Rapid Stabilization of Heat and Structure Interactions with Boundary Feedback Controllers
An interactive heat-structure model showing how thermal and mechanical dynamics can be stabilized together.
Overview
This Demonstration couples heat transfer in a copper rod to the longitudinal motion of a magnetizable piezoelectric beam. The live controls set time, the tip-velocity and total-current amplifiers, and the thermal diffusivity, then choose sinusoidal, hot, or cold initial heat distributions and linear, box, or pinch initial beam distributions.
Mathematical model
The copper rod follows the heat equation $z_t-\kappa z_{xx}=0$, while the beam displacement $v$ and electrode charge $p$ satisfy a coupled piezoelectric system. Temperature and velocity meet at the rod-beam interface, and tip gains $\xi_1$ and $\xi_2$ feed back velocity and total current; order-reduced finite differences preserve the continuous system’s exponential-stability behavior:
\[\begin{bmatrix} \rho & 0 \\ 0 & \mu \end{bmatrix} \begin{bmatrix} v \\ p \end{bmatrix}_{tt} = \begin{bmatrix} \alpha & -\gamma\beta \\ -\gamma\beta & \beta \end{bmatrix} \begin{bmatrix} v \\ p \end{bmatrix}_{xx}.\]Research project
This Demonstration is part of Piezoelectric Beam Stabilization, extending its boundary-feedback design to a coupled heat-structure system. It connects amplifier design and structure-preserving discretization to a multiphysics setting in which thermal and mechanical energy must decay together.
Reference
Walterman, Jacob, Ahmet Kaan Aydin, and Ahmet Özkan Özer. 2024. Rapid Stabilization of Heat and Structure Interactions with Boundary Feedback Controllers. Wolfram Demonstrations Project.