# Dynamic

:::{admonition} TODO @rwalkerlewis
This section will need to be updated for consistency with the dynamic prescribed slip formulation.
:::

For compatibility with PETSc TS algorithms, we want to turn the second order elasticity equation into two first order equations.
We introduce velocity as a unknown, $\vec{v}=\frac{\partial u}{\partial t}$, which leads to a slightly different three field problem,
%
\begin{gather}
% Solution
\vec{s}^{T} = \left(\vec{u} \quad p \quad \vec{v}\right) \\
% Displacement
\frac{\partial \vec{u}}{\partial t} = \vec{v} \text{ in } \Omega \\
% Pressure
\frac{\partial \zeta(\vec{u},p)}{\partial t } - \gamma(\vec{x},t) + \nabla \cdot \vec{q}(p) = 0 \text{ in } \Omega \\
% Velocity
\rho_{b} \frac{\partial \vec{v}}{\partial t} = \vec{f}(\vec{x},t) + \nabla \cdot \boldsymbol{\sigma}(\vec{u},p) \text{ in } \Omega \\
% Neumann traction
\boldsymbol{\sigma} \cdot \vec{n} = \vec{\tau}(\vec{x},t) \text{ on } \Gamma_{\tau} \\
% Dirichlet displacement
\vec{u} = \vec{u}_{0}(\vec{x}, t) \text{ on } \Gamma_{u} \\
% Neumann flow
\vec{q} \cdot \vec{n} = q_{0}(\vec{x}, t) \text{ on } \Gamma_{q} \\
% Dirichlet pressure
p = p_{0}(\vec{x},t) \text{ on } \Gamma_{p}
\end{gather}
%
For compatibility with PETSc TS explicit time stepping algorithms, we need the left hand side to be $F = (t,s,\dot{s}) = \dot{s}$.
We replace the variation of fluid content variable, $\zeta$, with its definition in the conservation of fluid mass equation and solve for the rate of change of pressure,
%
\begin{gather}
  \frac{\partial}{\partial t}\left(\alpha \epsilon_{v} + \frac{p}{M}\right) - \gamma\left(\vec{x},t\right) + \nabla \cdot \vec{q} = 0 \\
  \alpha \dot{\epsilon}_{v} + \frac{\dot{p}}{M} - \gamma \left(\vec{x},t\right) + \nabla \cdot \vec{q} = 0 \\
  \frac{\dot{p}}{M} = \gamma \left(\vec{x},t \right) - \alpha \dot{\epsilon}_{v} -\nabla \cdot \vec{q} \\
  \frac{\dot{p}}{M} = \gamma \left(\vec{x},t \right) - \alpha \left( \nabla \cdot \dot{\vec{u}} \right) -\nabla \cdot \vec{q}.
\end{gather}
%
We write the volumetric strain in terms of displacement, because this dynamic formulation does not include the volumetric strain as an unknown.

Using trial functions ${\vec{\psi}_\mathit{trial}^{u}}$, ${\psi_\mathit{trial}^{p}}$, and ${\vec{\psi}_\mathit{trial}^{v}}$, and incorporating the Neumann boundary conditions, the weak form may be written as:
%
\begin{align}
  % Displacement
  \int_{\Omega} {\vec{\psi}_\mathit{trial}^{u}} \cdot \left( \frac{\partial \vec{u}}{\partial t} \right)d \Omega &= \int_{\Omega} {\vec{\psi}_\mathit{trial}^{u}} \cdot \left( \vec{v} \right) d \Omega \\
  % Pressure
  \int_{\Omega} {\psi_\mathit{trial}^{p}} \left( \frac{1}{M}\frac{\partial p}{\partial t} \right) d\Omega &=
  \int_{\Omega} {\psi_\mathit{trial}^{p}} \left[\gamma(\vec{x},t) - \alpha \left(\nabla \cdot \dot{\vec{u}}\right) \right]  + \nabla {\psi_\mathit{trial}^{p}} \cdot \vec{q}(p) \ d\Omega +
  \int_{\Gamma_q} {\psi_\mathit{trial}^{p}} (-q_0(\vec{x},t)) \ d\Gamma, \\
  % Velocity
  \int_\Omega {\vec{\psi}_\mathit{trial}^{v}} \cdot \left( \rho_{b} \frac{\partial
  \vec{v}}{\partial t} \right) \,
  d\Omega &= \int_\Omega {\vec{\psi}_\mathit{trial}^{v}} \cdot \vec{f}(\vec{x},t) + \nabla {\vec{\psi}_\mathit{trial}^{v}} :
  -\boldsymbol{\sigma} (\vec{u},p_f) \, d\Omega + \int_{\Gamma_\tau} {\vec{\psi}_\mathit{trial}^{u}}
  \cdot \vec{\tau}(\vec{x},t) \, d\Gamma.
\end{align}
%
## Residual Pointwise Functions

