Purpose: Reference guide to the literature used in BELFEM development.

Note: The ./literature/ directory contains proprietary content and is maintained as an independent repository (in .gitignore). This document provides references so users can locate the cited works independently.

Last Updated: 2026-08-11

Paper Alias Index (retired — decoder for older records)

The paperN aliases were retired on 2026-08-11. New citations use author and year ("Messe et al. 2023, §2.7"); see CLAUDE.md. They were converted out of the source comments and the module documentation in the same pass, so the aliases now survive only in dated devlog entries, which are kept as written.

This table remains the authoritative alias → source file → citation map, so those older records stay readable and auditable. Source .txt files live in the proprietary literature/ repository; the citations below let anyone locate the works independently.

Alias	Source file	Citation
paper0	dular2021.txt	Dular et al. 2021, "Stability of Mixed FE Formulations for HTS", IEEE TASC, DOI:10.1109/TASC.2021.3098724
paper1	messe2023.txt	Messe et al. 2023, "BELFEM: a special purpose FE code for magnetodynamic modeling of HTS tapes", SUST, DOI:10.1088/1361-6668/acf7f9
paper2	messe2022.txt	Messe 2022, "A Special Purpose Finite-Element Framework for HTS Applications", HTS 2022 Conference, HAL:hal-03791404
paper3	arsenault2023.txt	Arsenault et al. 2023, "Magnetodynamic H-φ Formulation", IEEE TASC, DOI:10.1109/TASC.2023.3293449
paper4	riva2023.txt	Riva et al. 2023, "H-φ in Sparselizard with DDM", IEEE TASC, DOI:10.1109/TASC.2023.3240389
paper5	alves2022a.txt	Alves et al. 2022a, "3D FE Thin-Shell Model for HTS Tapes", IEEE TASC, DOI:10.1109/TASC.2022.3143076
paper6	alves2022b.txt	Alves et al. 2022b, "Thin-shell approach for modeling superconducting tapes in H-φ formulation", SUST, DOI:10.1088/1361-6668/ac3f9e
paper7	arsenault2021.txt	Arsenault et al. 2021, "Implementation of H-φ Formulation in COMSOL Multiphysics", IEEE TASC, DOI:10.1109/TASC.2020.3033998
paper8	alves2024.txt	Alves et al. 2024, "2-D Thin-Shell Model Based on H-φ-Formulation in COMSOL", IEEE TASC, DOI:10.1109/TASC.2024.3473850
paper9	schnaubelt2023.txt	Schnaubelt et al. 2023, "Electromagnetic Simulation of No-Insulation Coils Using H-φ TSA", IEEE TASC, DOI:10.1109/TASC.2023.3258905
paperA	schnaubelt2023.txt	Same work as paper9 — July 2026 devlogs use paperA for it (alias drift); both resolve here

Newer papers cited by author-year (no paperN alias assigned):

Source file	Citation
arsenault2026.txt	Arsenault et al. 2026, Erratum to "Magnetodynamic H-φ Formulation" (corrects Arsenault et al. 2023's air-domain coupling), IEEE TASC, DOI:10.1109/TASC.2026.3686487
denis2026.txt	Denis et al. 2026, "Simultaneous Multi-Scale Homogeneous H-Phi Thin-Shell Model for Stacked HTS Coils", IEEE TASC, DOI:10.1109/TASC.2026.3652981
dular1997.txt	Dular et al. 1997, "A Generalized Source Magnetic Field Calculation Method for Inductors of Any Shape", IEEE Trans. Magn., DOI:10.1109/20.582518
dular1999.txt	Dular et al. 1999, "A Natural Method for Coupling Magnetodynamic H-Formulations and Circuit Equations", IEEE Trans. Magn., DOI:10.1109/20.767308
luccini2025.txt	Lucchini 2025, "Evaluating Magnetization Losses in 3-D CORC Tapes With Integral and Finite-Element Methods", IEEE TASC, DOI:10.1109/TASC.2025.3544512 (filename luccini — cite as Lucchini)
wozniak2025.txt	Wozniak et al. 2025, "Influence of Critical Current Defect on Operation, Quench Detection and Protection of a Conduction-Cooled Pancake REBCO Coil", IEEE TASC 35(5):4604006, DOI:10.1109/TASC.2025.3532246
badel2021.txt	Badel et al. 2019, "Modeling of 'quench' or the occurrence and propagation of dissipative zones in REBCO high temperature superconducting coils", SUST 32(9):094001, DOI:10.1088/1361-6668/ab181f (filename badel2021 is the HAL deposit year, HAL:hal-02509494 — cite as Badel et al. 2019)

