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SimplePatch2.py
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SimplePatch2.py
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import sys
import numpy as np
from scipy import optimize as opt
import gmsh
from mpi4py import MPI as nMPI
import meshio
import dolfin
# We set up the mesh generation and finite element solution as prt of the
# objective function Cost(x)
def Cost(xp):
comm = nMPI.COMM_WORLD
mpi_rank = comm.Get_rank()
x1, x2 = xp #The two variables (length and feed offset)
rs = 8.0 # radiation boundary radius
l = x1 # Patch length
w = 4.5 # Patch width
s1 = x2 * x1 / 2.0 # Feed offset
h = 1.0 # Patch height
t = 0.05 # Metal thickness
lc = 1.0 # Coax length
rc = 0.25 # Coax shield radius
cc = 0.107 #Coax center conductor 50 ohm air diel
eps = 1.0e-4
tol = 1.0e-6
eta = 377.0 # vacuum intrinsic wave impedance
eps_c = 1.0 # dielectric permittivity
k0 = 2.45 * 2.0 * np.pi / 30.0 # Frequency in GHz
ls = 0.025 #Mesh density parameters for GMSH
lm = 0.8
lw = 0.06
lp = 0.3
# Run GMSH only on one MPI processor (process 0).
# We use the GMSH Python interface to generate the geometry and mesh objects
if mpi_rank == 0:
print("x[0] = {0:<f}, x[1] = {1:<f} ".format(xp[0], xp[1]))
print("length = {0:<f}, width = {1:<f}, feed offset = {2:<f}".format(l, w, s1))
gmsh.initialize()
gmsh.option.setNumber('General.Terminal', 1)
gmsh.model.add("SimplePatchOpt")
# Radiation sphere
gmsh.model.occ.addSphere(0.0, 0.0, 0.0, rs, 1)
gmsh.model.occ.addBox(0.0, -rs, 0.0, rs, 2*rs, rs, 2)
gmsh.model.occ.intersect([(3,1)],[(3,2)], 3, removeObject=True, removeTool=True)
# Patch
gmsh.model.occ.addBox(0.0, -l/2, h, w/2, l, t, 4)
# coax center
gmsh.model.occ.addCylinder(0.0, s1, -lc, 0.0, 0.0, lc+h, cc, 5, 2.0*np.pi)
# coax shield
gmsh.model.occ.addCylinder(0.0, s1, -lc, 0.0, 0.0, lc, rc, 7)
gmsh.model.occ.addBox(0.0, s1-rc, -lc, rc, 2.0*rc, lc, 8)
gmsh.model.occ.intersect([(3,7)], [(3,8)], 9, removeObject=True, removeTool=True)
gmsh.model.occ.fuse([(3,3)], [(3,9)], 10, removeObject=True, removeTool=True)
# cutout internal boundaries
gmsh.model.occ.cut([(3,10)], [(3,4),(3,5)], 11, removeObject=True, removeTool=True)
gmsh.option.setNumber('Mesh.MeshSizeMin', ls)
gmsh.option.setNumber('Mesh.MeshSizeMax', lm)
gmsh.option.setNumber('Mesh.Algorithm', 6)
gmsh.option.setNumber('Mesh.Algorithm3D', 1)
gmsh.option.setNumber('Mesh.MshFileVersion', 4.1)
gmsh.option.setNumber('Mesh.Format', 1)
gmsh.option.setNumber('Mesh.MinimumCirclePoints', 36)
gmsh.option.setNumber('Mesh.CharacteristicLengthFromCurvature', 1)
gmsh.model.occ.synchronize()
pts = gmsh.model.getEntities(0)
gmsh.model.mesh.setSize(pts, lm) #Set background mesh density
pts = gmsh.model.getEntitiesInBoundingBox(-eps, -l/2-eps, h-eps, w/2+eps, l/2+eps, h+t+eps)
gmsh.model.mesh.setSize(pts, ls)
pts = gmsh.model.getEntitiesInBoundingBox(-eps, s1-rc-eps, -lc-eps, rc+eps, s1+rc+eps, h+eps)
gmsh.model.mesh.setSize(pts, lw)
pts = gmsh.model.getEntitiesInBoundingBox(-eps, -rc-eps, -eps, rc+eps, rc+eps, eps)
gmsh.model.mesh.setSize(pts, lw)
# Embed points to reduce mesh density on patch faces
fce1 = gmsh.model.getEntitiesInBoundingBox(-eps, -l/2-eps, h+t-eps, w/2+eps, l/2+eps, h+t+eps, 2)
gmsh.model.occ.synchronize()
gmsh.model.geo.addPoint(w/4, -l/4, h+t, lp, 1000)
gmsh.model.geo.addPoint(w/4, 0.0, h+t, lp, 1001)
gmsh.model.geo.addPoint(w/4, l/4, h+t, lp, 1002)
gmsh.model.geo.synchronize()
gmsh.model.occ.synchronize()
print(fce1)
