bc_lines = elements[1] # Boundary condition is (bc) lines
triangles = elements[2]
potential = potential_2d("well", nodes, v0=0)
write_solver_input(
glo_tag + ".msh",
parameter=potential,
dimension=2,
bc_type="Dir",
sol_type="Stationary",
eq="Schro",
analysis_param=["y", "y", 20, 20, 15, 15, 1],
bc_filename=glo_tag + ".bc",
)
k = read_solver_input(glo_tag + ".msh")
v, solution = schroedinger(glo_tag + ".msh")
triangles = substract_1(triangles)
triangles = triangles[:, 1:]
# k, solution = schroedinger(glo_tag +'.msh')
# z = zeros((k[0].shape[0], 3))
# print k[0].shape, z[:, 0:1].shape
# z[:, 0:2] = k[0]
# nodes = z
# triangles = k[1]
print v
write_vtk(glo_tag + "_dir.vtk", "this shit", "", nodes, triangles, ["SCALARS", ["solution"], [solution]])
print solution.shape
sol = test_sol(nodes, pi, 3 * pi)
n = shape(nodes)[0] #Number of nodes
bc_lines = elements[1] # Boundary condition is (bc) lines
triangles = elements[2]
potential = potential_2d('well', nodes, v0 = 2)
write_solver_input(glo_tag +'.msh', parameter = potential, dimension = 2, \
bc_type = 'Dir', \
sol_type = 'Stationary', eq = 'Schro', \
analysis_param = ['y', 'y', 4, 20, 15, 15, 1], \
bc_filename = glo_tag+'.bc')
k = read_solver_input(glo_tag +'.msh')
v, solution = schroedinger(glo_tag +'.msh')
triangles = substract_1(triangles)
triangles = triangles[:, 1:]
#k, solution = schroedinger(glo_tag +'.msh')
#z = zeros((k[0].shape[0], 3))
#print k[0].shape, z[:, 0:1].shape
#z[:, 0:2] = k[0]
#nodes = z
#triangles = k[1]
print v
write_vtk(glo_tag + '_1.vtk', 'this shit', '', nodes, triangles, \
['SCALARS', ['solution'], [solution]])
def schroedinger(filename, nodes = 0, elements = 0, parameter = [], \
dimension = 1, bc_type = 'Dir', sol_type = 'Stationary', \
eq = 'Schro', analysis_param = ['y', 'y', 4, 4, 20, 20, 2], \
bc_filename = ''):
"""
Function Schroedinger computes the solution for Schroedinger
equation using the Finite Element Method.
For now it is able to solve the Stationary form with Dirichlet and
Bloch boundary conditions in 1D.
The function is made to be used either with direct assignment of
the input parameters, or by reading those parameters from a file.
Parameters:
-----------
filename: String with the name of the file that contains the
information regarding the geometry, mesh, border
conditions, and other parameters necessary for the
solution of the problem.
This file is a modified Gmsh output file with extension
.msh
nodes: Numpy array matrix of nodes containing the coordinates of
the nodes from the discretized domain.
nodes is an array like matrix of dimension (n,3).
Where each column represents the value of the nodes on
one of the three coordinate axes x,y,z.
elements: Numpy array matrix of elements containing the relations
between nodes from the discretized domain.
elements is an array like matrix of dimension (n_elements,2)
in the case of 1D, and (n_elements,3) for 2D problems.
parameter: Is an array that describes the potential actuating over the
the elements of the domain given by elements. For each element
in elements there is an associated potential value on the same
position in the array parameter.
The potential in Scroedinger equation defines the specific
nature of the problem to be solved. For more details on how
to define a potential and what does it mean pleas read the
documentation of the Potential1D function in the module
PrePro.
dimension: int parameter that tells the program wether to solve for a
1D problem or a 2D problem.
bc_type: String parameter for the selection of a border condition
that can be either:
'Dir' For the Dirichlet border condition
(Infinite potential well).
'Bloch' For the periodic formulation of the problem.
(Electron in a periodic material )
sol_type: String that tells wether to solve the stationary version of
the equation or another not yet suported.
