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10 Variational Problems of Functionals with Vector, Tensor …
− ∇ · uv(∇u × ∇ϕ)]δv}dV +
V
{∇ · ϕa − ∇ · [uv(∇v × ∇ϕ)δu] + ∇ · [uv(∇u × ∇ϕ)δv]}dV
(4)
The Euler equations of the functional are
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
∂ F
∂u
− (∇u × v∇v) · ∇
∂ F
∂∇ · ∇u × ∇v
+ ∇ · uv
∇v × ∇
∂ F
∂∇ · ∇u × ∇v
= 0
∂ F
∂v
− (u∇u × ∇v) · ∇
∂ F
∂∇ · ∇u × ∇v
− ∇ · uv
∇u × ∇
∂ F
∂∇ · ∇u × ∇v
= 0
(5)
The natural boundary conditions are
ϕ| S = 0, uv(∇v × ∇ϕ) · n| S = 0, uv(∇u × ∇ϕ) · n| S = 0
( 6 )
Example 10.4.18 The functional with the divergence of dyad of two vectors. Let
the functional
J =
V
F(a, b, ∇ · (ab))dV
(1)
where, the double inner product of the tensors of second order conforms to the law
of parallel operation, derive its Euler equations and natural boundary condition.
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂ a
· δa +
∂ F
∂ b
· δb +
∂ F
∂∇ · (ab)
· δ∇ · (ab)
dV
(2)
Let c =
∂ F
∂∇·(ab)
, T = δ(ab), and using Eq. (10.2.1), there is
c · δ∇ · (ab) = c · ∇ · δ(ab) = c · (∇ · T ) = ∇ · (T · c) − ∇c : T
= ∇ · (T · c) − ∇c : (δab + aδb) = ∇ · (T · c) − [(∇ c) · b] · δa − a · (∇ c) · δb (3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂ a
− (∇ c) · b
· δa +
∂ F
∂ b
− a · (∇ c)
· δb + ∇ · (T · c)
dV (4)
The Euler equations of the functional are
⎧
⎪ ⎨
⎪ ⎩
∂ F
∂ a
− ∇
∂ F
∂∇ · (ab)
· b = 0
∂ F
∂ b
− a · ∇
∂ F
∂∇ · (ab)
= 0
(5)