# Conditions for always positive gradient of heat field in evolutionary thermo-elastic system

I am investigating stability and convergence of series of approximations for coupled thermoelasticity problem yielded by one-step recurrent time-integration scheme.

I've managed to show that the one-step time integration scheme ( OSTIS ) is stable and convergent when the gradient of thermal field is positive (energy of heat of a body at every step is not less than energy on the previous time step of OSTIS). But now I wonder if this is sufficient result or this is unnatural condition?

• Generally speaking can someone tell the example of thermo-elastic process that occurs in nature (or in machinery) where the gradient of temperature of a body is always positive?

P.S. As I understand this occurs when the system/body is constantly heated by external source, so that more heat comes is than comes out in every time step.

UPDATE:

I need to know when this is true: \begin{align} & \tfrac{1}{2}\Delta {{t}^{-1}}s({{z}^{j+1}}-{{z}^{j}},{{u}^{j+1}}+{{u}^{j}})\ge 0 \\ \end{align} Here I user the following notation: \begin{align} & \,\theta =\theta (x,t) \\ \end{align} is the temperature field over the body in every time; \begin{align} & \frac{\partial }{\partial t}{{\left. [{{\theta }^{\Delta t}}(t)] \right|}_{{{t}_{j}}}}={{\left. {{[{{\theta }^{\Delta t}}(t)]}^{\prime }} \right|}_{{{t}_{j}}}}={{z}^{j}} \\ \end{align} and

\begin{align} & s(\theta ,\xi )=\int_{\Omega }{\theta \xi dx} \\ \end{align}

So basically, my question concerns term

\begin{align} \Delta {{t}^{-1}}[{{z}^{j+1}}-{{z}^{j}}]\ge 0 \end{align}

Are there physical processes where this occurs?

P.S. Meaning of indexes on the top of variables can be understood from here:

\begin{align} & {{\theta }^{\Delta t}}(t)={{\theta }^{j+1/2}}+\Delta t[{{\omega }_{j}}(t)-\tfrac{1}{2}]{{z}^{j+1/2}}+\tfrac{1}{2}\Delta {{t}^{2}}{{\omega }_{j}}(t)[{{\omega }_{j}}(t)-1]{{{\dot{z}}}^{j+1/2}}, \\ & {\theta }'(t)={{z}^{j+1/2}}+\Delta t[{{\omega }_{j}}(t)-\tfrac{1}{2}]{{{\dot{z}}}^{j+1/2}} \\ & {\theta }''(t)={{{\dot{z}}}^{j+1/2}}, \\ & \forall t\in [{{t}_{j}},{{t}_{j+1}}]. \\ \end{align}

• Is "the gradient of thermal field" a vector field or a matrix field? What does it mean that the gradient is positive? – Hui Zhang Apr 12 '15 at 20:18
• I think it would be useful if you stated the equations and conditions you derived in mathematical terms, rather than just using numbers. – Wolfgang Bangerth Apr 13 '15 at 3:44
• I've updated the question to include formulas. If they are not clear please tell in that specific place I will update them. – zmii Apr 15 '15 at 7:48
• I see this value is represents something like the acceleration of temperature change - when it is not less that 0? – zmii Apr 16 '15 at 18:33