def SpyDG(N=8, order=1):
mesh1D = Mesh1D(N)
fes = Discontinuous(H1(mesh1D, order=order)) #, dirichlet=)
u, v = fes.TnT()
a = BilinearForm(fes)
a += SymbolicBFI(grad(u) * grad(v))
a.Assemble()
rows, cols, vals = a.mat.COO()
A = sp.csr_matrix((vals, (rows, cols)))
plt.figure(figsize=(7, 7))
plt.spy(A)
plt.show()
def makeforms(mesh, p, F, q_zero, mu_zero, cwave, epsil=0):
d = mesh.dim
W = L2(mesh, order=p + d)
U = L2(mesh, order=p)
Zq = H1(mesh, order=p + 1, dirichlet=q_zero, orderinner=0)
Zmu = H1(mesh, order=p + 1, dirichlet=mu_zero, orderinner=0)
spacelist = [W] * d + [U] * d + [Zq] * (d - 1) + [Zmu]
X = FESpace(spacelist)
# separate W...W, U...U, Zq...Zq, Zmu
separators = [d, 2 * d, 2 * d + (d - 1)]
Xtrials = X.TrialFunction()
e = Xtrials[0:separators[0]]
u = Xtrials[separators[0]:separators[1]]
zu = Xtrials[separators[1]:]
Xtests = X.TestFunction()
w = Xtests[0:separators[0]]
v = Xtests[separators[0]:separators[1]]
zv = Xtests[separators[1]:]
n = specialcf.normal(d)
a = BilinearForm(X, symmetric=True, eliminate_internal=True)
a += SymbolicBFI(vec(e) * vec(w))
a += SymbolicBFI(waveA(e, cwave) * waveA(w, cwave))
a += SymbolicBFI(-vec(u) * waveA(w, cwave))
a += SymbolicBFI(-waveA(e, cwave) * vec(v))
a += SymbolicBFI(vec(e) * waveD(n, zv, cwave), element_boundary=True)
a += SymbolicBFI(waveD(n, zu, cwave) * vec(w), element_boundary=True)
a += SymbolicBFI(-epsil * vec(zu) * vec(zv))
f = LinearForm(X)
f += SymbolicLFI(F * vec(w))
return (a, f, X, separators)
def discretize_ngsolve():
from ngsolve import (ngsglobals, Mesh, H1, CoefficientFunction, LinearForm,
SymbolicLFI, BilinearForm, SymbolicBFI, grad,
TaskManager)
from netgen.csg import CSGeometry, OrthoBrick, Pnt
import numpy as np
ngsglobals.msg_level = 1
geo = CSGeometry()
obox = OrthoBrick(Pnt(-1, -1, -1), Pnt(1, 1, 1)).bc("outer")
b = []
b.append(
OrthoBrick(Pnt(-1, -1, -1), Pnt(0.0, 0.0,
0.0)).mat("mat1").bc("inner"))
b.append(
OrthoBrick(Pnt(-1, 0, -1), Pnt(0.0, 1.0, 0.0)).mat("mat2").bc("inner"))
b.append(
OrthoBrick(Pnt(0, -1, -1), Pnt(1.0, 0.0, 0.0)).mat("mat3").bc("inner"))
b.append(
OrthoBrick(Pnt(0, 0, -1), Pnt(1.0, 1.0, 0.0)).mat("mat4").bc("inner"))
b.append(
OrthoBrick(Pnt(-1, -1, 0), Pnt(0.0, 0.0, 1.0)).mat("mat5").bc("inner"))
b.append(
OrthoBrick(Pnt(-1, 0, 0), Pnt(0.0, 1.0, 1.0)).mat("mat6").bc("inner"))
b.append(
OrthoBrick(Pnt(0, -1, 0), Pnt(1.0, 0.0, 1.0)).mat("mat7").bc("inner"))
b.append(
OrthoBrick(Pnt(0, 0, 0), Pnt(1.0, 1.0, 1.0)).mat("mat8").bc("inner"))
box = (obox - b[0] - b[1] - b[2] - b[3] - b[4] - b[5] - b[6] - b[7])
geo.Add(box)
for bi in b:
geo.Add(bi)
