Hi all,
I've been trying to get the first variation of a functional that depends on
the function phi, and its first and second gradient. I'm probably doing
something wrong.
the function phi:
φ(x, y, t)
its first gradient gradient_1_phi:
⎡∂ ∂ ⎤
⎢──(φ(x, y, t)) ──(φ(x, y, t))⎥
⎣∂x ∂y ⎦
its second gradient gradient_2_phi:
⎡ 2 2 ⎤
⎢ ∂ ∂ ⎥
⎢ ───(φ(x, y, t)) ─────(φ(x, y, t))⎥
⎢ 2 ∂y ∂x ⎥
⎢ ∂x ⎥
⎢ ⎥
⎢ 2 2 ⎥
⎢ ∂ ∂ ⎥
⎢─────(φ(x, y, t)) ───(φ(x, y, t)) ⎥
⎢∂y ∂x 2 ⎥
⎣ ∂y ⎦
For the sake of simplicity, I split the functional in three parts, one
containing phi, another its first gradient, and the last containing the
second gradient. These are potential_PFC, regularizers_PFC_1, and
regularizers_PFC_2 respectively.
Problem: when computing the first variation of regularizers_PFC_2 wrt the
second gradient gradient_2_phi, the cross derivatives disappear.
In regularizers_PFC_2, I would expect to get
⎡ 4 4 4
⎤
⎢ ∂ ∂ ∂
⎥
⎢- 1.0⋅───(φ(x, y, t)) - 1.0⋅───(φ(x, y, t)) - 2.0⋅ ──────(φ(x, y, t))
= 0, True, True⎥
⎢ 4 4 2 2
⎥
⎣ ∂x ∂y ∂x ∂y
⎦
but I get
⎡ 4 4 ⎤
⎢ ∂ ∂ ⎥
⎢- 1.0⋅───(φ(x, y, t)) - 1.0⋅───(φ(x, y, t)) = 0, True, True⎥
⎢ 4 4 ⎥
⎣ ∂x ∂y ⎦
Here is the code, I appreciate very much and suggestion.
from sympy import Symbol, Function
from sympy.calculus.euler import euler_equations
from sympy import tensorcontraction, tensorproduct, eye
from sympy.tensor.array import derive_by_array
from sympy import init_printing
init_printing()
La = Lb = Lc = 1
t = Symbol('t')
x = Symbol('x')
y = Symbol('y')
phi = Function('phi')(x,y,t)
gradient_1_phi = Function('gradient_1_phi')(x,y,t)
gradient_2_phi = Function('gradient_2_phi')(x,y,t)
grad_1_phi = derive_by_array(phi,[x,y])
grad_2_phi = derive_by_array(grad_1_phi,[x,y])
regularizers_PFC_1 =
-Lb*tensorcontraction(tensorproduct(grad_1_phi,grad_1_phi), (0, 1))
regularizers_PFC_2 =
+0.5*Lc*tensorcontraction(tensorproduct(grad_2_phi,grad_2_phi),(0,1,2,3))
potential_PFC = La*(0.5*(1.0-0.2)*phi**2 + 0.25*phi*phi*phi*phi)
d_regularizers_PFC_1 = euler_equations(regularizers_PFC_1,
[phi,gradient_1_phi,gradient_2_phi], [x,y,t])
d_regularizers_PFC_2 = euler_equations(-regularizers_PFC_2,
[phi,gradient_1_phi,gradient_2_phi], [x,y,t])
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