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Updated 0D documentation by replacing resistor, capacitor, RCL, RC, RL and stenosis docs with BloodVessel doc
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‎documentation/rom_simulation/0d-solver/theory/readme.md‎

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@@ -35,7 +35,7 @@ We are interested in solving the DAE system for the solutions, $\textbf{y}\_{n+1
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$$\textbf{K}\left(\dot{\textbf{y}}\_{n+\alpha\_{m}}^{k}, \textbf{y}\_{n+\alpha\_{f}}^{k}, t\_{n+\alpha\_{f}}\right)\cdot\Delta \textbf{y}\_{n+\alpha\_{f}}^{k} = -\textbf{r}\left(\dot{\textbf{y}}\_{n+\alpha\_{m}}^{k}, \textbf{y}\_{n+\alpha\_{f}}^{k}, t\_{n+\alpha\_{f}}\right),$$
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where $\textbf{K}\left(\dot{\textbf{y}}\_{n+\alpha\_{m}}, \textbf{y}\_{n+\alpha\_{f}}, t\_{n+\alpha\_{f}}\right) = \frac{\partial \textbf{r}\left(\dot{\textbf{y}}\_{n+\alpha\_{m}}, \textbf{y}\_{n+\alpha\_{f}}, t\_{n+\alpha\_{f}}\right)}{\partial \textbf{y}\_{n+\alpha\_{f}}}$ is the consistent tangent matrix.
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where $\textbf{K}\left(\dot{\textbf{y}}\_{n+\alpha\_{m}}, \textbf{y}\_{n+\alpha\_{f}}, t\_{n+\alpha\_{f}}\right) = \frac{\partial \textbf{r}\left(\dot{\textbf{y}}\_{n+\alpha\_{m}}, \textbf{y}\_{n+\alpha\_{f}}, t\_{n+\alpha\_{f}}\right)}{\partial \textbf{y}\_{n+\alpha\_{f}}}$ is the tangent matrix.
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We solve this equation to find $\Delta \textbf{y}\_{n+\alpha\_{f}}^{k}$ and update our guess of $\dot{\textbf{y}}\_{n+\alpha\_{m}}$ and $\textbf{y}\_{n+\alpha\_{f}}$,
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@@ -45,7 +45,7 @@ We are interested in solving the DAE system for the solutions, $\textbf{y}\_{n+1
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$$\dot{\textbf{y}}\_{n+\alpha\_{m}}^{k+1} = \dot{\textbf{y}}\_{n+\alpha\_{m}}^{k} + \frac{\alpha\_{m}}{\Delta t\alpha\_{f}\gamma}\Delta \textbf{y}\_{n+\alpha\_{f}}^{k}.$$
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The consistent tangent matrix is
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The tangent matrix is
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$$\textbf{K}\left(\dot{\textbf{y}}\_{n+\alpha\_{m}}, \textbf{y}\_{n+\alpha\_{f}}, t\_{n+\alpha\_{f}}\right) = \underset{\text{Term 1}}{\underbrace{\frac{\partial \textbf{E}\left(\textbf{y}\_{n+\alpha\_{f}}, t\_{n+\alpha\_{f}}\right)}{\partial \textbf{y}\_{n+\alpha\_{f}}}\cdot\dot{\textbf{y}}\_{n+\alpha\_{m}}}} + \frac{\alpha\_{m}}{\Delta t\alpha\_{f}\gamma}\textbf{E}\left(\textbf{y}\_{n+\alpha\_{f}}, t\_{n+\alpha\_{f}}\right) + \underset{\text{Term 2}}{\underbrace{\frac{\partial \textbf{F}\left(\textbf{y}\_{n+\alpha\_{f}}, t\_{n+\alpha\_{f}}\right)}{\partial \textbf{y}\_{n+\alpha\_{f}}}\cdot\textbf{y}\_{n+\alpha\_{f}}}} + \textbf{F}\left(\textbf{y}\_{n+\alpha\_{f}}, t\_{n+\alpha\_{f}}\right) + \underset{\text{Term 3}}{\underbrace{\frac{\partial \textbf{c}\left(\textbf{y}\_{n+\alpha\_{f}}, t\_{n+\alpha\_{f}}\right)}{\partial \textbf{y}\_{n+\alpha\_{f}}}}}.$$
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@@ -93,285 +93,65 @@ $$\textbf{E}^{e}\left(\textbf{y}^{e}, t\right)\cdot\dot{\textbf{y}}^{e} + \textb
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<h4> Vessel elements </h4>
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<h5> Resistor </h5>
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<h5> BloodVessel </h5>
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<br>
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<figure>
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<img class="svImg svImgMd" src="/documentation/rom_simulation/0d-solver/images/resistor.png">
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<figcaption class="svCaption"> Resistor element.
