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Properly extend shape map into interior region #6477

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@knelli2 knelli2 commented Feb 7, 2025

This is only allowed when the inner surface is spherical and when we are using the "regular" transition (i.e. not reversed).

Proposed changes

Fixes #6451.

A previous PR (#6359) attempted to extend the shape map into the excision, but did so incorrectly. This PR implements this correctly. In order to do this, the meaning of the transition function had to be changed slightly. Previously, the shape map was

$$ \tilde{r} = r \left(1 - \frac{f(r, \theta, \phi)}{r} \sum_{\ell,m} \lambda_{\ell m}Y_{\ell m}(\theta, \phi)\right) $$

and the transition function was $f(r, \theta, \phi)$. The functional form of the shape map hasn't changed, but now it reads

$$ \tilde{r} = r \left(1 - G(r, \theta, \phi) \sum_{\ell,m} \lambda_{\ell m}Y_{\ell m}(\theta, \phi)\right) $$ $$ G(r, \theta, \phi) = \frac{f(r, \theta, \phi)}{r} $$

where $G(r, \theta, \phi)$ is the transition. The $1/r$ factor was just absorbed into the transition function because the full $G$ is different inside the excision, not just $f$.

Also, the transition function inside the excisions is

$$ G(r, \theta, \phi) = \frac{r^2}{r_{\text{min}}^3} $$

which is different than it is in SpEC. The transition defined in SpEC causes the jacobian to be undefined at the center of the shape map, which means the full frame velocity of all the maps can't be computed at the center of the shape map. This new transition inside the excision allows for this and has the correct behavior.

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    make doc to generate the documentation locally into BUILD_DIR/docs/html.
    Then open index.html.
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Further comments

@knelli2 knelli2 added this to the BBH larger q & higher spins milestone Feb 7, 2025
@knelli2 knelli2 added the in progress Don't review, used for sharing code and getting feedback label Feb 9, 2025
This is only allowed when the inner surface is spherical and when
we are using the "regular" transition (i.e. not reversed).
@knelli2 knelli2 removed the in progress Don't review, used for sharing code and getting feedback label Feb 11, 2025
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knelli2 commented Feb 11, 2025

Ok this is ready for review (maybe @nilsvu ?). Just an FYI for @nikwit that this adds the shape map to the ExcisionSpheres, but if you're only evaluating it at the center of the excision, then this shouldn't behave any differently than before when the shape map wasn't included.

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Shape map incorrect inside excision
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