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Heat Flow Visualization in 2D Surface

A MATLAB simulation for observing heat flow and temperature distribution across a 2D conductive surface using the finite difference method with Gauss-Seidel iteration.

Project Information

Field Details
Author Anik Tahabilder
Institution Western Carolina University, NC
Program Master of Science
Date Submitted November 27, 2018

Overview

This program simulates steady-state heat conduction on a 2D rectangular surface. The simulation models how heat distributes across a conductive material when temperature sources (probes) are applied at various positions.

Key Capabilities

  • Define custom grid dimensions (3x3 to 100x100)
  • Place temperature probes on all four edges (top, right, bottom, left)
  • Add an optional center probe for internal heat sources
  • Real-time animated visualization of heat propagation
  • Automatic convergence detection using RMS error analysis
  • Export simulation as MP4 video

Mathematical Background

Heat Equation

The simulation solves the 2D steady-state heat conduction equation (Laplace equation):

∂²T/∂x² + ∂²T/∂y² = 0

Finite Difference Approximation

Using the finite difference method, the temperature at each interior node is calculated as the average of its four neighboring nodes:

T(i,j) = [T(i-1,j) + T(i+1,j) + T(i,j-1) + T(i,j+1)] / 4

Iterative Solution (Gauss-Seidel Method)

The program uses the Gauss-Seidel iterative method to solve the system:

  1. Initialize the grid with boundary conditions
  2. Update each interior point using the 4-point average formula
  3. Calculate RMS error between consecutive iterations
  4. Repeat until RMS error < accuracy threshold OR maximum iterations reached

Convergence Criteria

RMS = sqrt( Σ(T_new - T_old)² / (rows × columns) )

The solution converges when RMS falls below the user-specified accuracy (0 to 1).

Boundary Conditions

Region Condition
Probe locations Fixed temperature (Dirichlet boundary)
Non-probe edges Heat insulated (Neumann boundary - zero flux)
Center probe Fixed temperature island (optional)

The heat-insulated boundary condition is implemented by copying the temperature from the adjacent interior cell:

% Example: Top edge insulation (before probe)
Matrix(1, 1:probeStart-1) = Matrix(2, 1:probeStart-1)

Installation & Requirements

Prerequisites

  • MATLAB R2015a or later
  • Image Processing Toolbox (for visualization)
  • No additional dependencies required

Setup

  1. Clone or download this repository
  2. Open MATLAB
  3. Navigate to the project directory
  4. Run the main script:
>> Project

Usage Guide

Step 1: Define Grid Size

Enter the number of rows (3 to 100): 60
Enter the number of columns (3 to 100): 50

Step 2: Configure Edge Probes

For each edge (upper, right, bottom, left), you will specify:

  • Start position: Where the probe begins
  • End position: Where the probe ends
  • Temperature: Temperature value in degrees Celsius
Enter start position for upper probe: 15
Enter end position for upper probe: 32
Enter temperature for upper probe: 500

Step 3: Optional Center Probe

Do you want to have a center probe? Enter Y for yes and N for No: Y
Enter center probe row start: 15
Enter center probe row end: 38
Enter center probe column start: 35
Enter center probe column end: 42
Enter temperature for center probe: 600

Step 4: Set Accuracy

Insert the value of minimum accuracy: 0.01

Step 5: View Results

The program will display:

  • Real-time heat map animation
  • Iteration count
  • Final RMS error
  • Execution time

Output

Console Output

*********************************************
Number of iterations = 287
RMS Error            = 9.85e-03
Run Time             = 12.45 seconds
*********************************************

Visual Output

  • Live Figure: Color-coded temperature distribution (jet colormap)
  • Color Scale: 0°C (blue) to 1000°C (red)
  • Axes: Row and column indices

Video Export

The simulation automatically generates TEMP.MP4 in the project directory:

