RateCondensation

Source: models/common/RateCondensation.py

Linearize \(f(\dot\gamma)\) at \(\dot\gamma = 0\), emitting \(b\) and \(A\).

CoordinateDescentPredictor solves \(\varphi(\dot\gamma) + A\dot\gamma = b\), which is the condensed system \(\dot\gamma = \varphi^{-1}(f(\dot\gamma))\) with \(f\) linearized:

\[f(\dot\gamma) \approx f(0) - A\dot\gamma, \qquad b = f(0), \qquad A = -\left.\frac{\partial f}{\partial \dot\gamma}\right|_0.\]

model supplies \(f\) – the return path: rate in, driving force out, with the flow direction frozen at the trial state. Both \(b\) and \(A\) come from a single jvp of it, which is one forward pass with a tangent riding the existing forward(v=) chain rule (see jvp). So this is neither a finite difference – no truncation error, no step size to pick – nor reverse-mode autograd.

Why this rather than a closed form. Writing \(A\) out by hand means one term per hardening mechanism: \(\Delta t\,N:\mathbb{C}:N\) for the elastic relaxation, \(\Delta t\,h'\) for isotropic hardening, \(\Delta t\,N:\partial \dot X_i/\partial\dot\gamma\) for each back stress. Every new hardening law is then a code change, and the terms are easy to get subtly wrong (the kinematic one is \(C_i\), not \(\tfrac{2}{3}C_i\), because \(N:N = \tfrac32\)). Differentiating the authored path costs nothing per physics: the hardening model is a block in model, and this leaf never learns any of it. All it needs is a first-order chain rule on each leaf in the path, which is the framework’s own invariant.

Dropping the hardening instead does not work. Freezing it leaves \(A = \Delta t\,N:\mathbb{C}:N\), whose relative error in \(\Delta\gamma\) is \(h'/(N:\mathbb{C}:N)\) – under a percent for the usual metal parameterizations. But the returned stress is a near-cancellation of the trial stress against the plastic relaxation, and it amplifies that error by \(\sigma^{\rm trial}/\sigma\), which grows with the step. Measured on the viscoplasticity scenarios, lagging halves the first-step Newton count below \(\Delta t \approx 1\) and buys nothing above it; the exact coupling converges at the predictor at every step size.

Only the trial-point tangent is taken, so a path that is strongly nonlinear over a step is linearized, not solved. That is a warm start, not an integrator: a predictor moves the initial guess and never the answer.

Outputs

coupling — output · Scalar · required

The coupling \(A = -\partial f/\partial\dot\gamma\) at zero rate

trial_driving_force — output · Scalar · required

The right-hand side \(b = f(0)\): the driving force with the rate at zero

Other options

model — str · required

The return path: the rate in, the driving force out, with the flow direction frozen at the trial state. Its rate input is named by rate and its driving-force output by driving_force; every other input is forwarded unchanged.

rate — str · required

Which of model’s inputs is the rate.

driving_force — str · required

Which of model’s outputs is the driving force.