Implementation of beta fixed point solve, based on Jon M work.

[grmtrkngtn]

Description

This PR replaces the iterative idempotence treatment of plasma beta with an explicit fixed-point solve.

Previously, beta_total_vol_avg could be treated as optimisation iteration variable ixc = 5, with equality constraint icc = 1 enforcing consistency between the total beta and its thermal, fast-alpha and beam components. This means that one of PROCESS's internal closure relationships is effectively being solved through the optimisation problem.

The change instead solves

$$ \beta = \beta_{\mathrm{fast\ alpha}} + \beta_{\mathrm{beam}} + \beta_{\mathrm{thermal}} $$

internally using scipy.optimize.fixed_point.

The thermal beta is calculated using the existing PlasmaBeta.calculate_plasma_beta() method in process.models.physics.physics.

The fixed-point calculation is performed within Caller for each model evaluation. The trial value of beta_total_vol_avg is applied after the optimisation variables have been set, ensuring that the fixed-point value is not overwritten by set_scaled_iteration_variable().

The existing MFILE/output idempotence handling is otherwise retained.

Validation

The low_aspect_ratio_DEMO.IN.DAT case was run with ixc = 5 removed while retaining beta consistency constraint icc = 1.

The case converges to essentially the same solution as main, while constraint 1 remains satisfied with zero residual. This demonstrates that the beta closure is being provided by the internal fixed-point calculation rather than by VMCON varying beta_total_vol_avg.

As a result, beta_total_vol_avg no longer needs to be exposed to the optimiser as ixc = 5. Once the fixed-point implementation is fully adopted, icc = 1 can also be removed from the optimisation problem because the same closure is enforced internally.

[codecov-commenter]

Codecov Report

❌ Patch coverage is 25.00000% with 18 lines in your changes missing coverage. Please review.
✅ Project coverage is 49.92%. Comparing base (4885172) to head (6c34f15).

Files with missing lines	Patch %	Lines
process/core/caller.py	25.00%	18 Missing ⚠️

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #4640   +/-   ##
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  Coverage   49.92%   49.92%           
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  Files         151      151           
  Lines       30236    30239    +3     
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+ Hits        15094    15098    +4     
+ Misses      15142    15141    -1     

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[CoronelBuendia]

Hmm.. this only happens when beta is used to calculate things that relate to how beta is calculated.

As far as I am aware, this was only happening in the Saamuli pedestal temperature scaling $\beta = f(T_{e,ped} = g(\beta))$ that was used for EU-DEMO, hence the constraint/Picard iteration. But this pedestal model no longer exists in open-source PROCESS.

Where is the fixed point problem now for the open-source PROCESS LAR point? I just think it's really important to identify what these "circular loops" are.

More importantly, are you sure this is a complete closure of the plasma calculation chain, or are there other fixed point issues?

[timothy-nunn]

Going to invite comments from as many @ukaea/process-model-review on this one, thanks @CoronelBuendia for already commenting. @grmtrkngtn I will do a code review once modeller questions/concerns have been answered/addressed.

[timothy-nunn]

Does this PR not remove the need for the beta consistency equation? Why does this need to wait for another PR?

[grmtrkngtn]

Does this PR not remove the need for the beta consistency equation?

Yes, that is the primary motivation.

Why does this need to wait for another PR?

I'm not sure what you mean, it doesn't need another PR - it's independent of anything else.