def objective_function(self, xs):
self.topology_factory.update_topology(xs)
if self.topology_factory.is_connected is True:
params = self.topology_factory.get_params()
cantilever = microfem.Cantilever(*params)
fem = microfem.PlateFEM(self.material, cantilever)
coords = (cantilever.xtip, cantilever.ytip)
opr = microfem.PlateDisplacement(fem, coords).get_operator()
mode_ident = microfem.ModeIdentification(fem, cantilever, 'plate')
try:
w, _, vall = fem.modal_analysis(1)
except RuntimeError:
print('singular')
kuu = fem.get_stiffness_matrix(free=False)
f1 = np.asscalar(np.sqrt(w) / (2 * np.pi))
phi1 = vall[:, [0]]
wtip1 = np.asscalar(opr @ phi1)
kfunc = lambda p, w: np.asscalar(p.T @ kuu @ p / w**2)
k1 = kfunc(phi1, wtip1)
type1 = mode_ident.is_mode_flexural(phi1)
if type1 is False:
cost = 1e8
else:
cost = -f1 * 1e-6 if k1 < self.k1 else k1
return (cost, )
return (self.topology_factory.connectivity_penalty, )
def objective_function(self, xs):
self.topology_factory.update_topology(xs)
if self.topology_factory.is_connected is True:
params = self.topology_factory.get_params()
cantilever = microfem.Cantilever(*params)
fem = microfem.PlateFEM(self.material, cantilever)
coords = (cantilever.xtip, cantilever.ytip)
opr = microfem.PlateDisplacement(fem, coords).get_operator()
mode_ident = microfem.ModeIdentification(fem, cantilever, 'plate')
w, _, vall = fem.modal_analysis(self.n_modes)
kuu = fem.get_stiffness_matrix(free=False)
fs = np.sqrt(w) / (2*np.pi)
phis = [vall[:, [i]] for i in range(self.n_modes)]
wtips = [opr @ p for p in phis]
kfunc = lambda p, w: np.asscalar(p.T @ kuu @ p / w ** 2)
ks = [kfunc(p, w) for p, w in zip(phis, wtips)]
types = [mode_ident.is_mode_flexural(p) for p in phis]
if types[0] is False:
cost = 1e8
elif types[1] is True:
cost = self.evaluation(fs[0], fs[1], ks[0], ks[1])
elif types[2] is True:
cost = self.evaluation(fs[0], fs[2], ks[0], ks[2])
else:
cost = 2e7
return (cost,)
return (self.topology_factory.connectivity_penalty,)
def console_output(self, xopt, image_file):
self.topology_factory.update_topology(xopt)
tup = (2*self.topology_factory.a, 2*self.topology_factory.b)
print('The element dimensions are (um): %gx%g' % tup)
params = self.topology_factory.get_params()
cantilever = microfem.Cantilever(*params)
microfem.plot_topology(cantilever, image_file)
if self.topology_factory.is_connected is True:
fem = microfem.PlateFEM(self.material, cantilever)
coords = (cantilever.xtip, cantilever.ytip)
opr = microfem.PlateDisplacement(fem, coords).get_operator()
mode_ident = microfem.ModeIdentification(fem, cantilever, 'plate')
w, _, vall = fem.modal_analysis(self.n_modes)
freq = np.sqrt(w) / (2*np.pi)
kuu = fem.get_stiffness_matrix(free=False)
phis = [vall[:, [i]] for i in range(self.n_modes)]
wtips = [opr @ p for p in phis]
kfunc = lambda p, w: np.asscalar(p.T @ kuu @ p / w ** 2)
ks = [kfunc(p, w) for p, w in zip(phis, wtips)]
types = [mode_ident.is_mode_flexural(p) for p in phis]
tup = ('Disp', 'Freq (Hz)', 'Stiffness', 'Flexural')
print('\n %-15s %-15s %-15s %-10s' % tup)
for i in range(self.n_modes):
tup = (i, wtips[i], freq[i], ks[i], str(types[i]))
print('%-2d: %-15g %-15g %-15g %-10s' % tup)
for i in range(self.n_modes):
microfem.plot_mode(fem, vall[:, i])
def objective_function(self, xs):
self.topology_factory.update_topology(xs)
a = self.topology_factory.a
b = self.topology_factory.b
if self.topology_factory.is_connected is True:
topology = self.topology_factory.topology
cantilever = microfem.Cantilever(topology, a, b)
fem = microfem.PlateFEM(self.material, cantilever)
coords = (cantilever.xtip, cantilever.ytip)
opr = PlateDisplacement(fem, coords).get_operator()
mode_ident = ModeIdentification(fem, cantilever)
