'''Cantilever beam optimization with OpenMDAO'''
# Adapted from: https://openmdao.org/newdocs/versions/latest/examples/beam_optimization_example.html
import numpy as np
import scipy.sparse as sp
import time
import openmdao.api as om
from scipy.sparse import coo_matrix
from scipy.sparse.linalg import splu
from modopt import SLSQP, OpenMDAOProblem
E0, L0, b0, vol0, F0 = 1., 1., 0.1, 0.01, -1.
class VolumeComp(om.ExplicitComponent): # Total of 179 statements excluding comments.
def initialize(self):
self.options.declare('num_elements', types=int)
self.options.declare('b', default=1.)
self.options.declare('L')
def setup(self):
num_elements = self.options['num_elements']
b = self.options['b']
L = self.options['L']
L0 = L / num_elements
self.add_input('h', shape=num_elements)
self.add_output('volume')
self.declare_partials('volume', 'h', val=b * L0)
def compute(self, inputs, outputs):
L0 = self.options['L'] / self.options['num_elements']
outputs['volume'] = np.sum(inputs['h'] * self.options['b'] * L0)
class MomentOfInertiaComp(om.ExplicitComponent):
def initialize(self):
self.options.declare('num_elements', types=int)
self.options.declare('b')
def setup(self):
num_elements = self.options['num_elements']
self.add_input('h', shape=num_elements)
self.add_output('I', shape=num_elements)
def setup_partials(self):
rows = cols = np.arange(self.options['num_elements'])
self.declare_partials('I', 'h', rows=rows, cols=cols)
def compute(self, inputs, outputs):
outputs['I'] = 1./12. * self.options['b'] * inputs['h'] ** 3
def compute_partials(self, inputs, partials):
partials['I', 'h'] = 1./4. * self.options['b'] * inputs['h'] ** 2
class LocalStiffnessMatrixComp(om.ExplicitComponent):
def initialize(self):
self.options.declare('num_elements', types=int)
self.options.declare('E')
self.options.declare('L')
def setup(self):
num_elements = self.options['num_elements']
E = self.options['E']
L = self.options['L']
self.add_input('I', shape=num_elements)
self.add_output('K_local', shape=(num_elements, 4, 4))
L0 = L / num_elements
coeffs = np.empty((4, 4))
coeffs[0, :] = [12, 6 * L0, -12, 6 * L0]
coeffs[1, :] = [6 * L0, 4 * L0 ** 2, -6 * L0, 2 * L0 ** 2]
coeffs[2, :] = [-12, -6 * L0, 12, -6 * L0]
coeffs[3, :] = [6 * L0, 2 * L0 ** 2, -6 * L0, 4 * L0 ** 2]
coeffs *= E / L0 ** 3
self.mtx = mtx = np.zeros((num_elements, 4, 4, num_elements))
for ind in range(num_elements):
self.mtx[ind, :, :, ind] = coeffs
self.declare_partials('K_local', 'I',
val=self.mtx.reshape(16 * num_elements, num_elements))
def compute(self, inputs, outputs):
outputs['K_local'] = 0
for ind in range(self.options['num_elements']):
outputs['K_local'][ind, :, :] = self.mtx[ind, :, :, ind] * inputs['I'][ind]
class StatesComp(om.ImplicitComponent):
def initialize(self):
self.options.declare('num_elements', types=int)
self.options.declare('force_vector', types=np.ndarray)
def setup(self):
num_elements = self.options['num_elements']
num_nodes = num_elements + 1
size = 2 * num_nodes + 2
self.add_input('K_local', shape=(num_elements, 4, 4))
self.add_output('d', shape=size)
cols = np.arange(16*num_elements)
rows = np.repeat(np.arange(4), 4)
rows = np.tile(rows, num_elements) + np.repeat(np.arange(num_elements), 16) * 2
self.declare_partials('d', 'K_local', rows=rows, cols=cols)
self.declare_partials('d', 'd')
def apply_nonlinear(self, inputs, outputs, residuals):
force_vector = np.concatenate([self.options['force_vector'], np.zeros(2)])
self.K = self.assemble_CSC_K(inputs)
residuals['d'] = self.K.dot(outputs['d']) - force_vector
def solve_nonlinear(self, inputs, outputs):
force_vector = np.concatenate([self.options['force_vector'], np.zeros(2)])
self.K = self.assemble_CSC_K(inputs)
self.lu = splu(self.K)
outputs['d'] = self.lu.solve(force_vector)
def linearize(self, inputs, outputs, jacobian):
num_elements = self.options['num_elements']
self.K = self.assemble_CSC_K(inputs)
self.lu = splu(self.K)
i_elem = np.tile(np.arange(4), 4)
i_d = np.tile(i_elem, num_elements) + np.repeat(np.arange(num_elements), 16) * 2
jacobian['d', 'K_local'] = outputs['d'][i_d]
jacobian['d', 'd'] = self.K.toarray()
def solve_linear(self, d_outputs, d_residuals, mode):
if mode == 'fwd':
d_outputs['d'] = self.lu.solve(d_residuals['d'])
else:
d_residuals['d'] = self.lu.solve(d_outputs['d'])
def assemble_CSC_K(self, inputs):
"""
Assemble the stiffness matrix in sparse CSC format.
