Isogeometric de Rham complex discretization in solid toroidal domains

TL;DR

This work develops an isogeometric discretization of the de Rham complex on toroidal solids by pushing forward restricted spline spaces from a polar parametric domain. It constructs a spline subcomplex with spaces $V_0\subset H^1(\Omega^{pol})$, $V_1\subset H(\mathrm{curl};\Omega^{pol})$, $V_2\subset H(\mathrm{div};\Omega^{pol})$, and $V_3\subset L^2(\Omega^{pol})$, using extraction operators to handle the polar singularity while preserving the cohomology of the continuous complex. The discrete complex satisfies $ ext{dim }\mathcal{H}^0=1$, $\text{dim }\mathcal{H}^1=1$, $\text{dim }\mathcal{H}^2=0$, and $\text{dim }\mathcal{H}^3=0$, ensuring the topological correctness of the numerical method. Numerical experiments validate optimal convergence rates for smooth fields under L2 projection, demonstrating robustness to the polar singularity and potential applicability to electromagnetics in magnetic confinement geometries such as tokamaks and stellarators.

Abstract

In this work we define a spline complex preserving the cohomological structure of the continuous de Rham complex when the underlying physical domain is a toroidal solid. In the spirit of the isogeometric analysis, the spaces involved will be defined as pushforward of suitable spline spaces on a parametric domain. The singularity of the parametrization of the solid will demand the imposition of smoothness constraints on the full tensor product spline spaces in the parametric domain to properly set up the discrete complex on the physical domain.

Figures (18)

• Figure 1: $C^1$ smooth volumetric geometric models of a tokamak (a) and a stellarator (b) created using polar splines. The blue dots are the control points of the mappings, obtained via interpolation at the Greville points of such magnetic hydrodynamics equilibria using the Python libraries Struphy struphy and Psydac psydac.
• Figure 2: The B-spline basis of the quadratic $C^1$ spline space (a) and the basis functions of the quadratic $C^1$ periodic spline space (b) on the uniform open knot vector of five distinct knots $\mathbf{t}=[0,0,0,1,2,3,4,4,4]$. We stress that the basis functions in (b) tie in at the endpoints of the interval with $C^1$ continuity.
• Figure 3: Comparison of the linear spline functions spanning the derivatives of the quadratic splines generated by the bases reported in Figure \ref{['periodicsplinebasis']} (a)-(b) respectively. These derivatives are $C^0$ (a) and $C^0$ periodic (b) respectively as the linear splines in (b) tie in at the endpoints of the interval with $C^0$ continuity.
• Figure 4: The transformation of the parametric domain $\Omega$ into the toroidal domain $\Omega^{pol}$ by the polar map $\pmb{F}$. The three-dimensional tensor mesh defined on $\Omega$ is mapped to a three-dimensional polar mesh in $\Omega^{pol}$. The latter can be seen in the right-most figure.
• Figure 5: The scheme of the different complexes we consider. The second bottom row is the spline sub-complex on $\Omega^{pol}$ we use to discretize the de Rham complex \ref{['continuousderham']} (bottom row) and which preserves the cohomological structure of the latter. The two top rows are complexes on the parametric domain $\Omega$. The top one is the spline complex of the full tensor product spline space. The second top is the complex of the reduced spline space in which the pushforward operators $\mathcal{F}^i$ related to the polar map $\pmb{F}$ can be applied.

Theorems & Definitions (17)

• Remark 2.1
• Remark 2.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Corollary 3.3
• proof
• Proposition 3.4
• Proposition 3.5