Poroelastic Network Model

Description

This benchmark is the poroelastic network model presented in AMU2021. The model is derived from Biot's consolidation model for poroelastic elasticity. Let $\Omega \subseteq \mathbb{R}^d$ be a Lipschitz domain with $d~\in~\{2,\,3\}$ and $\mathbb{T} = [0,T]$ for $T \in (0,\infty)$. Consider the system of coupled partial differential equations

\[\begin{aligned} \rho \frac{\partial^2}{\partial t^2} u(t,\xi) - \nabla \sigma(u(t,\xi)) + \nabla (\alpha p(t,\xi)) &= \widehat{f}(t,\xi) \quad \text{in } (0,T] \times \Omega, \\ \frac{\partial}{\partial t} \left( \alpha \nabla\cdot u(t,\xi) + \frac{1}{M} p(t,\xi)\right) - \nabla \cdot \left( \frac{\kappa}{\nu}\nabla p(t,\xi) \right) &= \widehat{g}(t,\xi) \quad \text{in } (0,T] \times \Omega. \end{aligned}\]

Here, the displacement field $u: \mathbb{T} \times \Omega \to \R$ and the pressure field $p: \mathbb{T} \times \Omega \to \R$ are searched solution functions. Moreover, the stress-strain constitute relation

\[\sigma(u(t,\xi)) = 2\mu\varepsilon(u(t,\xi)) + \lambda(\nabla \cdot u(t,\xi)) \mathcal{I} \quad \text{with} \quad \varepsilon(u(t,\xi)) = \frac{1}{2}\left( \nabla u(t,\xi) + (\nabla u(t,\xi))^\mathsf{T} \right)\]

is satisfied, where

• $\mu$ and $\lambda$ are the Lamé coefficients,
• $\mathcal{I}$ denotes the identity tensor,
• $\alpha$ is the Biot-Willes fluid solid coupling coefficient,
• $M$ is the Biot modulus,
• $\kappa$ is the permeability,
• $\rho$ is the density,
• $\mu$ is the fluid viscosity,
• $\widehat{f}: (0,T] \times \Omega \to \R^d$ are the volume-distributed forces,
• $\widehat{g}: (0,T] \times \Omega \to \R$ is the external injection.

The PDE system is equipped with homogeneous Dirichlet boundary conditions

\[ u(t,\xi) = 0, \quad p(t,\xi) = 0 \quad \text{on } (0,T] \times \partial \Omega\]

as well initial conditions $p(0,\cdot) = p^0 : \partial \Omega \to \R$, $u(0,\cdot) = u^0 : \partial \Omega \to \R^d$, and $\frac{\partial}{\partial t}u(0,\cdot) = \dot{u}^0 : \partial \Omega \to \R^d$. Define the Hilbert spaces

\[ \mathcal{V} := \left[H_0^1(\Omega)\right]^d,\quad \mathcal{H}_{\mathcal{V}} := \left[L^2(\Omega)\right]^d, \quad \mathcal{Q} := H_0^1(\Omega),\quad \mathcal{H}_{\mathcal{Q}} := L^2(\Omega)\]

and the operators

\[\begin{aligned} \mathcal{Y}: \mathcal{H}_{\mathcal{V}} \to \mathcal{H}_{\mathcal{V}}^*,& \quad \left\langle \mathcal{Y}u,v \right\rangle := \int_\Omega \rho u v\,\mathrm{d}\xi, \\ \mathcal{M}: \mathcal{H}_{\mathcal{Q}} \to \mathcal{H}_{\mathcal{Q}}^*,& \quad \left\langle \mathcal{M}p,q \right\rangle := \int_\Omega \frac{1}{M} pq\,\mathrm{d}\xi, \\ \mathcal{A}: \mathcal{V} \to \mathcal{V}^*,& \quad \left\langle \mathcal{A}u,v \right\rangle := \int_\Omega \sigma(u): \varepsilon(v)\,\mathrm{d}\xi, \\ \mathcal{K}: \mathcal{Q} \to \mathcal{Q}^*,& \quad \left\langle \mathcal{K}p,q \right\rangle := \int_\Omega \frac{\kappa}{\nu} \nabla p \cdot \nabla q\,\mathrm{d}\xi, \\ \mathcal{D}: \mathcal{V} \to \mathcal{H}_\mathcal{Q}^*,& \quad \left\langle \mathcal{D}u,q \right\rangle := \int_\Omega \alpha(\nabla \cdot u)q \,\mathrm{d}\xi. \end{aligned}\]

