def compute(self, inputs, outputs):
dtype = float
if self.under_complex_step:
dtype = complex
self.T = np.zeros((3, 3),dtype=dtype)
self.x_gl = np.array([1, 0, 0],dtype=dtype)
radius = inputs['radius']
disp = inputs['disp']
nodes = inputs['nodes']
T = self.T
E = self.E
G = self.G
x_gl = self.x_gl
num_elems = self.ny - 1
for ielem in range(num_elems):
P0 = nodes[ielem, :]
P1 = nodes[ielem+1, :]
L = norm(P1 - P0)
x_loc = unit(P1 - P0)
y_loc = unit(np.cross(x_loc, x_gl))
z_loc = unit(np.cross(x_loc, y_loc))
T[0, :] = x_loc
T[1, :] = y_loc
T[2, :] = z_loc
u0x, u0y, u0z = T.dot(disp[ielem, :3])
r0x, r0y, r0z = T.dot(disp[ielem, 3:])
u1x, u1y, u1z = T.dot(disp[ielem+1, :3])
r1x, r1y, r1z = T.dot(disp[ielem+1, 3:])
tmp = np.sqrt((r1y - r0y)**2 + (r1z - r0z)**2)
sxx0 = E * (u1x - u0x) / L + E * radius[ielem] / L * tmp
sxx1 = E * (u0x - u1x) / L + E * radius[ielem] / L * tmp
sxt = G * radius[ielem] * (r1x - r0x) / L
outputs['vonmises'][ielem, 0] = np.sqrt(sxx0**2 + 3 * sxt**2)
outputs['vonmises'][ielem, 1] = np.sqrt(sxx1**2 + 3 * sxt**2)
def compute(self, inputs, outputs):
radius = inputs['radius']
disp = inputs['disp']
nodes = inputs['nodes']
T = self.T
E = self.E
G = self.G
x_gl = self.x_gl
if fortran_flag:
vm = OAS_API.oas_api.calc_vonmises(nodes, radius, disp, E, G, x_gl)
outputs['vonmises'] = vm
else:
num_elems = self.ny - 1
for ielem in range(self.ny - 1):
P0 = nodes[ielem, :]
P1 = nodes[ielem + 1, :]
L = norm(P1 - P0)
x_loc = unit(P1 - P0)
y_loc = unit(np.cross(x_loc, x_gl))
z_loc = unit(np.cross(x_loc, y_loc))
T[0, :] = x_loc
T[1, :] = y_loc
T[2, :] = z_loc
u0x, u0y, u0z = T.dot(disp[ielem, :3])
r0x, r0y, r0z = T.dot(disp[ielem, 3:])
u1x, u1y, u1z = T.dot(disp[ielem + 1, :3])
r1x, r1y, r1z = T.dot(disp[ielem + 1, 3:])
tmp = np.sqrt((r1y - r0y)**2 + (r1z - r0z)**2)
sxx0 = E * (u1x - u0x) / L + E * radius[ielem] / L * tmp
sxx1 = E * (u0x - u1x) / L + E * radius[ielem] / L * tmp
sxt = G * radius[ielem] * (r1x - r0x) / L
outputs['vonmises'][ielem, 0] = np.sqrt(sxx0**2 + 3 * sxt**2)
outputs['vonmises'][ielem, 1] = np.sqrt(sxx1**2 + 3 * sxt**2)
def compute(self, inputs, outputs):
dtype = float
if self.under_complex_step:
dtype = complex
self.T = np.zeros((3, 3), dtype=dtype)
self.x_gl = np.array([1, 0, 0], dtype=dtype)
radius = inputs['radius']
disp = inputs['disp']
nodes = inputs['nodes']
T = self.T
E = self.E
G = self.G
x_gl = self.x_gl
num_elems = self.ny - 1
for ielem in range(num_elems):
P0 = nodes[ielem, :]
P1 = nodes[ielem + 1, :]
L = norm(P1 - P0)
x_loc = unit(P1 - P0)
y_loc = unit(np.cross(x_loc, x_gl))
z_loc = unit(np.cross(x_loc, y_loc))
T[0, :] = x_loc
T[1, :] = y_loc
T[2, :] = z_loc
u0x, u0y, u0z = T.dot(disp[ielem, :3])
r0x, r0y, r0z = T.dot(disp[ielem, 3:])
u1x, u1y, u1z = T.dot(disp[ielem + 1, :3])
r1x, r1y, r1z = T.dot(disp[ielem + 1, 3:])
tmp = np.sqrt((r1y - r0y)**2 + (r1z - r0z)**2)
sxx0 = E * (u1x - u0x) / L + E * radius[ielem] / L * tmp
sxx1 = E * (u0x - u1x) / L + E * radius[ielem] / L * tmp
sxt = G * radius[ielem] * (r1x - r0x) / L
outputs['vonmises'][ielem, 0] = np.sqrt(sxx0**2 + 3 * sxt**2)
outputs['vonmises'][ielem, 1] = np.sqrt(sxx1**2 + 3 * sxt**2)
def compute(self, inputs, outputs):
disp = inputs['disp']
nodes = inputs['nodes']
A_enc = inputs['A_enc']
Qy = inputs['Qz']
J = inputs['J']
htop = inputs['htop']
hbottom = inputs['hbottom']
hfront = inputs['hfront']
hrear = inputs['hrear']
spar_thickness = inputs['spar_thickness']
vonmises = outputs['vonmises']
# Only use complex type for these arrays if we're using cs to check derivs
dtype = type(disp[0, 0])
T = np.zeros((3, 3), dtype=dtype)
x_gl = np.array([1, 0, 0], dtype=dtype)
E = self.E
G = self.G
num_elems = self.ny - 1