FVM papers (aliases F0–F4):

Alias	Source file	Citation
F0	aavatsmark2002.txt	Aavatsmark 2002, "An Introduction to Multipoint Flux Approximations for Quadrilateral Grids", Comput. Geosci. 6:405-432, DOI:10.1023/A:1021291114475
F1	agelas2008.txt	Agélas, Di Pietro, Masson 2008, "A Symmetric and Coercive Finite Volume Scheme for Multiphase Porous Media Flow", FVCA V, DOI:10.1515/JNUM.2008.006
F2	klausen2006.txt	Klausen & Winther 2006, "Robust Convergence of Multi Point Flux Approximation on Rough Grids", Numer. Math. 104:317-337, DOI:10.1007/s00211-006-0023-4
F3	ingram2010.txt	Ingram, Wheeler, Yotov 2010, "A Multipoint Flux Mixed Finite Element Method on Hexahedra", SIAM J. Numer. Anal. 48(4):1281-1312, DOI:10.1137/090766176
F4	wheeler2011.txt	Wheeler, Xue, Yotov 2011, "A Family of Multipoint Flux Mixed Finite Element Methods for Elliptic Problems on General Grids", Procedia CS 4:918-927, DOI:10.1016/j.procs.2011.04.097

Overview

BELFEM development is informed by four complementary literature categories:

FEM Fundamentals — Authoritative textbooks and lecture notes on finite element theory, implementation, and numerical methods
BELFEM FEM Specialization — Research papers specific to HTS modeling, h-φ formulations, and thin-shell approximations
BELFEM FVM Methods — Research papers on finite volume methods, specifically MPFA for diffusion problems
Computational Topology and Optimal Cuts — Algebraic-topology foundations and optimization theory behind cut generation in multiply-connected domains (the homology/ module)

Quick Reference: Which Literature Do I Need?

For BELFEM-Specific Questions

Start with research papers if your question relates to:

h-φ formulation and interface coupling
Thin-shell models for HTS tapes
Transport current constraints and cuts (thin/thick)
Cohomology and multiply-connected domains
HTS material models (E-J power law, Kim model)
Validation benchmarks for superconductor applications

For BELFEM FVM Questions

Start with FVM research papers if your question relates to:

Multipoint Flux Approximation (MPFA) methods
Anisotropic diffusion on general grids
Transmissibility matrix computation
Physical vs reference space evaluation
Enhanced BDDF₁ spaces for 3D hexahedra
Coercivity and convergence conditions

For Cuts, Cohomology, and Topology Questions

Start with the topology references if your question relates to:

Why cuts exist and what they are topologically
How a cohomology basis is computed on an FE mesh (BELFEM's thick cuts)
Whether a minimal cut is computable, and at what cost
Reduction/coreduction preprocessing before Smith normal form
The graph and LP algorithms underneath any of the above

For General FEM Theory

Start with textbooks if your question relates to:

Weak formulations and Galerkin methods
Element technology (isoparametric, mixed, reduced integration)
Locking phenomena (volumetric, shear, membrane)
Inf-sup stability and mixed formulations
Solver strategies (direct, iterative, eigenvalue)
Nonlinear solvers (Newton-Raphson, arc-length)
Time integration and dynamic analysis

FEM Textbooks

Core References

Bathe, K.-J. Finite Element Procedures, 2nd ed., 4th printing. Prentice Hall, 2016.

Use for: Robustness, reliability, production code implementation
Key chapters: Ch. 4 (formulation), Ch. 5 (elements), Ch. 8 (solvers), Ch. 9 (time integration)
Personality: Disciplined engineer — emphasizes proven, robust methods

Hughes, T.J.R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover, 2000 (reprint of 1987 original).

