Tensor manipulation for custom vector equation source term

Diro7 · January 19, 2021, 04:59

Hi!

In a solver I'm working on I need to perform some algebraic manipulation of tensors in order to build a source term for a vector equation.
More specifically, I need to implement this expression:

$F_i \propto \sum_{\substack{k=1\\k \neq i}}^3 \left| \frac{\partial U_k}{\partial x_i} \right|^{\frac{1}{2}} (U_k - V_k)$

Where $\textbf{V}$ is an additional velocity field for which the equation is solved, and $\textbf{U}$ is the standard fluid velocity field.
I thought it could be useful to write the same expression in vector form in order to directly implement it as a vector equation. With some manipulation

$\mathbf{F} \propto \left| \nabla \textbf{U} - \textrm{diag}(\nabla \textbf{U}) \right|^{\frac{1}{2}} (\textbf{U} - \textbf{V})$

where the operation $\left| \cdot \right|^{\frac{1}{2}}$ is intended element-wise. Now I'm faced with at least a couple of issues:

1) assemble the tensor $\textbf{T} = \nabla \textbf{U} - \textrm{diag}(\nabla \textbf{U})$

2) perform the operation $\left| \textbf{T} \right|^{\frac{1}{2}}$

3) perform the matrix product between $\left| \textbf{T} \right|^{\frac{1}{2}}$ and $(\textbf{U} - \textbf{V})$

Due to my somewhat limited knowledge of the OpenFOAM library, for the moment I neglected 2).
What I came up with is:

Code:

fvVectorMatrix F(fvc::grad(U) - diag(fvc::grad(U)) & (U - V));

Unfortunately the extraction of the diagonal is not working as expected and I keep getting

Code:

error: no match for ‘operator-’ (operand types are ‘Foam::tmp<Foam::GeometricField<Foam::Tensor<double>, Foam::fvPatchField, Foam::volMesh> >’ and ‘Foam::tmp<Foam::Field<Foam::DiagTensor<double> > >’)
     fvVectorMatrix F(fvc::grad(U) - diag(fvc::grad(U)) & (U - V));

It seems like the diag(volTensorField) function returns a different type which is incompatible with volTensorField.

Does anyone know how to properly extract/subtract the diagonal contribution from a volTensorField object? In my limited experience, I couldn't find any other related function.
Furthermore, how can I perform the element-wise operations described in 2)?

Or maybe I'm just following the wrong road and someone knows a better way to perform my task.
Of course any advice or observation would be greatly appreciated!

Regards,
Andrea

January 19, 2021, 04:59	Tensor manipulation for custom vector equation source term	#1
Diro7 Member Andrea Di Ronco Join Date: Nov 2016 Location: Milano, Italy Posts: 55 Rep Power: 9	Hi! In a solver I'm working on I need to perform some algebraic manipulation of tensors in order to build a source term for a vector equation. More specifically, I need to implement this expression: $F_i \propto \sum_{\substack{k=1\\k \neq i}}^3 \left\| \frac{\partial U_k}{\partial x_i} \right\|^{\frac{1}{2}} (U_k - V_k)$ Where $\textbf{V}$ is an additional velocity field for which the equation is solved, and $\textbf{U}$ is the standard fluid velocity field. I thought it could be useful to write the same expression in vector form in order to directly implement it as a vector equation. With some manipulation $\mathbf{F} \propto \left\| \nabla \textbf{U} - \textrm{diag}(\nabla \textbf{U}) \right\|^{\frac{1}{2}} (\textbf{U} - \textbf{V})$ where the operation $\left\| \cdot \right\|^{\frac{1}{2}}$ is intended element-wise. Now I'm faced with at least a couple of issues: 1) assemble the tensor $\textbf{T} = \nabla \textbf{U} - \textrm{diag}(\nabla \textbf{U})$ 2) perform the operation $\left\| \textbf{T} \right\|^{\frac{1}{2}}$ 3) perform the matrix product between $\left\| \textbf{T} \right\|^{\frac{1}{2}}$ and $(\textbf{U} - \textbf{V})$ Due to my somewhat limited knowledge of the OpenFOAM library, for the moment I neglected 2). What I came up with is: Code: fvVectorMatrix F(fvc::grad(U) - diag(fvc::grad(U)) & (U - V)); Unfortunately the extraction of the diagonal is not working as expected and I keep getting Code: error: no match for ‘operator-’ (operand types are ‘Foam::tmp<Foam::GeometricField<Foam::Tensor<double>, Foam::fvPatchField, Foam::volMesh> >’ and ‘Foam::tmp<Foam::Field<Foam::DiagTensor<double> > >’) fvVectorMatrix F(fvc::grad(U) - diag(fvc::grad(U)) & (U - V)); It seems like the diag(volTensorField) function returns a different type which is incompatible with volTensorField. Does anyone know how to properly extract/subtract the diagonal contribution from a volTensorField object? In my limited experience, I couldn't find any other related function. Furthermore, how can I perform the element-wise operations described in 2)? Or maybe I'm just following the wrong road and someone knows a better way to perform my task. Of course any advice or observation would be greatly appreciated! Regards, Andrea

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