for ielem in range(num_elems):
P0 = nodes[ielem, :]
P1 = nodes[ielem + 1, :]
L = norm(P1 - P0)
x_loc = unit(P1 - P0)
y_loc = unit(np.cross(x_loc, x_gl))
z_loc = unit(np.cross(x_loc, y_loc))
T[0, :] = x_loc
T[1, :] = y_loc
T[2, :] = z_loc
u0x, u0y, u0z = T.dot(disp[ielem, :3])
r0x, r0y, r0z = T.dot(disp[ielem, 3:])
u1x, u1y, u1z = T.dot(disp[ielem + 1, :3])
r1x, r1y, r1z = T.dot(disp[ielem + 1, 3:])
# this is stress = modulus * strain; positive is tensile
axial_stress = E * (u1x - u0x) / L
# this is Torque / (2 * thickness_min * Area_enclosed)
torsion_stress = G * J[ielem] / L * (
r1x - r0x) / 2 / spar_thickness[ielem] / A_enc[ielem]
# this is moment * h / I
top_bending_stress = E / (L**2) * (
6 * u0y + 2 * r0z * L - 6 * u1y + 4 * r1z * L) * htop[ielem]
# this is moment * h / I
bottom_bending_stress = -E / (L**2) * (
6 * u0y + 2 * r0z * L - 6 * u1y + 4 * r1z * L) * hbottom[ielem]
# this is moment * h / I
front_bending_stress = -E / (L**2) * (
-6 * u0z + 2 * r0y * L + 6 * u1z + 4 * r1y * L) * hfront[ielem]
# this is moment * h / I
rear_bending_stress = E / (L**2) * (
-6 * u0z + 2 * r0y * L + 6 * u1z + 4 * r1y * L) * hrear[ielem]
# shear due to bending (VQ/It) note: the I used to get V cancels the other I
vertical_shear = E / (L**3) * (-12 * u0y - 6 * r0z * L + 12 * u1y -
6 * r1z * L) * Qy[ielem] / (
2 * spar_thickness[ielem])
# print("==========",ielem,"================")
# print("vertical_shear", vertical_shear)
# print("top",top_bending_stress)
# print("bottom",bottom_bending_stress)
# print("front",front_bending_stress)
# print("rear",rear_bending_stress)
# print("axial", axial_stress)
# print("torsion", torsion_stress)
# The 4 stress combinations:
vonmises[ielem, 0] = np.sqrt(
(top_bending_stress + rear_bending_stress + axial_stress)**2 +
3 * torsion_stress**2) / self.tssf
vonmises[ielem,
1] = np.sqrt((bottom_bending_stress +
front_bending_stress + axial_stress)**2 +
3 * torsion_stress**2)
vonmises[ielem,
2] = np.sqrt((front_bending_stress + axial_stress)**2 +
3 * (torsion_stress - vertical_shear)**2)
vonmises[ielem, 3] = np.sqrt(
(rear_bending_stress + axial_stress)**2 + 3 *
(torsion_stress + vertical_shear)**2) / self.tssf
def compute_partials(self, inputs, partials):
radius = inputs['radius']
disp = inputs['disp']
nodes = inputs['nodes']
T = self.T
E = self.E
G = self.G
x_gl = self.x_gl
num_elems = self.ny - 1
for ielem in range(num_elems):
# Compute the coordinate delta between the two element end points
P0 = nodes[ielem, :]
P1 = nodes[ielem+1, :]
dP = P1 - P0
# Compute the derivative of element length
L = norm(dP)
dLddP = norm_d(dP)
# unit function converts a vector to a unit vector
# calculate the transormation to the local element frame.
# We use x_gl to provide a reference axis to reference
x_loc = unit(dP)
dxdP = unit_d(dP)
y_loc = unit(np.cross(x_loc, x_gl))
dtmpdx, _ = cross_d(x_loc, x_gl)
dydtmp = unit_d(np.cross(x_loc, x_gl))
dydP = dydtmp.dot(dtmpdx).dot(dxdP)
z_loc = unit(np.cross(x_loc, y_loc))
dtmpdx, dtmpdy = cross_d(x_loc,y_loc)
dzdtmp = unit_d(np.cross(x_loc, y_loc))
dzdP = dzdtmp.dot(dtmpdx).dot(dxdP) + dzdtmp.dot(dtmpdy).dot(dydP)
T[0, :] = x_loc
T[1, :] = y_loc
T[2, :] = z_loc
u0x = x_loc.dot(disp[ielem, :3])
r0x, r0y, r0z = T.dot(disp[ielem, 3:])
u1x = x_loc.dot(disp[ielem+1, :3])
r1x, r1y, r1z = T.dot(disp[ielem+1, 3:])
# #$$$$$$$$$$$$$$$$$$$$$$$$$$
# The derivatives of the above code wrt displacement all boil down to sections of the T matrix
dxddisp = T[0,:]
dyddisp = T[1,:]
dzddisp = T[2,:]
#The derivatives of the above code wrt T all boil down to sections of the #displacement vector
du0dloc = disp[ielem, :3]
dr0dloc = disp[ielem, 3:]
