def eval(Zeta00, Zeta01, Zeta02, Zeta03, Uc):
'''
Returns a 4 x 3 array, containing the derivative of Wnc*Uc w.r.t the panel
vertices coordinates.
'''
R02 = Zeta02 - Zeta00
R13 = Zeta03 - Zeta01
crR02R13 = libalg.cross3d(R02, R13)
norm_crR02R13 = np.linalg.norm(crR02R13)
cub_crR02R13 = norm_crR02R13**3
Acr = libalg.cross3d(crR02R13, R13)
Bcr = libalg.cross3d(crR02R13, R02)
Cdot = np.dot(crR02R13, Uc)
uc_x, uc_y, uc_z = Uc
r02_x, r02_y, r02_z = R02
r13_x, r13_y, r13_z = R13
crR13Uc_x, crR13Uc_y, crR13Uc_z = libalg.cross3d(R13, Uc)
crR02Uc_x, crR02Uc_y, crR02Uc_z = libalg.cross3d(R02, Uc)
crR02R13_x, crR02R13_y, crR02R13_z = crR02R13
Acr_x, Acr_y, Acr_z = Acr
Bcr_x, Bcr_y, Bcr_z = Bcr
# dUnorm_dR.shape=(2,3)
dUnorm_dR = np.array(
[[
Acr_x * Cdot / cub_crR02R13 + crR13Uc_x / norm_crR02R13,
Acr_y * Cdot / cub_crR02R13 + crR13Uc_y / norm_crR02R13,
Acr_z * Cdot / cub_crR02R13 + crR13Uc_z / norm_crR02R13
],
[
-Bcr_x * Cdot / cub_crR02R13 - crR02Uc_x / norm_crR02R13,
-Bcr_y * Cdot / cub_crR02R13 - crR02Uc_y / norm_crR02R13,
-Bcr_z * Cdot / cub_crR02R13 - crR02Uc_z / norm_crR02R13
]])
# Allocate
dUnorm_dZeta = np.zeros((4, 3))
for vv in range(4): # loop through panel vertices
for cc_zeta in range(3): # loop panel vertices component
for rr in range(2): # loop segments R02, R13
for cc_rvec in range(3): # loop segment component
dUnorm_dZeta[vv,cc_zeta]+=\
dUnorm_dR[rr,cc_rvec]*dR_dZeta[vv,cc_zeta,rr,cc_rvec]
return dUnorm_dZeta
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 joukovski_qs_segment(zetaA,zetaB,v_mid,gamma=1.0,fact=0.5): ''' Joukovski force over vetices A and B produced by the segment A->B. The factor fact allows to compute directly the contribution over the vertices A and B (e.g. 0.5) or include DENSITY. ''' rab=zetaB-zetaA fs=libalg.cross3d(v_mid,rab) gfact=fact*gamma return gfact*fs
def panel_normal(ZetaPanel): ''' return normal of panel with vertiex coordinates ZetaPanel, where: ZetaPanel.shape=(4,3) ''' # build cross-vectors r02=ZetaPanel[2,:]-ZetaPanel[0,:] r13=ZetaPanel[3,:]-ZetaPanel[1,:] nvec=libalg.cross3d(r02,r13) nvec=nvec/libalg.norm3d(nvec) return nvec
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_seg_exp_loop(DerP, DerA, DerB, ZetaP, ZetaA, ZetaB, gamma_seg):
'''
Derivative of induced velocity Q w.r.t. collocation (DerC) and segment
coordinates in format.
To optimise performance, the function requires the derivative terms to be
pre-allocated and passed as input.
Each Der* term returns derivatives in the format
[ (x,y,z) of Zeta, (x,y,z) of Q]
Warning: to optimise performance, variables are scaled during the execution.
'''
RA = ZetaP - ZetaA
RB = ZetaP - ZetaB
RAB = ZetaB - ZetaA
Vcr = libalg.cross3d(RA, RB)
vcr2 = np.dot(Vcr, Vcr)
# numerical radious
vortex_radious_here = VORTEX_RADIUS * libalg.norm3d(RAB)
if vcr2 < vortex_radious_here**2:
return
# scaling
ra1, rb1 = libalg.norm3d(RA), libalg.norm3d(RB)
ra2, rb2 = ra1**2, rb1**2
rainv = 1. / ra1
rbinv = 1. / rb1
ra_dir, rb_dir = RA * rainv, RB * rbinv
ra3inv, rb3inv = rainv**3, rbinv**3
Vcr = Vcr / vcr2
diff_vec = ra_dir - rb_dir
vdot_prod = np.dot(diff_vec, RAB)
T2 = vdot_prod / vcr2
# Extract components
ra_x, ra_y, ra_z = RA
rb_x, rb_y, rb_z = RB
rab_x, rab_y, rab_z = RAB
vcr_x, vcr_y, vcr_z = Vcr
ra2_x, ra2_y, ra2_z = RA**2
rb2_x, rb2_y, rb2_z = RB**2
ra_vcr_x, ra_vcr_y, ra_vcr_z = 2. * libalg.cross3d(RA, Vcr)
rb_vcr_x, rb_vcr_y, rb_vcr_z = 2. * libalg.cross3d(RB, Vcr)
