Rapid Vacuum Response Calculations for Non-axisymmetric Plasma Geometries

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Kai: We're looking at the paper: "Rapid Vacuum Response Calculations for Non-axisymmetric Plasma Geometries".

Mira: Here's the gist of the paper. We present a fast, open-source calculation of the vacuum response matrix and surface inductance in non-axisymmetric geometries capable of running with up to 105 boundary points in tens of seconds on a laptop This work enables fast free boundary ideal MHD stability and perturbed equilibrium calculations in stellarators Pruning redundant reasoning tokens The paper presents a fast, open-source calculation of the vacuum response matrix and surface inductance in non-axisymmetric geometries capable of running with up to 105 boundary points in tens of seconds on a laptop It extends the VACUUM code to non-axisymmetric geometries in the new Julia implementation of the Generalized Perturbed Equilibrium Code, combining a collocation approach to the boundary integral equations with a high-order singularity correction scheme The authors find that the vacuum response matrix converges at fourth order or greater in the grid spacing, reaching a given accuracy on a coarser grid than previous implementations without the high-order method The non-axisymmetric calculation converges to the axisymmetric VACUUM result for a tokamak and the CAS3D result for a W7-X stellarator equilibrium Non-axisymmetric Vacuum Response Matrix The free boundary stability problem is solved by constructing the total response matrix from its plasma and vacuum components, defined as deltaW = deltaWP + deltaWV = Ξ⃗ † (1) To obtain the vacuum response matrix, the perturbed vacuum energy is expressed as an integral over the vacuum region This volume integral is recast as a surface integral over the boundary Γ of the vacuum region using Green’s functions The resulting Fredholm integral equation of the second kind relates the potential phi and the normal perturbed field b on the boundary The discrete form of this equation leads to a matrix form (12) where D is the double-layer potential operator and S is the single-layer potential operator The solution for the response function is given by R = (1/2I2N + D)−1S (13) Treatment of Green’s Function Singularities The singularity in the Green’s function as r − r′ → 0 means that standard numerical quadrature cannot supply the quadrature weights in eq.

Kai: (11) The authors adapt the higher-order correction scheme of Malhotra et al.

31, 32: to this problem They employ a partition of unity function eta = exp (-36rho8) to split the integrals into a smooth global integral and a local singular component The singular integral is then discretized by evaluating it on-grid contributions using the transpose of the interpolation operator prior implementations have used This approach allows the singular integral to collapse onto strictly on-grid contributions, yielding a matrix form (40) Field-Period Symmetry Field-period symmetry can be leveraged in the boundary integral formulation to reduce storage required by a factor of nf p and the cost of the linear solve by a factor of n2fp When the boundary possesses an nfp-fold symmetry, the boundary operators become block-circulant matrices with blocks indexed by d (41) This structure allows the operators to be block-diagonalized by the discrete Fourier transform over the period index into nf p decoupled families This results in nfp independent systems of size Nper, or 2Nper when a conducting wall is present Numerical Implementation and Results The implementation is in the new Julia GPEC version 2.0, which accepts the plasma boundary in Cartesian coordinates as input and computes surface position, tangent, and normal vectors using periodic bicubic splines The cost for the 3D boundary solve is dominated by the O(N3) matrix factorization required to invert the linear system The self-convergence of the 3D VACUUM code along with its runtime shows an error scaling proportional to h4 for both with and without a conducting wall The eigenspectra of the two codes are nearly indistinguishable across the 732 modes of the k = 1 mode family and nearly nine decades of eigenvalue magnitude Axisymmetric Benchmarking The comparison between the 3D version of the VACUUM code and its axisymmetric counterpart demonstrates proper implementation of both the high-order quadrature and the non-axisymmetric extension The relative error falls to O(10−5) for both the nowall and conformal-wall cases at the highest resolution scan point Non-axisymmetric Benchmarking The eigenspectra of the two codes are nearly indistinguishable across the 732 modes of the k = 1 mode family and nearly nine decades of eigenvalue magnitude The full matrices also agree well, with a relative Frobenius error between the two of ≈ 2% The convergence rate approximately agrees with the h4 of fig.

