🌀 Finite Pointset Method for Couette Flow (GPU-Accelerated)

This project solves the 2D Couette flow problem using both the Generalized Finite Difference Method (GFDM) — a meshfree approach — and a meshgrid-based method, with GPU acceleration via CuPy. The solution is computed using the Chorin projection scheme, and the results are validated against the analytical solution.

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1. 📘 Governing Equations (Incompressible Navier-Stokes)

1.1 Lagrangian Formulation

$$ \begin{aligned} &\frac{d\vec{x}}{dt} = \vec{v} &\text{(Particle motion)} \\\ &\frac{D\vec{v}}{Dt} = -\nabla p + \nu \Delta \vec{v} + \vec{g} &\text{(Momentum)} \\\ &\nabla \cdot \vec{v} = 0 &\text{(Incompressibility)} \end{aligned} $$

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2. 🧮 Core Numerical Steps (Chorin Projection + GFDM)

2.1 Particle Motion

$$ \vec{x}^{n+1} = \vec{x}^n + \Delta t \vec{v}^n $$

2.2 Intermediate Velocity

$$ \vec{v}^* = \vec{v}^n + \Delta t \left( \nu \Delta \vec{v}^n + \vec{g}^n \right) $$

2.3 Pressure Poisson Equation

$$ \Delta p^{n+1} = \frac{\nabla \cdot \vec{v}^*}{\Delta t}, \quad \frac{\partial p}{\partial n} = 0 \ (\text{BC}) $$

2.4 Velocity Correction

$$ \vec{v}^{n+1} = \vec{v}^* - \Delta t \nabla p^{n+1} $$

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3. 🧠 Meshfree Derivative Approximation (GFDM - Least Squares)

For a particle at position $\vec{x}$ with neighbors $\vec{x}_i$:

$$ \begin{pmatrix} p \ p_x \ p_y \ p_{xx} \ p_{xy} \ p_{yy} \end{pmatrix} = (M^T W M)^{-1} (M^T W) \begin{pmatrix} p_1 \ \vdots \ p_m \end{pmatrix} $$

Weights (Gaussian Kernel)

$$ w_i = \exp\left(-6.25 \frac{|\vec{x}_i - \vec{x}|^2}{h^2}\right) $$

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4. 📊 Analytical Solution for Couette Flow

The analytical steady-state velocity profile is:

$$ u(y) = \frac{U_0}{L} y $$

Time-dependent analytical solution:

$$ u(y, t) = u(y) + \frac{2U_0}{\pi} \sum_{n=1}^\infty \frac{(-1)^n}{n} \sin\left(\frac{n\pi y}{L}\right) e^{-\frac{n^2 \pi^2 \nu t}{L^2}} $$

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5. 🧪 Results

• ✅ Velocity field closely matches the analytical solution.
• ✅ Pressure satisfies Neumann boundary conditions.
• ⚡ Significant speedup observed with GPU (CuPy) over CPU (NumPy) implementation.

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🚀 6. Project Setup & Usage

🔧 Requirements

• Python 3.11+
• CUDA-enabled GPU (with CUDA 12.x drivers)
• Anaconda (recommended)

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🧱 Step 1: Clone the repository

git clone https://github.com/nsreelekha/poisson_equation.git
cd poisson_equation

🧪 Step 2A: Create and activate the Conda environment

conda env create -f environment.yml
conda activate cupy129env

🧪 Step 2B: (Alternative) Use pip + virtualenv

python -m venv venv
# On Windows:
venv\Scripts\activate
# On macOS/Linux:
source venv/bin/activate

pip install -r requirements.txt

🚀 Running the Code

You can run the notebooks using Jupyter.

▶️ Step 3: Launch Jupyter Notebook

jupyter notebook

Then open:

📘 couette_flow.ipynb — Simulates shear-driven flow between parallel plates using meshfree and meshgrid-based methods.

The notebook will:

• ✅ Generate a meshfree particle distribution (or structured grid)

• ✅ Assemble the GFDM operators using weighted least squares

• ✅ Apply boundary conditions for Couette flow

• ✅ Solve the incompressible Navier–Stokes equations using the Chorin projection method

• ✅ Visualize velocity and pressure fields using Matplotlib

📦 Dependencies

All dependencies are managed via Conda or requirements.txt

Key packages include:

• cupy-cuda12x — GPU array library (NumPy-compatible)
• numpy, scipy, matplotlib
• pycuda (optional)
• Python 3.11

Install automatically via:

conda env create -f environment.yml

🛠️ Troubleshooting

• No GPU? Replace:

import cupy as cp

with:

import numpy as cp 

• Environment errors? Re-create the environment:

conda env remove -n cupy129env
conda env create -f environment.yml

• CuPy errors or crashes? Ensure that:
  • CUDA 12.x is correctly installed
  • Your GPU is compatible
  • cupy-cuda12x matches your CUDA version

📜 License

This project is licensed under the MIT License.

👤 Author

Sreelekha Nampally
🎓 B.Tech in Mathematics and Computing, NIT Mizoram
🛠️ Project developed as part of Summer Internship at IIT Tirupati
🌐 LinkedIn | GitHub