fce2 = gmsh.model.getEntitiesInBoundingBox(-eps, -l/2-eps, h-eps, w/2+eps, l/2+eps, h+eps, 2)
gmsh.model.geo.addPoint(w/4, -9*l/32, h, lp, 1003)
gmsh.model.geo.addPoint(w/4, 0.0, h, lp, 1004)
gmsh.model.geo.addPoint(w/4, 9*l/32, h, lp, 1005)
gmsh.model.geo.synchronize()
for tt in fce1:
gmsh.model.mesh.embed(0, [1000, 1001, 1002], 2, tt[1])
for tt in fce2:
gmsh.model.mesh.embed(0, [1003, 1004, 1005], 2, tt[1])
print(fce2)
gmsh.model.occ.remove(fce1)
gmsh.model.occ.remove(fce2)
gmsh.model.occ.synchronize()
gmsh.model.addPhysicalGroup(3, [11], 1)
gmsh.model.setPhysicalName(3, 1, "Air")
gmsh.model.mesh.optimize("Relocate3D", niter=5)
gmsh.model.mesh.generate(3)
gmsh.write("SimplePatch.msh")
gmsh.finalize()
# Mesh generation is finished. We now use Meshio to translate GMSH mesh to xdmf file for
# importation into Fenics FE solver
msh = meshio.read("SimplePatch.msh")
for cell in msh.cells:
if cell.type == "tetra":
tetra_cells = cell.data
for key in msh.cell_data_dict["gmsh:physical"].keys():
if key == "tetra":
tetra_data = msh.cell_data_dict["gmsh:physical"][key]
tetra_mesh = meshio.Mesh(points=msh.points, cells={"tetra": tetra_cells},
cell_data={"VolumeRegions":[tetra_data]})
meshio.write("mesh.xdmf", tetra_mesh)
# Here we import the mesh into Fenics
mesh = dolfin.Mesh()
with dolfin.XDMFFile("mesh.xdmf") as infile:
infile.read(mesh)
mvc = dolfin.MeshValueCollection("size_t", mesh, 3)
with dolfin.XDMFFile("mesh.xdmf") as infile:
infile.read(mvc, "VolumeRegions")
cf = dolfin.cpp.mesh.MeshFunctionSizet(mesh, mvc)
# The boundary classes for the FE solver
class PEC(dolfin.SubDomain):
def inside(self, x, on_boundary):
return on_boundary
class InputBC(dolfin.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and dolfin.near(x[2], -lc, tol)
class OutputBC(dolfin.SubDomain):
def inside(self, x, on_boundary):
rr = np.sqrt(x[0]*x[0]+x[1]*x[1]+x[2]*x[2])
return on_boundary and dolfin.near(rr, 8.0, 1.0e-1)
class PMC(dolfin.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and dolfin.near(x[0], 0.0, tol)
# Volume domains
dolfin.File("VolSubDomains.pvd").write(cf)
dolfin.File("Mesh.pvd").write(mesh)
# Mark boundaries
sub_domains = dolfin.MeshFunction("size_t", mesh, mesh.topology().dim() - 1)
sub_domains.set_all(4)
pec = PEC()
pec.mark(sub_domains, 0)
in_port = InputBC()
in_port.mark(sub_domains, 1)
out_port = OutputBC()
out_port.mark(sub_domains, 2)
pmc = PMC()
pmc.mark(sub_domains, 3)
dolfin.File("BoxSubDomains.pvd").write(sub_domains)
# Set up function spaces
cell = dolfin.tetrahedron
ele_type = dolfin.FiniteElement('N1curl', cell, 2, variant="integral") # H(curl) element for EM
V2 = dolfin.FunctionSpace(mesh, ele_type * ele_type)
V = dolfin.FunctionSpace(mesh, ele_type)
(u_r, u_i) = dolfin.TrialFunctions(V2)
(v_r, v_i) = dolfin.TestFunctions(V2)
dolfin.info(mesh)
#surface integral definitions from boundaries
ds = dolfin.Measure('ds', domain = mesh, subdomain_data = sub_domains)
#volume regions
dx_air = dolfin.Measure('dx', domain = mesh, subdomain_data = cf, subdomain_id = 1)
dx_subst = dolfin.Measure('dx', domain = mesh, subdomain_data = cf, subdomain_id = 2)
# with source and sink terms
u0 = dolfin.Constant((0.0, 0.0, 0.0)) #PEC definition
# The incident field sources (E and H-fields)
h_src = dolfin.Expression(('-(x[1] - s) / (2.0 * pi * (pow(x[0], 2.0) + pow(x[1] - s,2.0)))', 'x[0] / (2.0 * pi *(pow(x[0],2.0) + pow(x[1] - s,2.0)))', 0.0), degree = 2, s = s1)
e_src = dolfin.Expression(('x[0] / (2.0 * pi * (pow(x[0], 2.0) + pow(x[1] - s,2.0)))', 'x[1] / (2.0 * pi *(pow(x[0],2.0) + pow(x[1] - s,2.0)))', 0.0), degree = 2, s = s1)
