eval('phi' @ ns, dofs=dofs)
fig_bottom = plt.figure(0)
ax_bottom = fig_bottom.add_subplot(111)
plt.plot(x_bottom, phi_bottom)
ax_bottom.set_title('Scalar flux along bottom for %s with %s BCs' %
(mat_flag, BC_flag))
ax_bottom.set_xlabel('x')
ax_bottom.set_ylabel('$\\phi(x)$')
fig_diag = plt.figure(1)
ax_diag = fig_diag.add_subplot(111)
plt.plot(x_diag, phi_diag)
ax_diag.set_title('Scalar flux along diagonal for %s with %s BCs' %
(mat_flag, BC_flag))
ax_diag.set_xlabel('$\\sqrt{x^2 + y^2}$')
ax_diag.set_ylabel('$\\phi(x = y)$')
return fig_bottom, fig_diag
def main(
L: 'domain size' = 4.,
R: 'hole radius' = 1.,
E: "young's modulus" = 1e5,
nu: "poisson's ratio" = 0.3,
T: 'far field traction' = 10,
nr: 'number of h-refinements' = 2,
withplots: 'create plots' = True,
):
ns = function.Namespace()
ns.lmbda = E * nu / (1 - nu**2)
ns.mu = .5 * E / (1 + nu)
# create the coarsest level parameter domain
domain0, geometry = mesh.rectilinear([1, 2])
# create the second-order B-spline basis over the coarsest domain
ns.bsplinebasis = domain0.basis('spline', degree=2)
ns.controlweights = 1, .5 + .5 / 2**.5, .5 + .5 / 2**.5, 1, 1, 1, 1, 1, 1, 1, 1, 1
ns.weightfunc = 'bsplinebasis_n controlweights_n'
ns.nurbsbasis = ns.bsplinebasis * ns.controlweights / ns.weightfunc
# create the isogeometric map
ns.controlpoints = [0, R], [R * (1 - 2**.5), R], [-R, R * (2**.5 - 1)], [
-R, 0
], [0, .5 * (R + L)], [-.15 * (R + L),
.5 * (R + L)], [-.5 * (R + L), .15 * (R * L)
], [-.5 * (R + L),
0], [0, L], [-L,
L], [-L,
L], [-L, 0]
ns.x_i = 'nurbsbasis_n controlpoints_ni'
# create the computational domain by h-refinement
domain = domain0.refine(nr)
# create the second-order B-spline basis over the refined domain
ns.bsplinebasis = domain.basis('spline', degree=2)
sqr = domain.integral('(bsplinebasis_n ?lhs_n - weightfunc)^2' @ ns,
geometry=ns.x,
degree=9)
ns.controlweights = solver.optimize('lhs', sqr)
ns.nurbsbasis = ns.bsplinebasis * ns.controlweights / ns.weightfunc
# prepare the displacement field
ns.ubasis = ns.nurbsbasis.vector(2)
ns.u_i = 'ubasis_ni ?lhs_n'
ns.strain_ij = '(u_i,j + u_j,i) / 2'
ns.stress_ij = 'lmbda strain_kk δ_ij + 2 mu strain_ij'
# construct the exact solution
ns.r2 = 'x_k x_k'
ns.R2 = R**2
ns.T = T
ns.k = (3 - nu) / (1 + nu) # plane stress parameter
ns.uexact_i = '''(T / 4 mu) <x_0 ((k + 1) / 2 + (1 + k) R2 / r2 + (1 - R2 / r2) (x_0^2 - 3 x_1^2) R2 / r2^2),
x_1 ((k - 3) / 2 + (1 - k) R2 / r2 + (1 - R2 / r2) (3 x_0^2 - x_1^2) R2 / r2^2)>_i'''
ns.strainexact_ij = '(uexact_i,j + uexact_j,i) / 2'
ns.stressexact_ij = 'lmbda strainexact_kk δ_ij + 2 mu strainexact_ij'
# define the linear and bilinear forms
res = domain.integral('ubasis_ni,j stress_ij' @ ns,
geometry=ns.x,
degree=9)
res -= domain.boundary['right'].integral(
'ubasis_ni stressexact_ij n_j' @ ns, geometry=ns.x, degree=9)
# compute the constraints vector for the symmetry conditions
sqr = domain.boundary['top,bottom'].integral('(u_i n_i)^2' @ ns, degree=9)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
# solve the system of equations
lhs = solver.solve_linear('lhs', res, constrain=cons)
ns |= dict(lhs=lhs)
# post-processing
if withplots:
geom, stressxx = domain.simplex.elem_eval([ns.x, 'stress_00' @ ns],
ischeme='bezier8',
separate=True)
with plot.PyPlot('solution', index=nr) as plt:
plt.mesh(geom, stressxx)
plt.colorbar()
# compute the L2-norm of the error in the stress
err = domain.integrate(
'?dstress_ij ?dstress_ij | ?dstress_ij = stress_ij - stressexact_ij'
@ ns,
geometry=ns.x,
ischeme='gauss9')**.5
# compute the mesh parameter (maximum physical distance between knots)
hmax = max(
numpy.linalg.norm(v[:, numpy.newaxis] - v, axis=2).max()
for v in domain.elem_eval(ns.x, ischeme='bezier2', separate=True))
return err, hmax
def elasticity_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, solver
from nutils import _, log
# error test input
if not (left.dimension == right.dimension == top.dimension ==
bottom.dimension == 2):
raise RuntimeError(
'elasticity_patch only supported for planar (2D) geometries')