# domain 0 is empty!
mesh = Mesh(geo.GenerateMesh(maxh=0.3))
# H1-conforming finite element space
V = H1(mesh, order=NGS_ORDER, dirichlet="outer")
v = V.TestFunction()
u = V.TrialFunction()
# Coeff as array: variable coefficient function (one CoefFct. per domain):
sourcefct = CoefficientFunction([1 for i in range(9)])
with TaskManager():
# the right hand side
f = LinearForm(V)
f += SymbolicLFI(sourcefct * v)
f.Assemble()
# the bilinear-form
mats = []
coeffs = [[0, 1, 0, 0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0, 0]]
for c in coeffs:
diffusion = CoefficientFunction(c)
a = BilinearForm(V, symmetric=False)
a += SymbolicBFI(diffusion * grad(u) * grad(v),
definedon=(np.where(np.array(c) == 1)[0] +
1).tolist())
a.Assemble()
mats.append(a.mat)
from pymor.bindings.ngsolve import NGSolveVectorSpace, NGSolveMatrixOperator, NGSolveVisualizer
space = NGSolveVectorSpace(V)
op = LincombOperator(
[NGSolveMatrixOperator(m, space, space) for m in mats], [
ProjectionParameterFunctional('diffusion', (len(coeffs), ), (i, ))
for i in range(len(coeffs))
])
h1_0_op = op.assemble([1] * len(coeffs)).with_(name='h1_0_semi')
F = space.zeros()
F._list[0].real_part.impl.vec.data = f.vec
F = VectorOperator(F)
return StationaryModel(op,
F,
visualizer=NGSolveVisualizer(mesh, V),
products={'h1_0_semi': h1_0_op},
parameter_space=CubicParameterSpace(
op.parameter_type, 0.1, 1.))
def Heat1DFEM(N=8,
order=1,
k1=1,
k2=1,
Q1=0,
Q2=10,
boundary_condition_left="Robin",
boundary_condition_right="Dirichlet",
value_left=0,
value_right=1,
q_value_left=0,
q_value_right=1,
r_value_left=0,
r_value_right=1,
intervalsize=0.14):
if (boundary_condition_left == "Neumann" or
(boundary_condition_left == "Robin" and r_value_left == 0)) and (
boundary_condition_right == "Neumann" or
(boundary_condition_right == "Robin" and r_value_right == 0)):
print("Temperatur ist nicht eindeutig bestimmt.")
#return
mesh1D = Mesh1D(N, interval=(0, intervalsize))
dbnds = []
if boundary_condition_left == "Dirichlet":
dbnds.append(1)
# print(True)
if boundary_condition_right == "Dirichlet":
dbnds.append(2)
fes = H1(mesh1D, order=order, dirichlet=dbnds)
gf = GridFunction(fes)
if boundary_condition_left == "Dirichlet":
gf.vec[0] = value_left
if boundary_condition_right == "Dirichlet":
gf.vec[N] = value_right
Q = IfPos(X - 0.5 * intervalsize, Q2, Q1)
k = IfPos(X - 0.5 * intervalsize, k2, k1)
u, v = fes.TnT()
a = BilinearForm(fes)
a += SymbolicBFI(k * grad(u) * grad(v))
if boundary_condition_left == "Robin":
a += SymbolicBFI(r_value_left * u * v,
definedon=mesh1D.Boundaries("left"))
if boundary_condition_right == "Robin":
a += SymbolicBFI(r_value_right * u * v,
definedon=mesh1D.Boundaries("right"))
a.Assemble()
f = LinearForm(fes)
f += SymbolicLFI(Q * v)
f.Assemble()
if boundary_condition_left == "Neumann":
f.vec[0] += q_value_left
elif boundary_condition_left == "Robin":
f.vec[0] += r_value_left * value_left
if boundary_condition_right == "Neumann":
f.vec[N] += q_value_right
elif boundary_condition_right == "Robin":
f.vec[N] += r_value_right * value_right
f.vec.data -= a.mat * gf.vec
gf.vec.data += a.mat.Inverse(fes.FreeDofs()) * f.vec
Draw1D(mesh1D, [(gf, "u_h")], n_p=5 * order**2)