Professor! Today's topic is AC loss in superconductors, right? What exactly is it?

It involves calculating the losses in superconducting wires and cables under AC magnetic fields. This includes the separate evaluation of hysteresis loss, coupling loss, and eddy current loss, and the optimization of filament diameter and twist pitch.

What kind of equations are involved in AC loss for superconductors?

The equations governing AC loss in superconductors.

How do we actually solve these equations using a computer?

We use spatial discretization via the Finite Element Method (FEM). This involves assembling the element stiffness matrices and constructing the global stiffness equation.

What specifically does the matrix solution algorithm entail?

It involves solving the system of equations using either direct methods (LU decomposition, Cholesky decomposition) or iterative methods (Conjugate Gradient method, GMRES method). For large-scale problems, preconditioned iterative methods are particularly effective.

AC Loss in Superconductors

Category: 電磁場解析 | Integrated 2026-04-06

CAE visualization for ac loss superconductor theory - technical simulation diagram

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Theory and Physics

Overview

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Professor! Today's topic is about AC losses in superconductors, right? What exactly are they?

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Loss calculation for superconducting wires/cables under AC magnetic fields. Separate evaluation of hysteresis loss, coupling loss, and eddy current loss. Optimization of filament diameter and twist pitch.

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Wow, the topic of superconducting wires/cables is super interesting! Please tell me more.

Governing Equations

$$ Q_h = \frac{2\mu_0 H_a^3}{3J_c d} \text{ (slab, } H_a < H_p\text{)} $$

$$ Q_{coupling} = \frac{2\mu_0 l_p^2 \dot{B}^2}{\rho_{eff}}\cdot\frac{\tau}{T} $$

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I see... AC losses in superconductors seem simple at first glance, but are actually very profound, aren't they?

Discretization Methods

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How do you actually solve these equations on a computer?

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We use spatial discretization via the Finite Element Method (FEM). We assemble the element stiffness matrices and construct the global stiffness equation.

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We perform transformation to the weak form (variational form) and use formulation via the Galerkin method using test functions and shape functions. The choice of element type (low-order elements vs. high-order elements, full integration vs. reduced integration) directly affects the trade-off between solution accuracy and computational cost.

Matrix Solution Algorithms

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What exactly are matrix solution algorithms?

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We solve the simultaneous equations using direct methods (LU decomposition, Cholesky decomposition) or iterative methods (CG method, GMRES method). For large-scale problems, preconditioned iterative methods are effective.

Solver	Classification	Memory Usage	Applicable Scale
LU decomposition	Direct Method	O(n²)	Small to Medium Scale
Cholesky decomposition	Direct Method (Symmetric Positive Definite)	O(n²)	Small to Medium Scale
PCG Method	Iterative Method	O(n)	Large Scale
GMRES Method	Iterative Method	O(n·m)	Large Scale / Non-symmetric
AMG Preconditioner	Preprocessing	O(n)	Very Large Scale

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So, if you cut corners on the finite element method part, you'll pay for it later. I'll keep that in mind!

Implementation in Commercial Tools

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So, what software can be used to handle AC losses in superconductors?

Tool Name	Developer/Current Status	Main File Format
COMSOL Multiphysics	COMSOL AB	.mph
Ansys Maxwell	Ansys Inc.	.aedt, .maxwell
JMAG-Designer	JSOL Corporation	.jmag, .jproj

Vendor Lineage and Product Integration History

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Do these software tools have dramatic origin stories?

COMSOL Multiphysics

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Tell me about "COMSOL Multiphysics"!

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Founded in Sweden in 1986. Started as FEMLAB with MATLAB integration, later renamed to COMSOL. Strong in multiphysics.

Current Affiliation: COMSOL AB

Ansys Maxwell

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Tell me about "Ansys Maxwell"!

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Ansoft Maxwell. Low-frequency electromagnetic field analysis. Integrated into Ansys in 2008.

Current Affiliation: Ansys Inc.

JMAG-Designer

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What exactly is JMAG?

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Developed by Japan's JSOL Corporation. An electromagnetic field analysis tool specialized for electrical equipment design.

Current Affiliation: JSOL Corporation

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Ah, I see! Founded in Sweden in 1986, that's how it was structured.

File Formats and Interoperability

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Are there any points to note when transferring data between different software?

Format	Extension	Type	Overview
STEP	.stp/.step	Neutral CAD	3D CAD data exchange format compliant with ISO 10303. Supports geometry + PMI.
IGES	.igs/.iges	Neutral CAD	Early CAD data exchange standard. Has issues with surface data compatibility. Migration to STEP is progressing.
VTK	.vtk/.vtu	Visualization	Visualization Toolkit format. Used by ParaView, etc.

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When converting models between different solvers, attention must be paid to the correspondence of element types, compatibility of material models, and differences in the representation of loads/boundary conditions. Particularly, high-order elements and special elements (cohesive elements, user-defined elements, etc.) often cannot be directly converted between solvers.

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I see... formats seem simple at first glance, but are actually very profound, aren't they?

Practical Considerations

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Are there any "field wisdom" things not found in textbooks?

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Verifying mesh convergence, validating the appropriateness of boundary conditions, and performing sensitivity analysis of material parameters are extremely important.

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Mesh Dependency Verification: Confirm convergence with at least 3 levels of mesh density.
Boundary Condition Validity: Setting physically meaningful constraint conditions.
Result Verification: Comparison with theoretical solutions, experimental data, and known benchmark problems.

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I've grasped the overall picture of AC losses in superconductors! I'll try to be mindful of it in my practical work from tomorrow.

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