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Draft: Streamline upwinding A-V,A (SUPG)

Lukas Schafferhofer requested to merge LukasSchaff/cfs:StreamlineUpwinding-A-V,A into master

Description

Artificial diffusion does not take advantage of the known direction of motion. More recent techniques, such as streamline diffusion (SD) and streamline upwind Petrov Galerkin (SUPG) consider this phenomenon.\ A vector form of streamline upwinding is proposed and applied in a formulation using a magnetic vector potential and an electric scalar potential. The formulation is implemented using edge elements for the vector potential. To deduce a vector form of the streamline upwinding Galerkin finite-element equation, the governing equation is tested by a modified vector weight function

\widetilde{\mathbf{A'}} = \mathbf{A'} - \tau \gamma \mathbf{v}\times (\nabla \times \mathbf{A'}),

it introduces upwinding only along velocity direction (By reducing the weights in the transverse direction to velocity).

\int_{\Omega_{\mathbf{v}}}(\nabla \times{\nu \nabla \times{\mathbf{A}}} + \gamma \nabla{V} {\color{purple}- \gamma(\mathbf{v} \times (\nabla \times{\mathbf{A}}))})\cdot {\color{cyan}\widetilde{\mathbf{A}^{'}}} = 0

After brief mathematical manipulation, the upwind terms look like

-\int_{\Omega_{\mathbf{v}}}( \gamma \nabla{V} {\color{purple}- \gamma(\mathbf{v} \times (\nabla \times{\mathbf{A}}))}) \cdot {\color{cyan}(\tau \gamma \mathbf{v}\times (\nabla \times \mathbf{A'}))},

Now, the only remaining task is to select the parameter \tau. For each element a stabilization factor is chosen which depends on the geometry of the element, its material properties and velocity

    \begin{equation*}
        \tau_e = \frac{h_e}{2 \gamma |\mathbf{v}|} (\coth(\frac{\text{Pe}}{2}) + \frac{2}{\text{Pe}}),
    \end{equation*}

where Pe and h are, respectively, the element Peclet number and element size along velocity direction.

    \begin{align*}
        \text{Pe} = \frac{\gamma |\mathbf{v}| h_e}{\nu}\\
        h_e = \frac{2}{\Sigma_n |\frac{\mathbf{v}}{|\mathbf{v}|} \cdot \nabla{w_n}|},
    \end{align*}

where n is each node of the element and w_n is the Galerkin weight function for the node n (weight function of electric scalar potential).

( Summarise the changes introduced in this merge request ) Testsuite merge request: ( no Testsuite changes | paste a link to the Testsuite merge request )

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