Graduate Partial of the divergence of a gradient?

[Hypatio]

I am dealing with an expression in a large amount of literature usually presented as:

\frac{\partial}{\partial \phi_i}\left(\nabla \phi_i \cdot \nabla \phi_j \right)

I'm looking at tables of vector calculus identities and cannot seem to find one for the exact expression given, even if I remove the outside partial. Is it correct to expand this as:

\frac{\partial}{\partial \phi_i}\left[\frac{\partial \phi_i}{\partial x}\left(\frac{\partial \phi_j}{\partial x}\right)\right]+\frac{\partial}{\partial \phi_i}\left[\frac{\partial \phi_i}{\partial y}\left(\frac{\partial \phi_j}{\partial y}\right)\right]

or this:

\frac{\partial}{\partial \phi_i}\left[\frac{\partial}{\partial x}\left(\phi_i \frac{\partial \phi_j}{\partial x}\right)\right]+\frac{\partial}{\partial \phi_i}\left[\frac{\partial}{\partial y}\left(\phi_i\frac{\partial \phi_j}{\partial y}\right)\right]

Or are these the same?

I'm trying to construct the correct forward explicit, space centered, finite-difference of this expression but I can't find the correct form. Any help is appreciated.

EDIT: Looking at the wiki on vector calculus identities, it looks like this is a possible answer for the expression in parentheses:

\nabla^2(\phi_i \phi_j) = \phi_i\nabla^2\phi_j+2\nabla\phi_j\cdot\nabla\phi_j+\phi_j\nabla^2\phi_i
rearranging:
\nabla\phi_j\cdot\nabla\phi_j = \frac{1}{2}\left(\nabla^2(\phi_i\phi_j)-\phi_i\nabla^2\phi_j-\phi_j\nabla^2\phi_i\right)

Also, there is:
\nabla\cdot\left(\phi_i\nabla\phi_j\right) = \phi_i\nabla^2\phi_j + \nabla\phi_i\cdot \nabla\phi_j
rearranging:
\nabla\phi_i\cdot \nabla\phi_j = \nabla\cdot\left(\phi_i\nabla\phi_j\right)- \phi_i\nabla^2\phi_j

 

[jedishrfu]

I think you're dealing with the gradient of a vector field here ie $\phi_{i}$ is a vector function so you're taking the gradient of a vector function and dotting itself before taking the partials,

https://math.stackexchange.com/questions/156880/gradient-of-a-vector-field

and that then implies tensor derivatives:

https://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics)#Curvilinear_coordinates

 

[Hypatio]

T

I think you're dealing with the gradient of a vector field here ie $\phi_{i}$ is a vector function so you're taking the gradient of a vector function and dotting itself before taking the partials,

https://math.stackexchange.com/questions/156880/gradient-of-a-vector-field

and that then implies tensor derivatives:

https://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics)#Curvilinear_coordinates

I am confident that they are both scalar fields.

 

[martinbn]

Can you cite some of the large amount of literature?

 

[Hypatio]

Can you cite some of the large amount of literature?

It's from phase field literature. For example, Eq 2 in Miyoshi and Takaki (2017).

Not all literature presents the term with the dot. For example, Eq. 9 in Steinbach and Pezzolla, 1999.

Apparently the term (without the outer partial) is equal to or generalized by the following expression (Eq. 6, Steinbach et al., 1996; Eq. 51, Moelans et al., 2008), but I can't see exactly how they differ:

|\phi_i\nabla\phi_j - \phi_j\nabla\phi_i |^2
References:
https://www.sciencedirect.com/science/article/pii/S0022024816308144?via=ihub
Miyoshi and Takaki (2017), Multi-phase-field study of the effects of anisotropic grain-boundary propertis on polycrystalline grain growth, Journal of Crystal Growth.

https://www.sciencedirect.com/science/article/pii/S0364591607000880
Moelans et al. (2008), An introduction to phase-field modeling of microstructure evolution, Computer coupling of phase diagrams and thermochemistry.

https://www.sciencedirect.com/science/article/pii/S0167278999001293?via=ihub
Steinbach and Pezzolla (1999), A generalized field method for multiphase transformations using interface fields, Physica D: Nonlinear Phenomena.

https://www.sciencedirect.com/science/article/pii/0167278995002987
Steinbach et al. (1996), A phase field concept for multiphase systems, Physica D: Nonlinear Phenomena.

 

[martinbn]

I cannot find the expression you wrote in any of these papers. Can you cite the equation number?

The expression that appears there is $\nabla\phi_i\cdot\nabla\phi_j$ (or without the dot). This is just the dot product of the two vectors.

 

[Hypatio]

I cannot find the expression you wrote in any of these papers. Can you cite the equation number?

The expression that appears there is $\nabla\phi_i\cdot\nabla\phi_j$ (or without the dot). This is just the dot product of the two vectors.

Consider Miyoshi and Takaki (2017). $\nabla\phi_i\cdot\nabla\phi_j$ appears in Eq. 2, then variational derivatives of a function including the term are shown in Eq. 4. The apparent result is $\nabla^2\phi_j$ in Eq. 5. It's not clear to me how it is obtained.

 

[martinbn]

I see. When you wrote $\frac{\partial F}{\partial \phi}$, you meant $\frac{\delta F}{\delta \phi}$. You need to look up calculus of variations.

Roughly it is the following. You have functional $F$ and you want to vary with respect to $\phi_i$. In your case

$F[\phi_i]=\int \left(-\nabla\phi_i\cdot\nabla\phi_j\right)dV.$

Then the variation is

$\frac{\delta F}{\delta \phi}=\frac{d}{d\varepsilon}F[\phi_i+\varepsilon\varphi]|_{\varepsilon=0}$

That leads to

$\int \left(-\nabla\varphi\cdot\nabla\phi_j\right)dV.$

Here you use the identity $\nabla(\varphi\nabla\phi_j)=\nabla\varphi\cdot\nabla\phi_j+\varphi\nabla^2\phi_j$, the divergence theorem, some boundary or decay conditions that make the boundary integral zero and you are left with.

$\int \varphi\nabla^2\phi_jdV$

and since $\varphi$ is any, and you are looking for stationary point, for the equations you have just $\nabla^2\phi_j=0$.