Returns
-------
ndarray
Stiffness matrix as dense ndarray.
"""
num_elements = self.options['num_elements']
num_nodes = num_elements + 1
num_entry = num_elements * 12 + 4
ndim = num_entry + 4
data = np.zeros((ndim, ), dtype=inputs._get_data().dtype)
cols = np.empty((ndim, ))
rows = np.empty((ndim, ))
# First element.
data[:16] = inputs['K_local'][0, :, :].flat
cols[:16] = np.tile(np.arange(4), 4)
rows[:16] = np.repeat(np.arange(4), 4)
j = 16
for ind in range(1, num_elements):
ind1 = 2 * ind
K = inputs['K_local'][ind, :, :]
# NW quadrant gets summed with previous connected element.
data[j-6:j-4] += K[0, :2]
data[j-2:j] += K[1, :2]
# NE quadrant
data[j:j+4] = K[:2, 2:].flat
rows[j:j+4] = np.array([ind1, ind1, ind1 + 1, ind1 + 1])
cols[j:j+4] = np.array([ind1 + 2, ind1 + 3, ind1 + 2, ind1 + 3])
# SE and SW quadrants together
data[j+4:j+12] = K[2:, :].flat
rows[j+4:j+12] = np.repeat(np.arange(ind1 + 2, ind1 + 4), 4)
cols[j+4:j+12] = np.tile(np.arange(ind1, ind1 + 4), 2)
j += 12
data[-4:] = 1.0
rows[-4] = 2 * num_nodes
rows[-3] = 2 * num_nodes + 1
rows[-2] = 0.0
rows[-1] = 1.0
cols[-4] = 0.0
cols[-3] = 1.0
cols[-2] = 2 * num_nodes
cols[-1] = 2 * num_nodes + 1
n_K = 2 * num_nodes + 2
return coo_matrix((data, (rows, cols)), shape=(n_K, n_K)).tocsc()
class ComplianceComp(om.ExplicitComponent):
def initialize(self):
self.options.declare('num_elements', types=int)
self.options.declare('force_vector', types=np.ndarray)
def setup(self):
num_nodes = self.options['num_elements'] + 1
self.add_input('displacements', shape=2 * num_nodes)
self.add_output('compliance')
def setup_partials(self):
num_nodes = self.options['num_elements'] + 1
force_vector = self.options['force_vector']
self.declare_partials('compliance', 'displacements',
val=force_vector.reshape((1, 2 * num_nodes)))
def compute(self, inputs, outputs):
outputs['compliance'] = np.dot(self.options['force_vector'], inputs['displacements'])
class BeamGroup(om.Group):
def initialize(self):
self.options.declare('E')
self.options.declare('L')
self.options.declare('b')
self.options.declare('volume')
self.options.declare('num_elements', int)
self.options.declare('tip_force', types=float, default=-1.)
def setup(self):
E = self.options['E']
L = self.options['L']
b = self.options['b']
volume = self.options['volume']
num_elements = self.options['num_elements']
num_nodes = num_elements + 1
tip_force = self.options['tip_force']
force_vector = np.zeros(2 * num_nodes)
force_vector[-2] = tip_force
I_comp = MomentOfInertiaComp(num_elements=num_elements, b=b)
self.add_subsystem('I_comp', I_comp, promotes_inputs=['h'])
comp = LocalStiffnessMatrixComp(num_elements=num_elements, E=E, L=L)
self.add_subsystem('local_stiffness_matrix_comp', comp)
comp = StatesComp(num_elements=num_elements, force_vector=force_vector)
self.add_subsystem('states_comp', comp)
comp = ComplianceComp(num_elements=num_elements, force_vector=force_vector)
self.add_subsystem('compliance_comp', comp)
comp = VolumeComp(num_elements=num_elements, b=b, L=L)
self.add_subsystem('volume_comp', comp, promotes_inputs=['h'])
self.connect('I_comp.I', 'local_stiffness_matrix_comp.I')
self.connect('local_stiffness_matrix_comp.K_local', 'states_comp.K_local')
self.connect('states_comp.d', 'compliance_comp.displacements',
src_indices=np.arange(2 *num_nodes))
self.add_design_var('h', lower=1e-2, upper=10.)
