def test_ScalarView_mpl_unknown():
mesh = Mesh()
mesh.load(domain_mesh)
mesh.refine_element(0)
shapeset = H1Shapeset()
pss = PrecalcShapeset(shapeset)
# create an H1 space
space = H1Space(mesh, shapeset)
space.set_uniform_order(5)
space.assign_dofs()
# initialize the discrete problem
wf = WeakForm(1)
set_forms(wf)
solver = DummySolver()
sys = LinSystem(wf, solver)
sys.set_spaces(space)
sys.set_pss(pss)
# assemble the stiffness matrix and solve the system
sys.assemble()
A = sys.get_matrix()
b = sys.get_rhs()
from scipy.sparse.linalg import cg
x, res = cg(A, b)
sln = Solution()
sln.set_fe_solution(space, pss, x)
view = ScalarView("Solution")
def test_example_03():
set_verbose(False)
mesh = Mesh()
mesh.load(domain_mesh)
mesh.refine_element(0)
shapeset = H1Shapeset()
pss = PrecalcShapeset(shapeset)
# create an H1 space
space = H1Space(mesh, shapeset)
space.set_uniform_order(5)
from hermes2d.examples.c03 import set_bc
set_bc(space)
space.assign_dofs()
# initialize the discrete problem
wf = WeakForm(1)
set_forms(wf)
solver = DummySolver()
sys = LinSystem(wf, solver)
sys.set_spaces(space)
sys.set_pss(pss)
# assemble the stiffness matrix and solve the system
sln = Solution()
sys.assemble()
sys.solve_system(sln)
assert abs(sln.l2_norm() - 0.25493) < 1e-4
assert abs(sln.h1_norm() - 0.89534) < 1e-4
def test_example_09():
from hermes2d.examples.c09 import set_bc, temp_ext, set_forms
# The following parameters can be changed:
INIT_REF_NUM = 4 # number of initial uniform mesh refinements
INIT_REF_NUM_BDY = 1 # number of initial uniform mesh refinements towards the boundary
P_INIT = 4 # polynomial degree of all mesh elements
TAU = 300.0 # time step in seconds
# Problem constants
T_INIT = 10 # temperature of the ground (also initial temperature)
FINAL_TIME = 86400 # length of time interval (24 hours) in seconds
# Global variable
TIME = 0;
# Boundary markers.
bdy_ground = 1
bdy_air = 2
# Load the mesh
mesh = Mesh()
mesh.load(get_cathedral_mesh())
# Perform initial mesh refinements
for i in range(INIT_REF_NUM):
mesh.refine_all_elements()
mesh.refine_towards_boundary(bdy_air, INIT_REF_NUM_BDY)
# Create an H1 space with default shapeset
space = H1Space(mesh, P_INIT)
set_bc(space)
# Set initial condition
tsln = Solution()
tsln.set_const(mesh, T_INIT)
# Initialize the weak formulation
wf = WeakForm()
set_forms(wf)
# Initialize the linear system.
ls = LinSystem(wf)
ls.set_spaces(space)
# Time stepping
nsteps = int(FINAL_TIME/TAU + 0.5)
rhsonly = False;
# Assemble and solve
ls.assemble()
rhsonly = True
ls.solve_system(tsln, lib="scipy")
def test_fe_solutions():
mesh = Mesh()
mesh.load(domain_mesh)
shapeset = H1Shapeset()
pss = PrecalcShapeset(shapeset)
space = H1Space(mesh, shapeset)
space.set_uniform_order(2)
space.assign_dofs()
a = array([1, 2, 3, 8, 0.1])
sln = Solution()
sln.set_fe_solution(space, pss, a)
def test_fe_solutions():
mesh = Mesh()
mesh.load(domain_mesh)
space = H1Space(mesh, 1)
space.set_uniform_order(2)
space.assign_dofs()
a = array([1, 2, 3, 8, 0.1])
sln = Solution()
def test_example_04():
from hermes2d.examples.c04 import set_bc
set_verbose(False)
mesh = Mesh()
mesh.load(domain_mesh)
# mesh.refine_element(0)
# mesh.refine_all_elements()
mesh.refine_towards_boundary(5, 3)
shapeset = H1Shapeset()
pss = PrecalcShapeset(shapeset)
# create an H1 space
space = H1Space(mesh, shapeset)
space.set_uniform_order(5)
set_bc(space)
space.assign_dofs()
xprev = Solution()
yprev = Solution()
# initialize the discrete problem
wf = WeakForm()
set_forms(wf, -4)
solver = DummySolver()
sys = LinSystem(wf, solver)
sys.set_spaces(space)
sys.set_pss(pss)
# assemble the stiffness matrix and solve the system
sys.assemble()
sln = Solution()
sys.solve_system(sln)
assert abs(sln.l2_norm() - 1.22729) < 1e-4
assert abs(sln.h1_norm() - 2.90006) < 1e-4
def test_example_05():
from hermes2d.examples.c05 import set_bc
from hermes2d.examples.c05 import set_forms as set_forms_surf
set_verbose(False)
mesh = Mesh()
mesh.load(domain_mesh)
mesh.refine_towards_vertex(3, 12)
shapeset = H1Shapeset()
pss = PrecalcShapeset(shapeset)
# create an H1 space
space = H1Space(mesh, shapeset)
space.set_uniform_order(4)
set_bc(space)
space.assign_dofs()
xprev = Solution()
yprev = Solution()
# initialize the discrete problem
wf = WeakForm(1)
set_forms(wf, -1)
set_forms_surf(wf)
sln = Solution()
solver = DummySolver()
sys = LinSystem(wf, solver)
sys.set_spaces(space)
sys.set_pss(pss)
sys.assemble()
sys.solve_system(sln)
assert abs(sln.l2_norm() - 0.535833) < 1e-4
assert abs(sln.h1_norm() - 1.332908) < 1e-4
def poisson_solver(mesh_tuple):
"""
Poisson solver.
