JuliaFEM.jl/docs/theory/lagrange_basis_functions.md

Lagrange Basis Functions in JuliaFEM

Date: November 9, 2025
Author: JuliaFEM Development Team

Introduction

Lagrange basis functions are the foundation of the Finite Element Method. They provide a systematic way to construct polynomial interpolation functions that satisfy the Kronecker delta property: the basis function associated with node i equals 1 at that node and 0 at all other nodes.

N_i(\mathbf{x}_j) = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}

This property makes it trivial to interpolate field values: u(\mathbf{x}) = \sum_i u_i N_i(\mathbf{x}) where u_i are nodal values.

Mathematical Foundation

Vandermonde Matrix Method

Given:

• n nodes with coordinates \{\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\} in reference element
• A polynomial basis (ansatz) \{p_1(\mathbf{x}), p_2(\mathbf{x}), \ldots, p_n(\mathbf{x})\}

We seek coefficients \alpha_{ij} such that:

N_i(\mathbf{x}) = \sum_{j=1}^{n} \alpha_{ij} p_j(\mathbf{x})

The Kronecker delta property gives us:

N_i(\mathbf{x}_k) = \sum_{j=1}^{n} \alpha_{ij} p_j(\mathbf{x}_k) = \delta_{ik}

This is a linear system: \mathbf{V} \boldsymbol{\alpha}_i = \mathbf{e}_i

Where the Vandermonde matrix is:

V_{kj} = p_j(\mathbf{x}_k)

And \mathbf{e}_i is the $i$-th unit vector.

Example: 1D Linear Element (Seg2)

Ansatz: p(\xi) = 1 + \xi (complete linear polynomial)

Nodes: \xi_1 = 0, \xi_2 = 1

Vandermonde matrix:

$$\mathbf{V} = \begin{bmatrix} p_1(\xi_1) & p_2(\xi_1) \ p_1(\xi_2) & p_2(\xi_2) \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix}$$

Solve for N_1: \mathbf{V} \boldsymbol{\alpha}_1 = [1, 0]^T

\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} \alpha_{11} \\ \alpha_{12} \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}

Solution: \alpha_{11} = 1, \alpha_{12} = -1

Therefore: N_1(\xi) = 1 \cdot 1 + (-1) \cdot \xi = 1 - \xi ✓

Solve for N_2: \mathbf{V} \boldsymbol{\alpha}_2 = [0, 1]^T

Solution: \alpha_{21} = 0, \alpha_{22} = 1

Therefore: N_2(\xi) = 0 \cdot 1 + 1 \cdot \xi = \xi ✓

Verification:

• N_1(0) = 1, N_1(1) = 0 ✓
• N_2(0) = 0, N_2(1) = 1 ✓
• N_1(\xi) + N_2(\xi) = 1 (partition of unity) ✓

Polynomial Completeness

The ansatz polynomial must be complete to the desired order:

Order	1D	2D	3D	Nodes Required
Linear	1 + \xi	1 + \xi + \eta	1 + \xi + \eta + \zeta	d+1
Quadratic	1 + \xi + \xi^2	1 + \xi + \eta + \xi^2 + \xi\eta + \eta^2	...	(d+1)(d+2)/2

Example for 2D Triangle (Tri3):

Ansatz: p(\xi, \eta) = 1 + \xi + \eta (complete linear in 2D)

This is the minimal complete polynomial for 3 nodes.

Implementation in JuliaFEM

Automatic Generation Process

# 1. Define element geometry
coords = [(0.0, 0.0), (1.0, 0.0), (0.0, 1.0)]  # Tri3 nodes

# 2. Define ansatz polynomial
ansatz = :(1 + u + v)  # Complete linear in 2D

# 3. Build Vandermonde matrix
V[i,j] = eval_polynomial_term(ansatz_terms[j], coords[i])

# 4. For each node i:
coeffs = V \ e_i  # Solve linear system
N_i = sum(coeffs[j] * ansatz_terms[j])  # Construct basis function

# 5. Symbolic differentiation
∂N_i/∂ξ = differentiate(N_i, :u)
∂N_i/∂η = differentiate(N_i, :v)

Why This Works

1. Completeness: Ansatz spans full polynomial space of given order
2. Linear Independence: Vandermonde matrix is non-singular for distinct nodes
3. Interpolation Property: Follows directly from \mathbf{V} \boldsymbol{\alpha}_i = \mathbf{e}_i

Derivatives

Once we have N_i(\xi, \eta, \zeta) symbolically, derivatives are straightforward:

\frac{\partial N_i}{\partial \xi}, \frac{\partial N_i}{\partial \eta}, \frac{\partial N_i}{\partial \zeta}

These are computed once symbolically, then pre-compiled into efficient Julia code.