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</figcaption>
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</figure>
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The governing equations for the local resistor element are
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<!-- https://github.com/mathjax/MathJax/issues/329 -- need to add a backslash before all underscore in an equation -->
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$$P\_{in}^{e} - P\_{out}^{e} - RQ\_{in}^{e} = 0$$
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$$Q\_{in}^{e} - Q\_{out}^{e} = 0.$$
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The local contributions to the global arrays are
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\begin{gather}
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\textbf{y}^{e} =
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\begin{bmatrix}
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P\_{in}^{e} & Q\_{in}^{e} & P\_{out}^{e} & Q\_{out}^{e}
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\end{bmatrix}^T,
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\end{gather}
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\begin{gather}
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\textbf{F}^{e} =
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\begin{bmatrix}
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1 & -R & -1 & 0 \ \cr
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0 & 1 & 0 & -1
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\end{bmatrix}.
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\end{gather}
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<h5> Capacitor </h5>
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<br>
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<figure>
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<img class="svImg svImgMd" src="/documentation/rom_simulation/0d-solver/images/capacitor.png">
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<figcaption class="svCaption"> Capacitor element.
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</figcaption>
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</figure>
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The governing equations for the local capacitor element are
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$$C\frac{d\left( P\_{in}^{e} - P\_{out}^{e} \right)}{dt} - Q\_{in}^{e} = 0$$
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$$Q\_{in}^{e} - Q\_{out}^{e} = 0.$$
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The local contributions to the global arrays are
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\begin{gather}
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\textbf{y}^{e} =
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\begin{bmatrix}
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P\_{in}^{e} & Q\_{in}^{e} & P\_{out}^{e} & Q\_{out}^{e}
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\end{bmatrix}^T,
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\end{gather}
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\begin{gather}
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\textbf{E}^{e} =
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\begin{bmatrix}
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C & 0 & -C & 0 \ \cr
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0 & 0 & 0 & 0
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\end{bmatrix},
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\end{gather}
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\begin{gather}
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\textbf{F}^{e} =
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\begin{bmatrix}
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0 & -1 & 0 & 0 \ \cr
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0 & 1 & 0 & -1
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\end{bmatrix}.
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\end{gather}
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<h5> Inductor </h5>
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<br>
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<figure>
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<img class="svImg svImgMd" src="/documentation/rom_simulation/0d-solver/images/inductor.png">
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<figcaption class="svCaption"> Inductor element.
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</figcaption>
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</figure>
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The governing equations for the local inductor element are
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$$P\_{in}^{e} - P\_{out}^{e} - L\frac{dQ\_{in}^{e}}{dt} = 0$$
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$$Q\_{in}^{e} - Q\_{out}^{e} = 0.$$
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The local contributions to the global arrays are
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\begin{gather}
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\textbf{y}^{e} =
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\begin{bmatrix}
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P\_{in}^{e} & Q\_{in}^{e} & P\_{out}^{e} & Q\_{out}^{e}
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\end{bmatrix}^T,
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\end{gather}
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\begin{gather}
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\textbf{E}^{e} =
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\begin{bmatrix}
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0 & -L & 0 & 0 \ \cr
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0 & 0 & 0 & 0
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\end{bmatrix},
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\end{gather}
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\begin{gather}
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\textbf{F}^{e} =
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\begin{bmatrix}
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1 & 0 & -1 & 0 \ \cr
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0 & 1 & 0 & -1
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\end{bmatrix}.
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\end{gather}
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<h5> Resistor-Capacitor </h5>
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<br>
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<figure>
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<img class="svImg svImgMd" src="/documentation/rom_simulation/0d-solver/images/RC.png">
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<figcaption class="svCaption"> Resistor-Capacitor element.