Note that $\mathcal{Y}$, $\mathcal{M}$, $\mathcal{A}$, and $\mathcal{K}$ are positive definite. To determine the weak form of the PDE, the first equation is multiplied by a test function $v \in \mathcal{V}$ while the second equation is multiplied with $q \in \mathcal{Q}$. Further we introduce the linear forms

\[ f(t) := \int_\Omega \widehat{f}(t) \cdot \,\mathrm{d} \xi \in \mathcal{H}_{\mathcal{V}}^*, \quad g(t) := \int_\Omega \widehat{g}(t) \cdot \,\mathrm{d} \xi \in \mathcal{H}_{\mathcal{Q}}^*.\]

Then for initial conditions $p^0 \in \mathcal{H}_\mathcal{Q}$, $u^0 \in \mathcal{V}$, and $\dot{u}^0 \in \mathcal{H}_{\mathcal{V}}$ and right-hand sides $f \in L^2(\mathbb{T},\mathcal{H}_\mathcal{V})$ and $f \in L^2(\mathbb{T},\mathcal{H}_\mathcal{Q})$ one aims to find $u \in L^2(\mathbb{T},\mathcal{V})$ and $p \in L^2(\mathbb{T},\mathcal{Q})$ with $\dot{u} \in L^2(\mathbb{T},\mathcal{H}_\mathcal{V})$, $\ddot{u} \in L^2(\mathbb{T},\mathcal{V}^*)$, and $\dot{p} \in L^2(\mathbb{T},\mathcal{Q}^*)$ such that

\[\begin{aligned} \mathcal{Y} \ddot{u}(t) + \mathcal{A} \dot{u}(t) - \mathcal{D}^* u(t) &= f(t) \quad \text{in } \mathcal{V}^*, \\ \mathcal{D} \dot{u}(t) + \mathcal{M} \dot{p}(t) + \mathcal{K} p(t) &= g(t) \quad \text{in } \mathcal{Q}^* \end{aligned}\]

for almost all $t \in (0,T)$, where $\mathcal{D}^*$ denotes the dual operator of $\mathcal{D}$. By introducing the auxiliary variable $w := \dot{u}$, this operator equation can be written in first-order form as

\[ \begin{bmatrix} \mathcal{Y} & 0 & 0 \\ 0 & \mathcal{A} & 0 \\ 0 & 0 & \mathcal{M} \end{bmatrix} \begin{pmatrix} \dot{w}(t) \\ \dot{u}(t) \\ \dot{p}(t) \end{pmatrix} = \begin{bmatrix} 0 & -\mathcal{A} & \mathcal{D}^* \\ \mathcal{A}^* & 0 & 0 \\ -\mathcal{D} & 0 & -\mathcal{K} \end{bmatrix} \begin{pmatrix} w(t) \\ u(t) \\ p(t) \end{pmatrix} + \begin{pmatrix} f(t) \\ 0 \\ g(t) \end{pmatrix}.\]

Writing the inhomogeneity as

\[ \begin{pmatrix} f(t) \\ 0 \\ g(t) \end{pmatrix} = \begin{bmatrix} \operatorname{id} & 0 \\ 0 & 0 \\ 0 & \operatorname{id} \end{bmatrix} \begin{pmatrix} f(t) \\ g(t) \end{pmatrix}\]

and defining the output

\[ \mathbf{y}(t) := \begin{pmatrix} w(t) \\ p(t) \end{pmatrix} = \begin{bmatrix} \operatorname{id} & 0 & 0 \\ 0 & 0 & \operatorname{id} \end{bmatrix} \begin{pmatrix} w(t) \\ u(t) \\ p(t) \end{pmatrix},\]