With explicit time stepping the PETSc TS assumes the LHS is $\dot{s}$ , so we only need the RHS residual functions:
%
\begin{align}
% Displacement
  G^u(t,s) &= \int_{\Omega} {\vec{\psi}_\mathit{trial}^{u}} \cdot {\color{blue} \underbrace{\color{black}\vec{v}}_{\color{blue}{\vec{g}_0^u}}} d \Omega, \\
% Pressure
  G^p(t,s) &= \int_\Omega {\psi_\mathit{trial}^{p}} {\color{blue} \underbrace{\color{black}\left(\gamma(\vec{x},t) - \alpha (\nabla \cdot \dot{\vec{u}})\right)}_{\color{blue}{g^p_0}}} + \nabla {\psi_\mathit{trial}^{p}} \cdot {\color{blue} \underbrace{\color{black}\vec{q}(p_f)}_{\color{blue}{\vec{g}^p_1}}} \, d\Omega + \int_{\Gamma_q} {\psi_\mathit{trial}^{p}} ({\color{blue} \underbrace{\color{black}-q_0(\vec{x},t)}_{\color{blue}{g^p_0}}}) \, d\Gamma, \\
% Velocity
 G^v(t,s) &= \int_\Omega {\vec{\psi}_\mathit{trial}^{v}} \cdot {\color{blue}  \underbrace{\color{black}\vec{f}(\vec{x},t)}_{\color{blue}{\vec{g}^v_0}}} + \nabla {\vec{\psi}_\mathit{trial}^{v}} : {\color{blue}  \underbrace{\color{black}-\boldsymbol{\sigma}(\vec{u},p_f)}_{\color{blue}{\boldsymbol{g}^v_1}}} \, d\Omega + \int_{\Gamma_\tau} {\vec{\psi}_\mathit{trial}^{u}} \cdot   {\color{blue}  \underbrace{\color{black}\vec{\tau}(\vec{x},t)}_{\color{blue}{\vec{g}^v_0}}} \, d\Gamma.
\end{align}
%

## Jacobians Pointwise Functions

These are the pointwise functions associated with $M_{u}$, $M_{p}$, and $M_{v}$ for computing the lumped LHS Jacobian.
We premultiply the RHS residual function by the inverse of the lumped LHS Jacobian while $s_\mathit{tshift}$ remains on the LHS with $\dot{s}$. As a result, we use LHS Jacobian pointwise functions, but set $s_\mathit{tshift} = 1$.
The LHS Jacobians are:
%
\begin{align}
% Displacement
  M_{u} &= J_F^{uu} = \frac{\partial F^u}{\partial u} + s_{tshift} \frac{\partial F^u}{\partial \dot{u}} =
  \int_{\Omega} {\psi_\mathit{trial}^{u}}_{i} {\color{blue}  \underbrace{\color{black}s_{tshift} \delta_{ij}}_{\color{blue}{J^{uu}_{f0}}}} {\psi_\mathit{basis}^{u}}_{j} \, d \Omega \\
% Pressure
  M_{p} &= J_F^{pp} = \frac{\partial F^p}{\partial p} + s_{tshift} \frac{\partial F^p}{\partial \dot{p}} =
  \int_{\Omega} {\psi_\mathit{trial}^{p}} {\color{blue} \underbrace{\color{black}\left(s_{tshift} \frac{1}{M}\right)}_{\color{blue}{J_{f0}^{pp}}}} {\psi_\mathit{basis}^{p}} \ d\Omega \\
% Velocity
  M_{v} &= J_F^{vv} = \frac{\partial F^v}{\partial v} + s_{tshift} \frac{\partial F^v}{\partial \dot{v}} =
  \int_{\Omega} {\psi_\mathit{trial}^{v}}_{i}{\color{blue}  \underbrace{\color{black}\rho_{b}(\vec{x}) s_{tshift} \delta_{ij}}_{\color{blue}{J^{vv}_{f0}}}} {\psi_\mathit{basis}^{v}}_{j} \  d \Omega
\end{align}