def solve(self, nproc=1, disp=0):
""" 2D and 3D Finite Element Code
Currently configured to run either plane strain in 2D or general 3D but
could easily be modified for plane stress or axisymmetry.
"""
# Local Variables
# ---------------
# du : array_like, (i,)
# Nodal displacements.
# Let wij be jth displacement component at ith node. Then du
# contains [w00, w01, w10, w11, ...] for 2D
# and [w00, w01, w02, w10, w11, w12, ...) for 3D
# dw : array_like, (i,)
# Correction to nodal displacements.
# K : array_like, (i, j,)
# Global stiffness matrix. Stored as
# [K_1111 K_1112 K_1121 K_1122...
# K_1211 K_1212 K_1221 K_1222...
# K_2111 K_2112 K_2121 K_2122...]
# for 2D problems and similarly for 3D problems
# F : array_like, (i, )
# Force vector.
# Currently only includes contribution from tractions acting on
# element faces (body forces are neglected)
# R : array_like, (i, )
# Volume contribution to residual
# b : array_like (i, )
# RHS of equation system
runid = self.runid
control = self.control_params()
X = self.mesh.nodes()
connect = self.mesh.connect()
elements = self.mesh.elements()
fixnodes = self.mesh.displacement_bcs()
nforces = self.mesh.nodal_forces()
tractions = self.mesh.traction_bcs()
t0 = time.time()
dim = elements[0].ndof
nelems = elements.shape[0]
nnode = X.shape[0]
ndof = elements[0].ndof
ncoord = elements[0].ncoord
u = np.zeros((nnode * ndof))
du = np.zeros((nnode * ndof))
nodal_stresses = np.zeros((nnode, 6))
# tjf: nodal_state will have to be adjusted for multi-material where
# each node may be connected to elements of different material.
# nodal_state = np.zeros((nnode, max(el.material.nxtra for el in elements)))
nproc = 1.
# Simulation setup
(tint, nsteps, tol, maxit, relax, tstart,
tterm, dtmult, verbosity) = control
nsteps, maxit = int(nsteps), int(maxit)
t = tstart
dt = (tterm - tstart) / float(nsteps) * dtmult
global_data.set_var("TIME", t)
global_data.set_var("TIME_STEP", dt)
nodal_data.set_var("DISPL", u)
findstiff = True
logger.write_intro("Implicit", runid, nsteps, tol,
maxit, relax, tstart, tterm,
ndof, nelems, nnode, elements)
logger.write(HEAD)
for step in range(nsteps):
loadfactor = float(step + 1) / float(nsteps)
err1 = 1.
t += dt
# Newton-Raphson loop
mult = 10 if step == 0 and ro.reducedint else 1
for nit in range(mult * maxit):
# --- Update the state of each element to end of Newton step
update_element_states(t, dt, X, elements, connect, u, du)
# --- Update nodal stresses
for (inode, els) in enumerate(self.mesh._node_el_map):
sig = np.zeros(6)
# nx = elements[els[0]].material.nxtra
# xtra = np.zeros(nx)
acc_volume = 0.