mesh_tuple ... a tuple of (nodes, elements, boundary, nurbs)
"""
set_verbose(False)
mesh = Mesh()
mesh.create(*mesh_tuple)
mesh.refine_element(0)
shapeset = H1Shapeset()
pss = PrecalcShapeset(shapeset)
# create an H1 space
space = H1Space(mesh, shapeset)
space.set_uniform_order(5)
space.assign_dofs()
# initialize the discrete problem
wf = WeakForm(1)
set_forms(wf)
solver = DummySolver()
sys = LinSystem(wf, solver)
sys.set_spaces(space)
sys.set_pss(pss)
# assemble the stiffness matrix and solve the system
sys.assemble()
A = sys.get_matrix()
b = sys.get_rhs()
from scipy.sparse.linalg import cg
x, res = cg(A, b)
sln = Solution()
sln.set_fe_solution(space, pss, x)
return sln
def test_example_08():
from hermes2d.examples.c08 import set_bc, set_forms
set_verbose(False)
mesh = Mesh()
mesh.load(cylinder_mesh)
#mesh.refine_element(0)
#mesh.refine_all_elements()
mesh.refine_towards_boundary(5, 3)
shapeset = H1Shapeset()
pss = PrecalcShapeset(shapeset)
# create an H1 space
xvel = H1Space(mesh, shapeset)
yvel = H1Space(mesh, shapeset)
press = H1Space(mesh, shapeset)
xvel.set_uniform_order(2)
yvel.set_uniform_order(2)
press.set_uniform_order(1)
set_bc(xvel, yvel, press)
ndofs = 0
ndofs += xvel.assign_dofs(ndofs)
ndofs += yvel.assign_dofs(ndofs)
ndofs += press.assign_dofs(ndofs)
xprev = Solution()
yprev = Solution()
xprev.set_zero(mesh)
yprev.set_zero(mesh)
# initialize the discrete problem
wf = WeakForm(3)
set_forms(wf, xprev, yprev)
solver = DummySolver()
sys = LinSystem(wf, solver)
sys.set_spaces(xvel, yvel, press)
sys.set_pss(pss)
#dp.set_external_fns(xprev, yprev)
# visualize the solution
EPS_LOW = 0.0014
for i in range(3):
psln = Solution()
sys.assemble()
sys.solve_system(xprev, yprev, psln)
def test_example_06():
from hermes2d.examples.c06 import set_bc, set_forms
set_verbose(False)
# The following parameters can be changed:
UNIFORM_REF_LEVEL = 2
# Number of initial uniform mesh refinements.
CORNER_REF_LEVEL = 12
# Number of mesh refinements towards the re-entrant corner.
P_INIT = 6
# Uniform polynomial degree of all mesh elements.
# Boundary markers
NEWTON_BDY = 1
# Load the mesh file
mesh = Mesh()
mesh.load(get_example_mesh())
# Perform initial mesh refinements.
for i in range(UNIFORM_REF_LEVEL):
mesh.refine_all_elements()
mesh.refine_towards_vertex(3, CORNER_REF_LEVEL)
# Create an H1 space with default shapeset
space = H1Space(mesh, P_INIT)
set_bc(space)
# Initialize the weak formulation
wf = WeakForm()
set_forms(wf)
# Initialize the linear system.
ls = LinSystem(wf)
ls.set_spaces(space)
# Assemble and solve the matrix problem
sln = Solution()
ls.assemble()
ls.solve_system(sln)
def test_matrix():
set_verbose(False)
mesh = Mesh()
mesh.load(domain_mesh)
mesh.refine_element_id(0)
# create an H1 space with default shapeset
space = H1Space(mesh, 1)
# initialize the discrete problem
wf = WeakForm(1)
set_forms(wf)
sys = LinSystem(wf)
sys.set_spaces(space)
# assemble the stiffness matrix and solve the system
sln = Solution()
sys.assemble()
A = sys.get_matrix()
# create an H1 space
space = H1Space(mesh, shapeset)
space.set_uniform_order(5)
space.assign_dofs()
# initialize the discrete problem
wf = WeakForm(1)
set_forms(wf)
solver = DummySolver()
sys = LinSystem(wf, solver)
sys.set_spaces(space)
sys.set_pss(pss)
# assemble the stiffness matrix and solve the system
sys.assemble()
A = sys.get_matrix()
b = sys.get_rhs()
from scipy.sparse.linalg import cg
x, res = cg(A, b)
sln = Solution()
sln.set_fe_solution(space, pss, x)
view = ScalarView("Solution")
view.show(sln, lib="mayavi")
# view.wait()
mview = MeshView("Hello world!", 100, 100, 500, 500)
mview.show(mesh, lib="mpl", method="orders", notebook=False)
mview.wait()
# Load the mesh
mesh = Mesh()
mesh.load(get_cathedral_mesh())
# Perform initial mesh refinements
for i in range(INIT_REF_NUM):
mesh.refine_all_elements()
mesh.refine_towards_boundary(bdy_air, INIT_REF_NUM_BDY)
# Create an H1 space with default shapeset
space = H1Space(mesh, P_INIT)
set_bc(space)
# Set initial condition
tsln = Solution()
tsln.set_const(mesh, T_INIT)
# Initialize the weak formulation
wf = WeakForm()
set_forms(wf)
