def test_bases(mu, case):
domain, vbasis, pbasis = case.domain, case.basis('v'), case.basis('p')
trfgeom = case.geometry(mu)
refgeom = case.refgeom
a_vbasis, a_pbasis = piola_bases(mu, case)
J = trfgeom.grad(case.geometry())
detJ = fn.determinant(J)
b_vbasis = fn.matmat(vbasis, J.transpose()) / detJ
b_pbasis = pbasis / detJ
Z = case.geometry(mu).grad(case.geometry())
detZ = fn.determinant(Z)
zdiff = np.sqrt(domain.integrate((Z - J)**2 * fn.J(refgeom), ischeme='gauss9').export('dense'))
np.testing.assert_almost_equal(zdiff, 0.0)
c_vbasis = fn.matmat(vbasis, Z.transpose()) / detZ
c_pbasis = pbasis
pdiff = np.sqrt(domain.integrate((a_pbasis - b_pbasis)**2 * fn.J(refgeom), ischeme='gauss9'))
np.testing.assert_almost_equal(pdiff, 0.0)
pdiff = domain.integrate((a_pbasis - c_pbasis)**2 * fn.J(refgeom), ischeme='gauss9')
np.testing.assert_almost_equal(pdiff, 0.0)
vdiff = np.sqrt(domain.integrate((a_vbasis - b_vbasis)**2 * fn.J(refgeom), ischeme='gauss9').export('dense'))
np.testing.assert_almost_equal(vdiff, 0.0)
vdiff = domain.integrate((a_vbasis - c_vbasis)**2 * fn.J(refgeom), ischeme='gauss9').export('dense')
np.testing.assert_almost_equal(vdiff, 0.0)
def setUp(self):
super().setUp()
domain, geom = mesh.rectilinear([numpy.linspace(0, 1, 9)] * 2)
ubasis, pbasis = function.chain([
domain.basis('std', degree=2).vector(2),
domain.basis('std', degree=1),
])
dofs = function.Argument('dofs', [len(ubasis)])
u = ubasis.dot(dofs)
p = pbasis.dot(dofs)
viscosity = 1e-3
self.inertia = domain.integral((ubasis * u).sum(-1) * function.J(geom),
degree=5)
stokesres = domain.integral(
(viscosity * (ubasis.grad(geom) *
(u.grad(geom) + u.grad(geom).T)).sum([-1, -2]) -
ubasis.div(geom) * p + pbasis * u.div(geom)) * function.J(geom),
degree=5)
self.residual = stokesres + domain.integral(
(ubasis *
(u.grad(geom) * u).sum(-1) * u).sum(-1) * function.J(geom),
degree=5)
self.cons = domain.boundary['top,bottom'].project([0,0], onto=ubasis, geometry=geom, ischeme='gauss4') \
| domain.boundary['left'].project([geom[1]*(1-geom[1]),0], onto=ubasis, geometry=geom, ischeme='gauss4')
self.lhs0 = solver.solve_linear('dofs',
residual=stokesres,
constrain=self.cons)
self.tol = 1e-10
def setUp(self):
super().setUp()
domain, geom = mesh.rectilinear([8,8])
basis = domain.basis('std', degree=1)
self.cons = domain.boundary['left'].project(0, onto=basis, geometry=geom, ischeme='gauss2')
dofs = function.Argument('dofs', [len(basis)])
u = basis.dot(dofs)
self.residual = domain.integral((basis.grad(geom) * u.grad(geom)).sum(-1)*function.J(geom), degree=2) \
+ domain.boundary['top'].integral(basis*function.J(geom), degree=2)
def setUp(self):
super().setUp()
domain, geom = mesh.rectilinear([numpy.linspace(0, 1, 9)] * 2)
ubasis = domain.basis('std', degree=2)
if self.vector:
u = ubasis.vector(2).dot(
function.Argument('dofs', [len(ubasis) * 2]))
else:
u = (ubasis[:, numpy.newaxis] *
function.Argument('dofs', [len(ubasis), 2])).sum(0)
Geom = geom * [1.1, 1] + u
self.cons = solver.optimize('dofs',
domain.boundary['left,right'].integral(
(u**2).sum(0), degree=4),
droptol=1e-15)
self.boolcons = ~numpy.isnan(self.cons)
