Finite Element from Scratch in Python

Introduction

The best way to learn FEM is to implement it yourself. This tutorial depends on no existing framework - starting from triangular elements, it builds a 2D linear-elastic solver in pure NumPy/SciPy, covering the full pipeline from meshing to post-processing. For engineers and students who want to deeply understand FEM internals.

Path

Chapter 1: Triangular elements & shape functions (2-3 days)

• Understand FEM core: from weak form to element matrix
• Implement shape functions and their Jacobians

import numpy as np

# Linear triangular element shape functions N1, N2, N3,
# expressed in area coordinates (L1, L2, L3)
def shape_functions(xi, eta):
    return [1 - xi - eta, xi, eta]

# Shape function derivatives (Jacobian transform)
def dN_dXY(xi, eta, coords):
    # coords: 3x2 nodal coordinates
    J = np.array([[coords[1,0]-coords[0,0], coords[2,0]-coords[0,0]],
                  [coords[1,1]-coords[0,1], coords[2,1]-coords[0,1]]])
    detJ = np.linalg.det(J)
    # ... assemble the B matrix

Chapter 2: Stiffness assembly (2-3 days)

• Global stiffness assembly logic
• Gaussian integration; sparse COO assembly

Chapter 3: Boundary conditions & solve (1-2 days)

• Dirichlet/Neumann handling
• SciPy sparse solver; stress recovery

Chapter 4: Meshing & post-processing (1-2 days)

• Delaunay triangulation; matplotlib contours

Who it's for

• Mech/civil/aero students who know FEM formulas but want to verify by hand
• Programmers wanting first-principles FEM, not just software operation
• A foundation before learning FEniCS / Abaqus