🔢 mathematics

A Cloud-Accessible Open-Source Framework for the Electromagnetic Modelling of Applied Superconductors

This paper introduces H-cloud, a cloud-accessible, open-source finite-element framework built on Firedrake that enables rigorous, reproducible, and portable electromagnetic modeling of applied superconductors with accuracy comparable to commercial software while supporting forward simulation, adjoint differentiation, and optimization in both local and browser-based environments.

Original authors: Yusen Guo, Alberto Paganini, Harold S. Ruiz

Published 2026-07-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yusen Guo, Alberto Paganini, Harold S. Ruiz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a superpower: you can predict exactly how electricity flows through a special kind of wire called a superconductor. These wires are magical because they carry electricity with zero resistance, but only when they are super cold. To design things like powerful magnets for MRI machines or future fusion reactors, scientists need to run complex computer simulations to see how these wires behave.

For a long time, doing these simulations was like trying to bake a cake using a secret recipe locked inside a expensive, heavy oven that only a few people own. You could see the cake come out, but you couldn't peek inside the oven to see exactly how the heat was moving, and you couldn't easily share your recipe with a friend who didn't have that specific oven.

Enter the H-cloud, a new way of doing these simulations that the authors, Yusen Guo, Alberto Paganini, and Harold S. Ruiz, have cooked up. Think of the H-cloud not as a new oven, but as a universal, open-source recipe book that anyone can read, copy, and run on their own kitchen counter—or even on a free, browser-based computer in the sky (like Google Colab).

The Magic Recipe: Writing Math Like a Story

The core of this new method is a way of writing down the physics of superconductors that is so clear and flexible, it can be run anywhere. The authors used a specific mathematical language (called the "H-formulation") to describe how magnetic fields push and pull on the superconductor.

Usually, scientists have to translate their math into a specific computer program's language, which is like translating a poem into a different dialect just to read it in one specific library. The H-cloud approach writes the math in a "universal dialect" (using a tool called Firedrake and UFL) that looks almost exactly like the equations you'd write on a chalkboard. Because it's written this way, the same script can run on a powerful laptop in a lab, or on a free cloud computer, without changing a single line of the math.

The Big Test: Does it Taste Like the Real Thing?

The authors didn't just write the recipe; they tested it to make sure it actually works. They ran two main experiments:

The "Bean" Cylinder Test: They simulated a long, round superconductor wire being hit by a magnetic field. There is a famous, classic answer to how this should behave (known as Bean's model). The H-cloud simulation matched this classic answer perfectly, showing that the magnetic field penetrates the wire exactly as physics predicts it should.
The "Real-World" Wire Test: They compared their open-source recipe against a simulation run on a top-tier, expensive commercial software called COMSOL. They used a realistic high-temperature superconductor wire (Bi2212) with specific properties: a critical current density ( $J_c$ ) of $6.64 \times 10^9$ A m $^{-2}$ , a critical electric field ( $E_c$ ) of $10^{-4}$ V m $^{-1}$ , and a power-law exponent ( $n$ ) of 25.

The Result: The H-cloud simulation produced results that were almost identical to the expensive commercial software. The difference in the magnetic "loops" (the shape of the data) was tiny—less than 1% error in the overall shape, and the peak errors stayed below 1.5%.

Running in the Cloud vs. Running Locally

Here is the most exciting part: The authors ran the exact same code on three different setups to see if the "cloud" version was any different from the "local" version.

Local (Apple M3): Ran on a laptop with 8 CPU cores.
Local (COMSOL): Ran on a Windows PC with 6 CPU cores.
Cloud (Google Colab): Ran on a free server with just 2 logical CPU cores.

Even though the cloud computer was much weaker (it had fewer "muscles" to do the work), the mathematical answer was exactly the same. The cloud version just took a bit longer to finish the job because it had fewer resources. For example, on a "medium" mesh (a grid of 7,664 elements), the local Apple laptop finished in 12 seconds, while the cloud took 189 seconds. On a "fine" mesh (30,656 elements), the local machine took 407 seconds, and the cloud took 1,752 seconds.

The authors emphasize that this isn't about the cloud being faster (it wasn't); it's about the cloud being portable. You don't need to buy expensive software or have a supercomputer in your basement to get the same rigorous scientific results.

What This Is NOT

It is important to note what the H-cloud is not. The authors are not saying this new method replaces all other ways of doing simulations. They explicitly state that choosing the right method depends on the specific problem (like the size of the device or the materials used). They are not claiming to have solved every problem in superconductivity, nor are they saying their method is the only way to do it. Instead, they are offering a transparent, reproducible, and shareable tool that makes advanced modeling accessible to everyone, from students to researchers, without needing a paid license.

The Bottom Line

The H-cloud framework is a bridge. It connects the complex, heavy math of superconducting physics to a simple, open, and portable workflow. It proves that you can get the same high-quality, scientifically rigorous results on a free cloud notebook as you can on a powerful local machine or with expensive commercial software. It's like taking a secret, locked recipe and turning it into a public cookbook that anyone can use, anywhere, to bake the same perfect cake.