# Initialize the linear system.
ls = LinSystem(wf)
ls.set_spaces(space)
# Visualisation
sview = ScalarView("Temperature", 0, 0, 450, 600)
#title = "Time %s, exterior temperature %s" % (TIME, temp_ext(TIME))
#Tview.set_min_max_range(0,20);
#Tview.set_title(title);
def minus(sols, i):
sln = Solution()
vec = sols[:, i]
sln.set_fe_solution(rspace, pss, -vec)
return sln
mesh = Mesh()
mesh.load(get_example_mesh())
# Perform initial mesh refinements.
for i in range(UNIFORM_REF_LEVEL):
mesh.refine_all_elements()
mesh.refine_towards_vertex(3, CORNER_REF_LEVEL)
# Create an H1 space with default shapeset
space = H1Space(mesh, P_INIT)
set_bc(space)
# Initialize the weak formulation
wf = WeakForm()
set_forms(wf)
# Initialize the linear system.
ls = LinSystem(wf)
ls.set_spaces(space)
# Assemble and solve the matrix problem
sln = Solution()
ls.assemble()
ls.solve_system(sln)
# Visualize the approximation
sln.plot()
# Visualize the mesh
mesh.plot(space=space)
def calc(threshold=0.3,
strategy=0,
h_only=False,
error_tol=1,
interactive_plotting=False,
show_mesh=False,
show_graph=True):
mesh = Mesh()
mesh.create([
[0, 0],
[1, 0],
[1, 1],
[0, 1],
], [
[2, 3, 0, 1, 0],
], [
[0, 1, 1],
[1, 2, 1],
[2, 3, 1],
[3, 0, 1],
], [])
mesh.refine_all_elements()
shapeset = H1Shapeset()
pss = PrecalcShapeset(shapeset)
space = H1Space(mesh, shapeset)
set_bc(space)
space.set_uniform_order(1)
wf = WeakForm(1)
set_forms(wf)
sln = Solution()
rsln = Solution()
solver = DummySolver()
selector = H1ProjBasedSelector(CandList.HP_ANISO, 1.0, -1, shapeset)
view = ScalarView("Solution")
iter = 0
graph = []
while 1:
space.assign_dofs()
sys = LinSystem(wf, solver)
sys.set_spaces(space)
sys.set_pss(pss)
sys.assemble()
sys.solve_system(sln)
dofs = sys.get_matrix().shape[0]
if interactive_plotting:
view.show(sln,
lib=lib,
notebook=True,
filename="a%02d.png" % iter)
rsys = RefSystem(sys)
rsys.assemble()
rsys.solve_system(rsln)
hp = H1Adapt([space])
hp.set_solutions([sln], [rsln])
err_est = hp.calc_error() * 100
err_est = hp.calc_error(sln, rsln) * 100
print "iter=%02d, err_est=%5.2f%%, DOFS=%d" % (iter, err_est, dofs)
graph.append([dofs, err_est])
if err_est < error_tol:
break
hp.adapt(selector, threshold, strategy)
iter += 1
if not interactive_plotting:
view.show(sln, lib=lib, notebook=True)
if show_mesh:
mview = MeshView("Mesh")
mview.show(mesh, lib="mpl", notebook=True, filename="b.png")
if show_graph:
from numpy import array
graph = array(graph)
import pylab
pylab.clf()
pylab.plot(graph[:, 0], graph[:, 1], "ko", label="error estimate")
pylab.plot(graph[:, 0], graph[:, 1], "k-")
pylab.title("Error Convergence for the Inner Layer Problem")
pylab.legend()
pylab.xlabel("Degrees of Freedom")
pylab.ylabel("Error [%]")
pylab.yscale("log")
pylab.grid()
pylab.savefig("graph.png")
# Perform initial mesh refinements.
for i in range(INIT_REF_NUM):
mesh.refine_all_elements()
# Create an H1 space with default shapeset
space = H1Space(mesh, P_INIT)
set_bc(space)
# Initialize the weak formulation
wf = WeakForm()
set_forms(wf)
# Initialize views
sview = ScalarView("Coarse solution", 0, 100, 798, 700)
oview = OrderView("Polynomial orders", 800, 100, 798, 700)
# Initialize the linear system.
ls = LinSystem(wf)
ls.set_spaces(space)
# Assemble and solve the matrix problem
sln = Solution()
ls.assemble()
ls.solve_system(sln)
# View the solution
sln.plot()
# View the mesh
mesh.plot(space=space)
def test_example_11():
from hermes2d.examples.c11 import set_bc, set_wf_forms, set_hp_forms
SOLVE_ON_COARSE_MESH = True # If true, coarse mesh FE problem is solved in every adaptivity step.
P_INIT_U = 2 # Initial polynomial degree for u
P_INIT_V = 2 # Initial polynomial degree for v
INIT_REF_BDY = 3 # Number of initial boundary refinements
MULTI = True # MULTI = true ... use multi-mesh,
# MULTI = false ... use single-mesh.
# Note: In the single mesh option, the meshes are
# forced to be geometrically the same but the
# polynomial degrees can still vary.
THRESHOLD = 0.3 # This is a quantitative parameter of the adapt(...) function and
# it has different meanings for various adaptive strategies (see below).
STRATEGY = 1 # Adaptive strategy:
# STRATEGY = 0 ... refine elements until sqrt(THRESHOLD) times total
# error is processed. If more elements have similar errors, refine
# all to keep the mesh symmetric.
# STRATEGY = 1 ... refine all elements whose error is larger
# than THRESHOLD times maximum element error.
# STRATEGY = 2 ... refine all elements whose error is larger
# than THRESHOLD.
# More adaptive strategies can be created in adapt_ortho_h1.cpp.
CAND_LIST = CandList.H2D_HP_ANISO # Predefined list of element refinement candidates.
