def eval_seg_comp_loop(DerP, DerA, DerB, ZetaP, ZetaA, ZetaB, gamma_seg):
'''
Derivative of induced velocity Q w.r.t. collocation and segment coordinates
in format:
[ (x,y,z) of Q, (x,y,z) of Zeta ]
Warning: function optimised for performance. Variables are scaled during the
execution.
'''
Cfact = cfact_biot * gamma_seg
RA = ZetaP - ZetaA
RB = ZetaP - ZetaB
RAB = ZetaB - ZetaA
Vcr = libalg.cross3d(RA, RB)
vcr2 = np.dot(Vcr, Vcr)
# numerical radious
if vcr2 < (VORTEX_RADIUS_SQ * libalg.normsq3d(RAB)):
return
### other constants
ra1, rb1 = libalg.norm3d(RA), libalg.norm3d(RB)
rainv = 1. / ra1
rbinv = 1. / rb1
Tv = RA * rainv - RB * rbinv
dotprod = np.dot(RAB, Tv)
### --------------------------------------------- cross-product derivatives
# lower triangular part only
vcr2inv = 1. / vcr2
vcr4inv = vcr2inv * vcr2inv
diag_fact = Cfact * vcr2inv * dotprod
off_fact = -2. * Cfact * vcr4inv * dotprod
Dvcross = [[diag_fact + off_fact * Vcr[0]**2],
[off_fact * Vcr[0] * Vcr[1], diag_fact + off_fact * Vcr[1]**2],
[
off_fact * Vcr[0] * Vcr[2], off_fact * Vcr[1] * Vcr[2],
diag_fact + off_fact * Vcr[2]**2
]]
### ------------------------------------------ difference terms derivatives
Vsc = Vcr * vcr2inv * Cfact
Ddiff = np.array([RAB * Vsc[0], RAB * Vsc[1], RAB * Vsc[2]])
dQ_dRAB = np.array([Tv * Vsc[0], Tv * Vsc[1], Tv * Vsc[2]])
### ---------------------------------------------- Final assembly (crucial)
# ps: calling Dvcross_by_skew3d does not slow down execution.
dQ_dRA=Dvcross_by_skew3d(Dvcross,-RB)\
+np.dot(Ddiff, der_runit(RA,rainv,-rainv**3))
dQ_dRB=Dvcross_by_skew3d(Dvcross, RA)\
-np.dot(Ddiff, der_runit(RB,rbinv,-rbinv**3))
DerP += dQ_dRA + dQ_dRB # w.r.t. P
DerA -= dQ_dRAB + dQ_dRA # w.r.t. A
DerB += dQ_dRAB - dQ_dRB # w.r.t. B
def biot_segment(zetaP,zetaA,zetaB,gamma=1.0): ''' Induced velocity of segment A_>B of circulation gamma over point P. ''' # differences ra=zetaP-zetaA rb=zetaP-zetaB rab=zetaB-zetaA ra_norm,rb_norm=libalg.norm3d(ra),libalg.norm3d(rb) vcross=libalg.cross3d(ra,rb) vcross_sq=np.dot(vcross,vcross) # numerical radious if vcross_sq<(VORTEX_RADIUS_SQ*libalg.normsq3d(rab)): return np.zeros((3,)) q=((cfact_biot*gamma/vcross_sq)*\ ( np.dot(rab,ra)/ra_norm - np.dot(rab,rb)/rb_norm)) * vcross return q
def biot_panel_fast(zetaC,ZetaPanel,gamma=1.0): ''' Induced velocity over point ZetaC of a panel of vertices coordinates ZetaPanel and circulaiton gamma, where: ZetaPanel.shape=(4,3)=[vertex local number, (x,y,z) component] ''' Cfact=cfact_biot*gamma q=np.zeros((3,)) R_list = zetaC-ZetaPanel Runit_list=[R_list[ii]/libalg.norm3d(R_list[ii]) for ii in svec] for aa,bb in LoopPanel: RAB=ZetaPanel[bb,:]-ZetaPanel[aa,:] # segment vector Vcr = libalg.cross3d(R_list[aa],R_list[bb]) vcr2=np.dot(Vcr,Vcr) if vcr2<(VORTEX_RADIUS_SQ*libalg.normsq3d(RAB)): continue q+=( (Cfact/vcr2)*np.dot(RAB,Runit_list[aa]-Runit_list[bb]) ) *Vcr return q
def eval_panel_fast_coll(zetaP, ZetaPanel, gamma_pan=1.0):
'''
Computes derivatives of induced velocity w.r.t. coordinates of target point,
zetaP, coordinates. Returns two elements:
- DerP: derivative of induced velocity w.r.t. ZetaP, with:
DerP.shape=(3,3) : DerC[ Uind_{x,y,z}, ZetaC_{x,y,z} ]
The function is based on eval_panel_fast, but does not perform operations
required to compute the derivatives w.r.t. the panel coordinates.
