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Jukka Aho
9a55257ba7
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