def vector_gradient(self, expr):
grad = []
for i in range(len(expr)):
print(self.ijk_to_list(spv.gradient(eval(expr[i]))))
grad.append(self.ijk_to_list(spv.gradient(eval(expr[i]))))
return grad
def makeMatrix(U,B,p,order,dic,vort =True):
# if order !=0:
buoy = Ri*(rho*g)
# else:
# r1 = rho0
# buoy = Ri*r1*g
Cor=((2+chi*y)*qRo*C.k).cross(U)
BgradB = (AgradB.xreplace({Ax:B&C.i,Ay:B&C.j,Az:B&C.k,Bx:B&C.i,By:B&C.j,Bz:B&C.k})).doit()
UgradU = (AgradB.xreplace({Ax:U&C.i,Ay:U&C.j,Az:U&C.k,Bx:U&C.i,By:U&C.j,Bz:U&C.k})).doit()
Eq_NS = diff(U,t)+Cor+UgradU-(-gradient(p)+qRe*laplacian(U)+buoy+BgradB)
Eq_vort=diff((Eq_NS&C.j),x)-diff((Eq_NS&C.i),y)
Eq_m=divergence(U)
Eq_b=diff(B,t)- (qRm*laplacian(B) + curl(U.cross(B)))
if vort == True:
eq = zeros(7,1)
for i,j in enumerate([Eq_NS&C.i,Eq_vort,Eq_NS&C.k,Eq_b&C.i,Eq_b&C.j,Eq_b&C.k,Eq_m]):
eq[i] = taylor(j,order,dic)
var = [Symbol('u'+str(order)+'x'),Symbol('u'+str(order)+'y'),Symbol('u'+str(order)+'z'),Symbol('p'+str(order)),Symbol('b'+str(order)+'x'),Symbol('b'+str(order)+'y'),Symbol('b'+str(order)+'z')]
M, rme = linear_eq_to_matrix(eq, var)
M = simplify((M/ansatz)).xreplace(dic)
print("Matrix OK")
return(M,rme,r1)
def test_doit(self):
v1, v2, zero, one, nabla, C, vn1, vn2, x, y, z = self._get_vars()
v = C.x * C.y * C.z
assert gradient(v) == Grad(v).doit()
expr = v * (VecDot(vn1, vn2))
assert isinstance(Grad(expr).doit(deep=False), Grad)
assert isinstance(Grad(expr).doit(), Vector)
def doit(self, **kwargs):
deep = kwargs.get('deep', True)
args = self.args
if deep:
args = [arg.doit(**kwargs) for arg in args]
# TODO: is the following condition ok?
if not isinstance(args[1], VectorExpr):
return gradient(args[1])
return self.func(*args)
def compute_true_res_formula():
"""Tool to compute true res from sympy"""
import sympy as sp
from sympy.vector import CoordSysCartesian, gradient
R = CoordSysCartesian('R')
u = (sp.sin(2*sp.pi*R.x)*sp.cos(2*sp.pi*R.y)*sp.cos(2*sp.pi*R.z)) * R.i + \
(-sp.cos(2*sp.pi*R.x)*sp.sin(2*sp.pi*R.y)*sp.cos(2*sp.pi*R.z)) * R.j
n = sp.symbols('n')
res = lambda r, c: sp.simplify(u.dot(gradient(u.components[r], R)) -
n * sp.diff(u.components[r], c, c))
for r, c in zip((R.i, R.j), (R.x, R.y)):
print str(res(r, c)).replace('R.', '')
def laplacian(funct):
"""
Parameters
----------
funct : CoordSys3D type
The function that we are finding the laplacian of
Returns
-------
CoordSys3D type
This returns the laplacian of the function, which is the divergence of the gradient of the function.