du1dloc = disp[ielem+1, :3]
dr1dloc = disp[ielem+1, 3:]
#$$$$$$$$$$$$$$$$$$$$$$$$$$
# Original code
# $$$$$$$$$$$$
tmp = np.sqrt((r1y - r0y)**2 + (r1z - r0z)**2) + 1e-50 #added eps to avoid 0 disp singularity
sxx0 = E * (u1x - u0x) / L + E * radius[ielem] / L * tmp
sxx1 = E * (u0x - u1x) / L + E * radius[ielem] / L * tmp
sxt = G * radius[ielem] * (r1x - r0x) / L
# $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
dtmpdr0y = 1/tmp * (r1y - r0y)*-1
dtmpdr1y = 1/tmp * (r1y - r0y)
dtmpdr0z = 1/tmp * (r1z - r0z)*-1
dtmpdr1z = 1/tmp * (r1z - r0z)
# Combine all of the derivtives for tmp
dtmpdDisp = np.zeros(12)
dr0xdDisp = np.zeros(12)
dr1xdDisp = np.zeros(12)
#r0 term
dr0xdDisp[3:6] = dxddisp
dr1xdDisp[9:12] = dxddisp
dtmpdDisp[3:6] = dtmpdr0y*dyddisp
dtmpdDisp[3:6] += dtmpdr0z*dzddisp
dtmpdDisp[9:12] = dtmpdr1y*dyddisp
dtmpdDisp[9:12] += dtmpdr1z*dzddisp
# x_loc, y_loc and z_loc terms
# (dttmpx_loc is zeros, so don't compute with it)
dtmpdy_loc = dtmpdr0y*dr0dloc + dtmpdr1y*dr1dloc
dtmpdz_loc = dtmpdr0z*dr0dloc + dtmpdr1z*dr1dloc
dtmpdP = dtmpdy_loc.dot(dydP) + dtmpdz_loc.dot(dzdP)
dsxx0dtmp = E * radius[ielem] / L
dsxx0du0x = -E / L
dsxx0du1x = E / L
dsxx0dL = -E * (u1x - u0x) / (L*L) - E * radius[ielem] / (L*L) * tmp
dsxx1dtmp = E * radius[ielem] / L
dsxx1du0x = E / L
dsxx1du1x = -E / L
dsxx1dL = -E * (u0x - u1x) / (L*L) - E * radius[ielem] / (L*L) * tmp
dsxx0dP = dsxx0dtmp * dtmpdP + \
dsxx0du0x*du0dloc.dot(dxdP) + dsxx0du1x*du1dloc.dot(dxdP)+\
dsxx0dL*dLddP
dsxx1dP = dsxx1dtmp * dtmpdP + \
dsxx1du0x*du0dloc.dot(dxdP)+dsxx1du1x*du1dloc.dot(dxdP)+\
dsxx1dL*dLddP
# Combine sxx0 and sxx1 terms
# Start with the tmp term
dsxx0dDisp = dsxx0dtmp * dtmpdDisp
dsxx1dDisp = dsxx1dtmp * dtmpdDisp
# Now add the direct u dep
dsxx0dDisp[0:3] = dsxx0du0x * dxddisp
dsxx0dDisp[6:9] = dsxx0du1x * dxddisp
dsxx1dDisp[0:3] = dsxx1du0x * dxddisp
dsxx1dDisp[6:9] = dsxx1du1x * dxddisp
# Combine sxt term
dsxtdr0x = -G * radius[ielem] / L
dsxtdr1x = G * radius[ielem] / L
dsxtdL = - G * radius[ielem] * (r1x - r0x) / (L*L)
dsxtdP = dsxtdr0x*(dr0dloc.dot(dxdP)) + dsxtdr1x*(dr1dloc.dot(dxdP)) + \
dsxtdL*dLddP
#disp
dsxtdDisp = dsxtdr0x * dr0xdDisp + dsxtdr1x * dr1xdDisp
#radius derivatives
dsxxdrad = E / L * tmp
dsxtdrad = G * (r1x - r0x)/L
fact = 1.0 / (np.sqrt(sxx0**2 + 3 * sxt**2))
dVm0dsxx0 = sxx0 * fact
dVm0dsxt = 3 * sxt * fact
fact = 1.0 / (np.sqrt(sxx1**2 + 3 * sxt**2))
dVm1dsxx1 = sxx1 * fact
dVm1dsxt = 3 * sxt * fact
ii = 2 * ielem
partials['vonmises', 'radius'][ii] = dVm0dsxx0*dsxxdrad + dVm0dsxt*dsxtdrad
partials['vonmises', 'radius'][ii+1] = dVm1dsxx1*dsxxdrad + dVm1dsxt*dsxtdrad
ii = 24 * ielem
partials['vonmises', 'disp'][ii:ii+12] = dVm0dsxx0*dsxx0dDisp + dVm0dsxt*dsxtdDisp
partials['vonmises', 'disp'][ii+12:ii+24] = dVm1dsxx1*dsxx1dDisp + dVm1dsxt*dsxtdDisp
# Compute terms for the nodes
ii = 12 * ielem
dVm0_dnode = dVm0dsxx0*dsxx0dP + dVm0dsxt*dsxtdP
partials['vonmises', 'nodes'][ii:ii+3] = -dVm0_dnode
partials['vonmises', 'nodes'][ii+3:ii+6] = dVm0_dnode
dVM1_dnode = dVm1dsxx1*dsxx1dP + dVm1dsxt*dsxtdP
partials['vonmises', 'nodes'][ii+6:ii+9] = -dVM1_dnode
partials['vonmises', 'nodes'][ii+9:ii+12] = dVM1_dnode
def compute_partials(self, inputs, partials):
radius = inputs['radius']
disp = inputs['disp']
nodes = inputs['nodes']
T = self.T
E = self.E
G = self.G
x_gl = self.x_gl
num_elems = self.ny - 1
for ielem in range(num_elems):
# Compute the coordinate delta between the two element end points
P0 = nodes[ielem, :]
P1 = nodes[ielem + 1, :]
dP = P1 - P0
# and its derivatives
ddPdP0 = -1.0 * np.eye(3)
ddPdP1 = 1.0 * np.eye(3)
# Compute the element length and its derivative
L = norm(dP)
dLddP = norm_d(dP)
