Treatment of polar grid singularities in the bi-cubic Hermite-Bézier approximations: Isoparametric finite element framework

JOREK team; Ashish Bhole; Boniface Nkonga; Stanislas Pamela; Guido Huijsmans; Matthias Hoelzl

doi:10.1016/j.jcp.2022.111611

Treatment of polar grid singularities in the bi-cubic Hermite-Bézier approximations: Isoparametric finite element framework

JOREK team, Ashish Bhole (Corresponding author), Boniface Nkonga, Stanislas Pamela, Guido Huijsmans, Matthias Hoelzl

Onderzoeksoutput: Bijdrage aan tijdschrift › Tijdschriftartikel › Academic › peer review

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We propose a numerical treatment for the geometric singularity at the polar grid center encountered in the application of the isoparametric bi-cubic Hermite Bézier finite element method. The treatment applies a set of new basis functions at the polar grid center in the numerical algorithm where the new basis functions are simply the linear transformations of the original basis functions. The linear transformation comes out naturally by analyzing the interpolation formula at the polar grid center. The proposed polar treatment enforces the C¹ regularity in the physical space and preserves the order of the accuracy of the interpolation. The treatment is applied in the nonlinear MHD code JOREK. With the help of a range of numerical tests, it is demonstrated that the polar treatment improves the stability and accuracy of the numerical approximation near the polar grid center. The polar treatment presented can be applied to the grid center of circular or non-circular polar grids and is also applicable for the bi-cubic Hermite finite element method.

Originele taal-2	Engels
Artikelnummer	111611
Aantal pagina's	20
Tijdschrift	Journal of Computational Physics
Volume	471
DOI's	https://doi.org/10.1016/j.jcp.2022.111611
Status	Gepubliceerd - 15 dec. 2022

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Publisher Copyright:
© 2022 Elsevier Inc.

Financiering

This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

Financiers	Financiernummer
European Union’s Horizon Europe research and innovation programme	633053
European Commission	101052200 — EUROfusion

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title = "Treatment of polar grid singularities in the bi-cubic Hermite-B{\'e}zier approximations: Isoparametric finite element framework",

abstract = "We propose a numerical treatment for the geometric singularity at the polar grid center encountered in the application of the isoparametric bi-cubic Hermite B{\'e}zier finite element method. The treatment applies a set of new basis functions at the polar grid center in the numerical algorithm where the new basis functions are simply the linear transformations of the original basis functions. The linear transformation comes out naturally by analyzing the interpolation formula at the polar grid center. The proposed polar treatment enforces the C1 regularity in the physical space and preserves the order of the accuracy of the interpolation. The treatment is applied in the nonlinear MHD code JOREK. With the help of a range of numerical tests, it is demonstrated that the polar treatment improves the stability and accuracy of the numerical approximation near the polar grid center. The polar treatment presented can be applied to the grid center of circular or non-circular polar grids and is also applicable for the bi-cubic Hermite finite element method.",

keywords = "Bi-cubic Hermite B{\'e}zier finite element method, Isoparametric mapping, Magnetic flux surface aligned grid, Magneto-hydrodynamics, MHD instabilities in tokamaks, Polar grid singularities",

author = "{JOREK team} and Ashish Bhole and Boniface Nkonga and Stanislas Pamela and Guido Huijsmans and Matthias Hoelzl",

note = "Funding Information: This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them. ",

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AU - Huijsmans, Guido

AU - Hoelzl, Matthias

N1 - Funding Information: This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

PY - 2022/12/15

Y1 - 2022/12/15

N2 - We propose a numerical treatment for the geometric singularity at the polar grid center encountered in the application of the isoparametric bi-cubic Hermite Bézier finite element method. The treatment applies a set of new basis functions at the polar grid center in the numerical algorithm where the new basis functions are simply the linear transformations of the original basis functions. The linear transformation comes out naturally by analyzing the interpolation formula at the polar grid center. The proposed polar treatment enforces the C1 regularity in the physical space and preserves the order of the accuracy of the interpolation. The treatment is applied in the nonlinear MHD code JOREK. With the help of a range of numerical tests, it is demonstrated that the polar treatment improves the stability and accuracy of the numerical approximation near the polar grid center. The polar treatment presented can be applied to the grid center of circular or non-circular polar grids and is also applicable for the bi-cubic Hermite finite element method.

AB - We propose a numerical treatment for the geometric singularity at the polar grid center encountered in the application of the isoparametric bi-cubic Hermite Bézier finite element method. The treatment applies a set of new basis functions at the polar grid center in the numerical algorithm where the new basis functions are simply the linear transformations of the original basis functions. The linear transformation comes out naturally by analyzing the interpolation formula at the polar grid center. The proposed polar treatment enforces the C1 regularity in the physical space and preserves the order of the accuracy of the interpolation. The treatment is applied in the nonlinear MHD code JOREK. With the help of a range of numerical tests, it is demonstrated that the polar treatment improves the stability and accuracy of the numerical approximation near the polar grid center. The polar treatment presented can be applied to the grid center of circular or non-circular polar grids and is also applicable for the bi-cubic Hermite finite element method.

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KW - Magnetic flux surface aligned grid

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KW - MHD instabilities in tokamaks

KW - Polar grid singularities

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Treatment of polar grid singularities in the bi-cubic Hermite-Bézier approximations: Isoparametric finite element framework

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