# Possible values are are attributes of the class CandList:
# P_ISO, P_ANISO, H_ISO, H_ANISO, HP_ISO, HP_ANISO_H, HP_ANISO_P, HP_ANISO
# See the Sphinx tutorial (http://hpfem.org/hermes2d/doc/src/tutorial-2.html#adaptive-h-fem-and-hp-fem) for details.
MESH_REGULARITY = -1 # Maximum allowed level of hanging nodes:
# MESH_REGULARITY = -1 ... arbitrary level hangning nodes (default),
# MESH_REGULARITY = 1 ... at most one-level hanging nodes,
# MESH_REGULARITY = 2 ... at most two-level hanging nodes, etc.
# Note that regular meshes are not supported, this is due to
# their notoriously bad performance.
CONV_EXP = 1 # Default value is 1.0. This parameter influences the selection of
# cancidates in hp-adaptivity. See get_optimal_refinement() for details.
MAX_ORDER = 10 # Maximum allowed element degree
ERR_STOP = 0.5 # Stopping criterion for adaptivity (rel. error tolerance between the
# fine mesh and coarse mesh solution in percent).
NDOF_STOP = 60000 # Adaptivity process stops when the number of degrees of freedom grows over
# this limit. This is mainly to prevent h-adaptivity to go on forever.
H2DRS_DEFAULT_ORDER = -1 # A default order. Used to indicate an unkonwn order or a maximum support order
# Load the mesh
umesh = Mesh()
vmesh = Mesh()
umesh.load(get_bracket_mesh())
if MULTI == False:
umesh.refine_towards_boundary(1, INIT_REF_BDY)
# Create initial mesh (master mesh).
vmesh.copy(umesh)
# Initial mesh refinements in the vmesh towards the boundary
if MULTI == True:
vmesh.refine_towards_boundary(1, INIT_REF_BDY)
# Create the x displacement space
uspace = H1Space(umesh, P_INIT_U)
vspace = H1Space(vmesh, P_INIT_V)
# Initialize the weak formulation
wf = WeakForm(2)
set_wf_forms(wf)
# Initialize refinement selector
selector = H1ProjBasedSelector(CAND_LIST, CONV_EXP, H2DRS_DEFAULT_ORDER)
# Initialize the coarse mesh problem
ls = LinSystem(wf)
ls.set_spaces(uspace, vspace)
u_sln_coarse = Solution()
v_sln_coarse = Solution()
u_sln_fine = Solution()
v_sln_fine = Solution()
# Assemble and Solve the fine mesh problem
rs = RefSystem(ls)
rs.assemble()
rs.solve_system(u_sln_fine, v_sln_fine, lib="scipy")
# Either solve on coarse mesh or project the fine mesh solution
# on the coarse mesh.
if SOLVE_ON_COARSE_MESH:
ls.assemble()
ls.solve_system(u_sln_coarse, v_sln_coarse, lib="scipy")
# Calculate element errors and total error estimate
hp = H1Adapt(ls)
hp.set_solutions([u_sln_coarse, v_sln_coarse], [u_sln_fine, v_sln_fine])
set_hp_forms(hp)
err_est = hp.calc_error() * 100
# create an H1 space
xvel = H1Space(mesh, shapeset)
yvel = H1Space(mesh, shapeset)
press = H1Space(mesh, shapeset)
xvel.set_uniform_order(2)
yvel.set_uniform_order(2)
press.set_uniform_order(1)
set_bc(xvel, yvel, press)
ndofs = 0
ndofs += xvel.assign_dofs(ndofs)
ndofs += yvel.assign_dofs(ndofs)
ndofs += press.assign_dofs(ndofs)
xprev = Solution()
yprev = Solution()
xprev.set_zero(mesh)
yprev.set_zero(mesh)
# initialize the discrete problem
wf = WeakForm(3)
set_forms(wf, xprev, yprev)
# visualize the solution
vview = VectorView("velocity [m/s]", 0, 0, 1200, 350)
pview = ScalarView("pressure [Pa]", 0, 500, 1200, 350)
vview.set_min_max_range(0, 1.9)
vview.show_scale(False)
pview.show_scale(False)
uoview = OrderView("Coarse mesh for u", 0, 0, 360, 300)
voview = OrderView("Coarse mesh for v", 370, 0, 360, 300)
uview = ScalarView("Coarse mesh solution u", 740, 0, 400, 300)
vview = ScalarView("Coarse mesh solution v", 1150, 0, 400, 300)
# Initialize refinement selector
selector = H1ProjBasedSelector(CAND_LIST, CONV_EXP, H2DRS_DEFAULT_ORDER)
# Initialize the coarse mesh problem
ls = LinSystem(wf)
ls.set_spaces(uspace, vspace)
# adaptivity loop
it = 1
done = False
u_sln_coarse = Solution()
v_sln_coarse = Solution()
u_sln_fine = Solution()
v_sln_fine = Solution()
while(not done):
print ("\n---- Adaptivity step %d ---------------------------------------------\n" % it)
it += 1
# Assemble and Solve the fine mesh problem
rs = RefSystem(ls)
rs.assemble()
rs.solve_system(u_sln_fine, v_sln_fine, lib="scipy")
# Either solve on coarse mesh or project the fine mesh solution
def test_example_09():
from hermes2d.examples.c09 import set_bc, temp_ext, set_forms
# The following parameters can be played with:
P_INIT = 1 # polynomial degree of elements
INIT_REF_NUM = 4 # number of initial uniform refinements
TAU = 300.0 # time step in seconds
# Problem constants
T_INIT = 10 # temperature of the ground (also initial temperature)
FINAL_TIME = 86400 # length of time interval (24 hours) in seconds
