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docs(laneEmdenVariationalForm): updated to match MFEM sign convention more closley

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Emily Boudreaux
2025-06-05 12:36:26 -04:00
parent cf153e0644
commit 4eb8b71271
5 changed files with 34 additions and 17 deletions
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20
docs/derivations/laneEmdenVariationalForm/laneEmdenVariationalBlockForm.tex
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@@ -84,7 +84,7 @@ Now we exploit the linearity of summation and integration to move the sums out o
\end{align}
We will now define $M_{kj}$, $D_{\ell j}$, and $Q_{\ell i}$ such that
\begin{align}
M_{kj} &\equiv \int_{\Omega}\nabla \psi_{k}^{\theta}\cdot \vec{N}_{j}^{\phi}dV \\
M_{kj} &\equiv -\int_{\Omega}\nabla \psi_{k}^{\theta}\cdot \vec{N}_{j}^{\phi}dV \\
D_{\ell j} &\equiv \int_{\Omega}\vec{\psi}_{\ell}^{\phi}\cdot\vec{N}_{j}^{\phi}dV \\
Q_{\ell i} &\equiv \int_{\Omega}\vec{\psi}_{\ell}^{\phi}\cdot\nabla N_{i}^{\theta} dV
\end{align}
@@ -96,18 +96,18 @@ f(\bar{\theta}) \equiv \int_{\Omega}\psi_{k}^{\theta}\left(\theta_{h}\right)^{n}
\end{align}
We can write the variational form of our system of equations as
\begin{align}
-\sum_{j=1}^{N_{dof}^{\phi}}\phi_{j}M_{kj} + f(\bar{\theta)} &= 0 \\
\sum_{j=1}^{N_{dof}^{\phi}}\phi_{j}M_{kj} + f(\bar{\theta)} &= 0 \\
\sum_{j=1}^{N^{\phi}_{dof}}\phi_{j}D_{\ell j} - \sum_{i=1}^{N_{dof}^{\theta}}\theta_{i}Q_{\ell i} &= 0
\end{align}
Or using the notation we defined
\begin{align}
-\mathbf{M}\bar{\phi} + f(\bar{\theta}) &= 0 \\
\mathbf{M}\bar{\phi} + f(\bar{\theta}) &= 0 \\
\mathbf{D}\bar{\phi} - \mathbf{Q}\bar{\theta} &= 0
\end{align}
We can then set this up as a matrix operation
\begin{align}
\begin{bmatrix}
0 & -\mathbf{M} \\
0 & \mathbf{M} \\
-\mathbf{Q} & \mathbf{D}
\end{bmatrix}
\begin{bmatrix}
@@ -126,7 +126,7 @@ From this form we can easily see that the residual matrix is
\begin{align}
R &= \begin{bmatrix}
f(\bar{\theta}) - M\bar{\phi} \\
f(\bar{\theta}) + M\bar{\phi} \\
D\bar{\phi} - Q\bar{\theta}
\end{bmatrix}
\end{align}
@@ -144,11 +144,11 @@ in our Newton-Raphson method. Generally the Jacobian is the matrix of partial de
So then the Jacobian is
\begin{align}
J &= \begin{bmatrix}
\frac{\partial}{\partial \theta}\left(f(\theta) - M\phi\right) & \frac{\partial}{\partial \phi}\left(f(\theta) - M\phi\right) \\
\frac{\partial}{\partial \theta}\left(f(\theta) + M\phi\right) & \frac{\partial}{\partial \phi}\left(f(\theta) + M\phi\right) \\
\frac{\partial}{\partial \theta}\left(D\phi - Q\theta\right) & \frac{\partial}{\partial \phi}\left(D\phi - Q\theta\right)
\end{bmatrix} \\
J &= \begin{bmatrix}
\frac{df}{d\theta} - \phi\frac{\partial M}{\partial \theta} & -M-\phi\frac{\partial M}{\partial \phi} \\
\frac{df}{d\theta} + \phi\frac{\partial M}{\partial \theta} & M+\phi\frac{\partial M}{\partial \phi} \\
-Q - \theta\frac{\partial Q}{\partial \theta} & D + \phi\frac{\partial D}{\partial \phi} - \theta\frac{\partial Q}{\partial \phi}
\end{bmatrix}
\end{align}
@@ -156,7 +156,7 @@ So then the Jacobian is
Finally, we know that the matrices $M$, $D$, and $Q$ are constant with respect to $\theta$ and $\phi$. Therefore, we can drop the partial derivatives with respect to $\theta$ and $\phi$ from the Jacobian. This gives us
\begin{align}
\mathbf{J} &= \begin{bmatrix}
\frac{df}{d\theta} & -M \\
\frac{df}{d\theta} & M \\
-Q & D
\end{bmatrix}
\end{align}
@@ -164,9 +164,9 @@ Finally, we know that the matrices $M$, $D$, and $Q$ are constant with respect t
\noindent In a fully assembled, distritized form this will look like
\begin{align}
\mathbf{J} = \begin{bmatrix} \frac{df}{d\theta}_{00} & \dots & \frac{df}{d\theta}_{0n_{\theta}} & -M_{00} & \dots & -M_{0n_{\phi}} \\
\mathbf{J} = \begin{bmatrix} \frac{df}{d\theta}_{00} & \dots & \frac{df}{d\theta}_{0n_{\theta}} & M_{00} & \dots & M_{0n_{\phi}} \\
\vdots & \ddots & & \vdots & \ddots & \\
\frac{df}{d\theta}_{n_{\theta}0} & & \frac{df}{d\theta}_{n_{\theta}n_{\theta}} & -M_{n_{\theta}0} & & -M_{n_{\theta}n_{\phi}} \\
\frac{df}{d\theta}_{n_{\theta}0} & & \frac{df}{d\theta}_{n_{\theta}n_{\theta}} & M_{n_{\theta}0} & & M_{n_{\theta}n_{\phi}} \\
-Q_{00} & \dots & -Q_{0n_{\theta}} & D_{00} & \dots & D_{0n_{\phi}} \\
\vdots & \ddots & & \vdots & \ddots & \\
-Q_{n_{\phi}0} & & -Q_{n_{\phi}n_{\theta}} & D_{n_{\phi}0} & & D_{n_{\phi}n_{\phi}}