Heat Transfer Simulation

Problem Statement

The goal is to simulate 2D transient heat conduction on a conductive metal plate. The simulation models how thermal energy spreads from a constant heat source across a surface until it reaches a state of thermal equilibrium.

Objective: Treat a metal plate as an $M \times N$matrix and calculate the propagation of heat from an initial ambient temperature $T_R$ to a constant heat source $T_S$ at coordinates $(x_0, y_0)$.

Solution Design

The simulation uses a discrete convolution approach. The temperature of any given point at time $t+1$ is calculated as the average of itself and its eight immediate neighbors (a 3x3 mean filter):

This iterative process continues until Convergence is met: the maximum change in temperature between iterations is less than the specified tolerance $\epsilon$:

Solution Implementation

Hardware Acceleration

This application utilizes NVIDIA CUDA for parallel processing. By offloading the mean filter calculations to the GPU, the simulation handles large matrices significantly faster than a CPU-bound approach.

System Requirements

• GPU: NVIDIA CUDA-compatible GPU.
• Software: C++ Development Environment (Visual Studio).
• Dependency: CUDA Toolkit.

Solution Analysis

Performance is primarily bound by two factors:

• Grid Resolution: Larger $M \times N$ dimensions increase the memory footprint and calculation time per iteration.
• Iteration Count: Lower tolerance $\epsilon$ values lead to higher precision but require more cycles to reach equilibrium.

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Heat Transfer Simulation Demo

Conclusion

This framework provides a robust starting point for educational thermal modeling. By adjusting parameters, users can observe real-time diffusion patterns and the efficiency of GPU-accelerated numerical methods.