Phase 3a — Time integration + LocalNewton lowering #34

Part of the NumSim compiler epic: #28.

Summary

Lower the Material IR's time-evolving + LNS structure into concrete generated code: replace Dt(α) = f(...) with a discrete time-stepping rule (Backward Euler default) and emit a Newton iteration loop that solves the resulting algebraic system at each call.

Scope

3a.1 — TimeIntegrationPass

• Schemes: BackwardEuler (default), ForwardEuler, BDF2
• Lowers Dt(α) = f(α, ε) → (α_new − α_old)/dt − f(α_new, ε) = 0 using CAS's existing substitute()
• Adds α_new and α_old as IR state-variable nodes if not already present

3a.2 — LocalNewtonLoweringPass

• Builds residual vector R(x, ε) from each LocalNewtonSystem's residual list
• Computes ∂R/∂x symbolically via existing tensor_differentiation
• Emits a Newton iteration loop with:
  • Configurable convergence tolerance + max iterations
  • Residual-norm criterion (energy norm later)
  • Divergence detection (NaN check, residual growth)
  • Non-singular-Jacobian fallback (minor diagonal regularisation)
• Line search: none initially (flagged for follow-up; see plan)

Acceptance

• J2 plasticity recipe with StateVariable ε_p lowers through TimeIntegrationPass + LocalNewtonLoweringPass to produce a working stress update
• At uniaxial tension, matches an analytic reference for σ within 1e-6
• The 1D spike from Phase 1.3 is now production code (no longer throwaway)

Files affected

• New: include/numsim_codegen/passes/TimeIntegrationPass.h
• New: include/numsim_codegen/passes/LocalNewtonLoweringPass.h
• Test recipes: tests/goldens/J2PlasticityFull.cpp

Dependencies

• Phase 2 (Material IR with StateVariable, Dt, LocalNewtonSystem)