'Stationary'
analysis_param: Array that contains the information regarding the
number of solutions to be computed and wether to save
the values or not.
analysis_param[0]: String answer to the question
save Eigen Values?
analysis_param[1]: String answer to the question
save Eigen Vectors?
analysis_param[2]: Integer number of Eigen Values
to save
analysis_param[3]: Integer number of Eigen Vectors
to save
analysis_param[4]: Integer number of wave numbers
in x to sweep
analysis_param[5]: Integer number of wave numbers
in y to sweep
analysis_param[6]: biggest value of k. it may be the lenght
of the dommain
Last modification: date 16/11/2011
"""
#------------------------ Load from file if given -------------------------
assert isinstance(filename, str)
if filename != '': # If type is not blank
solver_input = read_solver_input(filename) # Import input parameters for
if solver_input != []:
dimension = solver_input[0]
bc_type = solver_input[1]
parameter = solver_input[2] # Reasign
eq = solver_input[3]
sol_type = solver_input[4]
analysis_param = solver_input[5] # the solution
bc_filename = solver_input[6]
# If the file does not contain solution parameters, the ones given as
# arguments of the function are written in the file.
else:
write_solver_input(filename, dimension = dimension,
bc_type = bc_type, sol_type = sol_type, \
eq = eq, parameter = parameter, \
bc_filename = bc_filename)
nodes, elements = read_mesh(filename) # Import nodes and elements
n = shape(nodes)[0] #Number of nodes
# el_points = elements[0]
bc_lines = elements[1] # Boundary condition is (bc) lines
if dimension == 2:
triangles = elements[2]
#------------------------------ 1D problem -------------------------------
# Reinterpretation of the parameter as the potential
v = parameter
if dimension == 1:
K, M = mat_assembly_1d(nodes, n, v)
#---------------------- With dirichlet boundary conditions ------------
if 'Dir' in bc_type and 'Stationary' in sol_type:
# These two matrices are called the Dirichlet matrices
# of both the stiffness equivalent and mass equivalent matrices.
Kd = K[1:n-1, 1:n-1]
Md = M[1:n-1, 1:n-1]
print 'K shape is:\n', K.shape
if 'y'in analysis_param[0] and 'n' in analysis_param[1]:
n_vals = int(analysis_param[2])
v = linalg.eigvalsh(Kd, Md, eigvals = (0, n_vals-1))
# v = v/2
print 'The Eigenvalues are:\n', v
return v
elif 'y'in analysis_param[0] and 'y'in analysis_param[1]:
n_vals = int(analysis_param[2])
n_vects = int(analysis_param[3])
n_solutions = max(n_vals,n_vects)
v, Dd = linalg.eigh(Kd, Md, eigvals = (0, n_solutions-1))
# v = v/2
print 'The Eigenvalues are:\n', v
E1=zeros(4)
for i in range(0,4):
E1[i]=(i+1)**2*pi**2/((2*pi)**2)
error=linalg.norm(E1-v)/linalg.norm(E1)
print 'error', error
D = zeros((n, n_vects))
print 'D', D.shape, 'Dd', Dd.shape
D[1:n-1, :] = Dd
return v, D
elif 'n' in analysis_param[0] and 'y' in analysis_param[1]:
n_vects = int(analysis_param[3])
v, Dd = linalg.eigh(Kd, Md, eigvals = (0, n_vects-1))
D = zeros((n, n_vects))
D[1:n-1, :] = Dd
return D
else:
print 'error: If you dont want anything why do you solve?'