# unit function converts a vector to a unit vector
# calculate the transormation to the local element frame.
# We use x_gl to provide a reference axis to reference
x_loc = unit(dP)
dxdP = unit_d(dP)
y_loc = unit(np.cross(x_loc, x_gl))
dtmpdx, dummy = cross_d(x_loc, x_gl)
dydtmp = unit_d(np.cross(x_loc, x_gl))
dydP = dydtmp.dot(dtmpdx).dot(dxdP)
z_loc = unit(np.cross(x_loc, y_loc))
dtmpdx, dtmpdy = cross_d(x_loc, y_loc)
dzdtmp = unit_d(np.cross(x_loc, y_loc))
dzdP = dzdtmp.dot(dtmpdx).dot(dxdP) + dzdtmp.dot(dtmpdy).dot(dydP)
T[0, :] = x_loc
T[1, :] = y_loc
T[2, :] = z_loc
#$$$$$$$$$$$$$$$$$$$$$$$$$$
# Original code
# $$$$$$$$$$$$
u0x, u0y, u0z = T.dot(disp[ielem, :3])
r0x, r0y, r0z = T.dot(disp[ielem, 3:])
u1x, u1y, u1z = T.dot(disp[ielem + 1, :3])
r1x, r1y, r1z = T.dot(disp[ielem + 1, 3:])
# #$$$$$$$$$$$$$$$$$$$$$$$$$$
# The derivatives of the above code wrt displacement all boil down to sections of the T matrix
dxddisp = T[0, :]
dyddisp = T[1, :]
dzddisp = T[2, :]
#The derivatives of the above code wrt T all boil down to sections of the #displacement vector
# du0dT = disp[ielem, :3]
# dr0dT = disp[ielem, 3:]
# du1dT = disp[ielem+1, :3]
# dr1dT = disp[ielem+1, 3:]
du0dloc = disp[ielem, :3]
dr0dloc = disp[ielem, 3:]
du1dloc = disp[ielem + 1, :3]
dr1dloc = disp[ielem + 1, 3:]
#$$$$$$$$$$$$$$$$$$$$$$$$$$
# Original code
# $$$$$$$$$$$$
tmp = np.sqrt((r1y - r0y)**2 + (r1z - r0z)**2)
sxx0 = E * (u1x - u0x) / L + E * radius[ielem] / L * tmp
sxx1 = E * (u0x - u1x) / L + E * radius[ielem] / L * tmp
sxt = G * radius[ielem] * (r1x - r0x) / L
# $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
dtmpdr0y = 1 / tmp * (r1y - r0y) * -1
dtmpdr1y = 1 / tmp * (r1y - r0y)
dtmpdr0z = 1 / tmp * (r1z - r0z) * -1
dtmpdr1z = 1 / tmp * (r1z - r0z)
# Combine all of the derivtives for tmp
dtmpdDisp = np.zeros(2 * 6)
dr0xdDisp = np.zeros(2 * 6)
dr0ydDisp = np.zeros(2 * 6)
dr0zdDisp = np.zeros(2 * 6)
dr1xdDisp = np.zeros(2 * 6)
dr1ydDisp = np.zeros(2 * 6)
dr1zdDisp = np.zeros(2 * 6)
idx1 = 3
idx2 = 6 + 3
#r0 term
dr0xdDisp[idx1:idx1 + 3] = dxddisp
dr0ydDisp[idx1:idx1 + 3] = dyddisp
dr0zdDisp[idx1:idx1 + 3] = dzddisp
dr1xdDisp[idx2:idx2 + 3] = dxddisp
dr1ydDisp[idx2:idx2 + 3] = dyddisp
dr1zdDisp[idx2:idx2 + 3] = dzddisp
dtmpdDisp = (dtmpdr0y*dr0ydDisp+dtmpdr1y*dr1ydDisp+\
dtmpdr0z*dr0zdDisp+dtmpdr1z*dr1zdDisp)
# x_loc, y_loc and z_loc terms
dtmpdx_loc = np.array([0, 0, 0])
dtmpdy_loc = dtmpdr0y * dr0dloc + dtmpdr1y * dr1dloc
dtmpdz_loc = dtmpdr0z * dr0dloc + dtmpdr1z * dr1dloc
dtmpdP = dtmpdx_loc.dot(dxdP) + dtmpdy_loc.dot(
dydP) + dtmpdz_loc.dot(dzdP)
dsxx0dtmp = E * radius[ielem] / L
dsxx0du0x = -E / L
dsxx0du1x = E / L
dsxx0dL = -E * (u1x -
u0x) / (L * L) - E * radius[ielem] / (L * L) * tmp
dsxx1dtmp = E * radius[ielem] / L
dsxx1du0x = E / L
dsxx1du1x = -E / L
dsxx1dL = -E * (u0x -
u1x) / (L * L) - E * radius[ielem] / (L * L) * tmp
dsxx0dP = dsxx0dtmp * dtmpdP + \
dsxx0du0x*du0dloc.dot(dxdP) + dsxx0du1x*du1dloc.dot(dxdP)+\
dsxx0dL*dLddP
dsxx1dP = dsxx1dtmp * dtmpdP + \
dsxx1du0x*du0dloc.dot(dxdP)+dsxx1du1x*du1dloc.dot(dxdP)+\
dsxx1dL*dLddP
# Combine sxx0 and sxx1 terms
idx1 = 0
idx2 = 6
# Start with the tmp term
dsxx0dDisp = dsxx0dtmp * dtmpdDisp
dsxx1dDisp = dsxx1dtmp * dtmpdDisp
# Now add the direct u dep
dsxx0dDisp[idx1:idx1 + 3] = dsxx0du0x * dxddisp
dsxx0dDisp[idx2:idx2 + 3] = dsxx0du1x * dxddisp
dsxx1dDisp[idx1:idx1 + 3] = dsxx1du0x * dxddisp
dsxx1dDisp[idx2:idx2 + 3] = dsxx1du1x * dxddisp
# Combine sxt term
dsxtdDisp = np.zeros(2 * 6)
idx1 = 3
idx2 = 6 + 3
dsxtdr0x = -G * radius[ielem] / L
dsxtdr1x = G * radius[ielem] / L
dsxtdL = -G * radius[ielem] * (r1x - r0x) / (L * L)