"""
return divergence(gradient(funct))
def laplacian(self, expr):
out = []
for i in range(len(expr)):
out.append(spv.divergence(spv.gradient(eval(expr[i]))))
return self.post_process(out)
#divergence #1ere methode nabla.dot(c.x * c.y * c.z * (c.i + c.j + c.k)).doit() #C.x*C.y + C.x*C.z + C.y*C.z (nabla & c.x * c.y * c.z * (c.i + c.j + c.k)).doit() #C.x*C.y + C.x*C.z + C.y*C.z #2eme methode: divergence(c.x * c.y * c.z * (c.i + c.j + c.k)) #c.x*C.y + C.x*C.z + C.y*C.z #Gradient #1ere methode """"Consider a scalar field f(x,y,z) in 3D space. The gradient of this field is defined as the vector of the 3 partial derivatives of f with respect to x, y and z in the X, Y and Z axes respectively. In the 3D Cartesian system, the divergence of a scalar field f, denoted by ∇f is given by - ∇f=∂f∂xi^+∂f∂yj^+∂f∂zk^ Computing the divergence of a vector field in sympy.vector can be accomplished in two ways. One, by using the Del() class""" gradient(c.x * c.y * c.z) #c.y*c.z*c.i+c.x*c.z*c.j+c.x*x.y*c.k. #or nabla.gradient(c.x * c.y * c.z).doit() #C.y*C.z*C.i + C.x*C.z*C.j + C.x*C.y*C.k nabla(c.x * c.y * c.z).doit() #C.y*C.z*C.i + C.x*C.z*C.j + C.x*C.y*C.k
init_printing()
global R
global transformation
u1, u2, u3 = symbols('u1:4', type='Function')
f1 = symbols('f_0', type='Function')
omega1 = symbols(r'\Omega', type='Function')
nabla = Del()
R.i = getattr(R, vec_names[0])
R.j = getattr(R, vec_names[1])
R.k = getattr(R, vec_names[2])
R.x = getattr(R, var_names[0])
R.y = getattr(R, var_names[1])
R.z = getattr(R, var_names[2])
u = u1(R.x, R.y, R.z) * R.i + u2(R.x, R.y, R.z) * R.j + u3(R.x, R.y, R.z) * R.k
e = eta(R.x, R.y, R.z)
if transformation == 'spherical':
u_h = u2(R.x, R.y, R.z) * R.j + u3(R.x, R.y, R.z) * R.k
stream = psi(R.y, R.z)
pot = chi(R.y, R.z)
u_r = -curl(R.i * stream).doit()
elif transformation == 'cartesian':
u_h = u1(R.x, R.y, R.z) * R.i + u2(R.x, R.y, R.z) * R.j
stream = psi(R.x, R.y)
pot = chi(R.x, R.y)
u_r = -curl(R.k * stream).doit()
u_d = gradient(pot).doit()
u_helm = u_r + u_d
def Bound(U,B,sol,eig,dic,order,condB = "harm pot",condU = 'Inviscid'):
lso = len(sol)
for s in range(len(sol)):
globals() ['C'+str(s)] = Symbol('C'+str(s))
#################################
### Inviscid solution : ###
### lso=3 --> 1 boundary ###
### lso=6 --> 2 boundaries ###
#################################
nn = n.xreplace(dic).doit()
U = (U.xreplace(makedic(veigen(eig,sol),order))).xreplace({U0:1})
B = (B.xreplace(makedic(veigen(eig,sol),order)))
if (condB == "harm pot"):
bchx,bchy,bchz = symbols("bchx,bchy,bchz")
bcc =surfcond((bchx*C.i +bchy*C.j + bchz*C.k)*ansatz - gradient(psi),dic).doit()
sob = list(linsolve([bcc&C.i,bcc&C.j,bcc&C.k],(bchx,bchy,bchz)))[0]