#----------------- With Bloch periodic boundary conditions ------------
elif 'Bloch' in bc_type:
import cmath
from numpy import asarray
K = asarray(K, dtype = complex)
M = asarray(M, dtype = complex)
print 'K shape is:\n', K.shape
# Bloch analysis parameters
xi = nodes[0, 0] # initial x
xf = nodes[n-1, 0] # final x
n_vals = int(analysis_param[2]) # number of eigenvales to compute
nk = int(analysis_param[4]) # number of k to sweep
kmax = 4.*pi/xf
kmin = -0.0
k_range = linspace(kmin, kmax, num = nk)
omega = zeros( (len(k_range), n_vals) )
E = zeros( (len(k_range), n_vals) )
print 'Number of eigenvales to compute: ', n_vals, \
'\nNumber of wave numbers to sweep: ', nk, \
' in ', [k_range[0],k_range[nk-1]]
# Bloch-Periodicity imposition
ll = 0
Kaux = K.copy()
Maux = M.copy()
for k in k_range: # Loop over the different wave numbers k
fi = cmath.exp(1.0j*k*xi)
ff = cmath.exp(1.0j*k*xf)
K = Kaux.copy()
M = Maux.copy()
for i in range(0, n):
K[0, i] = K[0, i]*fi.conjugate()
K[i, 0] = K[i, 0]*fi
K[n-1, i] = K[n-1, i]*ff.conjugate()
K[i, n-1] = K[i, n-1]*ff
M[0, i] = M[0, i]*fi.conjugate()
M[i, 0] = M[i, 0]*fi
M[n-1 , i] = M[n-1, i]*ff.conjugate()
M[i, n-1] = M[i, n-1]*ff
K[n-1, :] = K[0, :] + K[n-1, :]
K[:, n-1] = K[:, 0] + K[:, n-1]
M[n-1, :] = M[0, :] + M[n-1, :]
M[:, n-1] = M[:, 0] + M[:, n-1]
Kd = K[1:n, 1:n]
Md = M[1:n, 1:n]
vals = linalg.eigvalsh(Kd, Md, eigvals = (0, n_vals-1) )
for i in range(0, n_vals):
omega[ll, i] = sqrt( abs(vals[i]) )
for i in range(0, n_vals):
E[ll, i] = vals[i]
ll = ll + 1
plt.figure(2)
plt.hold(True)
legend = []
for i in range(0,n_vals):
plt.plot(k_range, E[:, i])
legend.append('n = '+str(i+1))
plt.plot(k_range, (k_range)**2, '--k')
legend.append('Free Electron')
plt.title('Dispersion relation')
plt.legend(legend,loc=2)
plt.xlabel('Nondimensional wave number - $a\kappa/\pi$')
plt.ylabel('Nondimensional energy - $2ma^2E/\hslash^2$')
plt.xlim(xmax=2)
plt.grid()
plt.show()
else:
print 'Only Dirichlet and Bloch for now, sorry'
#------------------------------ 2D problem -------------------------------
elif dimension == 2:
from matrices import global_stiffness_matrix, global_mass_matrix, \
global_potential_matrix
from vectors import dirichlet_vector, build_solution
h = 1.
m = 1.
print 'Building matrices...\n'
boundary = read_bc(bc_filename)# Read boundary conditions from .bc file
#===================== Build stiffness matrix ==================
stif = global_stiffness_matrix(nodes, triangles)
stif = h/(2.*m)*stif
#---------------------- With dirichlet boundary conditions ------------
if 'Dir' in bc_type and 'Stationary' in sol_type:
dirichlet = boundary[0] # Dirichlet conditions are the first list
#===================== Build vectors ===========================
stif_d, d, remove, g = dirichlet_vector(bc_lines, dirichlet, \
stif)
# ==================== Build mass matrix =======================
mass_d = global_mass_matrix(nodes, triangles, remove)
#===================== Build potential matrix ==================
v_d = global_potential_matrix(nodes, triangles, v, remove)
# print "nodes",nodes
# print "stif",stif
# print "mass",mass_d
# print "v_d", v_d
print 'Solving eigenvalue problem...\n'
if 'y'in analysis_param[0] and 'n' in analysis_param[1]:
n_vals = int(analysis_param[2])
v = linalg.eigvalsh(stif_d + v_d, mass_d, \
eigvals = (0, n_vals-1))
# v = v/2
print 'The Eigenvalues are:\n', v
return v
elif 'y'in analysis_param[0] and 'y'in analysis_param[1]:
n_vals = int(analysis_param[2])
n_vects = int(analysis_param[3])
n_solutions = max(n_vals,n_vects)
v, dir_solution = linalg.eigh(stif_d + v_d, mass_d, \
eigvals = (0, n_solutions-1))
# v = v/2
solution = zeros((n, n_vects))
for i in range(n_vects):
solution[:,i] = build_solution(dir_solution[:, i], g, remove)
return v, solution
elif 'n'in analysis_param[0] and 'y'in analysis_param[1]:
n_vects = int(analysis_param[3])
v, dir_solution = linalg.eigh(stif_d + v_d, mass_d, \
eigvals = (0, n_vects-1))
solution = zeros((n, n_vects))
for i in range(n_vects):
solution[:,i] = build_solution(dir_solution[:, i], g, remove)
return solution
else:
print 'error: If you dont want anything why do you solve?'