Use for: Clear linear FEM implementations, algorithm design
Key chapters: Ch. 1 (weak forms), Ch. 4 (locking), Ch. 5 (isoparametric elements)
Personality: Algorithm designer — clean, systematic approach

Zienkiewicz, O.C., Taylor, R.L., and Zhu, J.Z. The Finite Element Method: Its Basis and Fundamentals, 7th ed. Butterworth-Heinemann, 2013.

Use for: Comprehensive reference, adaptivity
Personality: Encyclopedia — covers everything

Zienkiewicz, O.C., and Taylor, R.L. The Finite Element Method for Solid and Structural Mechanics, 7th ed. Butterworth-Heinemann, 2014.

Use for: Nonlinear implementation, exact shell elements
Key chapters: Ch. 5 (nonlinear theory), Ch. 8 (contact)

Advanced References

Belytschko, T., Liu, W.K., Moran, B., and Elkhodary, K. Nonlinear Finite Elements for Continua and Structures, 2nd ed. Wiley, 2014.

Use for: Nonlinear mechanics, XFEM, stability theory
Not for: First-time learning (dense presentation)

Brenner, S.C., and Scott, L.R. The Mathematical Theory of Finite Element Methods, 3rd ed. Springer, 2008.

Use for: Rigorous proofs, convergence theory
Key chapters: Ch. 4-5 (convergence), Ch. 8 (mixed methods)

Bronshtein, I.N., Semendyayev, K.A., Musiol, G., and Mühlig, H. Handbook of Mathematics, 6th ed. Springer, 2015.

Use for: Quick formula verification and lookup

Lecture Notes

Evans, J.A. Mathematical Foundations of the Finite Element Method, Lecture Notes, University of Colorado Boulder, 2017.

Use for: The functional-analysis layer under everything else — Sobolev spaces, distributions, Lax-Milgram, Babuška-Brezzi, interpolation error estimates
Cite as: "Evans 2017, Ch. N" or "Evans 2017, §N.M"
Pairs with: Brenner & Scott (proofs in book form), Boffi et al. (mixed extensions)

Felippa, C.A. Introduction to Finite Element Methods (IFEM), Course Notes, University of Colorado Boulder.

Use for: Step-by-step learning, clear derivations; the penalty-method and Lagrange-multiplier tutorials, and static condensation
Cite as: "Felippa IFEM, Ch. N, §X.Y"
Available: Online as course materials

Specialized Advanced References

Arnold, D.N., Falk, R.S., and Winther, R. Finite Element Exterior Calculus. SIAM, 2018.

Use for: FEEC theory, topological foundations of FEM spaces
Key topics: de Rham complex, structure preservation, why FEM spaces must exist
Personality: Topologist — explains the mathematical structure underlying FEM

Boffi, D., Brezzi, F., and Fortin, M. Mixed Finite Element Methods and Applications. Springer, 2013.

Use for: Mixed FEM theory, saddle-point problems, inf-sup stability
Key chapters: Ch. 3-5 (inf-sup conditions), Ch. 11 (Maxwell eigenvalues)
Personality: Mixed FEM theorist — comprehensive treatment of stability

Monk, P. Finite Element Methods for Maxwell's Equations. Oxford University Press, 2003.

Use for: Maxwell equations, edge elements (Nédélec), electromagnetic scattering
Key chapters: Ch. 5-6 (edge elements), Ch. 7 (convergence), Ch. 9-12 (scattering)
Personality: Maxwell specialist — definitive source for electromagnetics FEM

Element Distortion on Quadrilateral/Hexahedral Meshes

Why QUAD/HEX elements in the magnetic solve must be perfectly rectangular: see src/fem/interpolation/doc/nedelec.md §6.6 and the maxwell usage guide §1.7. On non-affine (bi-/trilinearly mapped) quads and hexes, the mapped H(curl)/H(div) spaces lose completeness, and the lowest-order elements can lose convergence entirely. Monk 2003 develops the hexahedral edge-element theory only for parallelepipeds (§6.1) and cites this line of work as the reason (§8.2–8.3). Boffi et al. 2013 summarize it in §2.2.4/§2.5.5.

