Finite Element Teaching Repository

This repository is written in Julia and is designed for teaching purposes. It introduces the fundamentals of the finite element method (FEM) and explores both assembled and matrix-free approaches. It has two parts:

• src/ — FESolid2D, a 2D solid mechanics finite element package (matrix-free, libCEED/Ratel-style), with a test/ suite. Hyperelasticity uses a hand-derived Jacobian; plasticity's Jacobian is instead obtained via Enzyme.jl automatic differentiation.
• notebooks/ — the teaching/exploratory notebooks the package is developed from.

Running

Package + tests:

using Pkg
Pkg.develop(path=".")     # or path="/path/to/fe-demo" from elsewhere
Pkg.test("FESolid2D")

Notebooks: notebooks are not tied to the package's own Project.toml — they run in your notebook kernel's own Julia environment, so install their dependencies there (FESolid2D itself, plus whichever of NLsolve/Plots a given notebook uses):

using Pkg
Pkg.develop(path="/path/to/fe-demo")
Pkg.add(["NLsolve", "Plots"])

then open any notebook under notebooks/ in Jupyter/IJulia.

FEM1D

• Begins with Ciarlet’s (1978) definition of finite elements.
• Discusses the ill-conditioning problem of evenly spaced nodes for higher-order polynomials.
• Shows how using Lobatto points resolves this issue.
• Implements L² projection and Poisson problems using assembled and matrix-free methods for both linear and nonlinear constitutive equations.

FEM2D

• Extends 1D elements to 2D quadrilateral elements using tensor products.
• Solves L² projection and Poisson problems in 2D.

Benchmark2D

• Includes the benchmark problem from libCEED.
• Compares the performance of assembled vs. matrix-free approaches.

Elasticity2D

• Solves linear elasticity problems (manufactured solutions).
• Demonstrates convergence orders for different element types using assembled and matrix-free approaches.
• Includes traction (Neumann) BCs via a face element restriction.

MixedElasticity2D

• Solves incompressible linear elasticity (manufactured solutions).
• Explores different mixed elements with continuous and discontinuous pressure spaces.
• Performs the inf-sup test to examine the stability of mixed elements.

Hyperelasticity2D

• Solves Neo-Hookean hyperelasticity (isochoric-volumetric split, plane strain), matrix-free.
• Demo: a beam clamped on one edge and pulled by a traction on the opposite edge, solved with a load-stepped Newton solve, with a deformed-vs-undeformed mesh visualization.

Plasticity2D

• Solves multiplicative Hencky plasticity (von Mises/J2 yield, linear + Voce hardening, plane strain), matrix-free, with the Jacobian obtained via Enzyme.jl automatic differentiation.
• Demo: the same clamped-beam problem as Hyperelasticity2D, run through both a purely elastic reference material and a yielding one, showing the load-displacement response diverge once yielding starts.