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

Research output: Contribution to journal › Article › Academic › peer-review

5 Citations (Scopus)

140 Downloads (Pure)

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 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.

Original language	English
Article number	111611
Number of pages	20
Journal	Journal of Computational Physics
Volume	471
DOIs	https://doi.org/10.1016/j.jcp.2022.111611
Publication status	Published - 15 Dec 2022

Bibliographical 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.

Funding

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.

Funders	Funder number
European Union's Horizon 2020 - Research and Innovation Framework Programme	633053
European Commission	101052200 — EUROfusion

Keywords

Bi-cubic Hermite Bézier finite element method
Isoparametric mapping
Magnetic flux surface aligned grid
Magneto-hydrodynamics
MHD instabilities in tokamaks
Polar grid singularities

Access to Document

10.1016/j.jcp.2022.111611

pdfFinal published version, 3.43 MBLicence: TAVERNE

Cite this

@article{64b77c0999f44945a2cda876708cdc6d,

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. ",

year = "2022",

month = dec,

day = "15",

doi = "10.1016/j.jcp.2022.111611",

language = "English",

volume = "471",

journal = "Journal of Computational Physics",

issn = "0021-9991",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Treatment of polar grid singularities in the bi-cubic Hermite-Bézier approximations

T2 - Isoparametric finite element framework

AU - JOREK team

AU - Bhole, Ashish

AU - Nkonga, Boniface

AU - Pamela, Stanislas

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.

KW - Bi-cubic Hermite Bézier finite element method

KW - Isoparametric mapping

KW - Magnetic flux surface aligned grid

KW - Magneto-hydrodynamics

KW - MHD instabilities in tokamaks

KW - Polar grid singularities

UR - https://www.scopus.com/pages/publications/85138412501

U2 - 10.1016/j.jcp.2022.111611

DO - 10.1016/j.jcp.2022.111611

M3 - Article

AN - SCOPUS:85138412501

SN - 0021-9991

VL - 471

JO - Journal of Computational Physics

JF - Journal of Computational Physics

M1 - 111611

ER -

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

Abstract

Bibliographical note

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this