for iel in els:
if iel == -1:
break
el = elements[iel]
vol = el._volume
sig += el.element_data(ave=True, item="STRESS") * vol
# xtra += el.element_data(ave=True, item="XTRA") * vol
acc_volume += vol
nodal_stresses[inode][:] = sig / acc_volume
# nodal_state[inode][0:nx] = xtra / acc_volume
continue
# --- Get global quantities
if findstiff:
gK = global_stiffness(t, dt, X, elements, connect, du, nproc)
findstiff = False
K = np.array(gK)
F = global_traction(t, dt, X, elements, connect,
tractions, nforces, du)
R = global_residual(t, dt, X, elements, connect, du)
b = loadfactor * F - R
apply_dirichlet_bcs(ndof, nnode, t, fixnodes, u, du,
loadfactor, K, b)
# --- Solve for the correction
c, dw, info = linsolve(K, b)
if info > 0:
logger.write("using least squares to solve system",
beg="*** ")
dw = np.linalg.lstsq(K, b)[0]
elif info < 0:
raise WasatchError(
"illegal value in %d-th argument of internal dposv" % -info)
# --- update displacement increment
du += relax * dw
# --- Check convergence
wnorm = np.dot(du, du)
err1 = np.dot(dw, dw)
if err1 > 1.E-10:
findstiff = True
if wnorm != 0.:
err1 = np.sqrt(err1 / wnorm)
err2 = np.sqrt(np.dot(b, b)) / float(ndof * nnode)
logger.write_formatted(step+1, nit+1, loadfactor, t, dt,
err1, err2, tol, fmt=ITER_FMT)
if err1 < tol:
break
continue
else:
raise WasatchError("Problem did not converge")
# Update the total displacecment
u += du
# Update the nodal coordinates
x = np.zeros((nnode, ndof))
for i in range(nnode):
for j in range(ndof):
x[i, j] = X[i, j] + u[ndof * i + j]
continue
continue
# Advance the state of each element
for element in elements:
element.advance_state()
global_data.set_var("TIME", t)
global_data.set_var("TIME_STEP", dt)
nodal_data.set_var("DISPL", u)
self.io.write_data_to_db()
continue
self.io.finish()
tf = time.time()
logger.write("\n{0}: execution finished".format(self.runid))
logger.write("{0}: total execution time: {1:8.6f}s".format(
runid, tf - t0))
retval = 0
if disp:
retval = {
"DISPLACEMENT": u, "TIME": t, "DT": dt,
"ELEMENT DATA": np.array([el.element_data() for el in elements]),
"NODAL STRESSES": nodal_stresses,
# "NODAL STATES": nodal_state,
"NODAL COORDINATES": x}
return retval
err1 = 1.
t += dt
logger.write(
"Step {0:.5f}, Time: {1}, Time step: {2}".format(step + 1, t, dt))
# --- Update the state of each element to end of step
du = np.diff(u, axis=0)[0]
update_element_states(dt, X, elements, connect, du)
# --- Get global quantities
K = global_stiffness(
t, dt, X, elements, connect, du, nproc)
F = global_traction(
t, dt, X, elements, connect, tractions, du)
MK = M + .5 * b[1] * dt * dt * K
apply_dirichlet_bcs(ndof, nnode, t, fixnodes, u[0], du, 1., K, F, M)
# --- Update kinematic variables
b = F - np.dot(K, u[0] + (v[0] + .5 * (1. - b[1]) * a[0] * dt) * dt)
a[1] = np.linalg.solve(MK, b)
v[1] = v[0] + (1. - b[0]) * a[0] * dt + b[0] * a[1] * dt
u[1] = u[0] + v[0] * dt + ((1. - b[1]) * a[0] + b[1] * a[1]) * dt * dt / 2.
# Advance kinematic variables
u[0] = u[1]
v[0] = v[1]
a[0] = a[1]
def fe_solve(exo_io, runid, control, X, connect, elements, fixnodes,
tractions, nforces, nproc=1):
""" 2D and 3D Finite Element Code
Currently configured to run either plane strain in 2D or general 3D but
could easily be modified for plane stress or axisymmetry.
Parameters
----------
control : array_like, (i,)
control[0] -> time integration scheme
control[1] -> number of steps
control[2] -> Newton tolerance
control[3] -> maximum Newton iterations
control[4] -> relax
control[5] -> starting time
control[6] -> termination time
control[7] -> time step multiplier
X : array like, (i, j,)
Nodal coordinates
X[i, j] -> jth coordinate of ith node for i=1...nnode, j=1...ncoord
connect : array_like, (i, j,)
Nodal connections
connect[i, j] -> jth node on on the ith element
elements : array_like, (i,)
Element class for each element
fixnodes : array_like, (i, j,)
List of prescribed displacements at nodes
fixnodes[i, 0] -> Node number
fixnodes[i, 1] -> Displacement component (x: 0, y: 1, or z: 2)
fixnodes[i, 2] -> Value of the displacement
tractions : array_like, (i, j,)
List of element tractions
tractions[i, 0] -> Element number
tractions[i, 1] -> face number
tractions[i, 2:] -> Components of traction as a function of time
Returns
-------
retval : init
0 on completion
failure otherwise
Notes
-----
Original code was adapted from [1].