dsxtdP = dsxtdr0x*(dr0dloc.dot(dxdP))+ dsxtdr1x*(dr1dloc.dot(dxdP))+\
dsxtdL*dLddP
#disp
dsxtdDisp = dsxtdr0x * dr0xdDisp + dsxtdr1x * dr1xdDisp
#radius derivatives
dsxx0drad = E / L * tmp
dsxx1drad = E / L * tmp
dsxtdrad = G * (r1x - r0x) / L
dVm0dsxx0 = (sxx0) / (np.sqrt(sxx0**2 + 3 * sxt**2))
dVm0dsxt = (3 * sxt) / (np.sqrt(sxx0**2 + 3 * sxt**2))
dVm1dsxx1 = (sxx1) / (np.sqrt(sxx1**2 + 3 * sxt**2))
dVm1dsxt = (3 * sxt) / (np.sqrt(sxx1**2 + 3 * sxt**2))
idx = ielem * 2
partials['vonmises', 'radius'][
idx, ielem] = dVm0dsxx0 * dsxx0drad + dVm0dsxt * dsxtdrad
partials['vonmises', 'radius'][
idx + 1, ielem] = dVm1dsxx1 * dsxx1drad + dVm1dsxt * dsxtdrad
idx2 = ielem * 6
partials['vonmises','disp'][idx,idx2:idx2+12] = partials['vonmises','disp'][idx,idx2:idx2+12]+\
(dVm0dsxx0*dsxx0dDisp+dVm0dsxt*dsxtdDisp )
partials['vonmises','disp'][idx+1,idx2:idx2+12] = partials['vonmises','disp'][idx+1,idx2:idx2+12]+\
(dVm1dsxx1*dsxx1dDisp+dVm1dsxt*dsxtdDisp )
# Compute terms for the nodes
idx3 = ielem * 3
partials['vonmises', 'nodes'][idx, idx3:idx3 + 3] = partials[
'vonmises',
'nodes'][idx, idx3:idx3 + 3] + dVm0dsxx0 * dsxx0dP.dot(
ddPdP0) + dVm0dsxt * dsxtdP.dot(ddPdP0)
partials['vonmises', 'nodes'][idx, idx3 + 3:idx3 + 6] = partials[
'vonmises',
'nodes'][idx, idx3 + 3:idx3 + 6] + dVm0dsxx0 * dsxx0dP.dot(
ddPdP1) + dVm0dsxt * dsxtdP.dot(ddPdP1)
partials['vonmises', 'nodes'][idx + 1, idx3:idx3 + 3] = partials[
'vonmises',
'nodes'][idx + 1, idx3:idx3 + 3] + dVm1dsxx1 * dsxx1dP.dot(
ddPdP0) + dVm1dsxt * dsxtdP.dot(ddPdP0)
partials['vonmises',
'nodes'][idx + 1, idx3 + 3:idx3 +
6] = partials['vonmises', 'nodes'][
idx + 1,
idx3 + 3:idx3 + 6] + dVm1dsxx1 * dsxx1dP.dot(
ddPdP1) + dVm1dsxt * dsxtdP.dot(ddPdP1)
def compute_partials(self, inputs, partials):
radius = inputs['radius']
disp = inputs['disp']
nodes = inputs['nodes']
T = self.T
E = self.E
G = self.G
x_gl = self.x_gl
num_elems = self.ny - 1
for ielem in range(num_elems):
# Compute the coordinate delta between the two element end points
P0 = nodes[ielem, :]
P1 = nodes[ielem + 1, :]
dP = P1 - P0
# Compute the derivative of element length
L = norm(dP)
dLddP = norm_d(dP)
# unit function converts a vector to a unit vector
# calculate the transormation to the local element frame.
# We use x_gl to provide a reference axis to reference
x_loc = unit(dP)
dxdP = unit_d(dP)
y_loc = unit(np.cross(x_loc, x_gl))
dtmpdx, _ = cross_d(x_loc, x_gl)
dydtmp = unit_d(np.cross(x_loc, x_gl))
dydP = dydtmp.dot(dtmpdx).dot(dxdP)
z_loc = unit(np.cross(x_loc, y_loc))
dtmpdx, dtmpdy = cross_d(x_loc, y_loc)
dzdtmp = unit_d(np.cross(x_loc, y_loc))
dzdP = dzdtmp.dot(dtmpdx).dot(dxdP) + dzdtmp.dot(dtmpdy).dot(dydP)
T[0, :] = x_loc
T[1, :] = y_loc
T[2, :] = z_loc
u0x = x_loc.dot(disp[ielem, :3])
r0x, r0y, r0z = T.dot(disp[ielem, 3:])
u1x = x_loc.dot(disp[ielem + 1, :3])
r1x, r1y, r1z = T.dot(disp[ielem + 1, 3:])
# #$$$$$$$$$$$$$$$$$$$$$$$$$$
# The derivatives of the above code wrt displacement all boil down to sections of the T matrix
dxddisp = T[0, :]
dyddisp = T[1, :]
dzddisp = T[2, :]
#The derivatives of the above code wrt T all boil down to sections of the #displacement vector
du0dloc = disp[ielem, :3]
dr0dloc = disp[ielem, 3:]
du1dloc = disp[ielem + 1, :3]
dr1dloc = disp[ielem + 1, 3:]
#$$$$$$$$$$$$$$$$$$$$$$$$$$
# Original code
# $$$$$$$$$$$$
tmp = np.sqrt(
(r1y - r0y)**2 +
(r1z - r0z)**2) + 1e-50 #added eps to avoid 0 disp singularity
sxx0 = E * (u1x - u0x) / L + E * radius[ielem] / L * tmp
sxx1 = E * (u0x - u1x) / L + E * radius[ielem] / L * tmp
sxt = G * radius[ielem] * (r1x - r0x) / L
# $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
dtmpdr0y = 1 / tmp * (r1y - r0y) * -1