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</figcaption>
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</figure>
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The governing equations for the local resistor-capacitor element are
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$$P\_{in}^{e} - P\_{out}^{e} - RQ\_{in}^{e} = 0$$
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$$Q\_{in}^{e} - Q\_{out}^{e} - C\frac{dP\_{out}^{e}}{dt} = 0.$$
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The local contributions to the global arrays are
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\begin{gather}
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\textbf{y}^{e} =
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\begin{bmatrix}
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P\_{in}^{e} & Q\_{in}^{e} & P\_{out}^{e} & Q\_{out}^{e}
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\end{bmatrix}^T,
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\end{gather}
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\begin{gather}
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\textbf{E}^{e} =
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\begin{bmatrix}
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0 & 0 & 0 & 0 \ \cr
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0 & 0 & -C & 0
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\end{bmatrix},
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\end{gather}
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\begin{gather}
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\textbf{F}^{e} =
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\begin{bmatrix}
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1 & -R & -1 & 0 \ \cr
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0 & 1 & 0 & -1
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\end{bmatrix}.
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\end{gather}
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<h5> Resistor-Inductor </h5>
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<br>
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<figure>
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<img class="svImg svImgMd" src="/documentation/rom_simulation/0d-solver/images/RL.png">
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<figcaption class="svCaption"> Resistor-Inductor element.
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<img class="svImg svImgMd" src="/documentation/rom_simulation/0d-solver/images/RCL.png">
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<figcaption class="svCaption"> BloodVessel element.
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</figcaption>
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</figure>
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The governing equations for the local resistor-inductor element are
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$$P\_{in}^{e} - P\_{out}^{e} - RQ\_{in}^{e} - L\frac{dQ\_{out}^{e}}{dt} = 0$$
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$$Q\_{in}^{e} - Q\_{out}^{e} = 0.$$
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The local contributions to the global arrays are
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\begin{gather}
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\textbf{y}^{e} =
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\begin{bmatrix}
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P\_{in}^{e} & Q\_{in}^{e} & P\_{out}^{e} & Q\_{out}^{e}
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\end{bmatrix}^T,
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\end{gather}
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The BloodVessel element is a general block for modeling blood vessels. To use this block, the Poiseuille-based resistance must be specified. All other components (capacitance, inductance, and stenosis-based resistance) are optional.
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\begin{gather}
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\textbf{E}^{e} =
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\begin{bmatrix}
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0 & 0 & 0 & -L \ \cr
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0 & 0 & 0 & 0
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\end{bmatrix},
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\end{gather}
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<!-- All have a default value of zero, except for the curvature coefficient, which has a default value of 1. -->
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\begin{gather}
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\textbf{F}^{e} =
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\begin{bmatrix}
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1 & -R & -1 & 0 \ \cr
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0 & 1 & 0 & -1
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\end{bmatrix}.
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\end{gather}
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The governing equations for the local BloodVessel element, including the stenosis contribution [3], are
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<h5> Resistor-Capacitor-Inductor </h5>
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<br>
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<figure>
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<img class="svImg svImgMd" src="/documentation/rom_simulation/0d-solver/images/RCL.png">
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<figcaption class="svCaption"> Resistor-Capacitor-Inductor element.