Arnold, D.N., Boffi, D., Falk, R.S., and Gastaldi, L. "Finite element approximation on quadrilateral meshes". Communications in Numerical Methods in Engineering, 17(11):805-812,

DOI:10.1002/cnm.450

Necessary and sufficient conditions for approximation order under bilinear maps

Arnold, D.N., Boffi, D., and Falk, R.S. "Approximation by quadrilateral finite elements". Mathematics of Computation, 71(239):909-922, 2002. DOI:10.1090/S0025-5718-02-01439-4

Scalar theory showing that distorted quads already break serendipity elements

Arnold, D.N., Boffi, D., and Falk, R.S. "Quadrilateral H(div) finite elements". SIAM Journal on Numerical Analysis, 42(6):2429-2451, 2005. DOI:10.1137/S0036142903431924

The vector-valued case: RT[0] divergence does not converge on general quad meshes

Falk, R.S., Gatto, P., and Monk, P. "Hexahedral H(div) and H(curl) finite elements". ESAIM: Mathematical Modelling and Numerical Analysis, 45(1):115-143, 2011. DOI:10.1051/m2an/2010034

Extends the negative results to 3D hexahedral H(curl), the case that bites h-φ directly

BELFEM Research Papers

Tier 1: Core BELFEM

Messe et al. 2023 — "BELFEM: a special purpose FE code for magnetodynamic modeling of HTS tapes" (primary reference)

BELFEM architecture, h-φ formulation, thin-shell implementation
Static condensation preferred over Lagrange multipliers
Solver strategies: STRUMPACK vs MUMPS
Nonlinear iteration: hybrid Picard/Quasi-Newton/Newton
Checkerboarding prevention: tight tolerances (ε < 10⁻¹¹) — the paper's recommendation, tightened per deck; BELFEM's default tolerance is 1e-6

Messe 2022 — "A Special Purpose Finite-Element Framework for HTS Applications"

Project motivation and scope

Tier 2: h-φ Thin-Shell Focus

Alves et al. 2022b — "Thin-shell approach for modeling superconducting tapes in H-φ formulation"

Full derivation with interface conditions
Cohomology basis theory (thick cuts)
N=1 vs N>1 analysis
Key finding: N>1 needed for stacked tapes to capture top/bottom losses

Alves et al. 2024 — "2-D Thin-Shell Model Based on H-φ-Formulation in COMSOL"

Thin cut implementation
Alternative approach to Alves et al. 2022b

Schnaubelt et al. 2023 — "Electromagnetic Simulation of No-Insulation Coils Using H-φ TSA"

Practical interpretation for coils
Transport current handling

Tier 3: Related Methods

Arsenault et al. 2023 — "Magnetodynamic H-φ Formulation"

Recommended variant: Magnetodynamic (H-φ/D) over quasi-static
Interface coupling methodology
Includes ∂(μh)/∂t term for correct transients

Alves et al. 2022a — "3D FE Thin-Shell Model for HTS Tapes"

Adaptable to h-φ
Relative homology background

Arsenault et al. 2021 — "Implementation of H-φ Formulation in COMSOL Multiphysics"

Performance comparison with pure H-formulation
Element order recommendations (quadratic for 2D)

Tier 4: Supporting Theory

Dular et al. 2021 — "Stability of Mixed FE Formulations for HTS"

Inf-sup conditions for h-φ
Hierarchical basis functions
HTS-specific stability analysis

Riva et al. 2023 — "H-φ in Sparselizard with DDM"

Domain decomposition methods
Note: Different software (NOT BELFEM)

Tier 5: HTS Material Models (E-J laws)

Rhyner 1993 — "Magnetic properties and AC-losses of superconductors with power-law current-voltage characteristics"

Physica C 212, pp. 292–300
The E-J power law underlying all three BELFEM resistivity laws
Cited in src/physics/materials/powerlaws.hpp

Plummer & Evetts 1987 — "Dependence of the shape of the resistive transition on composite inhomogeneity in multifilamentary wires"

IEEE Trans. Magn. 23 (2), pp. 1179–1182
Phenomenology of the n-value characterizing the transition sharpness

Duron et al. 2004 — "Modelling the E–J relation of high-Tc superconductors in an arbitrary current range"