dtmpdr1y = 1 / tmp * (r1y - r0y)
dtmpdr0z = 1 / tmp * (r1z - r0z) * -1
dtmpdr1z = 1 / tmp * (r1z - r0z)
# Combine all of the derivtives for tmp
dtmpdDisp = np.zeros(12)
dr0xdDisp = np.zeros(12)
dr1xdDisp = np.zeros(12)
#r0 term
dr0xdDisp[3:6] = dxddisp
dr1xdDisp[9:12] = dxddisp
dtmpdDisp[3:6] = dtmpdr0y * dyddisp
dtmpdDisp[3:6] += dtmpdr0z * dzddisp
dtmpdDisp[9:12] = dtmpdr1y * dyddisp
dtmpdDisp[9:12] += dtmpdr1z * dzddisp
# x_loc, y_loc and z_loc terms
# (dttmpx_loc is zeros, so don't compute with it)
dtmpdy_loc = dtmpdr0y * dr0dloc + dtmpdr1y * dr1dloc
dtmpdz_loc = dtmpdr0z * dr0dloc + dtmpdr1z * dr1dloc
dtmpdP = dtmpdy_loc.dot(dydP) + dtmpdz_loc.dot(dzdP)
dsxx0dtmp = E * radius[ielem] / L
dsxx0du0x = -E / L
dsxx0du1x = E / L
dsxx0dL = -E * (u1x -
u0x) / (L * L) - E * radius[ielem] / (L * L) * tmp
dsxx1dtmp = E * radius[ielem] / L
dsxx1du0x = E / L
dsxx1du1x = -E / L
dsxx1dL = -E * (u0x -
u1x) / (L * L) - E * radius[ielem] / (L * L) * tmp
dsxx0dP = dsxx0dtmp * dtmpdP + \
dsxx0du0x*du0dloc.dot(dxdP) + dsxx0du1x*du1dloc.dot(dxdP)+\
dsxx0dL*dLddP
dsxx1dP = dsxx1dtmp * dtmpdP + \
dsxx1du0x*du0dloc.dot(dxdP)+dsxx1du1x*du1dloc.dot(dxdP)+\
dsxx1dL*dLddP
# Combine sxx0 and sxx1 terms
# Start with the tmp term
dsxx0dDisp = dsxx0dtmp * dtmpdDisp
dsxx1dDisp = dsxx1dtmp * dtmpdDisp
# Now add the direct u dep
dsxx0dDisp[0:3] = dsxx0du0x * dxddisp
dsxx0dDisp[6:9] = dsxx0du1x * dxddisp
dsxx1dDisp[0:3] = dsxx1du0x * dxddisp
dsxx1dDisp[6:9] = dsxx1du1x * dxddisp
# Combine sxt term
dsxtdr0x = -G * radius[ielem] / L
dsxtdr1x = G * radius[ielem] / L
dsxtdL = -G * radius[ielem] * (r1x - r0x) / (L * L)
dsxtdP = dsxtdr0x*(dr0dloc.dot(dxdP)) + dsxtdr1x*(dr1dloc.dot(dxdP)) + \
dsxtdL*dLddP
#disp
dsxtdDisp = dsxtdr0x * dr0xdDisp + dsxtdr1x * dr1xdDisp
#radius derivatives
dsxxdrad = E / L * tmp
dsxtdrad = G * (r1x - r0x) / L
fact = 1.0 / (np.sqrt(sxx0**2 + 3 * sxt**2))
dVm0dsxx0 = sxx0 * fact
dVm0dsxt = 3 * sxt * fact
fact = 1.0 / (np.sqrt(sxx1**2 + 3 * sxt**2))
dVm1dsxx1 = sxx1 * fact
dVm1dsxt = 3 * sxt * fact
ii = 2 * ielem
partials['vonmises',
'radius'][ii] = dVm0dsxx0 * dsxxdrad + dVm0dsxt * dsxtdrad
partials['vonmises',
'radius'][ii +
1] = dVm1dsxx1 * dsxxdrad + dVm1dsxt * dsxtdrad
ii = 24 * ielem
partials[
'vonmises',
'disp'][ii:ii +
12] = dVm0dsxx0 * dsxx0dDisp + dVm0dsxt * dsxtdDisp
partials[
'vonmises',
'disp'][ii + 12:ii +
24] = dVm1dsxx1 * dsxx1dDisp + dVm1dsxt * dsxtdDisp
# Compute terms for the nodes
ii = 12 * ielem
dVm0_dnode = dVm0dsxx0 * dsxx0dP + dVm0dsxt * dsxtdP
partials['vonmises', 'nodes'][ii:ii + 3] = -dVm0_dnode
partials['vonmises', 'nodes'][ii + 3:ii + 6] = dVm0_dnode
dVM1_dnode = dVm1dsxx1 * dsxx1dP + dVm1dsxt * dsxtdP
partials['vonmises', 'nodes'][ii + 6:ii + 9] = -dVM1_dnode
partials['vonmises', 'nodes'][ii + 9:ii + 12] = dVM1_dnode
def compute(self, inputs, outputs):
disp = inputs['disp']
nodes = inputs['nodes']
A_enc = inputs['A_enc']
Qy = inputs['Qz']
Iz = inputs['Iz']
J = inputs['J']
htop = inputs['htop']
hbottom = inputs['hbottom']
hfront = inputs['hfront']
hrear = inputs['hrear']
spar_thickness = inputs['spar_thickness']
skin_thickness = inputs['skin_thickness']
vonmises = outputs['vonmises']
# Only use complex type for these arrays if we're using cs to check derivs
dtype = type(disp[0, 0])
T = np.zeros((3, 3), dtype=dtype)
x_gl = np.array([1, 0, 0], dtype=dtype)
E = self.E
G = self.G
num_elems = self.ny - 1
for ielem in range(num_elems):
P0 = nodes[ielem, :]
P1 = nodes[ielem+1, :]
L = norm(P1 - P0)
x_loc = unit(P1 - P0)
y_loc = unit(np.cross(x_loc, x_gl))
z_loc = unit(np.cross(x_loc, y_loc))
T[0, :] = x_loc
T[1, :] = y_loc
T[2, :] = z_loc