we obtain the port-Hamiltonian system

\[\begin{aligned} \mathcal{E} \dot{\mathbf{x}}(t) &= (\mathcal{J} - \mathcal{R}) \mathbf{x}(t) + \mathcal{B} \mathbf{v}(t), \\ \mathbf{y}(t) &= \mathcal{B}^* \mathbf{x}(t) \end{aligned}\]

with

\[ \mathcal{E} := \begin{bmatrix} \mathcal{Y} & 0 & 0 \\ 0 & \mathcal{A} & 0 \\ 0 & 0 & \mathcal{M} \end{bmatrix}, \quad \mathcal{J} := \begin{bmatrix} 0 & -\mathcal{A} & \mathcal{D}^* \\ \mathcal{A}^* & 0 & 0 \\ -\mathcal{D} & 0 & 0 \end{bmatrix}, \quad \mathcal{R} := \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \mathcal{K} \end{bmatrix}, \quad \mathcal{B} := \begin{bmatrix} \operatorname{id} & 0 \\ 0 & 0 \\ 0 & \operatorname{id} \end{bmatrix},\]

where $\mathbf{x}(t) := \left[\begin{smallmatrix} w(t) \\ u(t) \\ p(t) \end{smallmatrix}\right]$ and $\mathbf{v}(t) := \left[\begin{smallmatrix} f(t) \\ g(t) \end{smallmatrix}\right]$.

Discretizing this system with standard $\mathcal{P}_1$ Lagrange finite elements results in the finite-dimensional port-Hamiltonian system

\[\begin{aligned} E \dot{x}(t) &= (J - R) x(t) + Bv(t), \\ y(t) &= B^\mathsf{T} x(t) \end{aligned}\]

with

\[\begin{aligned} E &:= \begin{bmatrix} \rho M_u & 0 & 0 \\ 0 & K_u(\mu,\lambda) & 0 \\ 0 & 0 & \frac{1}{M} M_p \end{bmatrix}, \quad J := \begin{bmatrix} 0 & -K_u(\mu,\lambda) & \alpha D^\mathsf{T} \\ K_u(\mu,\lambda)^\mathsf{T} & 0 & 0 \\ -\alpha D & 0 & 0 \end{bmatrix}, \\ R &:= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \frac{\kappa}{\nu} K_p \end{bmatrix}, \quad \mathcal{B} := \begin{bmatrix} B_f & 0 \\ 0 & 0 \\ 0 & B_g \end{bmatrix}. \end{aligned}\]

Parameters

For this benchmark, the domain $\Omega = [0,1]^2$ with $d=2$ has been chosen and the volume-distributed forces $\widehat{f}$ and injection $\widehat{g}$ are spatially independent resulting in two inputs, i.\,e., $B \in \R^{n \times m}$ with $m = 2$. Moreover, different discretization levels are available, resulting in systems with state-space dimensions $n=320$, $n = 980$, and $n = 1805$. These discretizations have been obtained using the python interface of FEniCS. The following fixed parameters have been chosen:

• $\lambda = 12$,
• $\mu = 6$.

The following parameters are variable with the default values

• $\rho = 10^{-3}$,
• $\alpha = 0.79$,
• $\frac{1}{M} = 7.80\cdot 10^3$,
• $\frac{\kappa}{\nu} = 633.33$.

Interface

The system matrices $E, J, R,$ and $B$ can be generated by the following function call.

using PortHamiltonianBenchmarkSystems
E, J, R, B = poro_elasticity_model()

The free parameters are given as named arguments. Note that $n \in \{ 320, 980, 1805 \}$.

using PortHamiltonianBenchmarkSystems
E, J, R, B = poro_elasticity_model(n = 320, eta = 1e-3)
H(s) = B'*((s*E-(J-R))\B)

Here H is the transfer function.

References

@misc{AMU2021,
      title={Port-{H}amiltonian formulations of poroelastic network models}, 
      author={R. Altmann and V. Mehrmann and B. Unger},
      year={2021},
      eprint={2012.01949},
      archivePrefix={arXiv},
      primaryClass={math.DS}
}