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Jukka Aho
552d701c5a
Explains the correct pattern for matrix-vector products in Krylov methods:
- Fields accessed through element_set (not passed separately)
- GPU kernel computes y=K*x (not K itself)
- O(N) memory (vs O(N²) for stored matrix)
- Type-stable field access (compile-time types)
Key insights:
- GMRES needs matvec operation, not the matrix
- ElementSet contains elements + fields together
- Zero allocations with immutable connectivity/fields
- Natural pattern for contact mechanics (nodal updates)
- Material state separate from field parameters
Compares old vs new approach:
- Old: Dict{String,Any} in element (type-unstable)
- New: NamedTuple in ElementSet (type-stable)
- Old: O(N²) matrix storage
- New: O(N) matrix-free operator
Validated with gpu_elementset_matvec_demo.jl:
- GPU/CPU results match exactly
- Fields accessed naturally through element_set
- Returns y vector (what Krylov methods need)
title, subtitle, description, date, author, categories, keywords, audience, level, type, status
| title | subtitle | description | date | author | categories | keywords | audience | level | type | status | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| The JuliaFEM Book | A comprehensive manual mixing theory, software design, and personal experience | Deep dive into FEM theory, design philosophy, and research directions | 2025-11-09 | Jukka Aho |
|
|
researchers and theory enthusiasts | expert | book | work in progress |
The JuliaFEM Book
Audience: Advanced researchers, theory nerds, those who want to understand the "why" and "how" at a deep level. And Jukka.
This is the JuliaFEM Bible - a comprehensive manual mixing theory, philosophy, software design, and personal experience. It's educational, opinionated, and unapologetically deep.
What's Here
- Mathematical Foundations: Lagrange basis functions, weak forms, contact mechanics
- Design Philosophy: Why JuliaFEM exists, what problems it solves (and doesn't)
- Technical Vision: Strategic mistakes from 2015-2019, lessons learned
- Research Directions: Experimental ideas (nodal assembly, matrix-free, etc.)
- Personal Notes: The journey, the failures, the "aha!" moments
- Theory + Code: How mathematics becomes software
What's NOT Here
- "How do I install?" (see
docs/user/) - "How do I add a feature?" (see
docs/contributor/) - Short answers (everything here is DEEP)
Philosophy
"Let me show you how I think about FEM."
This is:
- Educational: Teach FEM through implementation
- Personal: Written in Jukka's voice, reflecting 8+ years of experience
- Opinionated: Strong views on what works and what doesn't
- Comprehensive: From first principles to cutting-edge research
- Honest: Documents failures as much as successes
We assume you:
- Love mathematics AND programming
- Want to understand WHY, not just HOW
- Have time to read deeply
- Are curious about unconventional approaches
- Might be me, 5 years from now, trying to remember why I did this
Structure
Part I: Foundations
- Finite Element Method (brief review)
- Lagrange Basis Functions (deep dive)
- Assembly and Solving
- Contact Mechanics
Part II: Software Design
- Type Stability and Performance
- Zero-Allocation Design
- Immutability and Composition
- Field System Architecture
Part III: History and Vision
- Strategic Mistakes (2015-2019)
- Why JuliaFEM is Different
- Contact Mechanics Focus
- Laboratory Philosophy
Part IV: Research
- Nodal Assembly (experimental)
- Matrix-Free Methods
- Automatic Differentiation
- GPU Acceleration
Part V: The Journey
- Personal Reflections
- Lessons Learned
- Future Directions
- Open Questions
Reading Guide
- For Theory: Start with Part I
- For Design Rationale: Start with Part II
- For History: Start with Part III
- For Research Ideas: Start with Part IV
- For Philosophy: Read Part V first, then everything else
Start here: Mathematical Foundations | Strategic Mistakes | Why JuliaFEM?