Efficient high-order singular quadrature schemes in magnetic fusion

Malhotra, Dhairya; Cerfon, Antoine J.; O’Neil, Michael; Toler, Evan

doi:10.1088/1361-6587/ab57f4

Title: Efficient high-order singular quadrature schemes in magnetic fusion

Journal Article · 01 December 2019 · Plasma Physics and Controlled Fusion

DOI: https://doi.org/10.1088/1361-6587/ab57f4 · OSTI ID:1800644

^[1]; Cerfon, Antoine J. ^[2]; O’Neil, Michael ^[2]; Toler, Evan ^[2]

New York Univ. (NYU), NY (United States). Courant Inst.; OSTI
New York Univ. (NYU), NY (United States). Courant Inst.

Several problems in magnetically confined fusion, such as the computation of exterior vacuum fields or the decomposition of the total magnetic field into separate contributions from the plasma and the external sources, are best formulated in terms of integral equation expressions. Based on Biot–Savart-like formulae, these integrals contain singular integrands. The regularization method commonly used to address the computation of various singular surface integrals along general toroidal surfaces is low-order accurate, and therefore requires a dense computational mesh in order to obtain sufficient accuracy. In this work, we present a fast, high-order quadrature scheme for the efficient computation of these integrals. Several numerical examples are provided demonstrating the computational efficiency and the high-order accurate convergence. A corresponding code for use in the community has been publicly released.

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Research Organization:: New York Univ. (NYU), NY (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Fusion Energy Sciences (FES)

Grant/Contract Number:: FG02-86ER53223; SC0012398

OSTI ID:: 1800644

Journal Information:: Plasma Physics and Controlled Fusion, Journal Name: Plasma Physics and Controlled Fusion Journal Issue: 2 Vol. 62; ISSN 0741-3335

Publisher:: IOP ScienceCopyright Statement

Country of Publication:: United States

Language:: English

References (66)