References
----------
1. solidmechanics.org
"""
# Local Variables
# ---------------
# du : array_like, (i,)
# Nodal displacements.
# Let wij be jth displacement component at ith node. Then du
# contains [w00, w01, w10, w11, ...] for 2D
# and [w00, w01, w02, w10, w11, w12, ...) for 3D
# dw : array_like, (i,)
# Correction to nodal displacements.
# K : array_like, (i, j,)
# Global stiffness matrix. Stored as
# [K_1111 K_1112 K_1121 K_1122...
# K_1211 K_1212 K_1221 K_1222...
# K_2111 K_2112 K_2121 K_2122...]
# for 2D problems and similarly for 3D problems
# F : array_like, (i, )
# Force vector.
# Currently only includes contribution from tractions acting on
# element faces (body forces are neglected)
# R : array_like, (i, )
# Volume contribution to residual
# b : array_like (i, )
# RHS of equation system
# number of processors
nproc = np.amin([mp.cpu_count(), nproc, elements.size])
# Set up timing and logging
t0 = time.time()
logger = Logger(runid)
# Problem dimensions
dim = elements[0].ndof
nelems = elements.shape[0]
nnode = X.shape[0]
ndof = elements[0].ndof
ncoord = elements[0].ncoord
# Setup kinematic variables
u = np.zeros((2, nnode * ndof))
v = np.zeros((2, nnode * ndof))
a = np.zeros((2, nnode * ndof))
# Simulation setup
tint, nsteps, tol, maxit, relax, tstart, tterm, dtmult = control
nsteps, maxit = int(nsteps), int(maxit)
t = tstart
dt = (tterm - tstart) / float(nsteps) * dtmult
# Newmark parameters
b = [.5, 0.]
# Global mass, find only once
M = global_mass(X, elements, connect, nproc)
# Determine initial accelerations
du = np.diff(u, axis=0)[0]
K = global_stiffness(0., 1., X, elements, connect, du, nproc)
findstiff = False
F = global_traction(
0., 1., X, elements, connect, tractions, nforces, du)
apply_dirichlet_bcs(ndof, nnode, 0., fixnodes, u[0], du, 1., K, F, M)
a[0] = np.linalg.solve(M, -np.dot(K, u[0]) + F)
logger.write_intro("Explicit", runid, nsteps, tol, maxit, relax, tstart,
tterm, ndof, nelems, nnode))
for step in range(nsteps):
err1 = 1.
t += dt
logger.write(
"Step {0:.5f}, Time: {1}, Time step: {2}".format(step + 1, t, dt))
# --- Update the state of each element to end of step
du = np.diff(u, axis=0)[0]
update_element_states(dt, X, elements, connect, du)
# --- Get global quantities
K = global_stiffness(
t, dt, X, elements, connect, du, nproc)
def solve(self, nproc=1, disp=0):
""" 2D and 3D Finite Element Code
Currently configured to run either plane strain in 2D or general 3D but
could easily be modified for plane stress or axisymmetry.
"""
# Local Variables
# ---------------
# du : array_like, (i,)
# Nodal displacements.
# Let wij be jth displacement component at ith node. Then du
# contains [w00, w01, w10, w11, ...] for 2D
# and [w00, w01, w02, w10, w11, w12, ...) for 3D
# dw : array_like, (i,)
# Correction to nodal displacements.
# K : array_like, (i, j,)
# Global stiffness matrix. Stored as
# [K_1111 K_1112 K_1121 K_1122...
# K_1211 K_1212 K_1221 K_1222...
# K_2111 K_2112 K_2121 K_2122...]
# for 2D problems and similarly for 3D problems
# F : array_like, (i, )
# Force vector.
# Currently only includes contribution from tractions acting on
# element faces (body forces are neglected)
# R : array_like, (i, )
# Volume contribution to residual
# b : array_like (i, )
# RHS of equation system
runid = self.runid
control = self.control_params()
X = self.mesh.nodes()
connect = self.mesh.connect()
elements = self.mesh.elements()
fixnodes = self.mesh.displacement_bcs()
nforces = self.mesh.nodal_forces()
tractions = self.mesh.traction_bcs()
t0 = time.time()
dim = elements[0].ndof
nelems = elements.shape[0]
nnode = X.shape[0]
ndof = elements[0].ndof
ncoord = elements[0].ncoord
u = np.zeros((nnode * ndof))
du = np.zeros((nnode * ndof))
nodal_stresses = np.zeros((nnode, 6))
# tjf: nodal_state will have to be adjusted for multi-material where
# each node may be connected to elements of different material.
# nodal_state = np.zeros((nnode, max(el.material.nxtra for el in elements)))
nproc = 1.