bbc = sob[0]*C.i + sob[1]*C.j + sob[2]*C.k
bb = B.xreplace(makedic(veigen(eig,sol),order)) - (bbc*ansatz)
Eq_b= surfcond(bb,dic)
Eq_bx = Eq_b&C.i; Eq_by = Eq_b&C.j; Eq_bz = Eq_b&C.k
if params.Bound_nb ==2:
bchx2,bchy2,bchz2 = symbols("bchx2,bchy2,bchz2")
bcc2 =surfcond_2((bchx2*C.i +bchy2*C.j +bchz2*C.k)*ansatz - gradient(psi_2b),dic)
sob2 = list(linsolve([bcc2&C.i,bcc2&C.j,bcc2&C.k],(bchx2,bchy2,bchz2)))[0]
bbc2 = sob2[0]*C.i + sob2[1]*C.j + sob2[2]*C.k
bb2 = B.xreplace(makedic(veigen(eig,sol),order)) - (bbc2*ansatz)
Eq_b2= surfcond_2(bb2,dic)
Eq_b2x = Eq_b2&C.i; Eq_b2y = Eq_b2&C.j; Eq_b2z = Eq_b2&C.k
if (condB == "thick"):
bchx,bchy,bchz,eta = symbols("bchx,bchy,bchz,eta")
kz_t = -sqrt(-kxl**2-kyl**2-I*omega/qRmm)
# kz_t = I*1e12
B_mant = (bchx*C.i +bchy*C.j + bchz*C.k)*ansatz.xreplace({kz:kz_t})
eq_ind = (surfcond((diff(B_mant,t)-qRmm*laplacian(B_mant)-diff(B,t) +qRm*laplacian(B) - curl(U.cross(B))).xreplace({kz:kz_t}),dic).xreplace(dic))
eq_E = (qRm*curl(B)-U.cross(B)-qRmm*curl(B_mant))
eq_Et = surfcond(((nn).cross(eq_E)).xreplace({kz:kz_t}),dic)
eq_B = surfcond(((B_mant.dot(nn)-B.dot(nn))).xreplace({kz:kz_t}),dic)
un = (U.dot(nn))
Eq_n1= surfcond((un).xreplace({kz:kz_t}),dic).xreplace(dic)
TEq = [(taylor(eq,order,dic)).xreplace({x:0,y:0,t:0}) for eq in [Eq_n1,eq_ind&C.i,eq_ind&C.j,eq_ind&C.k,eq_Et.dot(tx),eq_Et.dot(ty)]]
Mat, res = linear_eq_to_matrix(TEq,(C0,C1,C2,bchx,bchy,bchz))
U = (U.xreplace(makedic(veigen(eig,sol),order))).xreplace({U0:1})
un = U.dot(nn)
Eq_n1= surfcond(un,dic).xreplace(dic)
if condU == "Inviscid":
if params.Bound_nb ==2:
nn2 = n2.xreplace(dic).doit()
un2 = U.dot(nn2)
Eq_n2= surfcond_2(un2,dic)
TEq = [(taylor(eq,order,dic)).xreplace({x:0,y:0,t:0}) for eq in [Eq_n1,Eq_n2,Eq_bx,Eq_by,Eq_bz,Eq_b2x,Eq_b2y,Eq_b2z]]
Mat, res = linear_eq_to_matrix(TEq,(C0,C1,C2,C3,C4,C5,Symbol("psi"+str(order)),Symbol("psi"+str(order)+"_2b")))
elif params.Bound_nb ==1:
TEq = [(taylor(eq,order,dic)).xreplace({x:0,y:0,t:0}) for eq in [Eq_n1,Eq_bx,Eq_by,Eq_bz]]
Mat, res = linear_eq_to_matrix(TEq,(C0,C1,C2,Symbol("psi"+str(order))))
elif condU == 'noslip':
if params.Bound_nb ==1:
U = (U.xreplace(makedic(veigen(eig,sol),order)))
ut1 = U.dot(tx)
ut2 = U.dot(ty)
Eq_BU1= surfcond(ut1,dic)
Eq_BU2 = surfcond(ut2,dic)
TEq = [(taylor(eq,order,dic)).xreplace({x:0,y:0,t:0}) for eq in [Eq_n1,Eq_BU1,Eq_BU2,Eq_bx,Eq_by,Eq_bz]]
Mat, res = linear_eq_to_matrix(TEq,(C0,C1,C2,C3,C4,Symbol("psi"+str(order))))
elif params.Bound_nb ==2:
un1 = U.dot(tx2)
un2 = U.dot(ty2)
Eq2_BU1= surfcond_2(un1,dic)
Eq2_BU2 = surfcond_2(un2,dic)
TEq = [(taylor(eq,order,dic)).xreplace({x:0,y:0,t:0}) for eq in [Eq_n1,Eq_n2,Eq_BU1,Eq_BU2,Eq2_BU1,Eq2_BU2,Eq_bx,Eq_by,Eq_bz,Eq_b2x,Eq_b2y,Eq_b2z]]
