To implement the Finite Element Method (FEM) for electromagnetic field problems on a Raspberry Pi,

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To implement the Finite Element Method (FEM) for electromagnetic field problems on a Raspberry Pi, follow this structured approach:

1. Setup Raspberry Pi

  • Install Raspberry Pi OS: Ensure the latest OS is installed.
  • Install Required Packages:
    bash
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    sudo apt-get update
sudo apt-get install python3 python3-pip python3-matplotlib gmsh
pip3 install numpy scipy

2. Problem Definition

Choose a simple 2D electrostatic problem (e.g., parallel plate capacitor with potentials V and 0). Domain: Rectangular region with Dirichlet BCs on top/bottom plates.

3. Mesh Generation

  • Using Gmsh:
    1. Create a .geo file defining the geometry and mesh parameters.
    2. Generate a triangular mesh and export to mesh.msh.
    3. Convert mesh.msh to a format readable by Python (e.g., using meshio).

4. FEM Implementation in Python

Code Structure:

python
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import numpy as np
from scipy.sparse import lil_matrix, csr_matrix
from scipy.sparse.linalg import spsolve
import matplotlib.pyplot as plt
import meshio

# Read mesh
mesh = meshio.read("mesh.msh")
points = mesh.points[:, :2]  # 2D points
cells = mesh.cells_dict["triangle"]  # Triangular elements

# Parameters
epsilon = 8.85e-12  # Permittivity
V = 10  # Voltage on top plate

# Identify boundary nodes (e.g., y=0 and y=height)
top_nodes = np.where(points[:, 1] == np.max(points[:, 1]))[0]
bottom_nodes = np.where(points[:, 1] == np.min(points[:, 1]))[0]

# Initialize global matrix (sparse) and vector
n_nodes = points.shape[0]
K = lil_matrix((n_nodes, n_nodes))
F = np.zeros(n_nodes)

# Assemble stiffness matrix
for tri in cells:
    nodes = tri
    x = points[nodes, 0]
    y = points[nodes, 1]
    area = 0.5 * np.abs((x[1]-x[0])*(y[2]-y[0]) - (x[2]-x[0])*(y[1]-y[0]))
    B = np.array([[y[1]-y[2], y[2]-y[0], y[0]-y[1]],
                  [x[2]-x[1], x[0]-x[2], x[1]-x[0]]]) / (2 * area)
    Ke = (epsilon * area) * (B.T @ B)  # Element matrix
    for i in range(3):
        for j in range(3):
            K[nodes[i], nodes[j]] += Ke[i, j]

# Apply Dirichlet BCs
for node in top_nodes:
    K[node] = 0
    K[node, node] = 1
    F[node] = V
for node in bottom_nodes:
    K[node] = 0
    K[node, node] = 1
    F[node] = 0

# Solve
K = csr_matrix(K)
phi = spsolve(K, F)

# Compute electric field (E = -grad phi)
Ex = np.zeros_like(phi)
Ey = np.zeros_like(phi)
for tri in cells:
    nodes = tri
    x = points[nodes, 0]
    y = points[nodes, 1]
    B = np.array([[y[1]-y[2], y[2]-y[0], y[0]-y[1]],
                  [x[2]-x[1], x[0]-x[2], x[1]-x[0]]]) / (2 * area)
    grad_phi = B @ phi[nodes]
    Ex[nodes] -= grad_phi[0]
    Ey[nodes] -= grad_phi[1]

# Visualize
plt.tricontourf(points[:, 0], points[:, 1], cells, phi, levels=20)
plt.colorbar(label='Electric Potential (V)')
plt.quiver(points[:, 0], points[:, 1], Ex, Ey, scale=50, color='white')
plt.title('Electric Potential and Field')
plt.xlabel('x')
plt.ylabel('y')
plt.show()

5. Post-Processing

  • Visualize the electric potential and field using Matplotlib.
  • Compute derived quantities (e.g., capacitance).

6. Optimization Tips

  • Use sparse matrices to save memory.
  • Limit mesh size (e.g., <1000 nodes) for Raspberry Pi compatibility.
  • Prefer iterative solvers (e.g., Conjugate Gradient) for larger systems.

Example Results

  • Potential Distribution: Linear gradient between plates.
  • Electric Field: Uniform between plates, diverging at edges.

Extensions

  • 3D Problems: Use tetrahedral meshes (requires more computational power).
  • Time-Harmonic Analysis: Solve ∇×(μ−1∇×E)−ω2ϵE=0.
  • Nonlinear Materials: Iterate to update material properties.

This approach leverages the Raspberry Pi’s capabilities for educational or small-scale FEM projects in electromagnetics

Published by howdy