Computation of ideal MHD equilibria Lackner, K. Computer Physics Communications, Vol. 12, Issue 1 https://doi.org/10.1016/0010-4655(76)90008-4	journal	September 1976
An integral equation technique for the exterior and interior neumann problem in toroidal regions Merkel, Peter Journal of Computational Physics, Vol. 66, Issue 1 https://doi.org/10.1016/0021-9991(86)90055-0	journal	September 1986
On the solution of elliptic partial differential equations on regions with corners II: Detailed analysis Serkh, Kirill Applied and Computational Harmonic Analysis, Vol. 46, Issue 2 https://doi.org/10.1016/j.acha.2017.05.001	journal	March 2019
Taylor states in stellarators: A fast high-order boundary integral solver Malhotra, Dhairya; Cerfon, Antoine; Imbert-Gérard, Lise-Marie Journal of Computational Physics, Vol. 397 https://doi.org/10.1016/j.jcp.2019.06.067	journal	November 2019
Solution of stellarator boundary value problems with external currents Merkel, P. Nuclear Fusion, Vol. 27, Issue 5 https://doi.org/10.1088/0029-5515/27/5/018	journal	May 1987
Finite- magnetic field line tracing for Helias configurations Strumberger, E. Nuclear Fusion, Vol. 37, Issue 1 https://doi.org/10.1088/0029-5515/37/1/i03	journal	January 1997
The Exponentially Convergent Trapezoidal Rule Trefethen, Lloyd N.; Weideman, J. A. C. SIAM Review, Vol. 56, Issue 3 https://doi.org/10.1137/130932132	journal	January 2014
A Green’s Function Method for the Vacuum Contribution to the MHD Stability of Helically Symmetric Equilibria Merkel, Peter Zeitschrift für Naturforschung A, Vol. 37, Issue 8 https://doi.org/10.1515/zna-1982-0819	journal	August 1982
Fast algorithms for Quadrature by Expansion I: Globally valid expansions Rachh, Manas; Klöckner, Andreas; O'Neil, Michael arXiv https://doi.org/10.48550/arxiv.1602.05301	text	January 2016
An integral equation formulation for rigid bodies in Stokes flow in three dimensions Corona, Eduardo; Greengard, Leslie; Rachh, Manas arXiv https://doi.org/10.48550/arxiv.1606.07428	text	January 2016
An improved current potential method for fast computation of stellarator coil shapes Landreman, Matt arXiv https://doi.org/10.48550/arxiv.1609.04378	text	January 2016
Fast Ewald summation for Stokesian particle suspensions: FAST EWALD SUMMATION FOR STOKESIAN PARTICLE SUSPENSIONS Af Klinteberg, Ludvig; Tornberg, Anna-Karin International Journal for Numerical Methods in Fluids, Vol. 76, Issue 10 https://doi.org/10.1002/fld.3953	journal	September 2014
A Fast, High-Order Algorithm for the Solution of Surface Scattering Problems: Basic Implementation, Tests, and Applications Bruno, Oscar P.; Kunyansky, Leonid A. Journal of Computational Physics, Vol. 169, Issue 1 https://doi.org/10.1006/jcph.2001.6714	journal	May 2001
Ubiquitous evaluation of layer potentials using Quadrature by Kernel-Independent Expansion Rahimian, Abtin; Barnett, Alex; Zorin, Denis BIT Numerical Mathematics, Vol. 58, Issue 2 https://doi.org/10.1007/s10543-017-0689-2	journal	November 2017
Computation of ideal MHD equilibria Lackner, K. Computer Physics Communications, Vol. 12, Issue 1 https://doi.org/10.1016/0010-4655(76)90008-4	journal	September 1976
Erato stability code Gruber, R.; Troyon, F.; Berger, D. Computer Physics Communications, Vol. 21, Issue 3 https://doi.org/10.1016/0010-4655(81)90013-8	journal	January 1981
Hera and other extensions of Erato Gruber, R.; Semenzato, S.; Troyon, F. Computer Physics Communications, Vol. 24, Issue 3-4 https://doi.org/10.1016/0010-4655(81)90159-4	journal	December 1981
Three-dimensional free boundary calculations using a spectral Green's function method Hirshman, S. P.; van RIJ, W. I.; Merkel, P. Computer Physics Communications, Vol. 43, Issue 1 https://doi.org/10.1016/0010-4655(86)90058-5	journal	December 1986
An integral equation technique for the exterior and interior neumann problem in toroidal regions Merkel, Peter Journal of Computational Physics, Vol. 66, Issue 1 https://doi.org/10.1016/0021-9991(86)90055-0	journal	September 1986
On the polarizability and capacitance of the cube Helsing, Johan; Perfekt, Karl-Mikael Applied and Computational Harmonic Analysis, Vol. 34, Issue 3 https://doi.org/10.1016/j.acha.2012.07.006	journal	May 2013
On the solution of elliptic partial differential equations on regions with corners II: Detailed analysis Serkh, Kirill Applied and Computational Harmonic Analysis, Vol. 46, Issue 2 https://doi.org/10.1016/j.acha.2017.05.001	journal	March 2019
A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains Ying, Lexing; Biros, George; Zorin, Denis Journal of Computational Physics, Vol. 219, Issue 1 https://doi.org/10.1016/j.jcp.2006.03.021	journal	November 2006
