def scalar_deriv(a, xList):
'''Compute derivatives of a scalar w.r.t. a list of valirables'''
Nx = len(xList)
Der = []
for ii in range(Nx):
Der.append(a.diff(xList[ii]))
return smarr.MutableDenseNDimArray(Der)
def skew(Av):
'''
Compute skew matrix of vector Av, such that:
skew(Av)*Bv = Av x Bv
'''
assert len(Av) == 3, 'Av must be a 3 elements array'
ax, ay, az = Av[0], Av[1], Av[2]
Askew = smarr.MutableDenseNDimArray([[0, -az, ay], [az, 0, -ax],
[-ay, ax, 0]])
return Askew
def subs(Av, expr_old, expr_new):
'''
Iteratively apply the subs method to each element of tensor.
'''
Av_sub = []
if len(Av.shape) == 1:
for ii in range(len(Av)):
Av_sub.append(Av[ii].subs(expr_old, expr_new))
elif len(Av.shape) == 2:
for ii in range(Av.shape[0]):
row = []
for jj in range(Av.shape[1]):
row.append(Av[ii, jj].subs(expr_old, expr_new))
Av_sub.append(row)
else:
raise NameError('Method not developed for 3D arrays!')
return smarr.MutableDenseNDimArray(Av_sub)
def simplify(Av):
'''
Simplify each element of matrix/array
'''
Av_simple = []
if len(Av.shape) == 1:
for ii in range(len(Av)):
Av_simple.append(Av[ii].simplify())
elif len(Av.shape) == 2:
for ii in range(Av.shape[0]):
row = []
for jj in range(Av.shape[1]):
row.append(Av[ii, jj].simplify())
Av_simple.append(row)
else:
raise NameError('Method not developed for 3D arrays!')
return smarr.MutableDenseNDimArray(Av_simple)
'''
import numpy as np
import sympy as sm
#import sympy.tensor as smtens
import sympy.tensor.array as smarr
import linfunc
# 1,2 local index in the element. Node 1 is "ahead"
Zeta01,Zeta02=sm.symbols('Zeta01 Zeta02', real=True)
zeta01_x,zeta02_x,zeta01_y,zeta02_y=\
sm.symbols('zeta01_x,zeta02_x,zeta01_y,zeta02_y', real=True)
V0_x,V0_y=sm.symbols('V0_x,V0_y', real=True)
V0=smarr.MutableDenseNDimArray([V0_x,V0_y])
Kskew2D=smarr.MutableDenseNDimArray([[0,-1],[1,0]])
# Position vectors
Zeta01=smarr.MutableDenseNDimArray([zeta01_x,zeta01_y])
Zeta02=smarr.MutableDenseNDimArray([zeta02_x,zeta02_y])
# define order of derivation
ZetaAllList=[zeta01_x,zeta02_x,zeta01_y,zeta02_y]
# Delta vector (and derivative)
Dzeta=Zeta02-Zeta01
Dzeta_unit=Dzeta/sm.sqrt(linfunc.scalar_product(Dzeta,Dzeta))
#Dzeta_unit=Dzeta_unit.simplify()
for ii in range(Nx):
Der.append(a.diff(xList[ii]))
return smarr.MutableDenseNDimArray(Der)
if __name__ == '__main__':
import numpy as np
print('Verification matrix product:')
# numerical matrix product
Alist = [[1, 4], [2, 5], [7, 4], [1, 9]]
Blist = [[5, 8, 4], [1, 9, 7]]
Cref = np.dot(np.array(Alist), np.array(Blist))
Cref = smarr.MutableDenseNDimArray(list(Cref))
# symbolic matrix product
Asm = smarr.Array(Alist)
Bsm = smarr.Array(Blist)
Csm = matrix_product(Asm, Bsm)
Csm_simple = simplify(Csm)
print(Csm_simple - Cref)