strain = .5 * (function.outer(Geom.grad(geom), axis=1).sum(0) -
function.eye(2))
self.energy = domain.integral(
((strain**2).sum([0, 1]) + 20 *
(function.determinant(Geom.grad(geom)) - 1)**2) *
function.J(geom),
degree=6)
self.residual = self.energy.derivative('dofs')
self.tol = 1e-10
def _divs(fast: bool = False, piola: bool = False, num=8):
__, solutions, supremizers = get_ensemble(fast, piola, num)
case = get_case(fast, piola)
angles = np.linspace(-35, 35, 15) * np.pi / 180
eig_sol = reduction.eigen(case, solutions, fields=['v', 'p'])
rb_sol = reduction.reduced_bases(case, solutions, eig_sol, (20, 20))
results = []
for angle in angles:
log.user(f'angle = {angle}')
mu = case.parameter(angle=angle)
geom = case.physical_geometry(mu)
divs = []
for i in range(10):
vsol, = case.solution(rb_sol['v'][i], mu, ['v'], lift=True)
div = np.sqrt(
case.domain.integrate(vsol.div(geom)**2 * fn.J(geom),
ischeme='gauss9'))
log.user(f'{i}: {div:.2e}')
divs.append(div)
results.append([angle * 180 / np.pi, max(divs), np.mean(divs)])
results = np.array(results)
np.savetxt(
util.make_filename(_divs, 'airfoil-divs-{piola}.csv', piola=piola),
results)
def test_poly(self):
target = self.geom.sum(-1) if self.btype == 'bubble' \
else (self.geom**self.degree).sum(-1) + self.domain.f_index if self.btype == 'discont' \
else (self.geom**self.degree).sum(-1)
projection = self.domain.projection(target, onto=self.basis, geometry=self.geom, ischeme='gauss', degree=2*self.degree, droptol=0)
error2 = self.domain.integrate((target-projection)**2*function.J(self.geom), ischeme='gauss', degree=2*self.degree)
numpy.testing.assert_almost_equal(error2, 0, decimal=24)
class Mass(MuCallable):
_ident_ = 'Mass'
def __init__(self, n, basisname, *deps, scale=1):
super().__init__((n, n), deps, scale=scale)
self.basisname = basisname
def write(self, group):
super().write(group)
util.to_dataset(self.basisname, group, 'basisname')
def _read(self, group):
super()._read(group)
self.basisname = util.from_dataset(group['basisname'])
def evaluate(self, case, mu, cont):
geom = case['geometry'](mu)
basis = case.basis(self.basisname, mu)
itg = fn.outer(basis)
if basis.ndim > 1:
itg = itg.sum([-1])
itg = util.contract(itg, cont)
with matrix.Scipy():
return unwrap(case.domain.integrate(itg * fn.J(geom), ischeme='gauss9'))
def setUp(self):
super().setUp()
domain, geom = mesh.rectilinear([numpy.linspace(0,1,9)] * 2)
gauss = domain.sample('gauss', 5)
uin = geom[1] * (1-geom[1])
dx = function.J(geom, 2)
ubasis = domain.basis('std', degree=2)
pbasis = domain.basis('std', degree=1)
if self.single:
ubasis, pbasis = function.chain([ubasis.vector(2), pbasis])
dofs = function.Argument('dofs', [len(ubasis)])
u = ubasis.dot(dofs)
p = pbasis.dot(dofs)
dofs = 'dofs'
ures = gauss.integral((self.viscosity * (ubasis.grad(geom) * (u.grad(geom) + u.grad(geom).T)).sum([-1,-2]) - ubasis.div(geom) * p) * dx)
dres = gauss.integral((ubasis * (u.grad(geom) * u).sum(-1)).sum(-1) * dx)
else:
u = (ubasis[:,numpy.newaxis] * function.Argument('dofs', [len(ubasis), 2])).sum(0)
p = (pbasis * function.Argument('pdofs', [len(pbasis)])).sum(0)
