39-EFT.WP.Plasma.Confinement v1.0 | Chapter 11 — Magnets, Coils & Geometry: Implementation Binding (I10-)

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Chapter 11 — Magnets, Coils & Geometry: Implementation Binding (I10-)

I. Chapter Objectives & Structure

1. Objective: Establish the implementation binding layer I10-PC-* from magnets/coils/geometry to physics models, unifying interfaces for geometry–grid–metric–boundary–coil currents so that B(x), psi(R,Z), q(psi), metric factors, and the S30-/S60-/S70- computation chain are consistent and reproducible, with manufacturing/assembly tolerances and QA gates specified.
2. Structure: Symbols & domain → I10-PC-1 geometry/coil→field mapping → I10-PC-2 boundaries/metrics & grid → I10-PC-3 coupling interfaces to equilibrium/stability/waves → I10-PC-4 tolerances & alignment → Implementation & records → Falsifiability → Cross-chapter closure.
3. Shared arrival-time dialect (if phase/arrival diagnostics are involved, explicit gamma(ell) & d ell; record delta_form):
   • Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • General: T_arr = ( ∫ ( n_eff / c_ref ) d ell )

II. Symbols & Domain

• Geometry & coils: vacuum vessel/divertor/first-wall surface S_wall, coil centerlines C_i(s), conductor cross-sections A_i, coil-current vector I_coil.
• Fields & functions: magnetic field B(x)=∑_i B_i(x; I_i, C_i, A_i), poloidal flux psi(R,Z), toroidal-field function F(psi)=R B_φ.
• Coordinates & metrics: Cartesian/cylindrical (x,y,z)/(R,Z,φ) and magnetic (ψ,θ,ζ); volume derivative V'(ψ), metric tensor g_{ab}.
• Constraints & gates: ∇·B=0, check_dim=pass, topology consistency (LCFS/X-points/separatrix), S30- force-balance residual thresholds.

I10-PC-1 | Geometry/Coils → Magnetic Field (Minimal Binding)

Vacuum & coil fields

• In vacuum, use Biot–Savart/Green superposition:
• B(x) = ∑_i 𝒢_i[x; C_i(s), A_i] · I_i + B_ext(x)

where 𝒢_i is analytic/BEM/FEM kernel; release requirement: record kernel version & solver accuracy.

Plasma-current coupling (free boundary)

• For free-boundary, psi = G*I_coil + psi_plasma, with psi_plasma from equilibrium coupling (S30-); for fixed-boundary, prescribe psi_bc at LCFS.

Numerical stability & constraints

• Enforce ∇·B=0 and toroidal-field continuity via projection/cleaning or mixed FEM; publish residuals & mesh-convergence rates as QA fields.

I10-PC-2 | Boundaries/Metrics & Grid

Boundary surfaces & CAD/mesh consistency

• Generate volume mesh 𝒯 and boundary mesh 𝒮 from CAD surfaces S_wall and coil geometry; record mesh order, quality metrics, and geometric error u_geom.

Magnetic coordinates & Boozer/Hamada metrics

• Build (ψ,θ,ζ) from B(x) / psi(R,Z); output B(ψ,θ,ζ), V'(ψ), g_{ab} to S60-/S70-; must preserve S30- LCFS/separatrix topology.

Diagnostic-mapping geometry

• Provide diagnostics lines-of-sight/antenna geometry (ECE/CTS/reflectometry/magnetics) on the same grid/coordinates for M10 inputs.

I10-PC-3 | Coupling Interfaces to Equilibrium/Stability/Waves (I20-/S40-/S50-)

• Equilibrium (I20-/S30-):
  build_boundary(geometry, coils) -> psi_bc;
  solve_GS(psi_bc, p(psi), F(psi)) -> {psi, q(psi), J_∥}.
• Stability (S40-): provide vacuum/coil coupling operators for boundary term [ξ]_boundary, ensuring external-region consistency in the energy principle.
• Waves & heating (S50-): deliver B(x), n_s(x), and metrics to dispersion/ray modules; arrival-time computation uses unified paths & metrics.