print('Verification cross product:')
for ii in range(3):
for jj in range(3):
Av = smarr.MutableDenseNDimArray([0, 0, 0])
Bv = smarr.MutableDenseNDimArray([0, 0, 0])
Av[ii] = 1
Bv[jj] = 1
Cv = cross_product(Av, Bv)
import numpy as np
import sympy as sm
import sympy.tensor.array as smarr
import linfunc
##### Define symbols
### vertices vectors
# coordinates
zetaA_x, zetaA_y, zetaA_z = sm.symbols('zetaA_x,zetaA_y,zetaA_z', real=True)
zetaB_x, zetaB_y, zetaB_z = sm.symbols('zetaB_x,zetaB_y,zetaB_z', real=True)
zetaP_x, zetaP_y, zetaP_z = sm.symbols('zetaP_x,zetaP_y,zetaP_z', real=True)
# vectors
ZetaA, ZetaB, ZetaP = sm.symbols('ZetaA ZetaB ZetaP', real=True)
ZetaA = smarr.MutableDenseNDimArray([zetaA_x, zetaA_y, zetaA_z])
ZetaB = smarr.MutableDenseNDimArray([zetaB_x, zetaB_y, zetaB_z])
ZetaP = smarr.MutableDenseNDimArray([zetaP_x, zetaP_y, zetaP_z])
### Difference vectors
RA = ZetaP - ZetaA
RB = ZetaP - ZetaB
RAB = ZetaB - ZetaA
dRA = sm.derive_by_array(RA, [ZetaP, ZetaA, ZetaB])
dRB = sm.derive_by_array(RB, [ZetaP, ZetaA, ZetaB])
dRAB = sm.derive_by_array(RAB, [ZetaP, ZetaA, ZetaB])
################################################################################
### redefine R02,R13
ra_x, ra_y, ra_z = sm.symbols('ra_x ra_y ra_z', real=True)
##### Define symbols
### vertices vectors
# coordinates
zeta00_x, zeta00_y, zeta00_z = sm.symbols('zeta00_x,zeta00_y,zeta00_z',
real=True)
zeta01_x, zeta01_y, zeta01_z = sm.symbols('zeta01_x,zeta01_y,zeta01_z',
real=True)
zeta02_x, zeta02_y, zeta02_z = sm.symbols('zeta02_x,zeta02_y,zeta02_z',
real=True)
zeta03_x, zeta03_y, zeta03_z = sm.symbols('zeta03_x,zeta03_y,zeta03_z',
real=True)
# vectors
Zeta00, Zeta01, Zeta02, Zeta03 = sm.symbols('Zeta00 Zeta01 Zeta02 Zeta03',
real=True)
Zeta00 = smarr.MutableDenseNDimArray([zeta00_x, zeta00_y, zeta00_z])
Zeta01 = smarr.MutableDenseNDimArray([zeta01_x, zeta01_y, zeta01_z])
Zeta02 = smarr.MutableDenseNDimArray([zeta02_x, zeta02_y, zeta02_z])
Zeta03 = smarr.MutableDenseNDimArray([zeta03_x, zeta03_y, zeta03_z])
### external velocity at nodes - not required here
# coordinates
u00_x, u00_y, u00_z = sm.symbols('u00_x u00_y u00_z', real=True)
u01_x, u01_y, u01_z = sm.symbols('u01_x u01_y u01_z', real=True)
u02_x, u02_y, u02_z = sm.symbols('u02_x u02_y u02_z', real=True)
u03_x, u03_y, u03_z = sm.symbols('u03_x u03_y u03_z', real=True)
# vectors
U00, U01, U02, U03 = sm.symbols('U00 U01 U02 U03', real=True)
U01 = smarr.MutableDenseNDimArray([u00_x, u00_y, u00_z])
U02 = smarr.MutableDenseNDimArray([u01_x, u01_y, u01_z])
U03 = smarr.MutableDenseNDimArray([u02_x, u02_y, u02_z])
Scalar quantities are all lower case, e.g. zeta
Arrays begin with upper case, e.g. Zeta_i
2 D Matrices are all upper case, e.g. AW, ZETA=[Zeta_i]
