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polarUtilities.py
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polarUtilities.py
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import sympy as sp
r, theta = sp.symbols('r, theta')
#f = sp.Function('f')(r,theta)
def delr_delx():
return sp.cos(theta)
def delr_dely():
return sp.sin(theta)
def deltheta_delx():
return -sp.sin(theta)/r
def deltheta_dely():
return sp.cos(theta)/r
def del_delx(f):
return delr_delx()*sp.diff(f,r,1) + deltheta_delx()*sp.diff(f,theta,1)
def del_dely(f):
return delr_dely()*sp.diff(f,r,1) + deltheta_dely()*sp.diff(f,theta,1)
def del2_delx2(f):
return del_delx(del_delx(f))
def del2_dely2(f):
return del_dely(del_dely(f))
def polarLaplacian(f):
return (del2_delx2(f)+del2_dely2(f)).simplify()
def polarbiharmonic(f):
return polarLaplacian(polarLaplacian(f))
def sigma_xx(f):
return del2_dely2(f)
def sigma_yy(f):
return del2_delx2(f)
def sigma_xy(f):
return -del_delx(del_dely(f))
def sigma_rect(f):
return sp.Matrix([[sigma_xx(f), sigma_xy(f)],[sigma_xy(f), sigma_yy(f)]])
Q = sp.Matrix([[sp.cos(theta), sp.sin(theta)],[-sp.sin(theta), sp.cos(theta)]])
def sigma_polar(f):
return Q*sigma_rect(f)*(Q.T)
def sigma_rr(f):
return (sigma_polar(f)[0,0]).simplify().expand()
def sigma_tt(f):
return (sigma_polar(f)[1,1]).simplify().expand()
def sigma_rt(f):
return (sigma_polar(f)[0,1]).simplify().expand()
def sigma_tr(f):
return (sigma_polar(f)[1,0]).simplify().expand()