Finite Element Method (FEM) for Poisson's Equation

This repository contains an implementation of the Finite Element Method (FEM) to solve the Poisson's equation on a rectangular domain. The code is written in Python utilizing NumPy, SciPy and Matplotlib for calculations and visualizations.

Overview

The Finite Element Method (FEM) is a numerical technique used to approximate solutions to boundary value problems for partial differential equations (PDEs). It subdivides a large problem into smaller, simpler parts that are called finite elements.

This project specifically addresses the Poisson's equation, commonly used in electromagnetism, fluid dynamics, and other fields.

Dependencies

This project relies on several Python libraries to perform numerical calculations and data visualization. Below is a list of required dependencies:

• NumPy: for efficient numerical array operations.
• SciPy: used for sparse matrix operations and solving linear systems.
• Matplotlib: for generating 2D and 3D visualizations of the FEM results.

Installing Dependencies

To install these dependencies, you can use pip. First, ensure you have Python and pip installed on your system. Then, run the following command:

pip install numpy scipy matplotlib

Alternatively, you can use the included requirements file, requirements.txt, which lists all the necessary libraries. Simply run the following command to install all dependencies at once:

pip install -r requirements.txt

Project Structure

• get_nodes_and_triangles: generates the mesh of nodes and triangles for the specified domain.
• get_local_stiffness_matrix: computes the local stiffness matrix for a single triangular element.
• get_global_stiffness_matrix: assembles the local stiffness matrices into a global stiffness matrix.
• get_boundary_nodes: identifies nodes at the boundary of the domain to apply boundary conditions.
• apply_boundary_conditions: modifies the global stiffness matrix to enforce zero potential at the boundaries.
• construct_rhs: constructs the right-hand side vector that represents the source terms.
• solve_fem_system: solves the linear system using sparse matrix techniques.
• plot_solution_2D and plot_solution_3D: visualize the solution in 2D and 3D, respectively.

Step-by-Step Guide

Step 1: Mesh Generation

Theory: the domain is divided into a finite number of small elements, typically triangles, which together form a mesh. Each point where elements meet is called a node.

Explanation: think of this as cutting a piece of paper into tiny triangles. Each corner or point where the cuts meet is important because it helps us describe the shape of the paper.

Step 2: Local Stiffness Matrix

Theory: for each triangle, a local stiffness matrix is calculated based on the triangle's geometry and the physics of the problem. This matrix describes how the triangle's points (nodes) interact with each other mechanically.

Explanation: imagine each triangle as a mini trampoline. When you press down on one corner, the other corners react. The local stiffness matrix tells us how much each corner affects the others.

Step 3: Global Stiffness Matrix

Theory: the local stiffness matrices are assembled into a global matrix that represents the entire mesh. This step involves placing each local matrix into the global matrix according to the triangles' connections.

Explanation: now, connect all the mini trampolines together according to how the triangles touch each other. The global matrix tells us how pressing on any point of the entire mesh affects every other point.

Step 4: Boundary Conditions

Theory: boundary conditions are applied to the global stiffness matrix to reflect the physical constraints of the problem, such as fixed potentials at the boundaries.

Explanation: if the edges of the paper are taped down, those points can't move. We adjust our calculations to keep these points fixed.

Step 5: Solve the Linear System

Theory: the linear system, formed by the global stiffness matrix and the right-hand side vector (representing external forces or sources), is solved to find the potentials at each node.

Explanation: we now solve a big puzzle. Given the stiffness of each part and the forces applied, we find out how much each node moves or what its potential is.

Step 6: Visualization

Theory: the potentials at the nodes are visualized over the domain to analyze the behavior of the solution.

Explanation: we draw a map showing how high or low each point on the paper is, using colors to show different heights or potentials.

Examples

Gaussian Heat Source

This function simulates a localized heat source or sink, ideal for modeling phenomena such as concentrated heat or energy sources within a specified area.

Multi-Wave Interference

This function generates a multi-wave interference pattern using a combination of sine and cosine waves with varying frequencies and phases. It's designed to model phenomena where multiple wave sources interact, such as in water waves, sound waves, or electromagnetic fields.

Radial Symmetric Function

This function models radial effects like a pressure or gravitational field emanating from or concentrating towards a central point.

Sine Wave Pattern

This function creates a sinusoidal pattern over the domain, useful for simulating periodic processes like waves or oscillations across a surface.