J. Duron, F. Grilli, B. Dutoit, S. Stavrev — Physica C 401 (1-4), pp. 231–235
The parallel combination ρ = ρ_PL·ρ_n/(ρ_PL+ρ_n) used by the powerlaw and riva laws
DOI: 10.1016/j.physc.2003.09.044

Riva 2021 — "Quench behavior of high-temperature superconductor tapes for power applications: a strategy toward resilience"

N. Riva, EPFL doctoral thesis no. 8754 (2021)
Measured overcritical-regime resistivity of REBCO tapes 77–90 K (Ch. 3–5); the ρ_ηβ overcritical model and its continuity analysis (§5.1); impact of the law choice on simulated quench speed (Ch. 6)
Namesake of the resistivity type : riva law (which implements the Duron parallel form the thesis builds on, not the fitted ρ_ηβ model itself)
Local copy (ephemeral): ./tmp/EPFL_TH8754.pdf

See src/physics/materials/doc/resistivity_laws.md for how these map onto the three laws.

BELFEM FVM Papers

Primary References

Aavatsmark 2002 — "An Introduction to Multipoint Flux Approximations for Quadrilateral Grids"

DOI: 10.1023/A:1021291114475
Use for: MPFA O-method fundamentals, transmissibility matrices (start here)
Key topics: Interaction volumes, T = CA⁻¹B - D, K-orthogonality, monotonicity

Ingram et al. 2010 — "A Multipoint Flux Mixed Finite Element Method on Hexahedra"

DOI: 10.1137/090766176
Use for: Enhanced BDDF₁ for 3D hexahedra (critical for implementation)
Key topics: 4 DOF/face, curl enrichment, trapezoidal quadrature, superconvergence

Convergence and Stability

Agélas et al. 2008 — "A Symmetric and Coercive Finite Volume Scheme"

DOI: 10.1515/JNUM.2008.006
Use for: Coercivity conditions, when your mesh will converge
Key topics: Local coercivity coer(D,Λ), general meshes, L∞ coefficients

Klausen & Winther 2006 — "Robust Convergence of Multi Point Flux Approximation on Rough Grids"

DOI: 10.1007/s00211-006-0023-4
Use for: Physical vs reference space, robust convergence on rough grids
Key topics: Jacobian at cell center, stability condition (27), broken RT

Theory and Connection

Wheeler et al. 2011 — "A Family of Multipoint Flux Mixed Finite Element Methods"

DOI: 10.1016/j.procs.2011.04.097
Use for: MFMFE overview, variational framework
Key topics: BDM₁/BDDF₁ with quadrature, FEM-FVM connection, h²-perturbations

Computational Topology and Optimal Cuts

The references behind the homology/ module: why cuts exist, how BELFEM computes them, and what is and is not tractable if you want them minimal.

Theory and Method

Gross, P.W., and Kotiuga, P.R. Electromagnetic Theory and Computation: A Topological Approach. MSRI Publications, vol. 48, Cambridge University Press, 2004. ISBN 0-521-80160-5.

Use for: Why cuts must exist at all — cohomology for electromagnetics, branch cuts for the magnetic scalar potential, duality theorems
Cite as: "Gross & Kotiuga, §X" or "Gross & Kotiuga, Ch. N"
Answers: existence and construction in principle; not minimality

Pellikka, M., Suuriniemi, S., Kettunen, L., and Geuzaine, C. "Homology and Cohomology Computation in Finite Element Modeling", SIAM Journal on Scientific Computing, 2013.

DOI: 10.1137/130906556
Use for: BELFEM's actual method — chain complex → reduction → Smith normal form → thick cuts (the Gmsh solver)
Key topics: cohomology basis on an FE mesh, thick cuts vs thin cuts

Giard, G., et al. "Generalized Pellikka algorithm for cohomology computation", in preparation (drafted 2025, shelved, resumed 2026).

DOI: none — unpublished draft, not in ./literature/
Use for: the algorithm BELFEM actually runs by default — reduce_complexPellikkaGeneralized / coreduce_complexPellikkaGeneralized, selected at cl_MaxwellFactory.hpp:48
Key topics: generalized Pellikka reduction and coreduction; the cohomology-side 0-cochain omit loop
Note: this is the reason the cohomology core is closed to AI edits (doc/ai_collaboration_protocol.md §7.1). A modified algorithm with no published implementation gives a model no correct prior — its nearest neighbor is textbook Pellikka, so deliberate departures read as defects. Ask Gregory for the draft; do not reconstruct the method from the code.
Author's name is Giard. The module documentation carried "Giarda" in nine places until 2026-08-31; if it reappears, it is a typo.

