# LinFlowMechPDE

## LinFlow-Mechanic coupling

openCFS provides the coupling between incompressible/compressible viscous fluid with flexible structure by using the Nitsche method.

### Coupling conditions

In LinFlow-mechanic coupled problems, two transmission conditions along the interfaces have to be fulfilled:

• Dynamic coupling condition is the continuity of traction (force equilibrium) at the interface
\begin{equation} \mathbf \sigma_\mathrm{m} \cdot \mathbf n = \mathbf \sigma_\mathrm{v} \cdot \mathbf n \, , \label{eq:tractranscond-mechcoupl} \end{equation}

$\mathbf \sigma_\mathrm{m}$ and $\mathbf \sigma_\mathrm{v}$ are mechanic and LinFlow stress tensors, respectively and the surface normal vector is $\mathbf n = \mathbf {n}_{\rm m} = -\mathbf{n}_{\rm v}$

• Kinematic coupling condition is velocity continuity at the interface \begin{equation} \frac{\partial \mathbf u}{\partial t} = \mathbf v \, , \label{eq:veltranscond-mechcoupl} \end{equation} where $\mathbf u$ and $\mathbf v$ are mechanic displacement and LinFlow velocity vector, respectively.

### Weak formulation

To obtain the coupling PDEs in Nitsche method, the weak form of mechanic PDE and balance of momentum in LinFlow PDE will be combined and the traction continuity condition (\ref{eq:tractranscond-mechcoupl}) will be applied

\begin{equation} \int\limits_{\Omega_\mathrm{v}} \rho_0 \mathbf v' \cdot \frac{\partial \mathbf v}{\partial t}\, {\mathrm {d}} \Omega + \int\limits_{\Omega_\mathrm{v}} \nabla \mathbf v' : \mathbf \sigma_\mathrm{v} {\mathrm {d}}\Omega + \int\limits_{\Omega_\mathrm{m}} \mathbf u’ \cdot \rho_\mathrm{m} \frac{\partial^2 \mathbf u}{\partial t^2} {\mathrm {d}} \Omega+ \int\limits_{\Omega_\mathrm{m}} \nabla \mathbf u’ : \mathbf \sigma_\mathrm{m} {\mathrm {d}} \Omega - \int\limits_{\Gamma_\mathrm{i}} (\mathbf u’ - \mathbf v') \cdot \mathbf \sigma_\mathrm{v} \cdot \mathbf n {\mathrm {d}} \Gamma = \mathbf 0. \label{eq:mechcoupl1} \end{equation}

In this equation, $\mathbf{v'}$, $\mathbf{u’}$ are fluid velocity test functions, mechanical displacement test functions, $\rho_0$ and $\rho_\mathrm{m}$ are LinFlow viscous fluid and mechanic density, respectively. The final version of Nitsche coupling formulations will be obtained by adding the penalty term

\begin{eqnarray} \label{eq:NitscheWeak} \int\limits_{\Omega_\mathrm{v}} \rho_0 \mathbf v' \cdot \frac{\partial \mathbf v}{\partial t}\, {\mathrm {d}} \Omega + \int\limits_{\Omega_\mathrm{v}} \nabla \mathbf v' : \mathbf \sigma_\mathrm{v} {\mathrm {d}}\Omega + \int\limits_{\Omega_\mathrm{m}} \mathbf u’ \cdot \rho_\mathrm{m} \frac{\partial^2 \mathbf u}{\partial t^2} {\mathrm {d}} \Omega+ \int\limits_{\Omega_\mathrm{m}} \nabla \mathbf u’ : \mathbf \sigma_\mathrm{m} {\mathrm {d}} \Omega \nonumber \\ - \underbrace{\int\limits_{\Gamma_\mathrm{i}} (\mathbf u’ - \mathbf v') \cdot \mathbf \sigma_\mathrm{m} \cdot \mathbf n {\mathrm {d}} \Gamma}_{{Traction}}+ \underbrace{\beta \sum_\mathrm{e} \frac{1}{h_\mathrm{ e}}\int\limits_{\Gamma_\mathrm{i}^e}(\mathbf u’ -\mathbf v')\cdot (\frac{\partial \mathbf u}{\partial t}- \mathbf v) {\mathrm {d}} \Gamma }_{{Penalty}}= 0 \end{eqnarray}

In this formulation, the penalty term guarantees the velocity continuity, and the term traction guarantees the traction continuity at the interface.

The penalty term is considered dependent on the characteristic length $h_{\rm e}$ (which is mesh-dependent). This characteristic length is the smaller element length between LinFlow or mechanic regions $h_{\rm e} = \operatorname{min}(h_{\rm v}, h_{\rm m})$.

The penalty factor $\beta$ is defined as

\begin{eqnarray} \beta =\operatorname{max}(p_{\rm m},p_{\rm v})^2 \sqrt[]{\frac{ \rho_0}{\rho_{\rm m}}} (\mu + \lambda)\beta_0. \label{eq:penaltyfac} \end{eqnarray}

Where $\beta_0$ is Nitsche factor and $p_{\rm m}$ and $p_{\rm v}$ are the order of basic functions for mechanic and LinFlow fluid domains, respectively.

Assigning the correct value of the Nitsche factor $\beta_0$ is crucial for solving this formulation: A small value of the Nitsche factor does not meet velocity continuity at the interface. In contrast, a high value of the Nitsche factor makes numerical errors. A Nitsche factor between $10^7-10^{14}$ is appropriate for LinFlow-mechanic coupling problems.

### Analysis Types

Same as LinFlow PDE the coupling can be solved in harmonic transient studies.

• Harmonic

not validated yet: Transient

### Defining the coupling

LinFlow-mechanics coupling benefits form non-conforming interfaces option. Therefore, similar to other non-conforming interface couplings, in simulation xml-file the non-conforming interface option needs to be defined as

    <domain geometryType="">
<regionList>
<region name="" material=""/>
</regionList>
<surfRegionList>
<surfRegion name="LinFLow"/>
<surfRegion name="Mechanic"/>
</surfRegionList>
<ncInterfaceList>
<ncInterface name="LinFLow_mech_interface" masterSide="Mechanic" slaveSide="LinFLow"/>
</ncInterfaceList>
</domain>


Note that in this coupling, the LinFlow interface is considered as the slaveSide and the mechanic interface is considered as the masterSide. In the direct coupling section the non-conforming interface defined in <ncInterfaceList> will be entered as

        <couplingList>
<direct>

<linFlowMechDirect>
<ncInterfaceList>
<ncInterface name="LinFLow_mech_interface" formulation="Nitsche" nitscheFactor="10e10"/>
</ncInterfaceList>
</linFlowMechDirect>

</direct>
</couplingList>


In this coupling, defining nitscheFactor is optional and by default is set to $10^{10}$.

### Material and postprocessing results

In openCFS material-xml file beside the flow material input for the LinFlow PDE, a mechanic material input for the mechanic PDE should also be defined.

  <material name="Air">
<flow>
...
</flow>
</material>

<material name="Silicon">
<mechanical>
....
</mechanical>
</material>


The postprocessing results and the boundary conditions for each PDE are described in the "Singlefield" section.