dofs = 'dofs', 'pdofs'
ures = gauss.integral((self.viscosity * (ubasis[:,numpy.newaxis].grad(geom) * (u.grad(geom) + u.grad(geom).T)).sum(-1) - ubasis.grad(geom) * p) * dx)
dres = gauss.integral(ubasis[:,numpy.newaxis] * (u.grad(geom) * u).sum(-1) * dx)
pres = gauss.integral((pbasis * u.div(geom)) * dx)
cons = solver.optimize('dofs', domain.boundary['top,bottom'].integral((u**2).sum(), degree=4), droptol=1e-10)
cons = solver.optimize('dofs', domain.boundary['left'].integral((u[0]-uin)**2 + u[1]**2, degree=4), droptol=1e-10, constrain=cons)
self.cons = cons if self.single else {'dofs': cons}
stokes = solver.solve_linear(dofs, residual=ures + pres if self.single else [ures, pres], constrain=self.cons)
self.arguments = dict(dofs=stokes) if self.single else stokes
self.residual = ures + dres + pres if self.single else [ures + dres, pres]
inertia = gauss.integral(.5 * (u**2).sum(-1) * dx).derivative('dofs')
self.inertia = inertia if self.single else [inertia, None]
self.tol = 1e-10
self.dofs = dofs
self.frozen = types.frozenarray if self.single else solver.argdict
def mk_lift(case):
mu = case.parameter()
vbasis = case.basis('v', mu)
geom = case['geometry'](mu)
x, __ = geom
cons = case.domain.boundary['left'].project((0, 0),
onto=vbasis,
geometry=geom,
ischeme='gauss1')
cons = case.domain.boundary['right'].select(-x).project(
(1, 0),
onto=vbasis,
geometry=geom,
ischeme='gauss9',
constrain=cons,
)
mx = case['laplacian'](mu) + case['divergence'](mu, sym=True)
lhs = matrix.ScipyMatrix(mx).solve(np.zeros(case.ndofs), constrain=cons)
vsol = vbasis.dot(lhs)
vdiv = vsol.div(geom)**2
vdiv = np.sqrt(case.domain.integrate(vdiv * fn.J(geom), ischeme='gauss9'))
log.user('Lift divergence (ref coord):', vdiv)
lhs[case.bases['p'].indices] = 0.0
return lhs
def test_nonlinear_diagonalshift(self):
nelems = 10
domain, geom = mesh.rectilinear([nelems, 1])
geom *= [2 * numpy.pi / nelems, 1]
ubasis = domain.basis('spline', degree=2).vector(2)
u = ubasis.dot(function.Argument('dofs', [len(ubasis)]))
Geom = [.5 * geom[0], geom[1] + function.cos(geom[0])
] + u # compress by 50% and buckle
cons = solver.minimize('dofs',
domain.boundary['left,right'].integral(
(u**2).sum(0), degree=4),
droptol=1e-15).solve()
strain = .5 * (function.outer(Geom.grad(geom), axis=1).sum(0) -
function.eye(2))
energy = domain.integral(
((strain**2).sum([0, 1]) + 150 *
(function.determinant(Geom.grad(geom)) - 1)**2) *
function.J(geom),
degree=6)
nshift = 0
for iiter, (lhs, info) in enumerate(
solver.minimize('dofs', energy, constrain=cons)):
self.assertLess(iiter, 38)
if info.shift:
nshift += 1
if info.resnorm < self.tol:
break
self.assertEqual(nshift, 9)
def integrate(*args):
domain, geom, ischeme = args[0]._domain, args[0]._geometry, args[0]._ischeme
assert all(arg._domain is domain for arg in args[1:])
assert all(arg._geometry is geom for arg in args[1:])
assert all(arg._ischeme == ischeme for arg in args[1:])
with MaybeScipyBackend():
retval = domain.integrate([arg._obj*fn.J(geom) for arg in args], ischeme=ischeme)
return [r.core if isinstance(r, matrix.Matrix) else r for r in retval]
def test_pum_sum(self):
