Here, we outline a viscoelastic modification of the viscous contact
problem in the repository: https://github.com/agstub/grounding-line-methods.

Momentum balance and incompressibility are given by
\begin{eqnarray}
-\nabla \cdot \mathbf{\sigma}(\mathbf{u},p) &&= \rho_i\mathbf{g}  \\
\nabla \cdot \mathbf{u} &&= 0. 
\end{eqnarray}
The stress is related to pressure and the deviatoric stress through
\begin{eqnarray}
\mathbf{\sigma}(\mathbf{u},p)&&= -p\mathsf{I} + \mathbf{\tau},
\end{eqnarray}
where $\mathbf{\tau}$ is the deviatoric stress tensor. Conservation of angular momentum
implies that the deviatoric stress tensor must satisfy
\begin{equation}
\tau = \tau^T.
\end{equation}

The upper-convected Maxwell viscoelastic model takes the form
\begin{equation}
\mathbf{\tau} + \frac{\eta}{G} \overset{\triangledown}{\tau} = 2\eta(\tau)\mathsf{D}(\mathbf{u}) \label{maxwell}
\end{equation}
where
\begin{eqnarray}
\mathsf{D}(\mathbf{u}) &&= \frac{1}{2}\left[\nabla \mathbf{u} + (\nabla \mathbf{u})^\mathrm{T}  \right] 
\end{eqnarray}
is the strain rate and $\eta / G$ is the relaxation time
($G$ is the shear modulus and $\eta$ is viscosity).

The upper-convected time derivative is defined by
\begin{equation}
\overset{\triangledown}{\tau} = \frac{\partial \tau}{\partial t} + \mathbf{u}\cdot\nabla\tau -(\nabla\mathbf{u})\tau -\tau(\nabla\mathbf{u})^T.
\end{equation}

The viscosity is related to the deviatoric stress through Glen's law
\begin{equation}
\eta(\mathbf{\tau}) = \frac{1}{2A|\mathbf{\tau}|^{n-1}}.
\end{equation}

In the upper-convected time derivative, we make the backward Euler approximation 
\begin{equation}
\frac{\partial \mathbf{\tau}}{\partial t}(\mathbf{x},t)
\approx \frac{\mathbf{\tau}(\mathbf{x},t)-\mathbf{\tau}(\mathbf{x},t-\Delta t) }{\Delta t}. 
\end{equation}

For brevity, we let $(\cdot,\cdot)$ denote the inner product (or duality pairing) of scalars, vectors, and tensors over the domain $\Omega$.
We use a mixed finite element method to solve the above problem: Find $(\tau,\mathbf{u},p)$ such that
\begin{equation}
 (\tau,\nabla\mathbf{v})-(\rho_i\mathbf{g},\mathbf{v}) - (p,\nabla\cdot\mathbf{v})+(\nabla\cdot\mathbf{u},q) + (\mathbf{\tau} + \frac{\eta(\tau)}{G} \overset{\triangledown}{\tau} - 2\eta(\tau)\mathsf{D}(\mathbf{u}),\mu) + \text{BC's} = 0
\end{equation}
for all $(\mu,\mathbf{v},q)$ (where $\mu$ is a tensor-valued test function). 

The symmetry constraint $\tau=\tau^T$
is incorporated directly in the finite element function space.
The "BC's" terms include a sliding law at the base, water pressure condition at the ice-water boundary, and a penalty term for enforcing the contact conditions. See the following reference for more details:
>Stubblefield, A. G., Spiegelman, M., & Creyts, T. T. (2021). Variational formulation of marine ice-sheet and subglacial-lake grounding-line dynamics. Journal of Fluid Mechanics, 919.

82d2863c89b03ee1ceabfd3b69e4c774a2f2f5ec
ipynb
Jupyter Notebook
notes.ipynb
agstub/viscoelastic-glines
924230b2c79b6ec90177718417d26d7bf62c4b50
MIT
notes.ipynb
agstub/viscoelastic-glines
924230b2c79b6ec90177718417d26d7bf62c4b50
MIT
2021-12-16T20:27:35.000Z
2021-12-20T17:43:21.000Z
notes.ipynb
agstub/viscoelastic-glines
924230b2c79b6ec90177718417d26d7bf62c4b50
MIT
Qwen/Qwen-72B
1. YES
2. YES