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

Bhole, Ashish; Nkonga, Boniface; Pamela, S.; Huijsmans, G. T. A.; Hoelzl, M.

doi:10.1016/j.jcp.2022.111611

Cited by 3 publications

(5 citation statements)

References 31 publications

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“…In plasma simulations, the

𝒞^{1}

continuity is not enforced at particular points (such as the X‐point or the plasma axis in a flux aligned grid) shared by more than four elements, making impossible to impose enough conditions that assure it. A special treatment in the neighborhood of these points is necessary (see an example Reference 8). In this work, we have relied on composite meshes to bypass this problem, namely, we stop the mesh of quadrangles before it becomes unstructured, and cover this small neighborhood of critical points by a mesh

{\tau}_h

of triangles over which rHCT FEs are adopted.…”

Section: Discussionmentioning

confidence: 99%

“…, where |J| = det J(s * , t * ). The entries of row/column 1 of (Pγ•) * i * come from (8) and those of the row/column 2 and 3 are the entries of matrix (J(s * , t * ) ) −1 , according to (10). Finally…”

Section: 11mentioning

confidence: 99%

“…7 Recently, in curved bi-cubic Hermite-Bézier interpolation, we have overcome the singularity at the magnetic axis (polar axis) by a proper linear combination of basis functions. 8 Nevertheless, this solution does not cure the presence of stretched elements and the method suffers from a loss of accuracy. Therefore, it seems reasonable to use a non-aligned unstructured grid associated with a  1 finite element locally near the magnetic axis.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, the geometric singularity at the symmetry center gives rise to several stretched elements 7 . Recently, in curved bi‐cubic Hermite‐Bézier interpolation, we have overcome the singularity at the magnetic axis (polar axis) by a proper linear combination of basis functions 8 . Nevertheless, this solution does not cure the presence of stretched elements and the method suffers from a loss of accuracy.…”

Section: Introductionmentioning

confidence: 99%

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Coupling finite elements of class C1 on composite curved meshes for second order elliptic problems

Bhole,

Guillard,

Nkonga

et al. 2023

Numerical Methods in Fluids

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SummaryFinite elements of class are suitable for the computation of magnetohydrodynamics instabilities in tokamak plasmas. In addition, isoparametric approximations allow for a precise alignment of the mesh with the magnetic field line. Mesh alignment is crucial to achieve axisymmetric equilibria accurately. It is also helpful to deal with the anisotropy nature of magnetized plasma flows. In this numerical framework, several practical simulations are now available. They help to understand better the operation of existing devices and predict the optimal strategies for using the international ITER tokamak under construction. However, a mesh‐aligned isoparametric representation suffers from the presence of critical points of the magnetic field (magnetic axis, X‐point). We here explore a strategy that combines aligned mesh out of the critical points with non‐aligned unstructured mesh in a region containing these points. By this strategy, we can avoid highly stretched elements and the numerical difficulties that come with them. The mesh‐aligned interpolation uses bi‐cubic Hemite‐Bézier polynomials on a structured mesh of curved quadrangular elements. On the other hand, we assume reduced cubic Hsieh‐Clough‐Tocher finite elements on an unstructured triangular mesh. Both meshes overlap, and the resulting formulation is a coupled discrete problem solved iteratively by a suitable one‐level Schwarz algorithm. In this paper, we will focus on the Poisson problem on a two‐dimensional bounded regular domain. This elliptic equation is a simplified version of the axisymmetric tokamak equilibrium one at the asymptotic limit of infinite major radius (large aspect ratio).

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“…In plasma simulations, the

𝒞^{1}

{\tau}_h

of triangles over which rHCT FEs are adopted.…”

Section: Discussionmentioning

confidence: 99%

Section: 11mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Coupling finite elements of class C1 on composite curved meshes for second order elliptic problems

Bhole,

Guillard,

Nkonga

et al. 2023

Numerical Methods in Fluids

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show abstract

“…We can see that the polar grids constructed using the bi-cubic Hermite Bézier formulation have a geometric singularity as r 0 → 0 which is a point of concern numerically. The numerical treatment to be applied for such a singularity has been developed in [60].…”

Section: Meshing For Bi-cubic Hermite Bézier Femmentioning

confidence: 99%

Stabilized bi-cubic Hermite Bézier finite element method with application to gas-plasma interactions occurring during massive material injection in tokamaks

Bhole

Nkonga

Costa

et al. 2023

Computers & Mathematics with Applications

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Stabilized Bi-Cubic Hermite Bézier Finite Element Method with Application to Gas-Plasma Interactions Occurring During Massive Material Injection in Tokamaks

Bhole¹,

Nkonga²,

Costa³

et al. 2023

Preprint

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

Cited by 3 publications

References 31 publications

Coupling finite elements of class C1 on composite curved meshes for second order elliptic problems

Coupling finite elements of class C1 on composite curved meshes for second order elliptic problems

Stabilized bi-cubic Hermite Bézier finite element method with application to gas-plasma interactions occurring during massive material injection in tokamaks

Stabilized Bi-Cubic Hermite Bézier Finite Element Method with Application to Gas-Plasma Interactions Occurring During Massive Material Injection in Tokamaks

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