In many time-harmonic electromagnetic wave problems, the considered geometry exhibits an axial symmetry. In this case, by exploiting a Fourier expansion along the azimuthal direction, fully three-dimensional (3D) calculations can be carried out on a two-dimensional (2D) angular cross section of the problem, thus significantly reducing the computational effort. However, the transition from a full 3D problem to a 2D analysis introduces additional difficulties such as, among others, a singularity in the variational formulation. In this work, we compare and discuss different finite element formulations to deal with these obstacles. Particular attention is paid to spurious modes and to the convergence behavior when using high-order elements.

Superconducting electromagnets commonly exhibit thin layers with high aspect ratio such as insulation layers or turn-to-turn contacts. A finite element analysis of these devices can lead to unfavorable meshes in these thin layers, either because of a high number of degrees of freedom or mesh elements of poor quality which decrease the accuracy of the simulation results. To mitigate these issues when conducting a thermal finite element analysis solving the heat equation, this work proposes to collapse thin volume layers into surfaces by using a thermal thin shell approximation. The proposed method uses one-dimensional Lagrange elements across the thickness of the thin layer and can handle a variety of interface conditions, multi-layered structures, heat sources, nonlinear material behavior or coupling to physics other than heat transfer. The efficiency of the proposed approximation is highlighted by comparison with a reference model with a conventionally meshed insulation for a model problem exhibiting a brick wall structure where a stationary heat equation is solved. The formulation is then verified against reference models with meshed insulation solving a transient heat equation for an insulated high-temperature superconductor pancake coil exhibiting a local defect which causes a thermal runaway. The benefit of using the model with the thin shell approximation is studied by analyzing pancake coils with different ratios of the insulation layer to the coated conductor thickness. It is shown that the smaller the ratio, the shorter the solution time and the lower the number of unknowns of the thin shell model when compared to the conventionally meshed insulation in order to reach the same numerical accuracy. The method is implemented in an open-source finite element framework and a reference implementation for a simple model problem is shared alongside this paper.

When simulating no-insulation high-temperature superconducting pancake coils with the finite element (FE) method, the high aspect ratio of the thin turn-to-turn contact layer (T2TCL) leads to unfavorable meshes in these thin layers as manifested by a high number of degrees of freedom (DoF) or mesh elements of poor quality which decrease the accuracy of the simulation results. To mitigate this issue, we propose to collapse the T2TCL volume into a surface using a thin shell approximation (TSA) for three-dimensional FE analysis. A ⃗ H − ϕ formulation is used and solves for the magnetic field strength ⃗ H in conducting domains and the magnetic scalar potential ϕ in insulating domains. This formulation avoids spurious currents and reduces the number of DoF in insulating domains. Automatically created thick cuts are used to deal with multiply connected domains. Particular attention is paid to the interpretation of these cuts and the corresponding basis functions in the context of pancake coil geometries.The efficiency of the formulation facilitates the resolution of each turn. In this way, local phenomena such as quench can be captured in a straightforward way. The TSA formulation is verified by comparison against a reference model with volumetrically meshed T2TCL and is shown to be accurate and efficient, significantly reducing the solution time while reducing the effort for creating high-quality meshes. The TSA is implemented in an open-source FE framework and the source code is shared alongside this paper.

Axial symmetry in time‐harmonic electromagnetic wave problems can be exploited by considering a Fourier expansion along the angular direction, reducing fully three‐dimensional computations to two‐dimensional ones on an azimuthal cross section. While this transition leads to a significant decrease in computational effort, it introduces additional difficulties, which necessitate appropriate finite element (FE) formulations. By combining the latter with perfectly matched layers (PML), open problems can be considered. In this work, we compare and discuss the performance of different combinations of axisymmetric FE formulations and PMLs, using a dielectric sphere in open space as a test case. As an application example, a superconducting Fabry–Pérot photon trap is considered.