# Note that this test holds for btype 'lagrange' as well, although the
# basis functions are not confined to range [0,1].
error2 = self.domain.integrate(
(1 - self.basis.sum(0))**2 * function.J(self.geom),
ischeme='gauss',
degree=2 * self.degree)
numpy.testing.assert_almost_equal(error2, 0, decimal=22)
def test_mass_matrix(mu, case):
p_vbasis, p_pbasis = piola_bases(mu, case)
trfgeom = case.geometry(mu)
itg = fn.outer(p_vbasis.grad(trfgeom)).sum([-1, -2])
phys_mx = case.domain.integrate(itg * fn.J(trfgeom), ischeme='gauss9')
test_mx = case['v-h1s'](mu)
np.testing.assert_almost_equal(phys_mx.export('dense'), test_mx.toarray())
def test_laplacian_matrix(mu, case):
vbasis, pbasis = case.bases['v'].obj, case.bases['p'].obj
trfgeom = case.geometry(mu)
itg = fn.outer(vbasis.grad(trfgeom)).sum([-1, -2])
phys_mx = case.domain.integrate(itg * fn.J(trfgeom), ischeme='gauss9')
test_mx = case['laplacian'](mu)
np.testing.assert_almost_equal(phys_mx.export('dense'), test_mx.toarray())
def test_divergence_matrix(mu, case):
vbasis, pbasis = case.bases['v'].obj, case.bases['p'].obj
trfgeom = case.geometry(mu)
itg = -fn.outer(vbasis.div(trfgeom), pbasis)
phys_mx = case.domain.integrate(itg * fn.J(trfgeom), ischeme='gauss9')
test_mx = case['divergence'](mu)
np.testing.assert_almost_equal(phys_mx.export('dense'), test_mx.toarray())
def test_nutils_tensor():
domain, geom = mesh.rectilinear([[0, 1], [0, 1]])
basis = domain.basis('spline', degree=1)
itg = basis[:, _, _] * basis[_, :, _] * basis[_, _, :]
a = AffineIntegral(1, itg)
a.prop(domain=domain, geometry=geom, ischeme='gauss9')
a = a.cache_main(force=True)(None, {})
b = domain.integrate(itg * fn.J(geom), ischeme='gauss9')
np.testing.assert_almost_equal(a, b)
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 project(self, projection):
obj = self.obj
s = slice(None)
for i, p in enumerate(projection):
if p is None:
continue
obj = obj[(s,)*i + (_,s,Ellipsis)]
obj = obj * p[(_,)*i + (s,s) + (_,) * (self.ndim - i - 1)]
obj = obj.sum(i+1)
domain, geom, ischeme = self.prop('domain', 'geometry', 'ischeme')
with matrix.Numpy():
retval = domain.integrate(obj * fn.J(geom), ischeme=ischeme)
return NumpyArrayIntegrand(retval)
def cache(self, force=False, **kwargs):
if self.ndim >= 3 and force:
return self._highdim_cache(**kwargs)
elif self.ndim >= 3:
# Store properties for later integration
self.prop(**kwargs)
return self
domain, geom, ischeme = self.prop('domain', 'geometry', 'ischeme', **kwargs)
with MaybeScipyBackend():
value = domain.integrate(self.obj * fn.J(geom), ischeme=ischeme)
if isinstance(value, matrix.Matrix):
value = value.core
return Integrand.make(value)
def test_convection(mu, case):
vbasis, pbasis = case.bases['v'].obj, case.bases['p'].obj
trfgeom = case.geometry(mu)
a, b, c = [np.random.rand(vbasis.shape[0]) for __ in range(3)]
u = vbasis.dot(b)
v = vbasis.dot(c).grad(trfgeom)