#---------------------- With Bloch boundary conditions ------------
elif 'Bloch' in bc_type and 'Stationary' in sol_type:
from numpy import asarray,vstack
from vectors import reference_image_bloch_vectors
from utils import bloch_multiplication, bloch_sum
# ==================== Build mass matrix =======================
mass = global_mass_matrix(nodes, triangles)
#===================== Build potential matrix ====================
v = global_potential_matrix(nodes, triangles, v)
#================ Convert to complex matrices ====================
stif = asarray(stif, dtype = complex)
mass = asarray(mass, dtype = complex)
v = asarray(v, dtype = complex)
#===== Sum kinetic and potential matrices of the Hamiltonian ======
stif = stif + v
#= Retrieve the list of image and reference bloch boundary nodes ==
bloch = boundary[2]
ref_im = reference_image_bloch_vectors(bc_lines, bloch)
#============= Discretize the wavenumber dommain ==================
nk_x = int(analysis_param[4]) # number of k to sweep in x
nk_y = int(analysis_param[5]) # number of k to sweep in y
k_max = pi/float(analysis_param[6])
k_min = -k_max
k_range_x = linspace(k_min, k_max, num = nk_x)
k_range_y = linspace(k_min, k_max, num = nk_y)
n_vals = int(analysis_param[2])
#================ Initiate global variable =======================
energy = zeros( (nk_x * nk_y, n_vals) )
# print 'Number of eigenvales to compute: ', n_vals, \
# '\nNumber of wave numbers to sweep: ', 'in x: ', nk_x, \
# ' and in y: ', nk_y, ' in ', \
# [k_range_x[0],k_range_x[nk_x-1]], \
# [k_range_y[0],k_range_y[nk_y-1]], '\n'
#======== Defin<xe a new image reference ===========================
ref_im_aux = ref_im[0] # image(im) reference(ref) for complex
# multimplication (mul) operations
for bl in range(1, len(ref_im)):
ref_im_aux = vstack((ref_im_aux, ref_im[bl]))
#=============== Find who are the corners =======================
for corner in list(ref_im_aux[:, 0]):
if list(ref_im_aux[:, 0]).count(corner) == 2:
break
for corner2 in list(ref_im_aux[:, 1]):
if list(ref_im_aux[:, 1]).count(corner2) == 2:
break
#==== Relate the corners and delete their previous links===========
im_mul = list(set(list(ref_im_aux[:, 1])))
ref_mul = []
for i in im_mul:
ref_mul.append(list(ref_im_aux[:, 0])[list(ref_im_aux[:, 1]).index(i)])
ref_im_mul = array([ref_mul, im_mul]).T # Rearranged without
# repeated reference
# to corner
# ref_im_sum = ref_im_mul
# from numpy import delete
# for i in range(2):#Delete the links between conrner2 and other nodes
# ref_im_sum=delete(ref_im_sum, list(ref_im_sum[:, 1]).index(corner2), 0)
# # Make corner2 the image of corner
# ref_im_sum[list(ref_im_mul[:, 0]).index(corner), 1] = corner2
ref_im_mul[list(ref_im_mul[:, 1]).index(corner2), 0] = corner
print "ref_im_mul", ref_im_mul
#======================= Main cycles ===============================
i = 0
print 'Calculating each of the ', nk_x * nk_y, \
' points in k plane...\n'
for k_x in k_range_x: # Loop over the different wave numbers k in x
for k_y in k_range_y: # Loop over the different wave numbers k in y
# Multiply boundary nodes of each matrix by phase factor
new_stif, new_mass = bloch_multiplication(k_x, \
k_y, nodes, \
ref_im_mul, stif.copy(), \
mass.copy())
new_stif, new_mass = bloch_sum(ref_im_mul, new_stif, new_mass)
vals = linalg.eigvalsh(new_stif, new_mass, \
eigvals = (0, n_vals-1))
for val in range(0, n_vals):
energy[i, val] = vals[val]
print i+1, ' points out of :', nk_x * nk_y, '\n'
i = i+1
from meshUtils import meshtr2D
k_mesh = meshtr2D(k_min, k_max, k_min, k_max, nk_x, nk_y)
return k_mesh, energy
#------------------ If they ask for something we don't have-----------
else:
print 'Hang on, we are currently working on more options.'
else:
print 'Only 1D and 2D for now. Sorry'
print 'done'