u0x, u0y, u0z = T.dot(disp[ielem, :3])
r0x, r0y, r0z = T.dot(disp[ielem, 3:])
u1x, u1y, u1z = T.dot(disp[ielem+1, :3])
r1x, r1y, r1z = T.dot(disp[ielem+1, 3:])
axial_stress = E * (u1x - u0x) / L # this is stress = modulus * strain; positive is tensile
torsion_stress = G * J[ielem] / L * (r1x - r0x) / 2 / spar_thickness[ielem] / A_enc[ielem] # this is Torque / (2 * thickness_min * Area_enclosed)
top_bending_stress = E / (L**2) * (6 * u0y + 2 * r0z * L - 6 * u1y + 4 * r1z * L ) * htop[ielem] # this is moment * htop / I
bottom_bending_stress = - E / (L**2) * (6 * u0y + 2 * r0z * L - 6 * u1y + 4 * r1z * L ) * hbottom[ielem] # this is moment * htop / I
front_bending_stress = - E / (L**2) * (-6 * u0z + 2 * r0y * L + 6 * u1z + 4 * r1y * L ) * hfront[ielem] # this is moment * htop / I
rear_bending_stress = E / (L**2) * (-6 * u0z + 2 * r0y * L + 6 * u1z + 4 * r1y * L ) * hrear[ielem] # this is moment * htop / I
vertical_shear = E / (L**3) *(-12 * u0y - 6 * r0z * L + 12 * u1y - 6 * r1z * L ) * Qy[ielem] / (2 * spar_thickness[ielem]) # shear due to bending (VQ/It) note: the I used to get V cancels the other I
# print("==========",ielem,"================")
# print("vertical_shear", vertical_shear)
# print("top",top_bending_stress)
# print("bottom",bottom_bending_stress)
# print("front",front_bending_stress)
# print("rear",rear_bending_stress)
# print("axial", axial_stress)
# print("torsion", torsion_stress)
vonmises[ielem, 0] = np.sqrt((top_bending_stress + rear_bending_stress + axial_stress)**2 + 3*torsion_stress**2) / self.tssf
vonmises[ielem, 1] = np.sqrt((bottom_bending_stress + front_bending_stress + axial_stress)**2 + 3*torsion_stress**2)
vonmises[ielem, 2] = np.sqrt((front_bending_stress + axial_stress)**2 + 3*(torsion_stress-vertical_shear)**2)
vonmises[ielem, 3] = np.sqrt((rear_bending_stress + axial_stress)**2 + 3*(torsion_stress+vertical_shear)**2) / self.tssf
def compute_partials(self, inputs, partials):
radius = inputs['radius']
disp = inputs['disp']
nodes = inputs['nodes']
T = self.T
E = self.E
G = self.G
x_gl = self.x_gl
num_elems = self.ny - 1
for ielem in range(num_elems):
# Compute the coordinate delta between the two element end points
P0 = nodes[ielem, :]
P1 = nodes[ielem+1, :]
dP = P1 - P0
# and its derivatives
ddPdP0 = -1.0*np.eye(3)
ddPdP1 = 1.0*np.eye(3)
# Compute the element length and its derivative
L = norm(dP)
dLddP = norm_d(dP)
# unit function converts a vector to a unit vector
# calculate the transormation to the local element frame.
# We use x_gl to provide a reference axis to reference
x_loc = unit(dP)
dxdP = unit_d(dP)
y_loc = unit(np.cross(x_loc, x_gl))
dtmpdx,dummy = cross_d(x_loc,x_gl)
dydtmp = unit_d(np.cross(x_loc, x_gl))
dydP = dydtmp.dot(dtmpdx).dot(dxdP)
z_loc = unit(np.cross(x_loc, y_loc))
dtmpdx,dtmpdy = cross_d(x_loc,y_loc)
dzdtmp = unit_d(np.cross(x_loc, y_loc))
dzdP = dzdtmp.dot(dtmpdx).dot(dxdP)+dzdtmp.dot(dtmpdy).dot(dydP)
T[0, :] = x_loc
T[1, :] = y_loc
T[2, :] = z_loc
#$$$$$$$$$$$$$$$$$$$$$$$$$$
# Original code
# $$$$$$$$$$$$
u0x, u0y, u0z = T.dot(disp[ielem, :3])
r0x, r0y, r0z = T.dot(disp[ielem, 3:])
u1x, u1y, u1z = T.dot(disp[ielem+1, :3])
r1x, r1y, r1z = T.dot(disp[ielem+1, 3:])
# #$$$$$$$$$$$$$$$$$$$$$$$$$$
# The derivatives of the above code wrt displacement all boil down to sections of the T matrix
dxddisp = T[0,:]
dyddisp = T[1,:]
dzddisp = T[2,:]
#The derivatives of the above code wrt T all boil down to sections of the #displacement vector
# du0dT = disp[ielem, :3]
# dr0dT = disp[ielem, 3:]
# du1dT = disp[ielem+1, :3]
# dr1dT = disp[ielem+1, 3:]
du0dloc = disp[ielem, :3]
dr0dloc = disp[ielem, 3:]
du1dloc = disp[ielem+1, :3]