w = vbasis.dot(a)
itg = (w[:, _] * u[_, :] * v[:, :]).sum([-1, -2])
phys_conv = case.domain.integrate(itg * fn.J(trfgeom), ischeme='gauss9')
test_conv = affine.integrate(case['convection'](mu, cont=(a, b, c)))
np.testing.assert_almost_equal(phys_conv, test_conv)
def test_exact_stokes(e_exact, a_exact, mu):
mu = e_exact.parameter(*mu)
elhs = stokes(e_exact, mu)
alhs = stokes(a_exact, mu)
_check_exact(e_exact, mu, elhs, with_p=False) # TODO
# Size of physical geometry
pgeom = e_exact.geometry(mu)
size = e_exact.domain.integrate(fn.J(pgeom), ischeme='gauss9')
np.testing.assert_almost_equal(size, mu['w'] * mu['h'])
# Solenoidal in physical coordinates
pgeom = e_exact.geometry(mu)
vdiv = e_exact.basis('v', mu).dot(elhs + e_exact.lift(mu)).div(pgeom)
vdiv = np.sqrt(
e_exact.domain.integrate(vdiv**2 * fn.J(pgeom), ischeme='gauss9'))
np.testing.assert_almost_equal(0.0, vdiv)
pgeom = a_exact.geometry(mu)
vdiv = a_exact.basis('v', mu).dot(alhs + a_exact.lift(mu)).div(pgeom)
vdiv = np.sqrt(
a_exact.domain.integrate(vdiv**2 * fn.J(pgeom), ischeme='gauss9'))
np.testing.assert_almost_equal(0.0, vdiv)
# Solenoidal in reference coordinates
rgeom = e_exact.refgeom
vdiv = e_exact.basis('v').dot(elhs).div(rgeom)
vdiv = np.sqrt(
e_exact.domain.integrate(vdiv**2 * fn.J(rgeom), ischeme='gauss9'))
np.testing.assert_almost_equal(0.0, vdiv)
rgeom = a_exact.refgeom
vdiv = a_exact.basis('v').dot(alhs).div(rgeom)
vdiv = np.sqrt(
a_exact.domain.integrate(vdiv**2 * fn.J(rgeom), ischeme='gauss9'))
np.testing.assert_almost_equal(0.0, vdiv)
class NSDivergence(MuCallable):
_ident_ = 'NSDivergence'
def __init__(self, n, *deps, scale=1):
super().__init__((n, n), deps, scale=scale)
def evaluate(self, case, mu, cont):
geom = case['geometry'](mu)
vbasis = case.basis('v', mu)
pbasis = case.basis('p', mu)
itg = -fn.outer(vbasis.div(geom), pbasis)
itg = util.contract(itg, cont)
with matrix.Scipy():
return unwrap(case.domain.integrate(itg * fn.J(geom), ischeme='gauss9'))
class NSConvection(MuCallable):
_ident_ = 'NSConvection'
def __init__(self, n, *deps, scale=1):
super().__init__((n, n, n), deps, scale=scale)
def evaluate(self, case, mu, cont):
geom = case['geometry'](mu)
vbasis = case.basis('v', mu)
vgrad = vbasis.grad(geom)
itg = (vbasis[:,_,_,:,_] * vbasis[_,:,_,_,:] * vgrad[_,_,:,:,:]).sum([-1, -2])
itg = util.contract(itg, cont)
backend = matrix.Scipy if itg.ndim < 3 else COOTensorBackend
with backend():
return unwrap(case.domain.integrate(itg * fn.J(geom), ischeme='gauss9'))
def mu():
return {
'viscosity': 1.0,
'length': 10.0,
'height': 1.5,
'velocity': 1.0,
}
def test_divergence_matrix(case, mu):
domain, vbasis, pbasis = case.domain, case.bases['v'].obj, case.bases[
'p'].obj
geom = case.geometry(mu)
itg = -fn.outer(vbasis.div(geom), pbasis)
phys_mx = domain.integrate(itg * fn.J(geom),
ischeme='gauss9').export('dense')
test_mx = case['divergence'](mu).toarray()
