A Reduced Magnetic Vector Potential Approach with Higher-Order Splines

Abstract

This work presents a high-order isogeometric formulation for magnetoquasistatic eddy-current problems based on a decomposition into Biot-Savart-driven source fields and finite-element reaction fields. Building upon a recently proposed surface-only Biot-Savart evaluation, we generalize the reduced magnetic vector potential framework to the quasistatic regime and introduce a consistent high-order spline discretization. The resulting method avoids coil meshing, supports arbitrary winding paths, and enables high-order field approximation within a reduced computational domain. Beyond establishing optimal convergence rates, the numerical investigation identifies the requirements necessary to recover high-order accuracy in practice, including geometric regularity of the enclosing interface, accurate kernel quadrature, and compatible trace spaces for the source-reaction coupling.

Figures (7)

• Figure 1: Problem setup illustrated through cross‑sections in the $xy$‑ and $xz$‑planes. The closed surface $\Gamma$ partitions the computational domain $V$ d into the outer region $V_{\mathrm{ext}}$ and the inner region $V_{\mathrm{int}}$. The conductor (blue) is fully contained in $V_{\mathrm{int}}$, while the coil (red), shown here with five turns, carries the source current in $V_{\mathrm{ext}}$. This setup is analyzed in \ref{['sec:results']} with two different coil configurations.
• Figure 2: Convergence results for degree $p$.
• Figure 3: Convergence of the error in the source field for the different quadrature rules.
• Figure 4: Convergence results for $p=3$ for different number of quadrature points used in the trapezoidal rule to compute $\vec{A}_\mathrm{s}$.
• Figure 5: Convergence of the reduced field in the $\mathrm{H}^{}_{}\!\left(\operatorname{\iflanguage{ngerman}{rot}{curl}}\right)$–seminorm under trace space mismatch. The bulk for $\vec{A}_\mathrm{g}$ is $S_p^{1}(V)$ (here: $p=2$), while the interface current density $\vec{K}_\mathrm{g}$ is approximated either in the matching trace space $S_{p}^{1\ast}(\Gamma)$ (optimal) or in the degraded space $S_{p-1}^{1\ast}(\Gamma)$.