Calculation of the vacuum Green’s function valid even for high toroidal mode numbers in tokamaks Chance, M. S.; Turnbull, A. D.; Snyder, P. B. Journal of Computational Physics, Vol. 221, Issue 1 https://doi.org/10.1016/j.jcp.2006.06.025	journal	January 2007
Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning Helsing, Johan; Ojala, Rikard Journal of Computational Physics, Vol. 227, Issue 20 https://doi.org/10.1016/j.jcp.2008.06.022	journal	October 2008
A spectral boundary integral method for flowing blood cells Zhao, Hong; Isfahani, Amir H. G.; Olson, Luke N. Journal of Computational Physics, Vol. 229, Issue 10 https://doi.org/10.1016/j.jcp.2010.01.024	journal	May 2010
A Nyström method for weakly singular integral operators on surfaces Bremer, James; Gimbutas, Zydrunas Journal of Computational Physics, Vol. 231, Issue 14 https://doi.org/10.1016/j.jcp.2012.04.003	journal	May 2012
On the numerical evaluation of the singular integrals of scattering theory Bremer, James; Gimbutas, Zydrunas Journal of Computational Physics, Vol. 251 https://doi.org/10.1016/j.jcp.2013.05.048	journal	October 2013
Quadrature by expansion: A new method for the evaluation of layer potentials Klöckner, Andreas; Barnett, Alexander; Greengard, Leslie Journal of Computational Physics, Vol. 252 https://doi.org/10.1016/j.jcp.2013.06.027	journal	November 2013
An explicit kernel-split panel-based Nyström scheme for integral equations on axially symmetric surfaces Helsing, Johan; Karlsson, Anders Journal of Computational Physics, Vol. 272 https://doi.org/10.1016/j.jcp.2014.04.053	journal	September 2014
On the solution of elliptic partial differential equations on regions with corners Serkh, Kirill; Rokhlin, Vladimir Journal of Computational Physics, Vol. 305 https://doi.org/10.1016/j.jcp.2015.10.024	journal	January 2016
An integral equation formulation for rigid bodies in Stokes flow in three dimensions Corona, Eduardo; Greengard, Leslie; Rachh, Manas Journal of Computational Physics, Vol. 332 https://doi.org/10.1016/j.jcp.2016.12.018	journal	March 2017
Fast algorithms for Quadrature by Expansion I: Globally valid expansions Rachh, Manas; Klöckner, Andreas; O'Neil, Michael Journal of Computational Physics, Vol. 345 https://doi.org/10.1016/j.jcp.2017.04.062	journal	September 2017
An integral equation-based numerical solver for Taylor states in toroidal geometries O'Neil, Michael; Cerfon, Antoine J. Journal of Computational Physics, Vol. 359 https://doi.org/10.1016/j.jcp.2018.01.004	journal	April 2018
A local target specific quadrature by expansion method for evaluation of layer potentials in 3D Siegel, Michael; Tornberg, Anna-Karin Journal of Computational Physics, Vol. 364 https://doi.org/10.1016/j.jcp.2018.03.006	journal	July 2018
A fast algorithm for Quadrature by Expansion in three dimensions Wala, Matt; Klöckner, Andreas Journal of Computational Physics, Vol. 388 https://doi.org/10.1016/j.jcp.2019.03.024	journal	July 2019
Taylor states in stellarators: A fast high-order boundary integral solver Malhotra, Dhairya; Cerfon, Antoine; Imbert-Gérard, Lise-Marie Journal of Computational Physics, Vol. 397 https://doi.org/10.1016/j.jcp.2019.06.067	journal	November 2019
Erato stability code Gruber, R.; Troyon, F.; Berger, D. Computer Physics Communications, Vol. 35 https://doi.org/10.1016/s0010-4655(84)82825-8	journal	January 1984
Tokamak elongation – how much is too much? Part 2. Numerical results Lee, J. P.; Cerfon, A.; Freidberg, J. P. Journal of Plasma Physics, Vol. 81, Issue 6 https://doi.org/10.1017/S0022377815001300	journal	December 2015
Integral formulation for 3D eddy-current computation using edge elements Albanese, R.; Rubinacci, G. IEE Proceedings A Physical Science, Measurement and Instrumentation, Management and Education, Reviews, Vol. 135, Issue 7 https://doi.org/10.1049/ip-a-1.1988.0072	journal	January 1988
General formulation of the resistive wall mode coupling equations Pustovitov, V. D. Physics of Plasmas, Vol. 15, Issue 7 https://doi.org/10.1063/1.2943711	journal	July 2008
Magnetohydrodynamic stability of a sharp boundary model of tokamak Freidberg, J. P.; Grossmann, W. Physics of Fluids, Vol. 18, Issue 11 https://doi.org/10.1063/1.861035	journal	January 1975
Stability of a high-β, l=3 stellarator Freidberg, J. P.; Grossmann, W.; Haas, F. A. Physics of Fluids, Vol. 19, Issue 10 https://doi.org/10.1063/1.861366	journal	January 1976
Vacuum calculations in azimuthally symmetric geometry Chance, M. S. Physics of Plasmas, Vol. 4, Issue 6 https://doi.org/10.1063/1.872380	journal	June 1997