I10-PC-4 | Tolerances, Alignment & Engineering Summary (DFM/DFA)

• Manufacturing/assembly tolerances: coil centerline/cross-section tolerances ±σ_C/±σ_A, placement ±σ_pos, orientation ±σ_rot; inject tolerances into field solver & S30- regression to produce sensitivities ΔLCFS/Δq(ψ)/ΔX-point.
• Thermal/mechanical drifts: record current drift ΔI/I and thermal expansion ΔC_i(s) impacts on fields & equilibrium.
• Acceptance table: tabulate gates and remedies across geometry → fields → topology → equilibrium → diagnostics (linked to Chapter 16 sign-off).

VI. Implementation & Records (minimum execution dialect)

• Required fields:
  geometry:{S_wall, ports, divertor}, coils:{C_i(s), A_i, I_coil}, solver:{kernel, order, tol},
  B_field:{file, divB_residual}, boundary:{psi_bc, LCFS, Xpoints},
  metrics:{coords(ψ,θ,ζ), V'(ψ), g_ab}, diagnostics:{LOS, antennas},
  coupling:{G_operator, FEM/BEM flags}, tolerances:{σ_C,σ_A,σ_pos,σ_rot},
  qa_gates:{check_dim, divB_gate, topology_gate, force_balance}.
• Record template:
• i10_binding:
• geometry:
• wall_surface: "/cad/wall.step"
• ports: "/cad/ports.step"
• divertor: "/cad/divertor.step"
• coils:
• centerlines: "/cad/coils.centerlines.json"
• cross_sections: "/cad/coils.cs.json"
• I_coil_kA: [ ... ]
• solver:
• kernel: "BiotSavart-BEM"
• order: 2
• tol: 1.0e-6
• B_field:
• file: "/fields/B_vacuum.nc"
• divB_residual: 3.2e-5
• boundary:
• psi_bc: "/eq/psi_bc.nc"
• LCFS: "/eq/lcfs.curve"
• Xpoints:{coords:[[R1,Z1],[R2,Z2]]}
• metrics:
• coords: "/eq/coords_boozer.nc"
• Vprime: "/eq/Vprime.nc"
• g_ab: "/eq/metric.nc"
• diagnostics:
• LOS: "/diag/los.json"
• antennas: "/diag/antennas.json"
• coupling:
• G_operator: "/ops/green_kernel.h5"
• fem_bem: {fem:false, bem:true}
• tolerances:
• sigma_C_mm: 0.5
• sigma_A_mm: 0.2
• sigma_pos_mm: 0.8
• sigma_rot_mrad: 1.5
• qa_gates: {check_dim:"pass", divB_gate:"pass", topology_gate:"pass", force_balance:"pass"}

VII. Falsifiability (for I10-)

• J-I10-1 (∇·B mismatch): divB_residual exceeds gate or fails to converge with mesh refinement → reject field kernel/mesh/geometry.
• J-I10-2 (Topology inconsistency): LCFS/X-points/separatrix from psi disagree with imaging/magnetics → reject boundary/coil parametrization.
• J-I10-3 (Force-balance residual): S30- coupled solution exceeds thresholds → reject p(psi)/F(psi) or coupling operator.
• J-I10-4 (Tolerance sensitivity over limit): tolerance sweep yields Δq(ψ)/ΔLCFS beyond sign-off gates → reject manufacturing window or require geometric fixes.
• J-I10-5 (Diagnostic-mapping break): LOS/antenna geometry inconsistent with grid/metrics leading to spectral misplacement → reject coordinate mapping or calibration.

VIII. Cross-Chapter Links & Closure

• Dependencies: Chapter 2 (Terms & Symbols), Chapter 4 (S20- minimal equations), Chapter 5 (S30- equilibrium), Chapter 6 (S40- stability), Chapter 7 (S50- waves), Chapter 8 (S60- guiding centers), Chapter 9 (S70- transport).
• Downstream: Chapter 12 (Diagnostics & Metrology—geometry/LOS/antenna inputs), Chapter 14 (SimStack & benchmarks—field/boundary/metric baselines), Chapter 16 (Design Protocol & Checklist—tolerances & sign-off gates).