# Global variable
TIME = 0
# Load the mesh
mesh = Mesh()
mesh.load(get_cathedral_mesh())
# for i in range(INIT_REF_NUM):
# mesh.refine_all_elements()
# mesh.refine_towards_boundary(2, 5)
# Set up shapeset
shapeset = H1Shapeset()
pss = PrecalcShapeset(shapeset)
# Set up spaces
space = H1Space(mesh, shapeset)
set_bc(space)
space.set_uniform_order(P_INIT)
# Enumerate basis functions
space.assign_dofs()
# Set initial condition
tsln = Solution()
tsln.set_const(mesh, T_INIT)
# Weak formulation
wf = WeakForm(1)
set_forms(wf, tsln)
# Matrix solver
solver = DummySolver()
# Linear system
ls = LinSystem(wf, solver)
ls.set_spaces(space)
ls.set_pss(pss)
# Visualisation
sview = ScalarView("Temperature", 0, 0, 450, 600)
# title = "Time %s, exterior temperature %s" % (TIME, temp_ext(TIME))
# Tview.set_min_max_range(0,20);
# Tview.set_title(title);
# Tview.fix_scale_width(3);
# Time stepping
nsteps = int(FINAL_TIME / TAU + 0.5)
rhsonly = False
# Assemble and solve
ls.assemble()
rhsonly = True
ls.solve_system(tsln)
def test_example_22():
from hermes2d.examples.c22 import set_bc, set_forms
# The following parameters can be changed:
SOLVE_ON_COARSE_MESH = True # if true, coarse mesh FE problem is solved in every adaptivity step
INIT_REF_NUM = 1 # Number of initial uniform mesh refinements
P_INIT = 2 # Initial polynomial degree of all mesh elements.
THRESHOLD = 0.3 # This is a quantitative parameter of the adapt(...) function and
# it has different meanings for various adaptive strategies (see below).
STRATEGY = 0 # Adaptive strategy:
# STRATEGY = 0 ... refine elements until sqrt(THRESHOLD) times total
# error is processed. If more elements have similar errors, refine
# all to keep the mesh symmetric.
# STRATEGY = 1 ... refine all elements whose error is larger
# than THRESHOLD times maximum element error.
# STRATEGY = 2 ... refine all elements whose error is larger
# than THRESHOLD.
# More adaptive strategies can be created in adapt_ortho_h1.cpp.
CAND_LIST = CandList.H2D_HP_ANISO # Predefined list of element refinement candidates.
# Possible values are are attributes of the class CandList:
# P_ISO, P_ANISO, H_ISO, H_ANISO, HP_ISO, HP_ANISO_H, HP_ANISO_P, HP_ANISO
# See the Sphinx tutorial (http://hpfem.org/hermes2d/doc/src/tutorial-2.html#adaptive-h-fem-and-hp-fem) for details.
MESH_REGULARITY = -1 # Maximum allowed level of hanging nodes:
# MESH_REGULARITY = -1 ... arbitrary level hangning nodes (default),
# MESH_REGULARITY = 1 ... at most one-level hanging nodes,
# MESH_REGULARITY = 2 ... at most two-level hanging nodes, etc.
# Note that regular meshes are not supported, this is due to
# their notoriously bad performance.
CONV_EXP = 0.5
ERR_STOP = 0.1 # Stopping criterion for adaptivity (rel. error tolerance between the
# fine mesh and coarse mesh solution in percent).
NDOF_STOP = 60000 # Adaptivity process stops when the number of degrees of freedom grows
# over this limit. This is to prevent h-adaptivity to go on forever.
H2DRS_DEFAULT_ORDER = -1 # A default order. Used to indicate an unkonwn order or a maximum support order
# Problem parameters.
SLOPE = 60 # Slope of the layer.
# Load the mesh
mesh = Mesh()
mesh.create([
[0, 0],
[1, 0],
[1, 1],
[0, 1],
], [
[2, 3, 0, 1, 0],
], [
[0, 1, 1],
[1, 2, 1],
[2, 3, 1],
[3, 0, 1],
], [])
# Perform initial mesh refinements
mesh.refine_all_elements()
# Create an H1 space with default shapeset
space = H1Space(mesh, P_INIT)
set_bc(space)
# Initialize the weak formulation
wf = WeakForm()
set_forms(wf)
# Initialize refinement selector
selector = H1ProjBasedSelector(CAND_LIST, CONV_EXP, H2DRS_DEFAULT_ORDER)
# Initialize the coarse mesh problem
ls = LinSystem(wf)
ls.set_spaces(space)
# Adaptivity loop
iter = 0
done = False
sln_coarse = Solution()
sln_fine = Solution()
# Assemble and solve the fine mesh problem
rs = RefSystem(ls)
rs.assemble()
rs.solve_system(sln_fine)
# Either solve on coarse mesh or project the fine mesh solution
# on the coarse mesh.
if SOLVE_ON_COARSE_MESH:
ls.assemble()
ls.solve_system(sln_coarse)
# Calculate error estimate wrt. fine mesh solution
hp = H1Adapt(ls)
hp.set_solutions([sln_coarse], [sln_fine])
err_est = hp.calc_error() * 100
def schroedinger_solver(n_eigs=4, iter=2, verbose_level=1, plot=False,
potential="hydrogen", report=False, report_filename="report.h5",
force=False, sim_name="sim", potential2=None):
"""
One particle Schroedinger equation solver.
n_eigs ... the number of the lowest eigenvectors to calculate
iter ... the number of adaptive iterations to do
verbose_level ...
