CMPS Theory Manual › Nonlinear Correction, Relaxation and Admissibility

Nonlinear Correction, Relaxation and Admissibility

Why this matters. Solving the coupled block linear system does not by itself guarantee a physically admissible nonlinear state. CMPS applies the solved cell correction as one coupled direction and limits that direction when temperature, species or VOF composition constraints would be violated.

Global explicit correction relaxation

Let the linear solve return a cell correction \(\Delta\mathbf q_i\). Before cell-local admissibility checks, CMPS applies the explicit relaxation coefficient

\[\Delta\mathbf q_i\leftarrow \mathrm{ERC}\,\Delta\mathbf q_i.\]

The current default is \(\mathrm{ERC}=0.75\). Subsequent positivity/simplex controls can only reduce the cell correction further.

Temperature positivity rate limit

If a proposed negative temperature correction is too large, CMPS scales the complete cell correction block. With current temperature \(T_i\), correction \(\Delta T_i<0\), and positivity-rate parameter \(r_T\), limiting is triggered when

\[-\Delta T_i>r_TT_i.\]

The cell scale is

\[\alpha_T=\frac{r_TT_i}{-\Delta T_i},\qquad \Delta\mathbf q_i\leftarrow\alpha_T\Delta\mathbf q_i.\]

The current default is \(r_T=0.2\). This is a rate limit on the Newton correction; the committed primitive update subsequently also enforces configured absolute temperature bounds.

Species-simplex control

For \(N_s\) species, CMPS solves only the first \(N_s-1\) mass fractions and reconstructs the dependent species:

\[Y_{N_s}=1-\sum_{k=1}^{N_s-1}Y_k.\]

For each solved species with negative correction, the candidate scale is bounded by

\[\alpha_k\le 0.99\frac{Y_k}{-\Delta Y_k}\qquad(\Delta Y_k<0).\]

Let

\[Y_\Sigma=\sum_{k=1}^{N_s-1}Y_k,\qquad \Delta Y_\Sigma=\sum_{k=1}^{N_s-1}\Delta Y_k.\]

The dependent-species lower bound requires

\[Y_\Sigma+\alpha\Delta Y_\Sigma\le1.\]

Thus, for \(\Delta Y_\Sigma>0\),

\[\alpha\le\frac{1-Y_\Sigma}{\Delta Y_\Sigma}.\]

The source also guards the lower bound of the solved-fraction sum for negative \(\Delta Y_\Sigma\). The smallest admissible \(\alpha\) scales the full coupled cell correction, retaining the solved pressure/velocity/temperature/species direction.

VOF fraction-to-boundary update

With \(N-1\) independent phase fractions and reference phase

\[\alpha_r=1-\sum_{k=1}^{N-1}\alpha_k,\]

the complete phase composition must remain inside the closed unit simplex. For proposed independent-fraction corrections \(\Delta\alpha_k\), every negative component gives

\[\lambda_k\le\frac{\alpha_k}{-\Delta\alpha_k}\qquad(\Delta\alpha_k<0).\]

Let \(\Delta\alpha_\Sigma=\sum_k\Delta\alpha_k\). If that sum is positive, the reference phase gives the additional bound

\[\lambda_r\le\frac{\alpha_r}{\Delta\alpha_\Sigma}.\]

CMPS takes

\[\lambda_{max}=\min\!\left(1,\min_{\Delta\alpha_k<0}\lambda_k,\lambda_r\right)\]

and, when limiting is needed, moves slightly inside the feasible boundary using the fixed fraction-to-boundary factor 0.99:

\[\lambda=0.99\,\lambda_{max}.\]

If roundoff still prevents a valid candidate, the factor is applied again for a bounded number of retries. Crucially, the resulting \(\lambda\) multiplies the entire coupled correction block, not only the fraction entries.

Atomic VOF state commit

After correction limiting, the current source constructs a complete candidate VOF state before changing cell-owned data. Candidate velocity, pressure, temperature, independent fractions, reconstructed reference fraction, mixture properties and acoustic data are all checked first. A failed candidate leaves the old cell state intact. This prevents partially updated mixture thermodynamics from becoming visible to later assembly stages.

Absolute primitive limits

The ordinary primitive-commit and VOF primitive-commit paths clamp temperature to the configured interval

\[T_{min}\le T\le T_{max},\]

and, for compressible flow, explicitly floor pressure:

\[p\leftarrow\max(p,p_{min}).\]

The current general fluid defaults are \(T_{min}=5\,\mathrm K\), \(T_{max}=5000\,\mathrm K\), \(p_{min}=1\,\mathrm{Pa}\) and \(p_{max}=10^9\,\mathrm{Pa}\); case settings may override them. The audited ordinary/VOF primitive commit explicitly applies the pressure lower bound rather than a symmetric upper clamp. By contrast, the learned primitive correction map clamps its predicted pressure to the configured \([p_{min},p_{max}]\) interval before the state is accepted.

Why CMPS scales the complete block

Suppose the coupled linearization gives

\[\mathbf A_i\Delta\mathbf q_i\approx-\mathbf R_i.\]

Scaling only one variable after the solve would rotate the correction away from the coupled Newton direction. CMPS instead uses

\[\Delta\mathbf q_i^{adm}=\lambda_i\Delta\mathbf q_i,\qquad 0\le\lambda_i\le1,\]

so admissibility changes step length while preserving the local direction and its pressure-velocity-temperature-species/VOF coupling.