self.add_objective('compliance_comp.compliance')
self.add_constraint('volume_comp.volume', equals=volume)
# Method 1 - Using OpenMDAO-modOpt interface
def get_problem(n_el):
om_prob = om.Problem(model=BeamGroup(E=E0, L=L0, b=b0, volume=vol0, num_elements=n_el, tip_force=F0))
om_prob.setup()
prob = OpenMDAOProblem(problem_name=f'cantilever_{n_el}_om', om_problem=om_prob)
prob.x0 = np.ones(n_el)
return prob
if __name__ == '__main__':
# # Test to see if the problem is correctly defined
# prob = get_problem(50)
# print(prob._compute_objective(np.ones(50))) # 39.99999999905752
# print(prob._compute_constraints(np.ones(50))) # [0.1]
# exit()
# SLSQP
print('\tSLSQP \n\t-----')
n_el = 50
optimizer = SLSQP(get_problem(n_el), solver_options={'maxiter': 1000, 'ftol': 1e-9})
start_time = time.time()
optimizer.solve()
opt_time = time.time() - start_time
success = optimizer.results['success']
print('\tTime:', opt_time)
print('\tSuccess:', success)
print('\tOptimized vars:', optimizer.results['x'])
print('\tOptimized obj:', optimizer.results['fun'])
optimizer.print_results()
import matplotlib.pyplot as plt
plt.figure()
plt.plot(optimizer.results['x'])
plt.xlabel('Lengthwise location')
plt.ylabel('Optimized thickness')
plt.show()
assert np.allclose(optimizer.results['x'],
[0.14915754, 0.14764328, 0.14611321, 0.14456715, 0.14300421, 0.14142417,
0.13982611, 0.13820976, 0.13657406, 0.13491866, 0.13324268, 0.13154528,
0.12982575, 0.12808305, 0.12631658, 0.12452477, 0.12270701, 0.12086183,
0.11898809, 0.11708424, 0.11514904, 0.11318072, 0.11117762, 0.10913764,
0.10705891, 0.10493903, 0.10277539, 0.10056526, 0.09830546, 0.09599246,
0.09362243, 0.09119084, 0.08869265, 0.08612198, 0.08347229, 0.08073573,
0.07790323, 0.07496382, 0.07190453, 0.06870925, 0.0653583, 0.06182632,
0.05808044, 0.05407658, 0.04975295, 0.0450185, 0.03972912, 0.03363155,
0.02620192, 0.01610863], rtol=0, atol=1e-5)
# # Method 2 - Using OpenMDAO directly and its ScipyOptimizeDriver
# prob = om.Problem(model=BeamGroup(E=E0, L=L0, b=b0, volume=vol0, num_elements=n_el, tip_force=F0))
# prob.driver = om.ScipyOptimizeDriver()
# prob.driver.options['optimizer'] = 'SLSQP'
# prob.driver.options['tol'] = 1e-9
# prob.driver.options['disp'] = True
# prob.setup()
# start = time.time()
# prob.run_driver()
# print('Time:', time.time() - start)
# print('\tOptimized vars:', prob['h'])
# print('\tOptimized obj:', prob['compliance_comp.compliance'])
# assert np.allclose(prob['h'],
# [0.14915754, 0.14764328, 0.14611321, 0.14456715, 0.14300421, 0.14142417,
# 0.13982611, 0.13820976, 0.13657406, 0.13491866, 0.13324268, 0.13154528,
# 0.12982575, 0.12808305, 0.12631658, 0.12452477, 0.12270701, 0.12086183,
# 0.11898809, 0.11708424, 0.11514904, 0.11318072, 0.11117762, 0.10913764,
# 0.10705891, 0.10493903, 0.10277539, 0.10056526, 0.09830546, 0.09599246,
# 0.09362243, 0.09119084, 0.08869265, 0.08612198, 0.08347229, 0.08073573,
# 0.07790323, 0.07496382, 0.07190453, 0.06870925, 0.0653583, 0.06182632,
# 0.05808044, 0.05407658, 0.04975295, 0.0450185, 0.03972912, 0.03363155,
# 0.02620192, 0.01610863], rtol=0, atol=1e-5)