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</figcaption>
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</figure>
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The governing equations for the local resistor-capacitor-inductor element are
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$$P\_{in}^{e} - P\_{out}^{e} - RQ\_{in}^{e} - L\frac{dQ\_{out}^{e}}{dt} = 0$$
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$$P\_{in}^{e} - P\_{out}^{e} - R\left(Q\_{in}^{e}\right)Q\_{in}^{e} - L\frac{dQ\_{out}^{e}}{dt} = 0$$
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$$Q\_{in}^{e} - Q\_{out}^{e} - C\frac{dP\_{c}^{e}}{dt} = 0.$$
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$$P\_{in}^{e} - RQ\_{in}^{e} - P\_{c} = 0$$
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$$P\_{in}^{e} - R\left(Q\_{in}^{e}\right)Q\_{in}^{e} - P\_{c} = 0$$
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where $R\left(Q\right) = R\_{s}\left(Q\right) + R\_{p}$, $R\_{s}\left(Q\right) = K\_{t}\frac{\rho}{2A\_{o}^{2}}\left(\frac{A\_{o}}{A\_{s}} - 1\right)^{2}|Q|$, and $R\_{p} = \frac{8\mu L}{\pi r^{4}}.$
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The local contributions to the global arrays are
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\begin{gather}
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\textbf{y}^{e} =
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\begin{bmatrix}
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P\_{in}^{e} & Q\_{in}^{e} & P\_{out}^{e} & Q\_{out}^{e} & P\_{c}
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\end{bmatrix}^T,
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\textbf{y}^{e} =
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\begin{bmatrix}
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P\_{in}^{e} & Q\_{in}^{e} & P\_{out}^{e} & Q\_{out}^{e} & P\_{c}
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\end{bmatrix}^T,
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\end{gather}
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\begin{gather}
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\textbf{E}^{e} =
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\begin{bmatrix}
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0 & 0 & 0 & -L & 0 \ \cr
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0 & 0 & 0 & 0 & -C \ \cr
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0 & 0 & 0 & 0 & 0
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\end{bmatrix},
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\textbf{E}^{e} =
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\begin{bmatrix}
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0 & 0 & 0 & -L & 0 \ \cr
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0 & 0 & 0 & 0 & -C \ \cr
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0 & 0 & 0 & 0 & 0
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\end{bmatrix},
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\end{gather}
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\begin{gather}
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\textbf{F}^{e} =
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\begin{bmatrix}
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1 & -R & -1 & 0 & 0 \ \cr
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0 & 1 & 0 & -1 & 0 \ \cr
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1 & -R & 0 & 0 & -1
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\end{bmatrix}.
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\textbf{F}^{e} =
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\begin{bmatrix}
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1 & -R\left(Q\_{in}^{e}\right) & -1 & 0 & 0 \ \cr
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0 & 1 & 0 & -1 & 0 \ \cr
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1 & -R\left(Q\_{in}^{e}\right) & 0 & 0 & -1
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\end{bmatrix}.
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\end{gather}
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<h5> Stenosis </h5>
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<br>
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<figure>
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<img class="svImg svImgMd" src="/documentation/rom_simulation/0d-solver/images/stenosis.png">
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<figcaption class="svCaption"> Stenosis element.
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</figcaption>
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</figure>
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The governing equations for the local stenosis element [3] are
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$$P\_{in}^{e} - P\_{out}^{e} - R\_{pre}\left(Q\_{in}^{e}\right)Q\_{in}^{e} = 0$$
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$$Q\_{in}^{e} - Q\_{out}^{e} = 0,$$
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where $R\_{pre}\left(Q\right) = K\_{t}\frac{\rho}{2A\_{o}^{2}}\left(\frac{A\_{o}}{A\_{s}} - 1\right)^{2}|Q| + R\_{p}$ and $R\_{p} = \frac{8\mu L}{\pi r^{4}}.$
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The local contributions to the global arrays are
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\begin{gather}
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\textbf{y}^{e} =
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\begin{bmatrix}
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P\_{in}^{e} & Q\_{in}^{e} & P\_{out}^{e} & Q\_{out}^{e}
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\end{bmatrix}^T,
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\textbf{dF}^{e} =
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\begin{bmatrix}
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0 & -R\_{s}\left(Q\_{in}^{e}\right) & 0 & 0 & 0 \ \cr
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0 & 0 & 0 & 0 & 0 \ \cr
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0 & -R\_{s}\left(Q\_{in}^{e}\right) & 0 & 0 & 0
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\end{bmatrix}.
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\end{gather}
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\begin{gather}
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\textbf{F}^{e} =
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\begin{bmatrix}
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1 & -R\_{pre}|Q\_{in}^{e}| - R\_{p} & -1 & 0 \ \cr
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0 & 1 & 0 & -1
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\end{bmatrix}.
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\end{gather}
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\begin{gather}
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\textbf{dF}^{e} =
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\begin{bmatrix}
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1 & -R\_{pre}|Q\_{in}^{e}| & -1 & 0 \ \cr
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0 & 0 & 0 & 0
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\end{bmatrix}.
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\end{gather}
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<h5> Junction </h5>
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