AI 中文总结
该研究提出H-cloud开源框架用于超导体电磁建模，基于Nédélec有限元在Python工作流程中表达非线性电磁响应，经多模型验证，开源工作流程精度高且云执行可保留数值解，为超导模拟等提供严格可重复便携途径。
AI 中文摘要
我们提出了H-cloud形式主义，这是一种用于应用超导体电磁建模的可云访问的开源有限元框架。该方法基于Nédélec有限元在符合旋度的离散化中表达II型超导体的非线性电磁响应，在变分层面明确陈述切向应用场边界条件、非线性E-J功率定律和全隐式时间离散残差，均在脚本化的Python有限元工作流程内。弱形式用作正向模拟以及扩展到伴随微分和PDE约束优化的基础，同时让控制假设、边界条件和求解器结构对用户完全可见。该实现是在Firedrake中使用UFL和PETSc支持的非线性求解器完成的，允许相同的脚本在本地机器和诸如Google Colab等可通过浏览器访问的环境中运行而无需重新制定问题。该方法在类Bean穿透条件下圆柱形超导体的标准磁化基准上得到验证，然后与针对实际高温超导Bi2212线独立构建的COMSOL模型进行基准测试。在匹配网格研究中，开源工作流程将商业参考磁化回路再现到约1%以内，相对峰值误差低于1.5%，而云执行在使用（免费可用）减少的硬件资源的情况下，在相当适度的额外运行时保留相同的数值解。所提出的框架为应用和功能超导体的超导模拟、基准测试以及未来基于优化的建模提供了一条严格、可重复和便携的途径，可以共享并在开放云环境中执行。
英文摘要
We present the H-cloud formalism, a cloud-accessible and open-source finite-element framework for electromagnetic modelling of applied superconductors. The proposed method expresses the nonlinear electromagnetic response of type-II superconductors in a curl-conforming discretisation based on Nédélec finite elements, where the tangential applied-field boundary condition, nonlinear E-J power law, and fully implicit time-discrete residual are stated explicitly at the variational level, all within a scripted Python finite-element workflow. The weak form is used as the basis for forward simulation and for extension to adjoint differentiation and PDE-constrained optimisation, while keeping the governing assumptions, boundary conditions, and solver structure fully visible to the user. The implementation is realised in Firedrake with UFL and PETSc-backed nonlinear solvers, allowing the identical script to run on local machines and in browser-accessible environments such as Google Colab without reformulating the problem. The method is verified on the canonical magnetisation benchmark of a cylindrical superconductor under Bean-like penetration conditions and then benchmarked against an independently constructed COMSOL model for a practical high temperature superconducting Bi2212 wire. Across matched mesh studies, the open-source workflow reproduces the commercial-reference magnetisation loops to within approximately \(1\%\) , with relative peak errors below 1.5%, while cloud execution preserves the same numerical solution at rather modest additional runtime considering the use of (freely available) reduced hardware resources. The proposed framework provides a rigorous, reproducible, and portable route for superconducting simulation, benchmarking, and future optimisation-led modelling of applied and functional superconductors, shareable and executable into open cloud environments.
Comments12 pages, 2 figures. Supplementary material is available at http://doi.org/10.5281/zenodo.21294926