3 D arrays (tensors) will be labelled with a 3 in the name, e.g. A3
'''
import numpy as np
import sympy as sm
import sympy.tensor.array as smarr
import linfunc
### Position vectors of vortex element
Zeta01, Zeta02 = sm.symbols('Zeta01 Zeta02', real=True)
zeta01_x,zeta02_x,zeta01_y,zeta02_y=\
sm.symbols('zeta01_x,zeta02_x,zeta01_y,zeta02_y', real=True)
Zeta01 = smarr.MutableDenseNDimArray([zeta01_x, zeta01_y])
Zeta02 = smarr.MutableDenseNDimArray([zeta02_x, zeta02_y])
### Segment point
ZetaCA, ZetaCB = sm.symbols('ZetaCA,ZetaCB', real=True)
zetaCA_x, zetaCA_y, zetaCB_x, zetaCB_y = sm.symbols(
'zetaCA_x,zetaCA_y,zetaCB_x,zetaCB_y', real=True)
ZetaCA = smarr.MutableDenseNDimArray([zetaCA_x, zetaCA_y])
ZetaCB = smarr.MutableDenseNDimArray([zetaCB_x, zetaCB_y])
### Other symbles/constants
Kskew2D = smarr.MutableDenseNDimArray([[0, -1], [1, 0]])
CF, Gamma, gamma_s = sm.symbols('CF Gamma gamma_s', real=True)
# define order of derivation
ZetaAllList = [
Sign convention:
Scalar quantities are all lower case, e.g. zeta
Arrays begin with upper case, e.g. Zeta_i
2 D Matrices are all upper case, e.g. AW, ZETA=[Zeta_i]
3 D arrays (tensors) will be labelled with a 3 in the name, e.g. A3
'''
import numpy as np
import sympy as sm
import sympy.tensor.array as smarr
import linfunc
### Position vectors of vortex element
Zeta01 = sm.symbols('Zeta01', real=True)
zeta01_x, zeta01_y = sm.symbols('zeta01_x,zeta01_y', real=True)
Zeta01 = smarr.MutableDenseNDimArray([zeta01_x, zeta01_y])
### Segment point
ZetaC = sm.symbols('ZetaC', real=True)
zetaC_x, zetaC_y = sm.symbols('zetaC_x,zetaC_y', real=True)
ZetaC = smarr.MutableDenseNDimArray([zetaC_x, zetaC_y])
### Other symbles/constants
Kskew2D = smarr.MutableDenseNDimArray([[0, -1], [1, 0]])
CF, gamma01, gammaC = sm.symbols('CF gamma01 gammaC', real=True)
# define order of derivation
ZetaAllList = [
zeta01_x,
zeta01_y, # w.r.t. segment 01
zetaC_x,
'''
import sympy as sm
import sympy.tensor.array as smarr
import linfunc
# coordinates
# a00,a01,a02=sm.symbols('a00 a01 a02', real=True)
# a11,a12=sm.symbols('a11 a12', real=True)
# a22=sm.symbols('a22', real=True)
##### derivative w.r.t. Ra,Rb due to cross-product
# show derivative is symmetric
ra_x, ra_y, ra_z = sm.symbols('ra_x ra_y ra_z', real=True)
rb_x, rb_y, rb_z = sm.symbols('rb_x rb_y rb_z', real=True)
RA = smarr.MutableDenseNDimArray([ra_x, ra_y, ra_z])
RB = smarr.MutableDenseNDimArray([rb_x, rb_y, rb_z])
Vcross = linfunc.cross_product(RA, RB)
V2 = linfunc.scalar_product(Vcross, Vcross)
V4 = V2 * V2
Dv = sm.zeros(3)
for ii in range(3):
Dv[ii, ii] += 1 / V2
for jj in range(3):
Dv[ii, jj] += -2 / V4 * Vcross[ii] * Vcross[jj]
DvRAskew = linfunc.matrix_product(Dv, linfunc.skew(RA))
DvRBskew = linfunc.matrix_product(Dv, linfunc.skew(RB))
# verify symetry