np.testing.assert_almost_equal(phys_mx, test_mx)
def test_laplacian_matrix(case, mu):
domain, vbasis, pbasis = case.domain, case.bases['v'].obj, case.bases[
'p'].obj
geom = case.geometry(mu)
itg = fn.outer(vbasis.grad(geom)).sum([-1, -2]) / mu['viscosity']
phys_mx = domain.integrate(itg * fn.J(geom),
ischeme='gauss9').export('dense')
test_mx = case['laplacian'](mu).toarray()
def _highdim_cache(self, **kwargs):
domain, geom, ischeme = self.prop('domain', 'geometry', 'ischeme', **kwargs)
with COOTensorBackend():
value = domain.integrate(self.obj * fn.J(geom), ischeme=ischeme)
return value
import pytest
from aroma import cases
from aroma.solvers import stokes, navierstokes
def _check_exact(case, mu, lhs, with_p=True):
vsol = case.basis('v', mu).dot(lhs + case.lift(mu))
psol = case.basis('p', mu).dot(lhs + case.lift(mu))
vexc, pexc = case.exact(mu, ['v', 'p'])
vdiff = fn.norm2(vsol - vexc)
pdiff = (psol - pexc)**2
geom = case.geometry(mu)
verr, perr = case.domain.integrate(
[vdiff * fn.J(geom), pdiff * fn.J(geom)], ischeme='gauss9')
np.testing.assert_almost_equal(verr, 0.0)
if with_p:
np.testing.assert_almost_equal(perr, 0.0)
@pytest.fixture()
def cavity():
case = cases.cavity(nel=2)
case.precompute()
return case
@pytest.fixture()
def e_exact():
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 poisson_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 as fn
from nutils import _, log
# error test input
if left.rational or right.rational or top.rational or bottom.rational:
raise RuntimeError('poisson_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])
basis = domain.basis('spline', [p1 - 1, p2 - 1],
knotmultiplicities=[m1, m2])
# assemble system matrix
grad = basis.grad(geom)
outer = fn.outer(grad, grad)
integrand = outer.sum(-1)
matrix = domain.integrate(integrand * fn.J(geom),
ischeme='gauss' + str(max(p1, p2) + 1))
# initialize variables
controlpoints = np.zeros((n1, n2, dim))
rhs = np.zeros((n1 * n2))
constraints = np.array([[np.nan] * n2] * n1)
# treat all dimensions independently
for d in range(dim):
# add boundary conditions
constraints[0, :] = left[:, d]
constraints[-1, :] = right[:, d]
constraints[:, 0] = bottom[:, d]
constraints[:, -1] = top[:, d]
# solve system
lhs = matrix.solve(rhs, constrain=np.ndarray.flatten(constraints))
# wrap results into splipy datastructures
controlpoints[:, :, d] = np.reshape(lhs, (n1, n2), order='C')
return Surface(bottom.bases[0],
left.bases[0],
controlpoints,
bottom.rational,
raw=True)
def main(nrefine: int, traction: float, radius: float, poisson: float):
'''
Horizontally loaded linear elastic plate with IGA hole.
.. arguments::
nrefine [2]
Number of uniform refinements starting from 1x2 base mesh.
traction [.1]
Far field traction (relative to Young's modulus).
radius [.5]
Cut-out radius.
poisson [.3]
Poisson's ratio, nonnegative and strictly smaller than 1/2.