Mrozek, M., and Batko, B. "Coreduction Homology Algorithm", Discrete & Computational Geometry, 2009.

DOI: 10.1007/s00454-008-9073-y
Use for: Why the reduction step is fast — linear-time coreduction collapses the complex before Smith normal form

Optimality and Complexity

Dey, T.K., Hirani, A.N., and Krishnamoorthy, B. "Optimal Homologous Cycles, Total Unimodularity, and Linear Programming", SIAM Journal on Computing, 2011.

DOI: 10.1137/100800245
Use for: When a minimal cut is computable in polynomial time — over Z, with a totally unimodular boundary matrix, via LP

Chen, C., and Freedman, D. "Hardness Results for Homology Localization", Discrete & Computational Geometry, 2011 (also Proc. ACM-SIAM SODA 2010).

DOI: 10.1007/s00454-010-9322-8
Use for: The negative result — over Z₂ the problem is NP-hard to approximate within any constant factor. Read before proposing a "just minimize it" cut algorithm

Dunfield, N.M., and Hirani, A.N. "The Least Spanning Area of a Knot and the Optimal Bounding Chain Problem", Proc. 27th Annual Symposium on Computational Geometry (SoCG '11), 2011.

DOI: 10.1145/1998196.1998218
Use for: Minimal spanning surfaces — NP-complete in general, polynomial when H₂ = 0

Costantini, M. "A Novel Phase Unwrapping Method Based on Network Programming", IEEE Transactions on Geoscience and Remote Sensing, 1998.

DOI: 10.1109/36.673674
Use for: The concrete grid analogue — branch cuts as min-cost network flow, i.e. a TU-structured LP solved at scale

Haken, W. "Theorie der Normalflächen: Ein Isotopiekriterium für den Kreisknoten", Acta Mathematica, 1961 (German).

DOI: 10.1007/BF02559591
Use for: Foundational — normal surface theory, decidability of unknot recognition. Historical context for why surface-finding is hard

Algorithms Reference

Cormen, T.H., Leiserson, C.E., Rivest, R.L., and Stein, C. Introduction to Algorithms, 3rd ed. MIT Press, 2009. ISBN 978-0-262-03384-8.

Use for: The graph and optimization algorithms underneath the above — BFS/DFS, spanning trees, shortest paths, min-cost flow, LP duality, total unimodularity, NP-completeness
Cite as: "CLRS, §X.Y" or "Cormen et al., Ch. N"

Key Implementation Guidelines from Literature

FVM Implementations (from FVM papers)

MPFA O-Method (Aavatsmark 2002 - Aavatsmark 2002)

Use interaction volumes at vertices with continuity points at subedge midpoints
Compute transmissibility T = CA⁻¹B - D for local flux-pressure relationship
Check K-orthogonality for TPFA feasibility

Physical Space Evaluation (Klausen & Winther 2006 - Klausen 2006)

Evaluate Jacobian Dₖ = D(x̂ₖ) at cell CENTER, not at integration points
Accept non-symmetric system matrix for robustness on rough grids
Check stability condition det(Λ + Λᵀ) ≥ γ₀ > 0

Enhanced BDDF₁ for 3D (Ingram et al. 2010 - Ingram 2010)

Standard BDDF₁ has only 3 DOF/face → insufficient for MPFA elimination
Must use enhanced BDDF₁ with 6 curl terms → 4 DOF/face
Requires h²-parallelepipeds for full O(h) convergence
Superconvergence O(h²) at cell centers on regular grids

Convergence Check (Agélas et al. 2008 - Agélas 2008)

Compute coercivity condition coer(D,Λ) ≥ θ > 0 before running
Condition is computable for any mesh + tensor combination
Predicts convergence on general polygonal/polyhedral meshes

FEM Implementations (from FEM papers)

Static Condensation (Messe et al. 2023)

