Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 10

Problem 1: Part 10: Discrete evolution equations

The relevant equations are

$\begin{array}{cc}{f}_{\mathit{\sigma }}& =\sqrt{\frac{3}{2}}𝐧\\ {\dot{\mathit{\epsilon }}}^p& =\dot{\gamma }{f}_{\mathit{\sigma }}=\sqrt{\frac{3}{2}}\dot{\gamma }𝐧\\ \dot{\alpha }& =\sqrt{\frac{2}{3}}\dot{\gamma }\frac{{\mathit{\epsilon }}^p:f_{\mathit{\sigma }}}{\Vert {\mathit{\epsilon }}^p{\Vert }_{}}=\dot{\gamma }\frac{{\mathit{\epsilon }}^p:𝐧}{\Vert {\mathit{\epsilon }}^p{\Vert }_{}}\\ \dot{T}& =\frac{\chi \dot{\gamma }}{\rho C_p}\mathit{\sigma }:\frac{\partial f}{\partial \mathit{\sigma }}=\sqrt{\frac{3}{2}}\frac{\chi \dot{\gamma }}{\rho C_p}\mathit{\sigma }:𝐧=\sqrt{\frac{3}{2}}\frac{\chi \dot{\gamma }}{\rho C_p}\Vert 𝐬{\Vert }_{}\end{array}$

A Forward Euler time discretization gives us

$\begin{array}{cc}{\mathit{\epsilon }}_{n+1}^p& ={\mathit{\epsilon }}_n^p+\sqrt{\frac{3}{2}}\mathrm{\Delta }t{\dot{\gamma }}_n{𝐧}_n\\ {\alpha }_{n+1}& ={\alpha }_n+\mathrm{\Delta }t{\dot{\gamma }}_n\frac{{\mathit{\epsilon }}_n^p:{𝐧}_n}{\Vert {\mathit{\epsilon }}_n^p{\Vert }_{}}\\ T_{n+1}& =T_n+\sqrt{\frac{3}{2}}\frac{\chi \mathrm{\Delta }t{\dot{\gamma }}_n}{{\rho }_n{C}_p}\Vert {𝐬}_n{\Vert }_{}\end{array}$

or

$\begin{array}{cc}{\mathit{\epsilon }}_{n+1}^p& ={\mathit{\epsilon }}_n^p+\sqrt{\frac{3}{2}}\mathrm{\Delta }\gamma {𝐧}_n\\ {\alpha }_{n+1}& ={\alpha }_n+\mathrm{\Delta }\gamma \frac{{\mathit{\epsilon }}_n^p:{𝐧}_n}{\Vert {\mathit{\epsilon }}_n^p{\Vert }_{}}\\ T_{n+1}& =T_n+\sqrt{\frac{3}{2}}\frac{\chi \mathrm{\Delta }\gamma }{{\rho }_n{C}_p}\Vert {𝐬}_n{\Vert }_{}\end{array}$