'''
DerP = np.zeros((3, 3))
### ---------------------------------------------- Compute common variables
# these are constants or variables depending only on vertices and P coords
Cfact = cfact_biot * gamma_pan
# distance vertex ii-th from P
R_list = zetaP - ZetaPanel
r1_list = [libalg.norm3d(R_list[ii]) for ii in svec]
r1inv_list = [1. / r1_list[ii] for ii in svec]
Runit_list = [R_list[ii] * r1inv_list[ii] for ii in svec]
Der_runit_list = [
der_runit(R_list[ii], r1inv_list[ii], -r1inv_list[ii]**3)
for ii in svec
]
### ------------------------------------------------- Loop through segments
for aa, bb in LoopPanel:
RAB = ZetaPanel[bb, :] - ZetaPanel[aa, :] # segment vector
Vcr = libalg.cross3d(R_list[aa], R_list[bb])
vcr2 = np.dot(Vcr, Vcr)
if vcr2 < (VORTEX_RADIUS_SQ * libalg.normsq3d(RAB)):
continue
Tv = Runit_list[aa] - Runit_list[bb]
dotprod = np.dot(RAB, Tv)
### ----------------------------------------- cross-product derivatives
# lower triangular part only
vcr2inv = 1. / vcr2
vcr4inv = vcr2inv * vcr2inv
diag_fact = Cfact * vcr2inv * dotprod
off_fact = -2. * Cfact * vcr4inv * dotprod
Dvcross = [[diag_fact + off_fact * Vcr[0]**2],
[
off_fact * Vcr[0] * Vcr[1],
diag_fact + off_fact * Vcr[1]**2
],
[
off_fact * Vcr[0] * Vcr[2], off_fact * Vcr[1] * Vcr[2],
diag_fact + off_fact * Vcr[2]**2
]]
### ---------------------------------------- difference term derivative
Vsc = Vcr * vcr2inv * Cfact
Ddiff = np.array([RAB * Vsc[0], RAB * Vsc[1], RAB * Vsc[2]])
### ------------------------------------------ Final assembly (crucial)
# dQ_dRA=Dvcross_by_skew3d(Dvcross,-R_list[bb])\
# +np.dot(Ddiff, Der_runit_list[aa] )
# dQ_dRB=Dvcross_by_skew3d(Dvcross, R_list[aa])\
# -np.dot(Ddiff, Der_runit_list[bb] )
DerP += Dvcross_by_skew3d(Dvcross,RAB)+\
+np.dot(Ddiff, Der_runit_list[aa]-Der_runit_list[bb] )
return DerP
def eval_panel_fast(zetaP, ZetaPanel, gamma_pan=1.0):
'''
Computes derivatives of induced velocity w.r.t. coordinates of target point,
zetaP, and panel coordinates. Returns two elements:
- DerP: derivative of induced velocity w.r.t. ZetaP, with:
DerP.shape=(3,3) : DerC[ Uind_{x,y,z}, ZetaC_{x,y,z} ]
- DerVertices: derivative of induced velocity wrt panel vertices, with:
DerVertices.shape=(4,3,3) :
DerVertices[ vertex number {0,1,2,3}, Uind_{x,y,z}, ZetaC_{x,y,z} ]
The function is based on eval_panel_comp, but minimises operationsby
recycling variables.
'''
DerP = np.zeros((3, 3))
DerVertices = np.zeros((4, 3, 3))
### ---------------------------------------------- Compute common variables
# these are constants or variables depending only on vertices and P coords
Cfact = cfact_biot * gamma_pan
# distance vertex ii-th from P
R_list = zetaP - ZetaPanel
r1_list = [libalg.norm3d(R_list[ii]) for ii in svec]
r1inv_list = [1. / r1_list[ii] for ii in svec]
Runit_list = [R_list[ii] * r1inv_list[ii] for ii in svec]
Der_runit_list = [
der_runit(R_list[ii], r1inv_list[ii], -r1inv_list[ii]**3)
for ii in svec
]
### ------------------------------------------------- Loop through segments
for aa, bb in LoopPanel:
RAB = ZetaPanel[bb, :] - ZetaPanel[aa, :] # segment vector
Vcr = libalg.cross3d(R_list[aa], R_list[bb])
vcr2 = np.dot(Vcr, Vcr)
if vcr2 < (VORTEX_RADIUS_SQ * libalg.normsq3d(RAB)):
continue
Tv = Runit_list[aa] - Runit_list[bb]
dotprod = np.dot(RAB, Tv)
### ----------------------------------------- cross-product derivatives
# lower triangular part only
vcr2inv = 1. / vcr2
vcr4inv = vcr2inv * vcr2inv
diag_fact = Cfact * vcr2inv * dotprod
off_fact = -2. * Cfact * vcr4inv * dotprod
Dvcross = [[diag_fact + off_fact * Vcr[0]**2],
[
off_fact * Vcr[0] * Vcr[1],
diag_fact + off_fact * Vcr[1]**2
],
[
off_fact * Vcr[0] * Vcr[2], off_fact * Vcr[1] * Vcr[2],
diag_fact + off_fact * Vcr[2]**2
]]
### ---------------------------------------- difference term derivative
Vsc = Vcr * vcr2inv * Cfact
Ddiff = np.array([RAB * Vsc[0], RAB * Vsc[1], RAB * Vsc[2]])
### ---------------------------------------------------- RAB derivative
dQ_dRAB = np.array([Tv * Vsc[0], Tv * Vsc[1], Tv * Vsc[2]])
### ------------------------------------------ Final assembly (crucial)
dQ_dRA=Dvcross_by_skew3d(Dvcross,-R_list[bb])\
+np.dot(Ddiff, Der_runit_list[aa] )
dQ_dRB=Dvcross_by_skew3d(Dvcross, R_list[aa])\
-np.dot(Ddiff, Der_runit_list[bb] )
DerP += dQ_dRA + dQ_dRB # w.r.t. P
DerVertices[aa, :, :] -= dQ_dRAB + dQ_dRA # w.r.t. A
DerVertices[bb, :, :] += dQ_dRAB - dQ_dRB # w.r.t. B
# ### collocation point only
# DerP +=Dvcross_by_skew3d(Dvcross,RA-RB)+np.dot(Ddiff,
# der_runit(RA,rainv,minus_rainv3)-der_runit(RB,rbinv,minus_rbinv3))
return DerP, DerVertices