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periodicmaxwell.py
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periodicmaxwell.py
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""" Periodic Maxwell system, solved using libDPG:
TEST PROBLEM DETAILS:
We want to solve the boundary value problem
curl ( curl E ) - k * k * E = f, on Omega = [0, 1]^3,
with boundary condition
n x ( curl E ) + i * k * n x (E x n) = g, on {z=0} and {z=1},
and periodic b.c. elsewhere. (Here we are using the -i*om*t
time-harmonic convention.) Introducing the flux interface variable
M = -i * curl(E), the DPG variational formulation reads
Y(e;v) + b(E,M; v) = L(v), for all v
conj(b(F,W; e)) + c(E,M ; F,W) = G(F,W), for all F, W
where,
Y(e;v) = (curl e, curl v) + (e,v)
b(E,M; v) = (curl E, curl v) - (k*k*E,v) + i*<<n x M, v>>
c(E,M; F,W) = - <n x M + k * n x (E x n), n x W + k * n x (F x n)>
L(v) = (f,v)
G(F,W) = - <g, i * (n x W + k * n x (F x n)) >
= <i * g, n x W + k * n x (F x n) >.
To measure errors, we consider the case with the contrived exact solution:
E = [ z ]
[ z * z ]
[ z * z * z ]
{z=1} boundary, bc=3
_____________________
Omega = [0, 1]^3 / /|
z /____________________/ |
^ | | |
| | material 2 | | all
| | | | periodic
| | wavenumber k2 |/| boundaries,
|------>y {z=0.5}, |____________________/ | bc=1
/ bc=4 | | |
/ | material 1 | |
x | | |
| wavenumber k1 |/
|____________________/
{z=0} boundary, bc=2
"""
from ngsolve import *
from ctypes import CDLL
libDPG = CDLL("../libDPG.so")
mesh = Mesh("../pde/periodiclayers.vol.gz")
symbolic = False # If True, use NGSolve's symbolic forms.
# Else, use libDPG's precompiled forms.
# (The solution should be the same in either case.)
k1 = 1.0 # bot material
k2 = 10.0 # top material
k = CoefficientFunction( [k1, k2] )
bdry = CoefficientFunction( [0.0, 1.0, 1.0, 0.0] )
kbdry = CoefficientFunction( [0.0, k1, k2, 0.0] )
f = CoefficientFunction( (-k*k*z, -2-k*k*z*z, -k*k*z*z*z) )
Eex= CoefficientFunction( ( z, z*z, z*z*z ) )
igxn = CoefficientFunction([( 0j , 0j , 0j ),
( 0 , 1j , 0 ),
( -2j - k2 , 1j + k2 , 0 ),
( 0 , 0 , 0 )])
ikbarg=CoefficientFunction([(0 , 0 , 0),
( 1j * k1 , 0 , 0),
(-k2 * (1j+k2), -k2*(2*1j+k2) , 0),
(0 , 0 , 0)])
p = 3
S0 = FESpace("hcurlho", mesh, order=p+3, complex=True,
discontinuous=True)
S1 = FESpace("hcurlho_periodic", mesh, order=p, complex=True,
xends=[0,1], yends=[0,1] )
S2 = FESpace("hcurlho_periodic", mesh, order=p+1, complex=True,
orderinner=0, xends=[0,1], yends=[0,1])
S = FESpace( [S0,S1,S2], complex=True)
e,E,M = S.TrialFunction()
v,F,W = S.TestFunction()
b = LinearForm(S)
a = BilinearForm(S, symmetric=False, eliminate_internal=True)
if symbolic:
b+= SymbolicLFI( f * v ) # (f,v)
b+= SymbolicLFI(ikbarg * F.Trace(), BND) # <i*kbar* n x (g x n), F>
b+= SymbolicLFI(igxn * W.Trace(),BND) # <i * g x n, W>
def cross(G,N): # G x N
return CoefficientFunction( ( G[1]*N[2] - G[2]*N[1],
G[2]*N[0] - G[0]*N[2],
G[0]*N[1] - G[1]*N[0] ) )
def bvol( EE, vv):
return ( curl(EE) * curl(vv) - k*k* EE * vv )
n = specialcf.normal(mesh.dim)
a+= SymbolicBFI( bvol(E,v) + # (curl E, curl v) - k*k(E,v)
Conj(bvol(e,F)) ) # + c.c.
a+= SymbolicBFI( 1j * M * cross(v,n) - 1j * cross(e,n) * W,
element_boundary=True) # i<<M, v x n>>
a+= SymbolicBFI(-kbdry * Conj(kbdry) * E.Trace() *
F.Trace(), BND) # -<k*kbar E x n, F x n>
a+= SymbolicBFI(-bdry*M.Trace()*W.Trace(), # -<M x n, W x n>
BND)
a+= SymbolicBFI(kbdry * E.Trace() * cross(W.Trace(), n) +
Conj(kbdry) * cross( M.Trace(),n) * F.Trace(),
BND) # <k E, W x n> + c.c.
a+= SymbolicBFI(e*v + curl(e)*curl(v))
else:
b.components[0] += LFI("sourceedge", coef=f)
b.components[2] += LFI("neumannedge", coef=igxn )
b.components[1] += LFI("neumannedge", coef=ikbarg)
a+= BFI("curlcurlpg", coef=[2,1,1]) # (curl E, curl v)
a+= BFI("eyeeyeedge", coef=[2,1,-k*k]) # -(k*k E, v)
a+= BFI("trctrcxn", coef=[3,1,1j] ) # i<<M, v x n>>
a+= BFI("xnbdry", coef=[2,3,kbdry]) # <k E, W x n>
a.components[1]+=BFI("robinedge", coef=-kbdry * Conj(kbdry))
a.components[2]+=BFI("robinedge", coef=-bdry)
a.components[0]+= BFI("massedge", coef=1.0)
a.components[0]+= BFI("curlcurledge", coef=1.0)
b.Assemble()
eEM = GridFunction(S)
c = Preconditioner(a, type="direct")
SetHeapSize(int(5e8))
a.Assemble()
bvp = BVP(bf=a, lf=b, gf=eEM, pre=c).Do()
Eh = eEM.components[1]
Draw(Eh)
print("Error = ", sqrt(Integrate( (Eh - Eex) * Conj(Eh - Eex), mesh)))