Standard Lagrange Elements in JuliaFEM

1D Elements

• Seg2: Linear (2 nodes)
• Seg3: Quadratic (3 nodes, mid-edge node)

2D Elements

• Tri3: Linear triangle (3 corner nodes)
• Tri6: Quadratic triangle (6 nodes: 3 corners + 3 mid-edges)
• Quad4: Bilinear quadrilateral (4 corner nodes)
• Quad8: Serendipity quadrilateral (8 nodes: 4 corners + 4 mid-edges)
• Quad9: Biquadratic quadrilateral (9 nodes: 4 corners + 4 mid-edges + 1 center)

3D Elements

• Tet4: Linear tetrahedron (4 corner nodes)
• Tet10: Quadratic tetrahedron (10 nodes: 4 corners + 6 mid-edges)
• Hex8: Trilinear hexahedron (8 corner nodes)
• Hex20: Serendipity hexahedron (20 nodes: 8 corners + 12 mid-edges)
• Hex27: Triquadratic hexahedron (27 nodes: full tensor product)
• Pyr5: Linear pyramid (5 nodes)
• Wedge6: Linear wedge/prism (6 nodes)
• Wedge15: Quadratic wedge (15 nodes)

Pre-Generation vs Runtime Generation

Historical Approach (JuliaFEM ≤ 0.5.1)

# At package load time:
create_basis_and_eval(:Tet10, "...", coords, ansatz)
# - Builds Vandermonde matrix
# - Solves n linear systems
# - Symbolic differentiation
# - Simplification
# - Code generation with eval()
# Result: __precompile__(false) - slow loading

Problems:

• ❌ Symbolic math every package load (100+ ms)
• ❌ Cannot precompile (eval() at module scope)
• ❌ Opaque code generation
• ❌ Hard to debug

Modern Approach (JuliaFEM ≥ 1.0)

# Once, during development:
scripts/generate_lagrange_basis.jl
# - Computes all bases symbolically
# - Writes clean Julia code to src/basis/lagrange_generated.jl

# At package load time:
include("basis/lagrange_generated.jl")
# - Just parses pre-written Julia code
# - Fully precompilable
# - Zero symbolic computation

Benefits:

• ✅ Instant package loading
• ✅ Full precompilation
• ✅ Readable generated code
• ✅ Easy to debug
• ✅ Version controlled (can review changes)

Numerical Stability

Vandermonde Matrix Conditioning

The Vandermonde matrix can be ill-conditioned for:

• High-order polynomials (p > 5)
• Poorly distributed nodes
• Reference elements far from unit cube/simplex

JuliaFEM's approach:

• Use canonical reference elements (unit cube [-1,1]^d or unit simplex)
• Lagrange elements rarely exceed order 3 in practice
• For high-order: Consider hierarchical bases (not Lagrange)

Verification

Generated basis functions are verified by:

1. Kronecker delta property: N_i(\mathbf{x}_j) = \delta_{ij}
2. Partition of unity: \sum_i N_i(\mathbf{x}) = 1 everywhere
3. Derivative correctness: Compare symbolic vs AD

See test/test_basis_functions.jl for comprehensive tests.

References

1. Hughes, T.J.R., "The Finite Element Method: Linear Static and Dynamic Finite Element Analysis", Dover, 2000
2. Zienkiewicz, O.C. and Taylor, R.L., "The Finite Element Method", Volumes 1-3, Butterworth-Heinemann, 2000
3. Szabó, B. and Babuška, I., "Finite Element Analysis", Wiley, 1991

See Also

• scripts/generate_lagrange_basis.jl - Generation script
• src/basis/lagrange_generated.jl - Generated code (do not edit manually)
• src/basis/lagrange_generator.jl - Generator functions (symbolic engine)
• benchmarks/tet10_derivatives_benchmark.jl - Performance analysis (manual vs AD)