0 ... quiet
1 ... only moderate output (default)
2 ... lot's of output
plot ........ plot the progress (solutions, refined solutions, errors)
potential ... the V(x) for which to solve, one of:
well, oscillator, hydrogen
potential2 .. other terms that should be added to potential
report ...... it will save raw data to a file, useful for creating graphs
etc.
Returns the eigenvalues and eigenvectors.
"""
set_verbose(verbose_level == 2)
set_warn_integration(False)
pot = {"well": 0, "oscillator": 1, "hydrogen": 2, "three-points": 3}
pot_type = pot[potential]
if report:
from timeit import default_timer as clock
from tables import IsDescription, UInt32Col, Float32Col, openFile, \
Float64Col, Float64Atom, Col, ObjectAtom
class Iteration(IsDescription):
n = UInt32Col()
DOF = UInt32Col()
DOF_reference = UInt32Col()
cpu_solve = Float32Col()
cpu_solve_reference = Float32Col()
eig_errors = Float64Col(shape=(n_eigs,))
eigenvalues = Float64Col(shape=(n_eigs,))
eigenvalues_reference = Float64Col(shape=(n_eigs,))
h5file = openFile(report_filename, mode = "a",
title = "Simulation data")
if hasattr(h5file.root, sim_name):
if force:
h5file.removeNode(getattr(h5file.root, sim_name),
recursive=True)
else:
print "The group '%s' already exists. Use -f to overwrite it." \
% sim_name
return
group = h5file.createGroup("/", sim_name, 'Simulation run')
table = h5file.createTable(group, "iterations", Iteration,
"Iterations info")
h5eigs = h5file.createVLArray(group, 'eigenvectors', ObjectAtom())
h5eigs_ref = h5file.createVLArray(group, 'eigenvectors_reference',
ObjectAtom())
iteration = table.row
mesh = Mesh()
mesh.load("square.mesh")
if potential == "well":
# Read the width of the mesh automatically. This assumes there is just
# one square element:
a = sqrt(mesh.get_element(0).get_area())
# set N high enough, so that we get enough analytical eigenvalues:
N = 10
levels = []
for n1 in range(1, N):
for n2 in range(1, N):
levels.append(n1**2 + n2**2)
levels.sort()
E_exact = [pi**2/(2.*a**2) * m for m in levels]
elif potential == "oscillator":
E_exact = [1] + [2]*2 + [3]*3 + [4]*4 + [5]*5 + [6]*6
elif potential == "hydrogen":
Z = 1 # atom number
E_exact = [-float(Z)**2/2/(n-0.5)**2/4 for n in [1]+[2]*3+[3]*5 +\
[4]*8 + [5]*15]
else:
E_exact = [1.]*50
if len(E_exact) < n_eigs:
print n_eigs
print E_exact
raise Exception("We don't have enough analytical eigenvalues.")
#mesh.refine_element(0)
mesh.refine_all_elements()
#mesh.refine_all_elements()
#mesh.refine_all_elements()
#mesh.refine_all_elements()
#mview = MeshView()
#mview.show(mesh)
shapeset = H1Shapeset()
space = H1Space(mesh, shapeset)
space.set_uniform_order(2)
space.assign_dofs()
pss = PrecalcShapeset(shapeset)
#bview = BaseView()
#bview.show(space)
wf1 = WeakForm(1)
# this is induced by set_verbose():
#dp1.set_quiet(not verbose)
set_forms8(wf1, pot_type, potential2)
wf2 = WeakForm(1)
# this is induced by set_verbose():
#dp2.set_quiet(not verbose)
set_forms7(wf2)
solver = DummySolver()
w = 320
h = 320
views = [ScalarView("", i*w, 0, w, h) for i in range(4)]
viewsm = [ScalarView("", i*w, h, w, h) for i in range(4)]
viewse = [ScalarView("", i*w, 2*h, w, h) for i in range(4)]
#for v in viewse:
# v.set_min_max_range(0, 10**-4)
ord = OrderView("Polynomial Orders", 0, 2*h, w, h)
rs = None
precision = 30.0
if verbose_level >= 1:
print "Problem initialized. Starting calculation."
for it in range(iter):
if verbose_level >= 1:
print "-"*80
print "Starting iteration %d." % it
if report:
iteration["n"] = it
#mesh.save("refined2.mesh")
sys1 = LinSystem(wf1, solver)
sys1.set_spaces(space)
sys1.set_pss(pss)
sys2 = LinSystem(wf2, solver)
sys2.set_spaces(space)
sys2.set_pss(pss)
if verbose_level >= 1:
print "Assembling the matrices A, B."