Rrad = dolfin.Expression(('sqrt(x[0] * x[0] + x[1] * x[1] + x[2] * x[2])'), degree = 2)
#Boundary condition dictionary
boundary_conditions = {0: {'PEC' : u0},
1: {'InputBC': (h_src)},
2: {'OutputBC': Rrad}}
n = dolfin.FacetNormal(mesh)
#Build PEC boundary conditions for real and imaginary parts
bcs = []
for i in boundary_conditions:
if 'PEC' in boundary_conditions[i]:
bc = dolfin.DirichletBC(V2.sub(0), boundary_conditions[i]['PEC'], sub_domains, i)
bcs.append(bc)
bc = dolfin.DirichletBC(V2.sub(1), boundary_conditions[i]['PEC'], sub_domains, i)
bcs.append(bc)
# Build input BC source term and loading term
integral_source = []
integrals_load =[]
for i in boundary_conditions:
if 'InputBC' in boundary_conditions[i]:
r = boundary_conditions[i]['InputBC']
bb1 = 2.0 * (k0 * eta) * dolfin.inner(v_i, dolfin.cross(n, r)) * ds(i) #Factor of two from field equivalence principle
integral_source.append(bb1)
bb2 = dolfin.inner(dolfin.cross(n, v_i), dolfin.cross(n, u_r)) * k0 * np.sqrt(eps_c) * ds(i)
integrals_load.append(bb2)
bb2 = dolfin.inner(-dolfin.cross(n, v_r), dolfin.cross(n, u_i)) * k0 * np.sqrt(eps_c) * ds(i)
integrals_load.append(bb2)
for i in boundary_conditions:
if 'OutputBC' in boundary_conditions[i]:
r = boundary_conditions[i]['OutputBC']
bb2 = (dolfin.inner(dolfin.cross(n, v_i), dolfin.cross(n, u_r)) * k0 + 1.0 * dolfin.inner(dolfin.cross(n, v_i), dolfin.cross(n, u_i)) / r)* ds(i)
integrals_load.append(bb2)
bb2 = (dolfin.inner(-dolfin.cross(n, v_r), dolfin.cross(n, u_i)) * k0 + 1.0 * dolfin.inner(dolfin.cross(n, v_r), dolfin.cross(n, u_r)) / r)* ds(i)
integrals_load.append(bb2)
# for PMC, do nothing. Natural BC.
a = (dolfin.inner(dolfin.curl(v_r), dolfin.curl(u_r)) + dolfin.inner(dolfin.curl(v_i), dolfin.curl(u_i)) - eps_c * k0 * k0 * (dolfin.inner(v_r, u_r) + dolfin.inner(v_i, u_i))) * dx_subst + (dolfin.inner(dolfin.curl(v_r), dolfin.curl(u_r)) + dolfin.inner(dolfin.curl(v_i), dolfin.curl(u_i)) - k0 * k0 * (dolfin.inner(v_r, u_r) + dolfin.inner(v_i, u_i))) * dx_air + sum(integrals_load)
L = sum(integral_source)
u1 = dolfin.Function(V2)
vdim = u1.vector().size()
print("Solution vector size =", vdim)
dolfin.solve(a == L, u1, bcs, solver_parameters = {'linear_solver' : 'mumps'})
#Here we write files of the field solution for inspection
u1_r, u1_i = u1.split(True)
fp = dolfin.File("EField_r.pvd")
fp << u1_r
fp = dolfin.File("EField_i.pvd")
fp << u1_i
# Compute power relationships and reflection coefficient
H = dolfin.interpolate(h_src, V) # Get input field
P = dolfin.assemble((-dolfin.dot(u1_r,dolfin.cross(dolfin.curl(u1_i),n))+dolfin.dot(u1_i,dolfin.cross(dolfin.curl(u1_r),n))) * ds(2))
P_refl = dolfin.assemble((-dolfin.dot(u1_i,dolfin.cross(dolfin.curl(u1_r), n)) + dolfin.dot(u1_r, dolfin.cross(dolfin.curl(u1_i), n))) * ds(1))
P_inc = dolfin.assemble((dolfin.dot(H, H) * eta / (2.0 * np.sqrt(eps_c))) * ds(1))
print("Integrated power on port 2:", P/(2.0 * k0 * eta))
print("Incident power at port 1:", P_inc)
print("Integrated reflected power on port 1:", P_inc - P_refl / (2.0 * k0 * eta))
#Reflection coefficient is returned as cost function
rho_old = (P_inc - P_refl / (2.0 * k0 * eta)) / P_inc #Fraction of incident power reflected as objective function
return rho_old
#Optimization
x0 = np.array([5.0, 0.675]) # Starting point for optimization
res = opt.minimize(Cost, x0, method='Nelder-Mead', options={'maxiter':100, 'disp':True, 'fatol':0.003})
print(res)
sys.exit(0)