Mat, res = linear_eq_to_matrix(TEq,(C0,C1,C2,C3,C4,C5,C6,C7,C8,C9,Symbol("psi"+str(order)),Symbol("psi"+str(order)+"_2b")))
elif condU == 'stressfree':
if params.Bound_nb ==1:
eu = strain(U)*nn
eu1 = eu*tx
eu2 = eu*ty
Eq_BU1 = surfcond(eu1,dic,realtopo =False)
Eq_BU2 = surfcond(eu2,dic,realtopo =False)
TEq = [(taylor(eq,order,dic)).xreplace({x:0,y:0,t:0}) for eq in [Eq_n1,Eq_BU1,Eq_BU2,Eq_bx,Eq_by,Eq_bz]]
Mat, res = linear_eq_to_matrix(TEq,(C0,C1,C2,C3,C4,Symbol("psi"+str(order))))
elif params.Bound_nb ==2:
eu = strain(U)*nn2
eu1 = eu*tx2
eu2 = eu*ty2
Eq2_BU1 = surfcond2(eu1,dic,realtopo =False)
Eq2_BU2 = surfcond2(eu2,dic,realtopo =False)
TEq = [(taylor(eq,order,dic)).xreplace({x:0,y:0,t:0}) for eq in [Eq_n1,Eq_n2,Eq_BU1,Eq_BU2,Eq2_BU1,Eq2_BU2,Eq_bx,Eq_by,Eq_bz,Eq_b2x,Eq_b2y,Eq_b2z]]
Mat, res = linear_eq_to_matrix(TEq,(C0,C1,C2,C3,C4,C5,C6,C7,C8,C9,Symbol("psi"+str(order)),Symbol("psi"+str(order)+"_2b")))
Mat = Mat.evalf(mp.mp.dps)
res = res.evalf(mp.mp.dps)
Mat = mpmathM(Mat)
res = mpmathM(res)
try:
abc = mp.qr_solve(Mat,res)[0]
except:
abc = mp.lu_solve(Mat,res)
mantle =0 #In progress ...
solans = zeros(7,1)
for l in range(lso):
solans = solans + abc[l]*Matrix(eig[l])*(ansatz).xreplace({kz:sol[l]})
solans = solans.xreplace(dic)
return(abc,solans,mantle)
# Calculus for each order i
for i in range(1,order+1):
print("ORDER",i)
if Bound_nb ==2:
U,B,p,psi,psi_2b = makeVar()
if Bound_nb ==1:
U,B,p,psi = makeVar()
# Solve the mass conservation equation
if i ==1:
rho= rho0.xreplace({**dico0,**dico1})
rho = rho + e**i*Symbol('rho'+str(i))*ansatz0.xreplace({kxl:i*kxl,kyl:i*kyl,omega:i*omega})
print(rho)
Eq_rho = diff(rho,t)+ U.dot(gradient(rho))
print(Eq_rho)
Eq_rho1 = taylor(Eq_rho,i,{**dico0,**dico1})
print(Eq_rho1)
r1 = list(solveset(Eq_rho1,Symbol('rho'+str(i))))[0]
rho = rho.xreplace({Symbol('rho'+str(i)):r1})
ansatz = (exp(I*(i*omega*t+i*kxl*x+i*kyl*y+kz*z))).xreplace({**dico0,**dico1})
### TEST OF ORDER0 ###
# print("we test the order 0...")
# M0,rme0,r10 = makeMatrix(U0 *u0,qAl*b0,p0,0,{**dico0,**dico1},vort=True)
# print(rme0.xreplace({**dico0,**dico1}))
M,rme,r1 = makeMatrix(U,B,p,i,{**dico0,**dico1},vort=True)
def q(u):
"""Return nonlinear coefficient"""
return 1 + u * u
R = CoordSys3D("R")
def apply(f, coords):
x, y = symbols("x y")
return lambdify((x, y), f.subs({R.x: x, R.y: y}))(*coords)
u_exact = 1 + R.x + 2 * R.y # exact solution
f = -divergence(q(u_exact) * gradient(u_exact)) # manufactured RHS
mesh = MeshTri()
mesh.refine(3)
V = InteriorBasis(mesh, ElementTriP1())
boundary = V.get_dofs().all()
interior = V.complement_dofs(boundary)
@LinearForm
def load(v, w):
return v * apply(f, w.x)