FEM & Shape Functions

The Finite Element Method approximates continuous fields with a finite number of nodal values. Shape functions interpolate between these values to reconstruct the full field.

The FEM Idea

An unknown function $u(x)$ has infinitely many degrees of freedom. We can't compute infinite values, so we:

• Discretize: Place nodes at positions $x_0, x_1, ..., x_n$
• Solve for nodal values $u_0, u_1, ..., u_n$
• Interpolate between nodes using shape functions

$$u(x) = \sum_{i=0}^{n} N_i(x) \cdot u_i$$

where $N_i(x)$ is the shape function for node $i$.

Shape Function Properties

1. Partition of Unity

$$\sum_{i=0}^{n} N_i(x) = 1 \quad \text{for all } x$$

This ensures that a constant field is represented exactly.

2. Kronecker Delta

$$N_i(x_j) = \delta_{ij}$$

At node $j$, only shape function $N_j$ equals 1; all others are 0.

Visualization

When you adjust nodal values, the entire interpolated field changes:

Python Code

import numpy as np
import matplotlib.pyplot as plt

def linear_shape_function(x, xi, L):
    """Linear shape function for node at xi with element length L."""
    if xi <= x <= xi + L:
        return (x - xi) / L
    elif xi - L <= x < xi:
        return (xi - x) / L + 1
    else:
        return 0

# Create mesh with 5 nodes
nodes = np.linspace(0, 1, 5)
L = nodes[1] - nodes[0]

# Nodal values
u_nodes = np.array([0, 0.3, 0.5, 0.2, 0])

# Interpolate
x = np.linspace(0, 1, 100)
u = np.zeros_like(x)
for i, xi in enumerate(nodes):
    N = np.array([linear_shape_function(xp, xi, L) for xp in x])
    u += N * u_nodes[i]

plt.plot(x, u, 'b-', label='Interpolated')
plt.plot(nodes, u_nodes, 'ro', label='Nodal values')
plt.legend()
plt.show()

Automotive Application: Engine Bracket Mesh

When analyzing an engine bracket:

• Coarse mesh: Few nodes, fast but less accurate
• Fine mesh: Many nodes, accurate but slow
• Shape functions interpolate displacement between nodes
• Stress is computed from displacement gradients (derivatives of shape functions)

Key Takeaways

• FEM approximates continuous fields with nodal values
• Shape functions interpolate between nodes
• Shape functions satisfy partition of unity and Kronecker delta
• More nodes = better approximation (but more computation)

What's Next

In the next lesson, we'll explore mesh refinement — how increasing node density improves accuracy and when to stop refining.