Create 415-line blog post combining technical demonstration with philosophical
vision for JuliaFEM v1.0 nodal assembly architecture.
Content structure:
- Lines 1-32: Why Krylov+nodal is brilliant for contact mechanics
* Contact is inherently nodal (constraints at nodes, not elements)
* Krylov only needs matvec (never forms global matrix)
* Nodal assembly provides natural row-by-row interface
* O(N) memory vs O(N²) for traditional element assembly
- Lines 34-73: Controversial hypothesis about nodal material modeling
* Claims material state should be at nodes, not integration points
* Argues integration points constrain physics to numerical method
* Variational consistency, physical meaning, scalability arguments
* "I will show them they're wrong" - experimental vision
- Lines 75-95: GMRES advantage for unsymmetric systems
* Material nonlinearity, contact, large deformation all unsymmetric
* GMRES solves positive definite unsymmetric systems
* O(N·iter) time, O(N) memory vs O(N³)/O(N²) for direct solvers
- Lines 97-415: Working GMRES demonstration on 10×10 unsymmetric system
* Problem setup: Positive definite but unsymmetric matrix (lines 105-145)
* Nodal assembly pattern: get_row() interface (lines 147-193)
* Simplified GMRES implementation (lines 195-282)
* Execution and verification (lines 284-318)
* Results: Converged in 10 iterations, 6.28×10⁻¹⁶ relative error
* Key insights section explaining significance (lines 320-365)
* Development roadmap: Immediate → Near-term → Long-term → Vision (lines 367-401)
* Conclusion: Philosophical statement about nodal correctness (lines 403-415)
Technical validation:
- Matrix: 10×10, eigenvalues [9.94, 38.06], condition number 3.83
- Nodal matvec: 1.59×10⁻¹⁴ error vs direct computation
- GMRES: 10 iterations to convergence
- Solution accuracy: 1.23×10⁻¹⁴ absolute error, 6.28×10⁻¹⁶ relative
Dependencies: LinearAlgebra, Random, Printf
Format: Literate.jl (# # for section headers, # for narrative)
Target: Blog post for JuliaFEM v1.0 development documentation
Tone: Opinionated, controversial, technically rigorous
Jukka Aho