ElecFlowHeatPDE

Electric flow - heat conduction coupling

This coupling allows the simulation of Joul's heating by current flow in structures and its resulting temperature distribution.

Coupling conditions

We consider the following coupling terms: (1) Joule's (heating) loss \begin{equation} \dot q(\mathbf{x}, t) = \mathbf{J} \cdot \mathbf{E} = \gamma\,\mathbf{E} \cdot \mathbf{E} = \gamma\, \nabla {\phi}\cdot \nabla {\phi} \end{equation} with $\mathbf J$ the current density, $\mathbf E$ the electric field intensity as well as $\phi$ the electric potential and (2) the dependecy of the electric conductivity $\gamma$ on temperature $T$ \begin{equation} \gamma = \gamma(T) \end{equation}

Iterative coupling schemes

With this coupling terms, the following coupled system of PDEs for the electric flow and heat conduction is solved \begin{equation} - \nabla \gamma(T)\nabla {\phi} = 0\,; \ \ \ \ \ \phi = \phi_\mathrm{e} \text{ on }\Gamma\mathrm{e}\,; \ \ \ \ \mathbf{J}\cdot\mathbf{n} = -\gamma(T)\nabla {\phi} \cdot\mathbf{n} = J_\mathrm{n} \text{ on }\Gamma\mathrm{n} \end{equation}

$$\begin{eqnarray*} \rho c_{\mathrm m}(T) \frac{\partial T(\mathbf{x},t)}{\partial t} - \nabla {\left( k(T) \nabla{T(\mathbf{x},t)} \right)} &=& \big( \nabla{\phi} \big) \cdot \gamma(T) \nabla{\phi} \\ T(\mathbf{x},t) &=& T\mathrm{e}(\mathbf{x},t)\ \ \ \ \ \ \text{on }\Gamma\mathrm{e}\\ \mathbf{q}\cdot\mathbf{n} &=& q_\mathrm{n} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \,\text{on }\Gamma_\mathrm{n} \\ \mathbf{q}\cdot\mathbf{n} &=& \alpha \left(T - T\mathrm{r}\right) \ \text{on }\Gamma_\mathrm{tr} \end{eqnarray*}$$

For details towards boundary conditions for electirc flow PDE see and for heat condution PDE see.

The PDEs are sequentially solved starting with the electirc flow PDE, computing Joule's lossed and then use them when solving the heat condution PDE.

• Nonlinearity in heat conduction:Fix-point method with stopping criterion

$$\begin{equation} \frac{|| \mathbf{T}_{i+1} - \mathbf{T}_i ||_2}{|| \mathbf{RHS} ||_2} \le \varepsilon \approx 10^{-3} \end{equation}$$

• Stopping criterion for outer iteration: relative change of Joule losses

$$\begin{equation} \frac{|| \mathbf{f}_{k+1} - \mathbf{f}_k ||_2}{|| \mathbf{f}_{k+1} ||_2} \le \varepsilon \approx 10^{-3} \end{equation}$$

Analysis Types

Depending on the temporal setting, the coupled PDEs can be solved for the following analysis types:

• Static-case ($\partial / \partial t = 0$)

  <analysis>
      <static>
      </static>
  </analysis>
  

• Transient-case ($\partial / \partial t (\cdot) \neq 0$)

  <analysis>
    <transient>
          <numSteps>100</numSteps>
          <deltaT>1e-3</deltaT>
    </transient>
  </analysis>
  

Single PDEs

Within the command <pdeList> we define the induvidual PDEs with their boundary conditions, etc.

<pdeList>
    <elecConduction systemId="elecPDE">
        ...
    </elecConduction>
    ...
    <heatConduction  systemId="heatPDE">
        ...
    </heatConduction>
</pdeList>

Defining the coupling

• Coupling terms: The dependency of the electric conductivity $\gamma$ on the tempertaure is defined for each region within the electric flow PDE itself, e.g.

      <elecConduction systemId="elecPDE">
          <regionList>
                  <region name="Vol_poly_A1" matDependIds="cond"/>
                  .....           
          </regionList>
  
          <matDependencyList>
              <elecConductivity id="cond">
                  <coupling pdeName="heatConduction">
                      <quantity name="heatTemperature"/>
                  </coupling>
              </elecConductivity>
          </matDependencyList>    
          .....
      </elecConduction systemId="elecPDE">    
  

  In addition, the coupling term via Joule's losses are defined in the heat conduction PDE for each region in <bcsAndLoads>, e.g.

      <heatConduction  systemId="heatPDE"
          ......
          <bcsAndLoads>
              ....
  
              <elecPowerDensity name="Vol_poly_A1">
                  <coupling pdeName="elecConduction">
                      <quantity name="elecPowerDensity"/>
                      </coupling>           
              </elecPowerDensity>   
              .....
          </bcsAndLoads>                      
              .....
      </heatConduction>
  

• Iterative scheme: In order to achieve a converges solution, we define an interative coupling with corresponding stopping criterion within the <couplingList>:

        <couplingList>
            <iterative>
                <convergence logging="yes" maxNumIters="20" stopOnDivergence="yes">
                   <quantity name="elecPower" value="1e-3" normType="rel"/>
                </convergence>
            </iterative>
        </couplingList>

Material and postprocessing results

For the definition of the individual material as well as the postprocessing results please refer to the individual description of the PDE covered in the Singlefield-section. Here is an example of a material definition including nonlineartis for the heat capacity, heat conduction and temperature dependency of the electric conductivity

<material name="Si8res">
    <heatConduction>
      <density>
        <linear>
          <real>2330</real>
        </linear>
      </density>
      <heatCapacity>
        <linear>
          <real>
            733
          </real>
        </linear>
        <nonlinear>
          <dependency>temperature</dependency>
          <approxType>LinInterpolate</approxType>
          <measAccuracy>0.05</measAccuracy>
          <maxApproxVal>871.049</maxApproxVal>
          <dataName>Cap_OTS_Si_8.fnc</dataName>
        </nonlinear>
      </heatCapacity>
      <heatConductivity>
        <linear>
          <isotropic>
            <real>
            155.761
          </real>
          </isotropic>
        </linear>
        <nonlinear>
          <isotropic>
            <dependency>temperature</dependency>
            <approxType>LinInterpolate</approxType>
            <measAccuracy>0.05</measAccuracy>
            <maxApproxVal>283.269</maxApproxVal>
            <dataName>Cond_OTS_Si_8.fnc</dataName>
          </isotropic>
        </nonlinear>
      </heatConductivity>
    </heatConduction>
    <elecConduction>
      <electricConductivity>
        <linear>
          <isotropic>
            <real> 1.233E+05 </real>
          </isotropic>
        </linear>
        <nonlinear>
          <isotropic>
            <dependency>temperature</dependency>
            <approxType>LinInterpolate</approxType>
            <measAccuracy>0.05</measAccuracy>
            <maxApproxVal>1.63209349E+05</maxApproxVal>
            <dataName>ElecCond_OTS_Si_8.fnc</dataName>
          </isotropic>
        </nonlinear>
      </electricConductivity>
    </elecConduction>
</material>