Technical Summary: The H-Cloud Formalism for Superconducting Modelling

Problem Statement
Electromagnetic modelling is critical for the design of superconducting devices, ranging from magnetic cloaks to high-temperature superconducting (HTS) cables. While the H-formulation is a standard approach for modelling type-II superconductors due to its direct representation of the magnetic field and compatibility with power-law E–J relations, existing workflows often face a trade-off between transparency and portability. Commercial multiphysics platforms (e.g., COMSOL) offer efficient forward solutions but often obscure the variational structure, limiting user control over solver configurations and adjoint differentiation. Conversely, open-source implementations often suffer from installation difficulties and lack portability across different computing environments. Furthermore, while various formulations exist (T–A, A–V, integral methods), there is a gap in providing a fully transparent, cloud-accessible workflow that allows the governing assumptions, boundary conditions, and solver structures to remain visible and modifiable without relying on local licensed software.

Methodology: The H-Cloud Formalism
The paper introduces the H-cloud formalism, a cloud-accessible, open-source finite-element framework implemented in Python using the Firedrake library. The methodology is built upon the following pillars:

Variational Formulation: The method employs the magneto-quasi-steady (MQS) H-formulation. It utilizes a curl-conforming discretisation based on Nédélec finite elements (edge elements) within the Hilbert space $H(\text{curl}; \Omega)$ . This choice ensures the tangential continuity of the magnetic field across element boundaries, preventing spurious solutions common with Lagrange elements.
Weak Form and Discretisation: The governing equations (Faraday's and Ampère's laws) are expressed in a weak form. The nonlinear electromagnetic response of type-II superconductors is captured via a power-law E–J relation ( $E = \rho(J)J$ ), while non-superconducting regions use a linear resistive law. The applied magnetic field is imposed explicitly through the tangential trace boundary condition.
Time-Stepping and Solvers: The weak form is converted into a fully implicit, nonlinear residual using backward-Euler time stepping. This residual serves as the central computational object, solved via Newton–Kantorovich algorithms with line search, backed by PETSc.
Cloud Portability and Optimisation: The entire workflow is scripted in the Unified Form Language (UFL), allowing the identical Python script to run on local machines or in browser-accessible environments like Google Colab without reformulation. The framework is designed to support PDE-constrained optimisation and adjoint differentiation (via Pyadjoint), enabling the calculation of gradients for inverse problems and design optimisation without explicitly differentiating the state with respect to every degree of freedom.

Key Contributions

Transparent Weak Formulation: The authors state the superconducting H-formulation problem in a self-contained weak form within a curl-conforming function space, making the variational structure, boundary conditions, and nonlinear laws fully explicit and modifiable.
Cloud-Executable Workflow: The paper demonstrates that the same mathematical model can be executed locally or in a cloud notebook environment (Google Colab) without altering the formulation, solver configuration, or mesh hierarchy.
Adjoint Integration: The residual formulation is shown to be directly compatible with PDE-constrained optimisation frameworks, providing a natural entry point for adjoint-based design and inverse problems.

Results
The framework was validated through two primary benchmarks:

Canonical Magnetisation Benchmark: The model was tested on an infinitely long cylindrical superconductor under a transverse AC magnetic field (Bean-like penetration conditions). Using a high power-law exponent ( $n=96$ ) and slow frequency, the H-cloud model accurately reproduced the analytical full-penetration scale ( $B_p$ ) and saturation magnetization ( $M_p$ ). The results confirmed that the numerical solution correctly predicts the disappearance of the flux-free core at the analytically predicted field strength.
Comparative Benchmarking: The open-source workflow was compared against an independently constructed COMSOL Multiphysics model for a practical Bi2212 wire. Across matched mesh studies (coarse, medium, and fine), the H-cloud model reproduced the commercial-reference magnetisation loops with high fidelity:
- Accuracy: The open-source workflow reproduced the magnetisation loops to within approximately 1% (measured by $L_2$ norm error), with relative peak errors remaining below 1.5%.
- Portability: The same script ran on local hardware (Apple M3, Intel i5) and cloud hardware (Google Colab Xeon). While cloud execution on reduced hardware (2 logical cores vs. 6–8 cores) resulted in longer runtimes (consistent with hardware limitations), the numerical solution remained identical. The peak process memory usage on Colab was minimal (0.46–0.51 GB), indicating the examples were CPU-time limited rather than memory limited.

Significance and Claims
The paper claims that the H-cloud formalism provides a rigorous, reproducible, and portable route for superconducting simulation. Its primary significance lies not in outperforming commercial solvers in raw speed on equivalent hardware, but in its transparency and accessibility. By decoupling the mathematical formulation from the execution environment, the framework allows researchers to:

Share and execute exact computational references without requiring proprietary licenses or complex local installations.
Maintain full visibility and control over the variational structure, solver tolerances, and constitutive laws.
Transition seamlessly from forward simulation to adjoint-based optimisation.

The authors position H-cloud as a tool to broaden access to advanced modelling in applied superconductivity, facilitating benchmark sharing, collaborative development, and teaching. They explicitly state that the framework is not intended to replace commercial software but to offer a complementary, open, and scientifically transparent alternative that is inherently extensible to more complex geometries, constitutive laws, and high-performance computing environments.