# Simulation setup
(tint, nsteps, tol, maxit, relax, tstart, tterm, dtmult,
verbosity) = control
nsteps, maxit = int(nsteps), int(maxit)
t = tstart
dt = (tterm - tstart) / float(nsteps) * dtmult
global_data.set_var("TIME", t)
global_data.set_var("TIME_STEP", dt)
nodal_data.set_var("DISPL", u)
findstiff = True
logger.write_intro("Implicit", runid, nsteps, tol, maxit, relax,
tstart, tterm, ndof, nelems, nnode, elements)
logger.write(HEAD)
for step in range(nsteps):
loadfactor = float(step + 1) / float(nsteps)
err1 = 1.
t += dt
# Newton-Raphson loop
mult = 10 if step == 0 and ro.reducedint else 1
for nit in range(mult * maxit):
# --- Update the state of each element to end of Newton step
update_element_states(t, dt, X, elements, connect, u, du)
# --- Update nodal stresses
for (inode, els) in enumerate(self.mesh._node_el_map):
sig = np.zeros(6)
# nx = elements[els[0]].material.nxtra
# xtra = np.zeros(nx)
acc_volume = 0.
for iel in els:
if iel == -1:
break
el = elements[iel]
vol = el._volume
sig += el.element_data(ave=True, item="STRESS") * vol
# xtra += el.element_data(ave=True, item="XTRA") * vol
acc_volume += vol
nodal_stresses[inode][:] = sig / acc_volume
# nodal_state[inode][0:nx] = xtra / acc_volume
continue
# --- Get global quantities
if findstiff:
gK = global_stiffness(t, dt, X, elements, connect, du,
nproc)
findstiff = False
K = np.array(gK)
F = global_traction(t, dt, X, elements, connect, tractions,
nforces, du)
R = global_residual(t, dt, X, elements, connect, du)
b = loadfactor * F - R
apply_dirichlet_bcs(ndof, nnode, t, fixnodes, u, du,
loadfactor, K, b)
# --- Solve for the correction
c, dw, info = linsolve(K, b)
if info > 0:
logger.write("using least squares to solve system",
beg="*** ")
dw = np.linalg.lstsq(K, b)[0]
elif info < 0:
raise WasatchError(
"illegal value in %d-th argument of internal dposv" %
-info)
# --- update displacement increment
du += relax * dw
# --- Check convergence
wnorm = np.dot(du, du)
err1 = np.dot(dw, dw)
if err1 > 1.E-10:
findstiff = True
if wnorm != 0.:
err1 = np.sqrt(err1 / wnorm)
err2 = np.sqrt(np.dot(b, b)) / float(ndof * nnode)
logger.write_formatted(step + 1,
nit + 1,
loadfactor,
t,
dt,
err1,
err2,
tol,
fmt=ITER_FMT)
if err1 < tol:
break
continue
else:
raise WasatchError("Problem did not converge")
# Update the total displacecment
u += du
# Update the nodal coordinates
x = np.zeros((nnode, ndof))
for i in range(nnode):
for j in range(ndof):
x[i, j] = X[i, j] + u[ndof * i + j]
continue
continue
# Advance the state of each element
for element in elements:
element.advance_state()
global_data.set_var("TIME", t)
global_data.set_var("TIME_STEP", dt)
nodal_data.set_var("DISPL", u)
self.io.write_data_to_db()
continue
self.io.finish()
tf = time.time()
logger.write("\n{0}: execution finished".format(self.runid))
logger.write("{0}: total execution time: {1:8.6f}s".format(
runid, tf - t0))
retval = 0
if disp:
retval = {
"DISPLACEMENT": u,
"TIME": t,
"DT": dt,
"ELEMENT DATA":
np.array([el.element_data() for el in elements]),
"NODAL STRESSES": nodal_stresses,
# "NODAL STATES": nodal_state,
"NODAL COORDINATES": x
}
return retval