'''
# create the coarsest level parameter domain
domain, geom0 = mesh.rectilinear([1, 2])
bsplinebasis = domain.basis('spline', degree=2)
controlweights = numpy.ones(12)
controlweights[1:3] = .5 + .25 * numpy.sqrt(2)
weightfunc = bsplinebasis.dot(controlweights)
nurbsbasis = bsplinebasis * controlweights / weightfunc
# create geometry function
indices = [0, 2], [1, 2], [2, 1], [2, 0]
controlpoints = numpy.concatenate([
numpy.take([0, 2**.5 - 1, 1], indices) * radius,
numpy.take([0, .3, 1], indices) * (radius + 1) / 2,
numpy.take([0, 1, 1], indices)
])
geom = (nurbsbasis[:, numpy.newaxis] * controlpoints).sum(0)
radiuserr = domain.boundary['left'].integral(
(function.norm2(geom) - radius)**2 * function.J(geom0),
degree=9).eval()**.5
treelog.info('hole radius exact up to L2 error {:.2e}'.format(radiuserr))
# refine domain
if nrefine:
domain = domain.refine(nrefine)
bsplinebasis = domain.basis('spline', degree=2)
controlweights = domain.project(weightfunc,
onto=bsplinebasis,
geometry=geom0,
ischeme='gauss9')
nurbsbasis = bsplinebasis * controlweights / weightfunc
ns = function.Namespace()
ns.x = geom
ns.lmbda = 2 * poisson
ns.mu = 1 - poisson
ns.ubasis = nurbsbasis.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['top,bottom'].integral('(u_i n_i)^2 J(x)' @ ns,
degree=9)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
sqr = domain.boundary['right'].integral('du_k du_k J(x)' @ ns, degree=20)
cons = solver.optimize('lhs', sqr, droptol=1e-15, constrain=cons)
# construct residual
res = domain.integral('d(ubasis_ni, x_j) stress_ij J(x)' @ ns, degree=9)
# solve system
lhs = solver.solve_linear('lhs', res, constrain=cons)
# vizualize result
bezier = domain.sample('bezier', 9)
X, stressxx = bezier.eval(['X', 'stress_00'] @ ns, lhs=lhs)
export.triplot('stressxx.png',
X,
stressxx,
tri=bezier.tri,
hull=bezier.hull,
clim=(numpy.nanmin(stressxx), numpy.nanmax(stressxx)))
# evaluate error
err = domain.integral('<du_k du_k, sum:ij(d(du_i, x_j)^2)>_n J(x)' @ ns,
degree=9).eval(lhs=lhs)**.5
treelog.user('errors: L2={:.2e}, H1={:.2e}'.format(*err))
return err, cons, lhs
pressure = pressure[:npts]
linbasis = case.domain.basis('spline', degree=1)
vsol = linbasis.dot(velocity[:,0])[_] * (1,0) + linbasis.dot(velocity[:,1])[_] * (0,1)
psol = linbasis.dot(pressure)
geom = case.physical_geometry(param)
vbasis = case.basis('v', param)
vgrad = vbasis.grad(geom)
pbasis = case.basis('p', param)
lhs = case.domain.project(psol, onto=pbasis, geometry=geom, ischeme='gauss9')
vind = case.basis_indices('v')
itg = fn.outer(vbasis).sum([-1]) + fn.outer(vgrad).sum([-1,-2])
with matrix.Scipy():
mx = case.domain.integrate(itg * fn.J(geom), ischeme='gauss9').core[np.ix_(vind,vind)]
itg = (vbasis * vsol[_,:]).sum([-1]) + (vgrad * vsol.grad(geom)[_,:,:]).sum([-1,-2])
rhs = case.domain.integrate(itg * fn.J(geom), ischeme='gauss9')[vind]
lhs[vind] = matrix.ScipyMatrix(mx).solve(rhs)
lift = case._lift(param)
solutions.append((lhs - lift) * weight)
solutions = np.array(solutions)
supremizers = ens.make_ensemble(
case, solvers.supremizer, scheme, weights=False, parallel=False, args=[solutions],
)
return scheme, solutions, supremizers
@util.pickle_cache('airfoil-{piola}-{imported}-{nred}.rcase')