sys1.assemble()
sys2.assemble()
if verbose_level == 2:
print "converting matrices A, B"
A = sys1.get_matrix()
B = sys2.get_matrix()
if verbose_level >= 1:
n = A.shape[0]
print "Solving the problem Ax=EBx (%d x %d)." % (n, n)
if report:
n = A.shape[0]
iteration["DOF"] = n
if report:
t = clock()
eigs, sols = solve(A, B, n_eigs, verbose_level == 2)
if report:
t = clock() - t
iteration["cpu_solve"] = t
iteration["eigenvalues"] = array(eigs)
#h5eigs.append(sols)
if verbose_level >= 1:
print " \-Done."
print_eigs(eigs, E_exact)
s = []
n = sols.shape[1]
for i in range(n):
sln = Solution()
vec = sols[:, i]
sln.set_fe_solution(space, pss, vec)
s.append(sln)
if verbose_level >= 1:
print "Matching solutions."
if rs is not None:
def minus2(sols, i):
sln = Solution()
vec = sols[:, i]
sln.set_fe_solution(space, pss, -vec)
return sln
pairs, flips = make_pairs(rs, s, d1, d2)
#print "_"*40
#print pairs, flips
#print len(rs), len(s)
#from time import sleep
#sleep(3)
#stop
s2 = []
for j, flip in zip(pairs, flips):
if flip:
s2.append(minus2(sols,j))
else:
s2.append(s[j])
s = s2
if plot:
if verbose_level >= 1:
print "plotting: solution"
ord.show(space)
for i in range(min(len(s), 4)):
views[i].show(s[i])
views[i].set_title("Iter: %d, eig: %d" % (it, i))
#mat1.show(dp1)
if verbose_level >= 1:
print "reference: initializing mesh."
rsys1 = RefSystem(sys1)
rsys2 = RefSystem(sys2)
if verbose_level >= 1:
print "reference: assembling the matrices A, B."
rsys1.assemble()
rsys2.assemble()
if verbose_level == 2:
print "converting matrices A, B"
A = rsys1.get_matrix()
B = rsys2.get_matrix()
if verbose_level >= 1:
n = A.shape[0]
print "reference: solving the problem Ax=EBx (%d x %d)." % (n, n)
if report:
n = A.shape[0]
iteration["DOF_reference"] = n
if report:
t = clock()
eigs, sols = solve(A, B, n_eigs, verbose_level == 2)
if report:
t = clock() - t
iteration["cpu_solve_reference"] = t
iteration["eigenvalues_reference"] = array(eigs)
#h5eigs_ref.append(sols)
if verbose_level >= 1:
print " \-Done."
print_eigs(eigs, E_exact)
rs = []
rspace = rsys1.get_ref_space(0)
n = sols.shape[1]
for i in range(n):
sln = Solution()
vec = sols[:, i]
sln.set_fe_solution(rspace, pss, vec)
rs.append(sln)
if verbose_level >= 1:
print "reference: matching solutions."
def minus(sols, i):
sln = Solution()
vec = sols[:, i]
sln.set_fe_solution(rspace, pss, -vec)
return sln
# segfaults
#mat2.show(rp1)
def d1(x, y):
return (x-y).l2_norm()
def d2(x, y):
return (x+y).l2_norm()
from pairs import make_pairs
pairs, flips = make_pairs(s, rs, d1, d2)
rs2 = []
for j, flip in zip(pairs, flips):
if flip:
rs2.append(minus(sols,j))
else:
rs2.append(rs[j])
rs = rs2
if plot:
if verbose_level >= 1:
print "plotting: solution, reference solution, errors"
for i in range(min(len(s), len(rs), 4)):
#views[i].show(s[i])
#views[i].set_title("Iter: %d, eig: %d" % (it, i))
viewsm[i].show(rs[i])
viewsm[i].set_title("Ref. Iter: %d, eig: %d" % (it, i))
viewse[i].show((s[i]-rs[i])**2)
viewse[i].set_title("Error plot Iter: %d, eig: %d" % (it, i))
if verbose_level >= 1:
print "Calculating errors."
hp = H1OrthoHP(space)
if verbose_level == 2:
print "-"*60
print "calc error (iter=%d):" % it
eig_converging = 0
errors = []
for i in range(min(len(s), len(rs))):
error = hp.calc_error(s[i], rs[i]) * 100
errors.append(error)
prec = precision
if verbose_level >= 1:
print "eig %d: %g%% precision goal: %g%%" % (i, error, prec)
if report:
iteration["eig_errors"] = array(errors)
if errors[0] > precision:
eig_converging = 0
elif errors[3] > precision:
eig_converging = 3
elif errors[1] > precision:
eig_converging = 1
elif errors[2] > precision:
eig_converging = 2
else:
precision /= 2
# uncomment the following line to only converge to some eigenvalue:
#eig_converging = 3
if verbose_level >= 1:
print "picked: %d" % eig_converging
error = hp.calc_error(s[eig_converging], rs[eig_converging]) * 100
if verbose_level >= 1:
print "Adapting the mesh."