Determination of the vacuum field resulting from the perturbation of a toroidally symmetric plasma Atanasiu, C. V.; Boozer, A. H.; Zakharov, L. E. Physics of Plasmas, Vol. 6, Issue 7 https://doi.org/10.1063/1.873235	journal	July 1999
Use of the virtual-casing principle in calculating the containing magnetic field in toroidal plasma systems Shafranov, V. D.; Zakharov, L. E. Nuclear Fusion, Vol. 12, Issue 5 https://doi.org/10.1088/0029-5515/12/5/009	journal	September 1972
Solution of stellarator boundary value problems with external currents Merkel, P. Nuclear Fusion, Vol. 27, Issue 5 https://doi.org/10.1088/0029-5515/27/5/018	journal	May 1987
Finite- magnetic field line tracing for Helias configurations Strumberger, E. Nuclear Fusion, Vol. 37, Issue 1 https://doi.org/10.1088/0029-5515/37/1/I03	journal	January 1997
Magnetic diagnostics: General principles and the problem of reconstruction of plasma current and pressure profiles in toroidal systems Pustovitov, V. D. Nuclear Fusion, Vol. 41, Issue 6 https://doi.org/10.1088/0029-5515/41/6/307	journal	June 2001
PIES free boundary stellarator equilibria with improved initial conditions Drevlak, M.; Monticello, D.; Reiman, A. Nuclear Fusion, Vol. 45, Issue 7 https://doi.org/10.1088/0029-5515/45/7/022	journal	July 2005
Decoupling in the problem of tokamak plasma response to asymmetric magnetic perturbations Pustovitov, V. D. Plasma Physics and Controlled Fusion, Vol. 50, Issue 10 https://doi.org/10.1088/0741-3335/50/10/105001	journal	August 2008
The virtual-casing principle for 3D toroidal systems Lazerson, S. A. Plasma Physics and Controlled Fusion, Vol. 54, Issue 12 https://doi.org/10.1088/0741-3335/54/12/122002	journal	November 2012
The virtual-casing principle and Helmholtz’s theorem Hanson, J. D. Plasma Physics and Controlled Fusion, Vol. 57, Issue 11 https://doi.org/10.1088/0741-3335/57/11/115006	journal	September 2015
Free-boundary simulations with the XTOR-2F code Marx, A.; Lütjens, H. Plasma Physics and Controlled Fusion, Vol. 59, Issue 6 https://doi.org/10.1088/1361-6587/aa6d1f	journal	May 2017
An improved current potential method for fast computation of stellarator coil shapes Landreman, Matt Nuclear Fusion, Vol. 57, Issue 4 https://doi.org/10.1088/1741-4326/aa57d4	journal	February 2017
Surface scattering in three dimensions: an accelerated high–order solver Bruno, Oscar P.; Kunyansky, Leonid A. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 457, Issue 2016 https://doi.org/10.1098/rspa.2001.0882	journal	December 2001
A Fast and Stable Solver for Singular Integral Equations on Piecewise Smooth Curves Helsing, Johan SIAM Journal on Scientific Computing, Vol. 33, Issue 1 https://doi.org/10.1137/090779218	journal	January 2011
The Exponentially Convergent Trapezoidal Rule Trefethen, Lloyd N.; Weideman, J. A. C. SIAM Review, Vol. 56, Issue 3 https://doi.org/10.1137/130932132	journal	January 2014
On the Numerical Solution of Elliptic Partial Differential Equations on Polygonal Domains Hoskins, Jeremy G.; Rokhlin, Vladimir; Serkh, Kirill SIAM Journal on Scientific Computing, Vol. 41, Issue 4 https://doi.org/10.1137/18M1199034	journal	January 2019
A Green’s Function Method for the Vacuum Contribution to the MHD Stability of Helically Symmetric Equilibria Merkel, Peter Zeitschrift für Naturforschung A, Vol. 37, Issue 8 https://doi.org/10.1515/zna-1982-0819	journal	August 1982
Nearly Singular Integrals in 3D Stokes Flow Tlupova, Svetlana; Beale, J. Thomas Communications in Computational Physics, Vol. 14, Issue 5 https://doi.org/10.4208/cicp.020812.080213a	journal	November 2013
On the polarizability and capacitance of the cube Helsing, Johan; Perfekt, Karl-Mikael arXiv https://doi.org/10.48550/arxiv.1203.5997	preprint	January 2012
Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials Klöckner, Andreas; Barnett, Alexander; Greengard, Leslie arXiv https://doi.org/10.48550/arxiv.1207.4461	text	January 2012
Fast algorithms for Quadrature by Expansion I: Globally valid expansions Rachh, Manas; Klöckner, Andreas; O'Neil, Michael arXiv https://doi.org/10.48550/arxiv.1602.05301	text	January 2016
An integral equation formulation for rigid bodies in Stokes flow in three dimensions Corona, Eduardo; Greengard, Leslie; Rachh, Manas arXiv https://doi.org/10.48550/arxiv.1606.07428	text	January 2016
An integral equation-based numerical solver for Taylor states in toroidal geometries O'Neil, Michael; Cerfon, Antoine J. arXiv https://doi.org/10.48550/arxiv.1611.01420	preprint	January 2016
Taylor States in Stellarators: A Fast High-order Boundary Integral Solver Malhotra, Dhairya; Cerfon, Antoine; Imbert-Gérard, Lise-Marie arXiv https://doi.org/10.48550/arxiv.1902.01205	text	January 2019

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