w, _, vall = fem.modal_analysis(self.n_modes)
kuu = fem.get_stiffness_matrix(free=False)
phi = vall[:, [0]]
wtip = opr @ phi
f1 = np.asscalar(np.sqrt(w) / (2 * np.pi))
k1 = np.asscalar(phi.T @ kuu @ phi / wtip**2)
type_ = mode_ident.is_mode_flexural(phi)
cost = (-f1, k1) if type_ is True else (1e8, 1e8)
return cost
return (self.topology_factory.connectivity_penalty,
self.topology_factory.connectivity_penalty)
def console_output(self, xopt, image_file):
self.topology_factory.update_topology(xopt)
topology = self.topology_factory.topology
cantilever = microfem.Cantilever(topology, self.a, self.b)
microfem.plot_topology(cantilever, image_file)
if self.topology_factory.is_connected is True:
fem = microfem.PlateFEM(self.material, cantilever)
coords = (cantilever.xtip, cantilever.ytip)
opr = PlateDisplacement(fem, coords).get_operator()
mode_ident = ModeIdentification(fem, cantilever)
w, _, vall = fem.modal_analysis(self.n_modes)
freq = np.sqrt(w) / (2*np.pi)
kuu = fem.get_stiffness_matrix(free=False)
phis = [vall[:, [i]] for i in range(self.n_modes)]
wtips = [opr @ p for p in phis]
ks = [np.asscalar(p.T @ kuu @ p / w ** 2) for p, w in zip(phis, wtips)]
costs = [k / ks[0] for k in ks]
types = [mode_ident.is_mode_flexural(p) for p in phis]
tup = ('Disp', 'Freq (Hz)', 'Stiffness', 'Ratio', 'Flexural')
print('\n %-15s %-15s %-15s %-15s %-10s' % tup)
for i in range(self.n_modes):
tup = (i, wtips[i], freq[i], ks[i], costs[i], str(types[i]))
print('%-2d: %-15g %-15g %-15g %-15g %-10s' % tup)
for i in range(self.n_modes):
microfem.plot_mode(fem, vall[:, i])
def console_output(self, xopt, image_file):
self.topology_factory.update_topology(xopt)
params = self.topology_factory.get_params()
cantilever = microfem.Cantilever(*params)
microfem.plot_topology(cantilever, image_file)
if self.topology_factory.is_connected is True:
fem = microfem.PlateFEM(self.material, cantilever)
w, _, vall = fem.modal_analysis(1)
f = np.asscalar(np.sqrt(w) / (2 * np.pi))
print('The first modal frequency is (Hz): %g' % f)
microfem.plot_mode(fem, vall[:, 0])
def objective_function(self, xs):
self.topology_factory.update_topology(xs)
if self.topology_factory.is_connected is True:
params = self.topology_factory.get_params()
cantilever = microfem.Cantilever(*params)
fem = microfem.PlateFEM(self.material, cantilever)
w, _, _ = fem.modal_analysis(1)
f = np.asscalar(np.sqrt(w) / (2 * np.pi))
cost = abs(f - self.f0)
return (cost, )
return (self.topology_factory.connectivity_penalty, )
# should be on a solid element. xtip = 250 ytip = 495 # Combine the above parameters into a Cantilever object in the microfem # package. Information about the cantilever can be displayed and the topology # can be plotted. cantilever = microfem.Cantilever(topology, a, b, xtip, ytip) cantilever.to_console() microfem.plot_topology(cantilever) # The material properties for the SOI Mumps manufacturing process has been # stored in an object. The plate finite element model requires both the # cantilever object and material object to initialize. material = microfem.SoiMumpsMaterial() fem = microfem.PlateFEM(material, cantilever) # With the FEM model built, modal analysis can be performed. This returns the # resonance frequencies and mode shapes of the cantilever. These can then be # plotted. n_modes = 3 ws, _, vall = fem.modal_analysis(n_modes=n_modes) # Using the results of the modal analysis, the frequency, stiffness, # tip displacement, and mode type (flexural, torsional), can be determined. coords = (cantilever.xtip, cantilever.ytip) opr = microfem.PlateDisplacement(fem, coords).get_operator() mode_ident = microfem.ModeIdentification(fem, cantilever, type_='plate') freq = np.sqrt(ws) / (2 * np.pi) kuu = fem.get_stiffness_matrix(free=False)