Preferred method for hanging DOFs
Avoids zero diagonal blocks in system matrix
Better conditioning for direct solvers (MUMPS/STRUMPACK)

Magnetodynamic Coupling (Arsenault et al. 2023)

Use H-φ/D formulation (not quasi-static)
Must include ∂(μh)/∂t term
Better convergence than pure quasi-static

Thin-Shell Element Order (Alves et al. 2022b, Arsenault et al. 2021)

N=1: Linear through-thickness distribution (faster, similar to T-A)
N>1: Needed when tangential field penetration matters
Use N>1 for closely-packed tapes

Convergence Tolerance (Messe et al. 2023)

Drive ε < 10⁻¹¹ to prevent checkerboarding
Tighter than typical FEM tolerances
Critical for nonlinear HTS problems

Solver Selection (Messe et al. 2023)

STRUMPACK: Typically 2× faster
MUMPS: More robust for difficult problems
Try STRUMPACK first, fall back to MUMPS if needed

Question Type → Literature Mapping

Question	Primary Source	Secondary Sources
How does BELFEM implement X?	Messe et al. 2023, Messe 2022	Specific papers
Theory behind h-φ formulation?	Arsenault et al. 2023, Arsenault et al. 2021	Bathe Ch. 4, Hughes Ch. 3
Why isn't this converging?	Messe et al. 2023 Section 2.7, Bathe Ch. 8	Dular et al. 2021 (stability)
How do thin-shell models work?	Messe et al. 2023, Alves et al. 2022b	Alves et al. 2022a, Alves et al. 2024, Schnaubelt et al. 2023
What are cuts for?	Alves et al. 2022b, Alves et al. 2024, Schnaubelt et al. 2023	Alves et al. 2022a (homology)
How does MPFA work?	Aavatsmark 2002	Wheeler et al. 2011
Will my FVM mesh converge?	Agélas et al. 2008, Klausen & Winther 2006	Aavatsmark 2002 (fundamentals)
How to implement 3D MPFA?	Ingram et al. 2010	Aavatsmark 2002, Klausen & Winther 2006
Physical vs reference space?	Klausen & Winther 2006	Ingram et al. 2010 (3D details)
How to implement element X?	Hughes Ch. 5, Bathe Ch. 5	Zienkiewicz Vol. 1
What's the weak form for Y?	BELFEM papers	Bathe Ch. 3, Hughes Ch. 1
Why is my element locking?	Hughes Ch. 4, Bathe §4.3-4.5	Zienkiewicz Ch. 10
How to prove convergence?	Brenner Ch. 4-5	Bathe Ch. 4
Which solver should I use?	Messe et al. 2023, Bathe Ch. 8	Hughes Ch. 11
How much Ic defect can a coil tolerate?	Wozniak et al. 2025	Badel et al. 2019
Will quench detection see a defect in time?	Wozniak et al. 2025 §IV-C	Badel et al. 2019 §5
How do I control the time step in a nonlinear transient?	Badel et al. 2019 §4.3.2 (integral controller)	Bathe Ch. 9
How do I warm-start Newton-Raphson between steps?	Badel et al. 2019 §4.3.1	Messe et al. 2023 §3
How is current shared between the tape and its stabilizer?	Badel et al. 2019 §3.1	Wozniak et al. 2025 §III

Common Pitfalls and Solutions

Pitfall	Symptom	Solution	Reference
Loose convergence tolerance	Checkerboarding	Drive ε < 10⁻¹¹	Messe et al. 2023
N=1 for stacked tapes	Missing losses	Use N > 1	Alves et al. 2022b
Identical FE orders at interface	Oscillations	Hierarchical enrichment	Dular et al. 2021
Missing cuts	Wrong current flow	Add cuts (thin/thick)	Alves et al. 2022b, Alves et al. 2024
Lagrange multipliers	Zero diagonal	Use static condensation	Messe et al. 2023
QS formulation + linear elements	Poor convergence	Use magnetodynamic	Arsenault et al. 2023
Neglecting ∂(μh)/∂t	Incorrect transient	Include time derivative	Arsenault et al. 2023, Alves et al. 2022b
TPFA on non-K-orthogonal grids	O(1) error	Use MPFA	Aavatsmark 2002
Reference space on rough grids	Divergence	Use physical space Jacobian	Klausen & Winther 2006
Standard BDDF₁ for hexahedra	Only 3 DOF/face	Use enhanced BDDF₁	Ingram et al. 2010
Violating coercivity	No convergence	Check coer(D,Λ) ≥ θ	Agélas et al. 2008