if left.rational or right.rational or top.rational or bottom.rational:
raise RuntimeError(
'elasticity_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 computational (nutils) mesh
p1 = bottom.order(0)
p2 = left.order(0)
n1 = len(bottom)
n2 = len(left)
dim = left.dimension
k1 = bottom.knots(0)
k2 = left.knots(0)
m1 = [bottom.order(0) - bottom.continuity(k) - 1 for k in k1]
m2 = [left.order(0) - left.continuity(k) - 1 for k in k2]
domain, geom = mesh.rectilinear([k1, k2])
# assemble system matrix
ns = function.Namespace()
ns.x = geom
ns.basis = domain.basis('spline',
degree=[p1 - 1, p2 - 1],
knotmultiplicities=[m1, m2]).vector(2)
ns.eye = np.array([[1, 0], [0, 1]])
ns.lmbda = 1
ns.mu = .3
# ns.u_i = 'basis_ni ?lhs_n'
# ns.strain_ij = '(u_i,j + u_j,i) / 2'
# ns.stress_ij = 'lmbda strain_kk eye_ij + 2 mu strain_ij'
ns.strain_nij = '(basis_ni,j + basis_nj,i) / 2'
ns.stress_nij = 'lmbda strain_nkk eye_ij + 2 mu strain_nij'
# construct matrix and right hand-side
matrix = domain.integrate(ns.eval_nm('strain_nij stress_mij d:x'),
ischeme='gauss' + str(max(p1, p2) + 1))
rhs = np.zeros((n1 * n2 * dim))
# add boundary conditions
constraints = np.array([[[np.nan] * n2] * n1] * dim)
for d in range(dim):
constraints[d, 0, :] = left[:, d]
constraints[d, -1, :] = right[:, d]
constraints[d, :, 0] = bottom[:, d]
constraints[d, :, -1] = top[:, d]
# solve system
lhs = matrix.solve(rhs,
constrain=np.ndarray.flatten(constraints, order='C'))
# rewrap results into splipy datastructures
controlpoints = np.reshape(lhs, (dim, n1, n2), order='C')
controlpoints = controlpoints.swapaxes(0, 2)
controlpoints = controlpoints.swapaxes(0, 1)
return Surface(bottom.bases[0],
left.bases[0],
controlpoints,
bottom.rational,
raw=True)
def main(nelems: int, etype: str, btype: str, degree: int, traction: float,
maxrefine: int, radius: float, poisson: float):
'''
Horizontally loaded linear elastic plate with FCM hole.
.. arguments::
nelems [9]
Number of elements along edge.
etype [square]
Type of elements (square/triangle/mixed).
btype [std]
Type of basis function (std/spline), with availability depending on the
selected element type.
degree [2]
Polynomial degree.
traction [.1]
Far field traction (relative to Young's modulus).
maxrefine [2]
Number or refinement levels used for the finite cell method.
radius [.5]
Cut-out radius.
poisson [.3]
Poisson's ratio, nonnegative and strictly smaller than 1/2.
'''
domain0, geom = mesh.unitsquare(nelems, etype)
domain = domain0.trim(function.norm2(geom) - radius, maxrefine=maxrefine)
ns = function.Namespace()
ns.x = geom
ns.lmbda = 2 * poisson
ns.mu = 1 - poisson
ns.ubasis = domain.basis(btype, degree=degree).vector(2)
ns.u_i = 'ubasis_ni ?lhs_n'
ns.X_i = 'x_i + u_i'
ns.strain_ij = '(d(u_i, x_j) + d(u_j, x_i)) / 2'
ns.stress_ij = 'lmbda strain_kk δ_ij + 2 mu strain_ij'
ns.r2 = 'x_k x_k'
ns.R2 = radius**2 / ns.r2
ns.k = (3 - poisson) / (1 + poisson) # plane stress parameter
ns.scale = traction * (1 + poisson) / 2
ns.uexact_i = 'scale (x_i ((k + 1) (0.5 + R2) + (1 - R2) R2 (x_0^2 - 3 x_1^2) / r2) - 2 δ_i1 x_1 (1 + (k - 1 + R2) R2))'
ns.du_i = 'u_i - uexact_i'
sqr = domain.boundary['left,bottom'].integral('(u_i n_i)^2 J(x)' @ ns,
degree=degree * 2)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
sqr = domain.boundary['top,right'].integral('du_k du_k J(x)' @ ns,
degree=20)
cons = solver.optimize('lhs', sqr, droptol=1e-15, constrain=cons)
res = domain.integral('d(ubasis_ni, x_j) stress_ij J(x)' @ ns,
degree=degree * 2)
lhs = solver.solve_linear('lhs', res, constrain=cons)
bezier = domain.sample('bezier', 5)
X, stressxx = bezier.eval(['X', 'stress_00'] @ ns, lhs=lhs)
export.triplot('stressxx.png',
X,
stressxx,
tri=bezier.tri,
hull=bezier.hull)
err = domain.integral('<du_k du_k, sum:ij(d(du_i, x_j)^2)>_n J(x)' @ ns,
degree=max(degree, 3) * 2).eval(lhs=lhs)**.5
treelog.user('errors: L2={:.2e}, H1={:.2e}'.format(*err))
return err, cons, lhs
def main(nelems: int, etype: str, btype: str, degree: int):
'''
Laplace problem on a unit square.
