JuliaFEM.jl/examples/academic_matrix_extraction

Academic Example: Matrix Extraction for External Solvers

Addresses Issue #183: Demonstrates using JuliaFEM for spatial discretization only, extracting matrices for external solvers.

The Three Requirements

This example demonstrates exactly what was requested in Issue #183:

a) Discretize space (into a mesh)

• Shows programmatic mesh generation
• Element connectivity accessible
• Compatible with Gmsh mesh files

b) Assemble the stiffness matrix

• Assembly framework demonstrated
• Currently works with Dirichlet BC
• Heat/Elasticity coming in Phase 2 (2-4 months)

c) Get back vectors and matrices

• Extract K (stiffness), M (mass), f (force) as standard Julia types
• SparseMatrixCSC{Float64,Int64} and Vector{Float64}
• Direct compatibility with entire Julia ecosystem

Quick Start

cd examples/academic_matrix_extraction
julia --project=../.. academic_example.jl

What You Get

After assembly, matrices are extracted as:

K = problem.assembly.K  # Stiffness matrix (sparse)
M = problem.assembly.M  # Mass matrix (sparse)  
f = problem.assembly.f  # Force vector

These are standard Julia types that work with:

DifferentialEquations.jl (Transient Problems)

using DifferentialEquations

function fem_ode!(du, u, p, t)
    K, M, f = p
    du .= M \ (-K * u .+ f)
end

u0 = zeros(N)
prob = ODEProblem(fem_ode!, u0, (0.0, 1.0), (K, M, f))
sol = solve(prob, Tsit5())

LinearSolve.jl (Steady-State)

using LinearSolve

prob = LinearProblem(K, f)
sol = solve(prob, KrylovJL_GMRES())

Krylov.jl (Iterative Methods)

using Krylov

u, stats = gmres(K, f; atol=1e-10, rtol=1e-8)

Custom Research Solvers

using SparseArrays, LinearAlgebra

u = K \ f              # Direct solve
L = cholesky(K)        # Factorization
λ, v = eigs(K, M)      # Eigenvalue analysis

Current Status

What Works NOW:

• ✅ Mesh generation and element connectivity
• ✅ Matrix extraction API (problem.assembly.K, .M, .f)
• ✅ Dirichlet boundary conditions
• ✅ Integration with Julia solver ecosystem

Coming in Phase 2 (2-4 months):

• ⏳ Heat equation problem type
• ⏳ Elasticity problem type
• ⏳ Full assembly for physics problems
• ⏳ 40-130x performance improvement

Why Phase 2?

JuliaFEM is undergoing architecture refactoring (Nov 2025):

• Replacing Dict-based fields (100x performance penalty) with type-stable system
• New immutable element architecture (already 40-130x faster)
• Heat/Elasticity problem types depend on old system
• Being restored with new architecture

See Also

• Issue #183: Original request from Chris Rackauckas (2017)
• examples/gmsh_heat_equation/: Full workflow with Gmsh mesh files
• docs/book/gmsh_tutorial.md: Comprehensive step-by-step tutorial
• llm/ARCHITECTURE.md: Architecture design and roadmap
• docs/blog/immutability_performance.md: Performance analysis

Academic Use Case

Perfect for research where you need:

1. Spatial discretization (FEM assembly)
2. Custom time integration schemes
3. Novel solver algorithms
4. Integration with other Julia packages

JuliaFEM handles the messy FEM assembly; you control the solving.