Kai: 2 at coarser grid sizes but rises toward the finest grids, with a least-squares fit over the entire scan giving h6.6, h7.0, and h5.2 for the 8, 16, and 24 maximum mode number cases respectively We have extended the vacuum energy calculations in the VACUUM code

5: to non-axisymmetric geometries, freely available through the open-source GPEC (https://github.com/OpenFUSIONToolkit/GPEC), soon to be released as version 2.0 This new boundary integral equation solver uses a high-order singularity correction scheme

31, 32: and leverages the block-circulant matrix structure under fieldperiod symmetry to improve performance The theory applies to plasmas both with and without a nearby conducting wall We will then use the 3D surface inductance matrix to calculate perturbed stellarator equilibria

10, 11: and quantify the effect of externally applied fields The vacuum response matrix is given by WV = chi′2/2mu04pi2NQΦ†RpΦQ (18) The development is the first step toward enabling several critical MHD calculations in stellarators Acknowledgements This material is based upon work supported by the U.S.

Paper summary: Kai: To recap, this paper presents a fast, open-source calculation of the vacuum response matrix and surface inductance in non-axisymmetric geometries capable of running with up to one hundred five boundary points in tens of seconds on a laptop <ref:2610.00536#pg1,a fast, open-source calculation of the vacuum response matrix and surface>.

Mira: They extended the VACUUM code to non-axisymmetric geometries in the new Julia implementation of the Generalized Perturbed Equilibrium Code, combining a collocation approach to the boundary integral equations with a high-order singularity correction scheme.

Lev: The authors derive both the non-axisymmetric vacuum response matrix and surface inductance matrix in section two showing that both reduce to the same pair of discretized layer potential operators <ref:2610.00536#pg2,the non-axisymmetric vacuum response matrix and surface inductance matrix in section>.

Kai: Because those kernels are singular, they need that specific singularity correction scheme noted from Malhotra et al., which is what enables the fast and accurate calculations.

Mira: They found that this approach means the boundary quadrature requires the singularity correction scheme noted above, which allows them to get their matrix form in equation forty.

Lev: They also show how field-period symmetry can be leveraged to reduce storage by a factor of nfp and the cost of the linear solve by a factor of n2fp when you have that kind of boundary symmetry.

Kai: So, they’ve shown that they can handle these geometries efficiently, and it’s built as part of the new open-source Julia version of GPEC which they plan to release as version two point zero soon <ref:2610.00536#pg2,built as part of the new open-source Julia version of GPEC>.

Mira: And what this means for the theory is that the theory applies to plasmas both with and without a nearby conducting wall.

Lev: It’s interesting that while they get fast convergence at coarser grids, their least-squares fit over the entire scan gives h6 point 6, h7 point 0, and h5 point 2 for the eight sixteen and twenty-four maximum mode number cases respectively as they go finer.

Conclusion: Kai: Looking at the paper "Rapid Vacuum Response Calculations for Non-axisymmetric Plasma Geometries," it really boils down to making the calculation of vacuum response and surface inductance practical for complex plasma shapes.

Mira: The authors are focusing on extending the VACUUM code into Julia version two point zero, giving us an open-source tool that can handle non-axisymmetric geometries quickly, up to one hundred five boundary points in under a minute on a laptop.

Lev: They use a high-order singularity correction scheme and leverage field-period symmetry to reduce the computational overhead for solving these boundary integral equations.

Kai: It means we can do fast free boundary ideal MHD stability and perturbed equilibrium calculations in stellarators now, which is what this work enables.

Mira: The implication is that we have a way to quickly quantify the effect of externally applied fields using three dee surface inductance matrices for things like W7-X stellarator equilibria <ref:2610.00536#pg3>.

Lev: It’s a step toward enabling several critical MHD calculations in stellarators, and they’ve shown how the convergence rates behave depending on how fine you make your grid spacing.

Kai: That's all we have on this paper, focusing on how they built this tool and what it lets us calculate for magnetic confinement.