LSTM PINN for transient heat conduction

0. Introduction

This repository contains google colab based FEniCS implementation of the transient heat conduction problem. This model will be further extended to include Physics Informed Neural Network (PINN) using Long Short-Term Memory (LSTM).

1. Prerequisites

We use python 3.6.9 as the programming language. In this project we use the libraries :

• FEniCS 2019.1.0 (www.fenicsproject.org)
• Matplolib 3.1.2 (www.matplotlib.org)
• numpy 1.17.4 (www.numpy.org)

The solutions are stored in .xdmf format, which can later be viewed with Paraview (www.paraview.org).

2. Installation

Simply clone the public repository:

git clone https://github.com/niravshah241/transient_heat_conduction_LSTM_PINN.git

3. Problem statement

The domain $\omega$ is divided into $7$ non-overlapping subdomains: $\bar{\omega} = \bigcup\limits_{i=1}^{7} \bar{\omega}_{i}$. The heat conduction equation in strong form at a given source term $Q$ is given by: $$\rho c \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + Q \ \text{in} \ \omega \times (0,\mathcal{T}] \ .$$
Above equation is solved using Finite Element Method in space and backward difference in time to compute the temperature field $T(x,t)$ at given point $x \in \omega$ and time $t \in (0,\mathcal{T}]$. The material properties (Density $\rho$, Heat capacity $c$ and Thermal conductivity $k$) vary across each subdomain characterising different material constituting the subdomain, i.e.:

$$ \rho=\rho_{i} \ , \ c = c_{i} \ , \ k = k_{i} \ \text{for} \ x \in \omega_{i}.$$

The boundary conditions are given by:

$$\text{Specified heat flux (Neumann BC):} \ \overrightarrow{q} \cdot \overrightarrow{n} = 0 \ \text{on} \ \gamma_{+} \cup \gamma_{s} \times (0,\mathcal{T}] \ .$$

$$\text{Heat exchanger (cold) boundary (Robin BC):} \ \overrightarrow{q} \cdot \overrightarrow{n} = h (T - T_{out}) \ \text{on} \ \gamma_{out} \cup \gamma_{-} \times (0,\mathcal{T}] \ \text{with} \ T_{out} = 300 \ K \ .$$

$$\text{Heat exchanger (hot) boundary (Robin BC):} \ \overrightarrow{q} \cdot \overrightarrow{n} = h (T - T_{sf}) \ \text{on} \ \gamma_{sf} \times (0,\mathcal{T}] \ \text{with} \ T_{sf} = 300 + 600t \ K \ .$$

The initial condition corressponds to the body at enviromental temperature:

$$T = 300 K \ \text{in} \ \omega \times \lbrace 0 \rbrace \ .$$

4. Numerical results

• Temperature evolution

5. Authors and contributors

This code has been developed by [Nirav Shah] email.

6. How to cite

@misc{transient_heat_conduction_LSTM_PINN,
key          = {TransientHeatCondductionLSTMPINN},
author       = {Shah, N.V.},
title        = {{Long Short Term Memory based Physics Informed Neural Network for transient heat conduction problem, Version 0.1}},
month        = July,
url          = {https://github.com/niravshah241/transient_heat_conduction_LSTM_PINN.git},
year         = 2022
}

6. License

• FEniCS is freely available under the GNU LGPL, version 3.

• Matplotlib only uses BSD compatible code, and its license is based on the PSF license. Non-BSD compatible licenses (e.g., LGPL) are acceptable in matplotlib toolkits.

Accordingly, this code is freely available under the GNU LGPL, version 3 and BSD-license.

7. Disclaimer

In downloading this SOFTWARE you are deemed to have read and agreed to the following terms: This SOFT- WARE has been designed with an exclusive focus on civil applications. It is not to be used for any illegal, deceptive, misleading or unethical purpose or in any military applications. This includes ANY APPLICATION WHERE THE USE OF THE SOFTWARE MAY RESULT IN DEATH, PERSONAL INJURY OR SEVERE PHYSICAL OR ENVIRONMENTAL DAMAGE. Any redistribution of the software must retain this disclaimer. BY INSTALLING, COPYING, OR OTHERWISE USING THE SOFTWARE, YOU AGREE TO THE TERMS ABOVE. IF YOU DO NOT AGREE TO THESE TERMS, DO NOT INSTALL OR USE THE SOFTWARE.