JuliaFEM.jl/docs/book/lagrange_basis_functions.md

title, subtitle, description, date, author, categories, keywords, audience, level, type, series, chapter, math, prerequisites

title	subtitle	description	date	author	categories	keywords	audience	level	type	series	chapter	math	prerequisites
Lagrange Basis Functions	Mathematical foundations of finite element interpolation	Complete derivation of Lagrange basis functions using Vandermonde matrix method	2025-11-09	Jukka Aho	theory mathematics fem	lagrange basis shape functions interpolation vandermonde matrix fem theory	researchers and advanced users	expert	theory	The JuliaFEM Book	Part I: Foundations	true	linear algebra numerical analysis fem basics

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)