Sharp bits

This section covers implementation details which are relevant to computational efficiency.

Computational mechanics aspects

Enforcing plastic consistency conditions

In computational plasticity, ensuring the plastic consistency condition is crucial for accurate stress updates. Traditionally, this is done using the elastic predictor / plastic corrector scheme, where an initial elastic trial stress is corrected iteratively to satisfy the yield condition. While effective, this approach often requires conditional flow control and makes behavior implementation more cumbersome, especially when handling multiple plastic surfaces.

An alternative is to enforce consistency directly using complementarity \(C\)-functions such as the Fischer-Burmeister (\(\text{FB}\)) function defined as:

\[\text{FB}(x, y) = x + y - \sqrt{x^2 + y^2}\]

This function provides a semi-smooth [1] reformulation of the complementarity conditions:

\[x \geq 0, y \geq 0, xy = 0 \quad \Leftrightarrow \text{FB}(x, y) = 0.\]

In the context of plasticity, complementarity conditions of plastic evolution generally write:

\[ f(\bsig, p) \leq 0, \quad \dot{p} \geq 0, \quad \dot{p}f(\sigma,p) = 0 \]

where \(f(\bsig, p)\) is the yield function and \(p\) the cumulated plastic strain acting as a plastic multiplier.

Using the $\text{FB} function allows to encode both elastic and plastic evolution into a single non-linear equation:

\[\text{FB}(-f(\bsig,p),\dot{p}) = 0\]

where the negative sign ensures that \(f(\bsig,p) \leq 0\) as common practice in plasticity.

The advantages are that:

1. The FB function transforms the non-smooth complementarity problem into a semi-smooth differentiable system, improving numerical robustness.

2. Both the yield condition and non-negativity of the plastic multiplier are enforced simultaneously, avoiding case disjunctions or active-set methods.

3. Newton-type solvers can be applied directly to the FB function, often leading to faster and more stable convergence than traditional elastic predictor/plastic corrector schemes.

JAX-related sharp bits

optimistix and lineax solver options

By default lineax and optimistix check that the solve was successful, verifying that the return doesn’t have NaNs etc. This extra check may induce additional costs. This can be disabled by passing solver(..., throw=False).

Recompilation due to changing dtype

If not careful, you may induce recompilation of jitted constitutive models when inadvertly changing dtypes between function calls. For instance, if calling first:

dt = 0.
material.constitutive_update(eps, state, dt)

and then doing:

for dt in jnp.ones((10,)):
    material.constitutive_update(eps, state, dt)

JIT compilation will be triggered once more at the start of the loop since dt changed dtype from float to jnp.float64 for instance.

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[1]

The Fischer-Burmeister is not differentiable at the origin \((0,0)\), but it is semi-smooth, meaning directional derivatives exist and Newton-type methods can still converge robustly.