dr1dloc = disp[ielem+1, 3:]
#$$$$$$$$$$$$$$$$$$$$$$$$$$
# Original code
# $$$$$$$$$$$$
tmp = np.sqrt((r1y - r0y)**2 + (r1z - r0z)**2) + 1e-50 #added eps to avoid 0 disp singularity
sxx0 = E * (u1x - u0x) / L + E * radius[ielem] / L * tmp
sxx1 = E * (u0x - u1x) / L + E * radius[ielem] / L * tmp
sxt = G * radius[ielem] * (r1x - r0x) / L
# $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
dtmpdr0y = 1/tmp * (r1y - r0y)*-1
dtmpdr1y = 1/tmp * (r1y - r0y)
dtmpdr0z = 1/tmp * (r1z - r0z)*-1
dtmpdr1z = 1/tmp * (r1z - r0z)
# Combine all of the derivtives for tmp
dtmpdDisp = np.zeros(2 * 6)
dr0xdDisp = np.zeros(2*6)
dr0ydDisp = np.zeros(2*6)
dr0zdDisp = np.zeros(2*6)
dr1xdDisp = np.zeros(2*6)
dr1ydDisp = np.zeros(2*6)
dr1zdDisp = np.zeros(2*6)
idx1 = 3
idx2 = 6 + 3
#r0 term
dr0xdDisp[idx1:idx1+3] = dxddisp
dr0ydDisp[idx1:idx1+3] = dyddisp
dr0zdDisp[idx1:idx1+3] = dzddisp
dr1xdDisp[idx2:idx2+3] = dxddisp
dr1ydDisp[idx2:idx2+3] = dyddisp
dr1zdDisp[idx2:idx2+3] = dzddisp
dtmpdDisp = (dtmpdr0y*dr0ydDisp+dtmpdr1y*dr1ydDisp+\
dtmpdr0z*dr0zdDisp+dtmpdr1z*dr1zdDisp)
# x_loc, y_loc and z_loc terms
dtmpdx_loc = np.array([0,0,0])
dtmpdy_loc = dtmpdr0y*dr0dloc + dtmpdr1y*dr1dloc
dtmpdz_loc = dtmpdr0z*dr0dloc + dtmpdr1z*dr1dloc
dtmpdP = dtmpdx_loc.dot(dxdP) + dtmpdy_loc.dot(dydP) + dtmpdz_loc.dot(dzdP)
dsxx0dtmp = E * radius[ielem] / L
dsxx0du0x = -E / L
dsxx0du1x = E / L
dsxx0dL = -E * (u1x - u0x) / (L*L) - E * radius[ielem] / (L*L) * tmp
dsxx1dtmp = E * radius[ielem] / L
dsxx1du0x = E / L
dsxx1du1x = -E / L
dsxx1dL = -E * (u0x - u1x) / (L*L) - E * radius[ielem] / (L*L) * tmp
dsxx0dP = dsxx0dtmp * dtmpdP + \
dsxx0du0x*du0dloc.dot(dxdP) + dsxx0du1x*du1dloc.dot(dxdP)+\
dsxx0dL*dLddP
dsxx1dP = dsxx1dtmp * dtmpdP + \
dsxx1du0x*du0dloc.dot(dxdP)+dsxx1du1x*du1dloc.dot(dxdP)+\
dsxx1dL*dLddP
# Combine sxx0 and sxx1 terms
idx1 = 0
idx2 = 6
# Start with the tmp term
dsxx0dDisp = dsxx0dtmp * dtmpdDisp
dsxx1dDisp = dsxx1dtmp * dtmpdDisp
# Now add the direct u dep
dsxx0dDisp[idx1:idx1+3] = dsxx0du0x * dxddisp
dsxx0dDisp[idx2:idx2+3] = dsxx0du1x * dxddisp
dsxx1dDisp[idx1:idx1+3] = dsxx1du0x * dxddisp
dsxx1dDisp[idx2:idx2+3] = dsxx1du1x * dxddisp
# Combine sxt term
dsxtdDisp = np.zeros(2 * 6)
idx1 = 3
idx2 = 6 + 3
dsxtdr0x = -G * radius[ielem] / L
dsxtdr1x = G * radius[ielem] / L
dsxtdL = - G * radius[ielem] * (r1x - r0x) / (L*L)
dsxtdP = dsxtdr0x*(dr0dloc.dot(dxdP))+ dsxtdr1x*(dr1dloc.dot(dxdP))+\
dsxtdL*dLddP
#disp
dsxtdDisp= dsxtdr0x * dr0xdDisp + dsxtdr1x * dr1xdDisp
#radius derivatives
dsxx0drad = E / L * tmp
dsxx1drad = E / L * tmp
dsxtdrad = G * (r1x - r0x)/L
dVm0dsxx0 = (sxx0)/(np.sqrt(sxx0**2 + 3 * sxt**2))
dVm0dsxt = (3*sxt)/(np.sqrt(sxx0**2 + 3 * sxt**2))
dVm1dsxx1 = (sxx1)/(np.sqrt(sxx1**2 + 3 * sxt**2))
dVm1dsxt = (3*sxt)/(np.sqrt(sxx1**2 + 3 * sxt**2))
idx = ielem*2
partials['vonmises','radius'][idx,ielem] = dVm0dsxx0*dsxx0drad+dVm0dsxt*dsxtdrad
partials['vonmises','radius'][idx+1,ielem] = dVm1dsxx1*dsxx1drad+dVm1dsxt*dsxtdrad
idx2 = ielem*6
partials['vonmises','disp'][idx,idx2:idx2+12] = (dVm0dsxx0*dsxx0dDisp+dVm0dsxt*dsxtdDisp )
partials['vonmises','disp'][idx+1,idx2:idx2+12] = (dVm1dsxx1*dsxx1dDisp+dVm1dsxt*dsxtdDisp )
# Compute terms for the nodes
idx3 = ielem*3
partials['vonmises','nodes'][idx,idx3:idx3+3] = dVm0dsxx0*dsxx0dP.dot(ddPdP0)+dVm0dsxt*dsxtdP.dot(ddPdP0)
partials['vonmises','nodes'][idx,idx3+3:idx3+6] = dVm0dsxx0*dsxx0dP.dot(ddPdP1)+dVm0dsxt*dsxtdP.dot(ddPdP1)
partials['vonmises','nodes'][idx+1,idx3:idx3+3] = dVm1dsxx1*dsxx1dP.dot(ddPdP0)+dVm1dsxt*dsxtdP.dot(ddPdP0)
partials['vonmises','nodes'][idx+1,idx3+3:idx3+6] = dVm1dsxx1*dsxx1dP.dot(ddPdP1)+dVm1dsxt*dsxtdP.dot(ddPdP1)
