backward_facing_step_EEA

This script compares the rate of convergence for a quantity of interest with both uniform and residual-based adaptive refinements using a known problem called backward facing step.

Problem definition

Strong form

$$ \begin{equation*} \begin{aligned} &\text{Given constants}, \mu , \text{and}, \gamma, &&\\ &\text{find}, \mathbf{u}: \Omega \in \mathbb{R}^2, \text{such that:} && \\ & - {\rm div}(\boldsymbol{\sigma}) = 0 && \text{in}, \Omega\\ &\phantom{-div(u)}\mathbf{u} = \mathbf{0} && \text{on}, \Gamma_{D} \\ &\phantom{-div(u} u_{x} = u_{\rm max}(x_1 - h) \cfrac{H - (x_1 - h)}{\left(\frac{H}{2}\right)^2} && \text{on}, \Gamma_{in} \\ &\phantom{-div} \boldsymbol{\sigma} \cdot \mathbf{n} = \mathbf{0} && \text{on}, \Gamma_{out} \\ \end{aligned} \end{equation*} $$

with

$$ \begin{align*} \boldsymbol{\sigma} = \mu (2 \boldsymbol{\varepsilon} + \gamma {\rm tr}(\boldsymbol{\varepsilon}) \mathbf{I}) \quad \quad \boldsymbol{\varepsilon} = \frac{1}{2} \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^{\rm T} \right) \end{align*} $$

Weak form

$$ \begin{equation*} \begin{aligned} &\text{Find}, \mathbf{u} \in \mathcal{S} \text{such that:} &&\\ & \mathcal{B}(\mathbf{u},\mathbf{v}) = \mathcal{F} (\mathbf{v}) && \forall \mathbf{v} \in \mathcal{V} \end{aligned} \end{equation*} $$

with

$$ \begin{align*} \mathcal{B}(\mathbf{u},\mathbf{v}) &= \int_{\Omega} \left[ 2 \mu \boldsymbol{\varepsilon} \left( \frac{1}{2} \left( \nabla \mathbf{v} + (\nabla \mathbf{v})^{\rm T} \right) \right) + \gamma \mu , {\rm div}(\mathbf{u}) , {\rm div}(\mathbf{v}) \right], {\rm dV}\\ \mathcal{F}(\mathbf{v}) &= \int_{\Gamma_{out}} \mathbf{v} \cdot (\boldsymbol{\sigma} \cdot \mathbf{n}) , {\rm dS} = 0 \end{align*} $$

Residual-based error

$$ \begin{equation*} \mathcal{R}^h(\mathbf{v}) := B(\mathbf{u}^h, \mathbf{v}) - \mathcal{F}(\mathbf{v}) \phantom{ {\sup_{\mathbf{v} \in \mathcal{V}\setminus 0}} \frac{\mathcal{R}^H(\mathbf{v})}{| \mathbf{v} |_{\mathcal{V}}}} \end{equation*} $$