[Paper Review] Simulation of high temperature superconductors and experimental validation

Marc Olm, Santiago Badia+1 more|arXiv (Cornell University)|Jul 31, 2017

Advanced Numerical Methods in Computational Mathematics58 references20 citations

TL;DR

This paper presents a parallel, adaptive finite element framework based on the H-formulation and Nédélec elements to simulate the low-frequency electromagnetic response of high-temperature superconductors (HTS) on HPC platforms. It achieves up to 52.5× speed-up in time-to-solution via h-adaptive mesh refinement and a tailored parallel nonlinear solver, validated against experimental data and a 3D benchmark.

ABSTRACT

In this work, we present a parallel, fully-distributed finite element numerical framework to simulate the low-frequency electromagnetic response of superconducting devices, which allows to efficiently exploit HPC platforms. We select the so-called H-formulation, which uses the magnetic field as a state variable. N\\'ed\\'elec elements (of arbitrary order) are required for an accurate approximation of the H-formulation for modelling electromagnetic fields along interfaces between regions with high contrast medium properties. An h-adaptive mesh refinement technique customized for N\\'ed\\'elec elements leads to a structured fine mesh in areas of interest whereas a smart coarsening is obtained in other regions. The composition of a tailored, robust, parallel nonlinear solver completes the exposition of the developed tools to tackle the problem. First, a comparison against experimental data is performed to show the availability of the finite element approximation to model the physical phenomena. Then, a selected state-of-the-art 3D benchmark is reproduced, focusing on the parallel performance of the algorithms.

Motivation & Objective

• To develop a scalable, high-performance numerical framework for simulating the electromagnetic behavior of high-temperature superconductors (HTS) under realistic conditions.
• To address the computational challenges of modeling HTS devices, including multiscale physics, strong material contrasts, and nonlinearities in the current-voltage relation.
• To enable efficient simulation on HPC platforms through advanced mesh adaptation and parallel nonlinear solvers.
• To validate the numerical model against experimental data and reproduce a state-of-the-art 3D benchmark for credibility and performance assessment.

Proposed method

• Adopting the H-formulation with the magnetic field as the primary variable, enabling direct application of external magnetic fields and current injection via Ampère’s law.
• Using curl-conforming Nédélec edge elements of arbitrary order to accurately capture field discontinuities across high-contrast interfaces (e.g., superconductor/air).
• Implementing h-adaptive mesh refinement on octree-based meshes with 2:1 refinement balance to concentrate resolution in regions of interest while coarsening in dielectric areas.
• Employing a fully-implicit time discretization with Backward Euler and adaptive time stepping guided by nonlinear solver convergence history.
• Designing a parallel, fully-distributed nonlinear solver using Newton-Raphson linearization with exact Jacobian and domain decomposition preconditioners for H(curl) spaces.
• Integrating the framework within the open-source FEMPAR finite element software for extensibility and community use.

Experimental results

Research questions

• RQ1 Can a fully-distributed, parallel finite element framework based on the H-formulation accurately simulate the electromagnetic response of HTS devices with complex geometry and material contrasts?
• RQ2 How effective is h-adaptive mesh refinement using Nédélec elements in reducing computational cost while maintaining accuracy in HTS simulations?
• RQ3 To what extent can a tailored parallel nonlinear solver with adaptive time stepping and domain decomposition preconditioning improve time-to-solution on HPC platforms?
• RQ4 How well does the numerical model reproduce experimental data for real HTS devices in terms of magnetic field and current distribution?
• RQ5 What level of strong scalability is achievable in 3D HTS simulations using the proposed framework, and where do performance bottlenecks emerge?

Key findings

• The framework achieves up to 52.5× speed-up in time-to-solution for the largest problem (144 MPI processes) compared to a serial run, reducing simulation time from 23 days to under 11 hours.
• For Mesh 2, a 35.8× speed-up was achieved, with wall clock time decreasing from 23d 6h 14’ to 17h 16’ across 48 processors.
• The nonlinear solver shows robust convergence across all simulations, with average linear solver iterations increasing only mildly with processor count.
• The coarse problem solver, limited to 48 cores, becomes a performance bottleneck beyond 144 cores, indicating a need for multilevel preconditioning to extend scalability.
• The numerical results show excellent agreement with experimental data, validating the model’s ability to capture physical phenomena such as flux penetration and current distribution.
• The 3D benchmark simulation demonstrates strong scalability up to 144 cores, with the framework successfully handling complex, adaptive, and heterogeneous meshes on HPC systems.