vcr_sca_x, vcr_sca_y, vcr_sca_z = Vcr * ra3inv
vcr_scb_x, vcr_scb_y, vcr_scb_z = Vcr * rb3inv
# # ### derivatives indices:
# # # the 1st is the component of the vaiable w.r.t derivative are taken.
# # # the 2nd is the component of the output
dQ_dRA = np.array([
[
-vdot_prod * rb_vcr_x * vcr_x + vcr_sca_x *
(rab_x *
(ra2 - ra2_x) - ra_x * ra_y * rab_y - ra_x * ra_z * rab_z),
-T2 * rb_z - vdot_prod * rb_vcr_x * vcr_y + vcr_sca_y *
(rab_x *
(ra2 - ra2_x) - ra_x * ra_y * rab_y - ra_x * ra_z * rab_z),
T2 * rb_y - vdot_prod * rb_vcr_x * vcr_z + vcr_sca_z *
(rab_x * (ra2 - ra2_x) - ra_x * ra_y * rab_y - ra_x * ra_z * rab_z)
],
[
T2 * rb_z - vdot_prod * rb_vcr_y * vcr_x + vcr_sca_x *
(rab_y *
(ra2 - ra2_y) - ra_x * ra_y * rab_x - ra_y * ra_z * rab_z),
-vdot_prod * rb_vcr_y * vcr_y + vcr_sca_y *
(rab_y *
(ra2 - ra2_y) - ra_x * ra_y * rab_x - ra_y * ra_z * rab_z),
-T2 * rb_x - vdot_prod * rb_vcr_y * vcr_z + vcr_sca_z *
(rab_y * (ra2 - ra2_y) - ra_x * ra_y * rab_x - ra_y * ra_z * rab_z)
],
[
-T2 * rb_y - vdot_prod * rb_vcr_z * vcr_x + vcr_sca_x *
(rab_z *
(ra2 - ra2_z) - ra_x * ra_z * rab_x - ra_y * ra_z * rab_y),
T2 * rb_x - vdot_prod * rb_vcr_z * vcr_y + vcr_sca_y *
(rab_z *
(ra2 - ra2_z) - ra_x * ra_z * rab_x - ra_y * ra_z * rab_y),
-vdot_prod * rb_vcr_z * vcr_z + vcr_sca_z *
(rab_z * (ra2 - ra2_z) - ra_x * ra_z * rab_x - ra_y * ra_z * rab_y)
]
])
dQ_dRB = np.array([[
vdot_prod * ra_vcr_x * vcr_x + vcr_scb_x *
(rab_x * (-rb2 + rb2_x) + rab_y * rb_x * rb_y + rab_z * rb_x * rb_z),
T2 * ra_z + vdot_prod * ra_vcr_x * vcr_y + vcr_scb_y *
(rab_x * (-rb2 + rb2_x) + rab_y * rb_x * rb_y + rab_z * rb_x * rb_z),
-T2 * ra_y + vdot_prod * ra_vcr_x * vcr_z + vcr_scb_z *
(rab_x * (-rb2 + rb2_x) + rab_y * rb_x * rb_y + rab_z * rb_x * rb_z)
],
[
-T2 * ra_z + vdot_prod * ra_vcr_y * vcr_x +
vcr_scb_x * (rab_x * rb_x * rb_y + rab_y *
(-rb2 + rb2_y) + rab_z * rb_y * rb_z),
vdot_prod * ra_vcr_y * vcr_y + vcr_scb_y *
(rab_x * rb_x * rb_y + rab_y *
(-rb2 + rb2_y) + rab_z * rb_y * rb_z),
T2 * ra_x + vdot_prod * ra_vcr_y * vcr_z +
vcr_scb_z * (rab_x * rb_x * rb_y + rab_y *
(-rb2 + rb2_y) + rab_z * rb_y * rb_z)
],
[
T2 * ra_y + vdot_prod * ra_vcr_z * vcr_x +
vcr_scb_x *
(rab_x * rb_x * rb_z + rab_y * rb_y * rb_z + rab_z *
(-rb2 + rb2_z)), -T2 * ra_x +
vdot_prod * ra_vcr_z * vcr_y + vcr_scb_y *
(rab_x * rb_x * rb_z + rab_y * rb_y * rb_z + rab_z *
(-rb2 + rb2_z)),
vdot_prod * ra_vcr_z * vcr_z + vcr_scb_z *
(rab_x * rb_x * rb_z + rab_y * rb_y * rb_z + rab_z *
(-rb2 + rb2_z))
]])
dQ_dRAB = np.array(
[[vcr_x * diff_vec[0], vcr_y * diff_vec[0], vcr_z * diff_vec[0]],
[vcr_x * diff_vec[1], vcr_y * diff_vec[1], vcr_z * diff_vec[1]],
[vcr_x * diff_vec[2], vcr_y * diff_vec[2], vcr_z * diff_vec[2]]])
DerP += (cfact_biot * gamma_seg) * (dQ_dRA + dQ_dRB).T # w.r.t. P
DerA += (cfact_biot * gamma_seg) * (-dQ_dRAB - dQ_dRA).T # w.r.t. A
DerB += (cfact_biot * gamma_seg) * (dQ_dRAB - dQ_dRB).T # w.r.t. B
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