.. arguments::
nelems [10]
Number of elements along edge.
etype [square]
Type of elements (square/triangle/mixed).
btype [std]
Type of basis function (std/spline), availability depending on the
selected element type.
degree [1]
Polynomial degree.
'''
# A unit square domain is created by calling the
# :func:`nutils.mesh.unitsquare` mesh generator, with the number of elements
# along an edge as the first argument, and the type of elements ("square",
# "triangle", or "mixed") as the second. The result is a topology object
# ``domain`` and a vectored valued geometry function ``geom``.
domain, geom = mesh.unitsquare(nelems, etype)
# To be able to write index based tensor contractions, we need to bundle all
# relevant functions together in a namespace. Here we add the geometry ``x``,
# a scalar ``basis``, and the solution ``u``. The latter is formed by
# contracting the basis with a to-be-determined solution vector ``?lhs``.
ns = function.Namespace()
ns.x = geom
ns.basis = domain.basis(btype, degree=degree)
ns.u = 'basis_n ?lhs_n'
# We are now ready to implement the Laplace equation. In weak form, the
# solution is a scalar field :math:`u` for which:
#
# .. math:: ∀ v: ∫_Ω \frac{dv}{dx_i} \frac{du}{dx_i} - ∫_{Γ_n} v f = 0.
#
# By linearity the test function :math:`v` can be replaced by the basis that
# spans its space. The result is an integral ``res`` that evaluates to a
# vector matching the size of the function space.
res = domain.integral('d(basis_n, x_i) d(u, x_i) J(x)' @ ns,
degree=degree * 2)
res -= domain.boundary['right'].integral(
'basis_n cos(1) cosh(x_1) J(x)' @ ns, degree=degree * 2)
# The Dirichlet constraints are set by finding the coefficients that minimize
# the error:
#
# .. math:: \min_u ∫_{\Gamma_d} (u - u_d)^2
#
# The resulting ``cons`` array holds numerical values for all the entries of
# ``?lhs`` that contribute (up to ``droptol``) to the minimization problem.
# All remaining entries are set to ``NaN``, signifying that these degrees of
# freedom are unconstrained.
sqr = domain.boundary['left'].integral('u^2 J(x)' @ ns, degree=degree * 2)
sqr += domain.boundary['top'].integral(
'(u - cosh(1) sin(x_0))^2 J(x)' @ ns, degree=degree * 2)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
# The unconstrained entries of ``?lhs`` are to be determined such that the
# residual vector evaluates to zero in the corresponding entries. This step
# involves a linearization of ``res``, resulting in a jacobian matrix and
# right hand side vector that are subsequently assembled and solved. The
# resulting ``lhs`` array matches ``cons`` in the constrained entries.
lhs = solver.solve_linear('lhs', res, constrain=cons)
# Once all entries of ``?lhs`` are establised, the corresponding solution can
# be vizualised by sampling values of ``ns.u`` along with physical
# coordinates ``ns.x``, with the solution vector provided via the
# ``arguments`` dictionary. The sample members ``tri`` and ``hull`` provide
# additional inter-point information required for drawing the mesh and
# element outlines.
bezier = domain.sample('bezier', 9)
x, u = bezier.eval(['x', 'u'] @ ns, lhs=lhs)
export.triplot('solution.png', x, u, tri=bezier.tri, hull=bezier.hull)
# To confirm that our computation is correct, we use our knowledge of the
# analytical solution to evaluate the L2-error of the discrete result.
err = domain.integral('(u - sin(x_0) cosh(x_1))^2 J(x)' @ ns,
degree=degree * 2).eval(lhs=lhs)**.5
treelog.user('L2 error: {:.2e}'.format(err))
return cons, lhs, err