hp.adapt(0.3)
space.assign_dofs()
if report:
iteration.append()
table.flush()
if report:
h5file.close()
return s
# Create the x- and y- displacement space using the default H1 shapeset
xdisp = H1Space(mesh, P_INIT)
ydisp = H1Space(mesh, P_INIT)
set_bc(xdisp, ydisp)
# Initialize the weak formulation
wf = WeakForm(2)
set_forms(wf)
# Initialize the linear system.
ls = LinSystem(wf)
ls.set_spaces(xdisp, ydisp)
# Assemble and solve the matrix problem
xsln = Solution()
ysln = Solution()
ls.assemble()
ls.solve_system(xsln, ysln, lib="scipy")
# Visualize the solution
view = ScalarView("Von Mises stress [Pa]", 50, 50, 1200, 600)
E = float(200e9)
nu = 0.3
l = (E * nu) / ((1 + nu) * (1 - 2 * nu))
mu = E / (2 * (1 + nu))
stress = VonMisesFilter(xsln, ysln, mu, l)
view.show(stress)
# Visualize the mesh
mesh.plot(space=xdisp)
def test_example_10():
from hermes2d.examples.c10 import set_bc, set_forms
from hermes2d.examples import get_motor_mesh
# The following parameters can be changed:
SOLVE_ON_COARSE_MESH = True # If true, coarse mesh FE problem is solved in every adaptivity step
P_INIT = 2 # Initial polynomial degree of all mesh elements.
THRESHOLD = 0.2 # This is a quantitative parameter of the adapt(...) function and
# it has different meanings for various adaptive strategies (see below).
STRATEGY = 1 # Adaptive strategy:
# STRATEGY = 0 ... refine elements until sqrt(THRESHOLD) times total
# error is processed. If more elements have similar errors, refine
# all to keep the mesh symmetric.
# STRATEGY = 1 ... refine all elements whose error is larger
# than THRESHOLD times maximum element error.
# STRATEGY = 2 ... refine all elements whose error is larger
# than THRESHOLD.
# More adaptive strategies can be created in adapt_ortho_h1.cpp.
CAND_LIST = CandList.H2D_HP_ANISO_H # Predefined list of element refinement candidates.
# Possible values are are attributes of the class CandList:
# H2D_P_ISO, H2D_P_ANISO, H2D_H_ISO, H2D_H_ANISO, H2D_HP_ISO, H2D_HP_ANISO_H, H2D_HP_ANISO_P, H2D_HP_ANISO
# See User Documentation for details.
MESH_REGULARITY = -1 # Maximum allowed level of hanging nodes:
# MESH_REGULARITY = -1 ... arbitrary level hangning nodes (default),
# MESH_REGULARITY = 1 ... at most one-level hanging nodes,
# MESH_REGULARITY = 2 ... at most two-level hanging nodes, etc.
# Note that regular meshes are not supported, this is due to
# their notoriously bad performance.
ERR_STOP = 1.0 # Stopping criterion for adaptivity (rel. error tolerance between the
# fine mesh and coarse mesh solution in percent).
CONV_EXP = 1.0
# Default value is 1.0. This parameter influences the selection of
# cancidates in hp-adaptivity. See get_optimal_refinement() for details.
# fine mesh and coarse mesh solution in percent).
NDOF_STOP = 60000 # Adaptivity process stops when the number of degrees of freedom grows
# over this limit. This is to prevent h-adaptivity to go on forever.
H2DRS_DEFAULT_ORDER = -1 # A default order. Used to indicate an unkonwn order or a maximum support order
# Load the mesh
mesh = Mesh()
mesh.load(get_motor_mesh())
# Create an H1 space with default shapeset
space = H1Space(mesh, P_INIT)
set_bc(space)
# Initialize the discrete problem
wf = WeakForm()
set_forms(wf)
# Initialize refinement selector.
selector = H1ProjBasedSelector(CAND_LIST, CONV_EXP, H2DRS_DEFAULT_ORDER)
# Initialize the linear system.
ls = LinSystem(wf)
ls.set_spaces(space)
sln_coarse = Solution()
sln_fine = Solution()
# Assemble and solve the fine mesh problem
rs = RefSystem(ls)
rs.assemble()
rs.solve_system(sln_fine)
# Either solve on coarse mesh or project the fine mesh solution
# on the coarse mesh.
if SOLVE_ON_COARSE_MESH:
ls.assemble()
ls.solve_system(sln_coarse)
# Calculate element errors and total error estimate
hp = H1Adapt(ls)
hp.set_solutions([sln_coarse], [sln_fine])
err_est = hp.calc_error() * 100
sview = ScalarView("Solution")
mview = MeshView("Mesh")
graph = []
# Initialize refinement selector
selector = H1ProjBasedSelector(CAND_LIST, CONV_EXP, H2DRS_DEFAULT_ORDER)
# Initialize the coarse mesh problem
ls = LinSystem(wf)
ls.set_spaces(space)
# Adaptivity loop
iter = 0
done = False
print "Calculating..."
sln_coarse = Solution()
sln_fine = Solution()
while (not done):
# Assemble and solve the fine mesh problem
rs = RefSystem(ls)
rs.assemble()
rs.solve_system(sln_fine)
# Either solve on coarse mesh or project the fine mesh solution
# on the coarse mesh.
if SOLVE_ON_COARSE_MESH:
ls.assemble()
ls.solve_system(sln_coarse)
else:
mesh.refine_towards_boundary(2, 5) # Set up shapeset shapeset = H1Shapeset() pss = PrecalcShapeset(shapeset) # Set up spaces space = H1Space(mesh, shapeset) set_bc(space) space.set_uniform_order(P_INIT) # Enumerate basis functions space.assign_dofs() # Set initial condition tsln = Solution() tsln.set_const(mesh, T_INIT) # Weak formulation wf = WeakForm(1) set_forms(wf, tsln) # Matrix solver solver = DummySolver() # Linear system ls = LinSystem(wf, solver) ls.set_spaces(space) ls.set_pss(pss) # Visualisation