Citation Guidelines

When Referencing Papers in Code/Documentation

DO:

Cite by author and year, with the exact section or equation: "Messe et al. 2023, §2.7", "Bathe, §8.6"
Use full academic citations with author names, journal, volume, pages, year in reference lists
Include DOIs as clickable links: DOI: [10.xxxx/xxxxx](https://doi.org/10.xxxx/xxxxx)
Reference the published work directly
Example: "Following the approach of Arsenault et al. 2023 (IEEE Trans. Appl. Supercond.)..."

DO NOT:

Write a new paperN or FN alias — they were retired on 2026-08-11 and are decoded above for older records only
Reference local file paths (e.g., "alves2022a.txt")
Include extracted text content
Assume literature files will be available in public repository

Rationale: Proper citations ensure academic attribution and provide public access via DOIs.

How to Access These References

Research Papers

Most BELFEM-related papers are published in:

IEEE Transactions on Applied Superconductivity
Superconductor Science and Technology
COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering
SIAM Journal on Scientific Computing

Search for papers using DOIs (when provided) or author names and keywords like:

"h-phi formulation superconductor"
"thin-shell HTS tape"
"cohomology cuts transport current"

Textbooks

Standard references available from:

University libraries
Publishers: Springer, Wiley, Dover, Prentice Hall
Some older editions available as Dover reprints (Hughes, Zienkiewicz)
Felippa's IFEM notes available online as course materials

Workflow Examples

Implementing a New Feature

Understand physics → BELFEM papers for HTS-specific, books for general FEM
Find weak form → Research papers
Check element technology → Hughes Ch. 5, Bathe Ch. 5
Verify stability → Dular et al. 2021 (h-φ specific) or Bathe/Brenner (general)
Implement → Cross-reference papers (equations) with books (algorithms)
Validate → Papers contain benchmark cases

Debugging Convergence Issues

Identify symptom → Oscillations (Dular et al. 2021, Hughes Ch. 4) or Divergence (Bathe Ch. 8, Messe et al. 2023)
Check formulation → Interface coupling (Messe et al. 2023, Arsenault et al. 2023, Alves et al. 2022b)
Verify implementation → Compare with cited equations
Adjust solver strategy → Messe et al. 2023 (nonlinear), Bathe Ch. 8 (linear)

Special Topics

h-φ Formulation

Consult in this order:

Messe et al. 2023 — BELFEM's implementation choices
Arsenault et al. 2023 — Magnetodynamic coupling (recommended)
Arsenault et al. 2021 — Performance comparison
Alves et al. 2022b — Interface conditions and cuts
Bathe Ch. 4-5 — Mixed formulation theory
Dular et al. 2021 — Stability analysis

Thin-Shell Implementation

Messe et al. 2023 — BELFEM's macro-element approach
Alves et al. 2022b — Full derivation
Alves et al. 2022a — 3D extension
Alves et al. 2024 — Alternative COMSOL approach
Hughes Ch. 5 — Isoparametric element foundations

Cuts and Cohomology

Alves et al. 2022b — Cohomology basis theory (thick cuts)
Alves et al. 2024 — Thin cut implementation
Schnaubelt et al. 2023 — Practical interpretation for coils
Alves et al. 2022a — Relative homology background

Tags for Searching

BELFEM FEM

#h-phi-formulation #thin-shell #cuts #cohomology #transport-current #hts-material #belfem-architecture #magnetodynamic #checkerboarding

BELFEM FVM

#mpfa #transmissibility #coercivity #physical-space #enhanced-bddf1 #k-orthogonal #mfmfe #anisotropic-diffusion

Computational Topology

#cohomology-computation #thick-cuts #coreduction #smith-normal-form #total-unimodularity #np-hardness #min-cost-flow #normal-surfaces

General FEM

#weak-form #isoparametric #locking #mixed-methods #inf-sup #newton-raphson #time-integration #eigenvalue #convergence