  • Format: MPEG-4
  • Frame Rate: 20 fps
  • Contains: Complete iteration history

Project Structure

Heat-Flow-Visualization/
│
├── Project.m                    # Main program file
├── figure1.m                    # Results display function
├── README.md                    # This file
│
├── Grid Input Functions/
│   ├── getRowMatrix.m           # Row count input (3-100)
│   └── getColMatrix.m           # Column count input (3-100)
│
├── Probe Position Functions/
│   ├── getupperProbStart.m      # Upper probe start position
│   ├── getupperProbEnd.m        # Upper probe end position
│   ├── getrightProbStart.m      # Right probe start position
│   ├── getrightProbEnd.m        # Right probe end position
│   ├── getbottomProbStart.m     # Bottom probe start position
│   ├── getbottomProbEnd.m       # Bottom probe end position
│   ├── getleftProbStart.m       # Left probe start position
│   ├── getleftProbEnd.m         # Left probe end position
│   ├── getcenterProbRowStart.m  # Center probe row start
│   ├── getcenterProbRowEnd.m    # Center probe row end
│   ├── getcenterProbColStart.m  # Center probe column start
│   └── getcenterProbColEnd.m    # Center probe column end
│
├── Temperature Input Functions/
│   ├── getTempUpper.m           # Upper edge temperature
│   ├── getTempRight.m           # Right edge temperature
│   ├── getTempBottom.m          # Bottom edge temperature
│   ├── getTempLeft.m            # Left edge temperature
│   └── getTempCenter.m          # Center probe temperature
│
├── Simulation Parameters/
│   ├── getAccuracy.m            # Convergence threshold (0-1)
│   └── getMaxIter.m             # Maximum iterations
│
└── Output/
    └── TEMP.MP4                 # Generated video (after run)

Example Configuration

Here is a sample configuration for testing:

Parameter Value
Grid Size 60 rows × 50 columns
Upper Probe Position 15-32, Temperature 500°C
Right Probe Position 36-43, Temperature 1000°C
Bottom Probe Position 17-25, Temperature 800°C
Left Probe Position 22-38, Temperature 387°C
Center Probe Rows 15-38, Cols 35-42, Temperature 600°C
Accuracy 0.01
Max Iterations 300

Algorithm Flowchart

┌─────────────────────────────────┐
│         Start Program           │
└─────────────┬───────────────────┘
              ▼
┌─────────────────────────────────┐
│    Input Grid Dimensions        │
│    (Rows: 3-100, Cols: 3-100)   │
└─────────────┬───────────────────┘
              ▼
┌─────────────────────────────────┐
│  Configure Edge Probes          │
│  (Position + Temperature)       │
└─────────────┬───────────────────┘
              ▼
┌─────────────────────────────────┐
│  Optional: Add Center Probe     │
└─────────────┬───────────────────┘
              ▼
┌─────────────────────────────────┐
│  Set Accuracy Threshold         │
└─────────────┬───────────────────┘
              ▼
┌─────────────────────────────────┐
│  Initialize Zero Matrix         │
│  Apply Boundary Conditions      │
└─────────────┬───────────────────┘
              ▼
┌─────────────────────────────────┐
│  ┌───────────────────────────┐  │
│  │  Iteration Loop           │  │
│  │  ────────────────────     │  │
│  │  1. Update interior nodes │  │
│  │  2. Apply insulation BC   │  │
│  │  3. Calculate RMS error   │  │
│  │  4. Update visualization  │  │
│  │  5. Capture video frame   │  │
│  └───────────────────────────┘  │
│         │                       │
│         ▼                       │
│  RMS < Accuracy OR              │
│  Iterations >= MaxIter?         │
│         │ Yes                   │
└─────────┼───────────────────────┘
          ▼
┌─────────────────────────────────┐
│  Display Results                │
│  Export Video (TEMP.MP4)        │
└─────────────┬───────────────────┘
              ▼
┌─────────────────────────────────┐
│  Run Again? (Y/N)               │
└─────────────────────────────────┘

Applications

This simulation can be used to study:

  • Heat distribution in electronic components
  • Thermal analysis of building materials
  • Heat sink design optimization
  • Understanding thermal equilibrium concepts
  • Educational demonstrations of heat transfer principles

Limitations

  • 2D simulation only (no 3D heat flow)
  • Assumes homogeneous material properties
  • Fixed colormap range (0-1000°C)
  • Maximum grid size limited to 100×100

Future Improvements

  • Add support for non-rectangular geometries
  • Implement variable thermal conductivity
  • Add 3D visualization option
  • Include transient (time-dependent) heat analysis
  • Export temperature data to CSV

License

Academic Project - Western Carolina University, 2018

Acknowledgments

This project was completed as part of the Master of Science program at Western Carolina University, Cullowhee, North Carolina.