02-EFT.WP.Core.Equations v1.1 | Chapter 4 — Minimal Equations for the Tension Field

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Chapter 4 — Minimal Equations for the Tension Field

I. Aims and Scope

• Establish the minimal families of equations for T_fil(x,t) (steady and unsteady), with unified statements, boundary conditions, and weak forms, and couple them consistently to the constitutive mapping F_map(•) for n_eff (see Chapter 3).
• All formulas and symbols are English plain text and appear inline in backticks; any expression with division, integrals, or composite operators must use parentheses, and any path-related quantity must make gamma(ell) and the measure d ell explicit.

II. Field, Domain, and Boundary Declarations (P40 series)

• P40-1 (domain and time)
  Working domain Ω ⊂ R^d with d ∈ {1,2,3}, time interval t ∈ (0, t_end].
• P40-2 (unknown and sources)
  Unknown field T_fil(x,t); source S_src(x,t); the flux J_T(x,t) in this context is dedicated to the tension field.
• P40-3 (boundary split and normal)
  ∂Ω = ∂Ω_D ∪ ∂Ω_N, disjoint; the outward normal is nu (to avoid conflict with n(x,t)).

III. Strong Mother Form and Steady Subform (S40-1 / S40-2)

• S40-1 (unsteady conservative strong form)
  ∂_t T_fil(x,t) + div[ J_T(x,t) ] = S_src(x,t) on Ω × (0,t_end]
• S40-2 (steady strong form)
  div[ J_T(x) ] = S_src(x) on Ω

IV. Flux Closures and Anisotropy (S40-3)

• Isotropic diffusion closure
  J_T def= - kappa_T * grad[T_fil], with kappa_T ≥ 0.
• Anisotropic diffusion closure (orientation p(x,t)–based split)
  grad_parallel[T_fil] def= ( p • grad[T_fil] ) * p
  grad_perp[T_fil] def= grad[T_fil] - grad_parallel[T_fil]
  J_T def= - K_parallel * grad_parallel[T_fil] - K_perp * grad_perp[T_fil], with K_parallel, K_perp ≥ 0.
• Poisson-type subform (isotropic steady case)
  - kappa_T * lap[T_fil] = S_src on Ω

V. Boundary and Initial Condition Templates (S40-4)

• Dirichlet
  T_fil(x,t) = g_D(x,t) on ∂Ω_D × (0,t_end]
• Neumann
  J_T(x,t) • nu = g_N(x,t) on ∂Ω_N × (0,t_end]
• Initial condition
  T_fil(x,0) = T_init(x) on Ω

VI. Dimensional Closure and Non-Dimensionalization

• Dimensional closure
  For S40-*, require dim( ∂_t T_fil ) = dim( div[J_T] ) = dim( S_src ). Under isotropic closure, dim( kappa_T ) = dim( L^2 / t ) relative to the dimension of T_fil.
• Non-dimensional mapping (aligned with Chapter 6)
  bar_x := ( x / L0 ), bar_t := ( t / t0 ), bar_T := ( T_fil / T0 )
  Forms can be written as:
  ∂_{bar_t} bar_T + div[ - Pe_T^{-1} * grad[bar_T] ] = bar_S,
  where Pe_T^{-1} := ( kappa_T * t0 / L0^2 ), bar_S := ( S_src * t0 / T0 ).

VII. Consistent Coupling to n_eff (S40-5)

• Constitutive coupling
  n_eff(x,t) def= F_map( z(x,t); theta ) (see S30-1), where z may include T_fil and its gradients.
• Arrival-time consistency constraint (see Chapter 2)
  If the arrival-time convention is used, T_arr = ( ∫_{gamma} ( n_eff / c_ref ) d ell ), then any change in T_fil must propagate through delta[n_eff] to satisfy
  delta[T_arr] = ( ∫_{gamma} ( ( ∂ n_eff / ∂ T_fil ) * delta[T_fil] / c_ref ) d ell ).

VIII. Weak and Variational Forms (S40-6)

• Test space and measure
  Choose w(x) in a suitable space V; volume integrals use the measure d V.
• Unsteady weak form
  weak= ( ∫_{Ω} w * ∂_t T_fil d V ) + ( ∫_{Ω} grad[w] • J_T d V ) = ( ∫_{Ω} w * S_src d V ) + ( ∫_{∂Ω_N} w * g_N d A )
  where boundary terms arise from integration by parts; d A is the surface measure.
• Steady weak form
  weak= ( ∫_{Ω} grad[w] • J_T d V ) = ( ∫_{Ω} w * S_src d V ) + ( ∫_{∂Ω_N} w * g_N d A )

IX. Minimal Consistency and Stability Conditions (P41 series)

• P41-1 (elliptic consistency)
  In the isotropic steady case, kappa_T ≥ kappa_min > 0 (or an equivalent condition for the anisotropic tensor) to ensure coercivity of the weak form.
• P41-2 (integrability of flux and source)
  J_T ∈ L^2(Ω)^d, S_src ∈ L^2(Ω), and boundary data g_N ∈ L^2(∂Ω_N).
• P41-3 (causality)
  If J_T or n_eff depends on a history kernel, the temporal kernel must satisfy K_tau(s) = 0 for s < 0.

X. Canonical Equation Cards (S40 Summary)

• S40-1 — unsteady strong form: ∂_t T_fil + div[J_T] = S_src
• S40-2 — steady strong form: div[J_T] = S_src
• S40-3 — flux closures (isotropic/anisotropic):
  J_T = - kappa_T * grad[T_fil]
  J_T = - K_parallel * grad_parallel[T_fil] - K_perp * grad_perp[T_fil]
• S40-4 — boundary/initial templates: T_fil = g_D; J_T • nu = g_N; T_fil(•,0) = T_init
• S40-5 — coupling to arrival time: delta[T_arr] = ( ∫_{gamma} ( ( ∂ n_eff / ∂ T_fil ) * delta[T_fil] / c_ref ) d ell )
• S40-6 — weak forms (unsteady/steady): as written above.

XI. Numerical Realization and Assembly Alignment (I20-2 / I20-3)

• Semi-discrete mother form (FEM/FVM)
  M_T * d/dt T_vec + K_T * T_vec = f_T
  where M_T is assembled from ( ∫_{Ω} w_i * w_j d V ), K_T from ( ∫_{Ω} grad[w_i] • ( kappa_T * grad[w_j] ) d V ) or its anisotropic extension, and f_T includes volumetric sources and Neumann terms.
• Implementation binding
  assemble_operator yields K_T; bc_dirichlet / bc_neumann apply boundary data; solve_* and adjoint_sensitivity are reused (see I20-2 / I20-3).
• Arrival-time regression interface
  From the solution T_fil, evaluate n_eff, then compute T_arr via propagate_time(n_eff_path, ds, c_ref) for consistency regression.

XII. Lint Rules and Forbidden Patterns

• Do not use T_fil as a symbol for time or temperature; time is always t.
• Never use bare "c", "T", or "n"; write c_ref, T_fil, n or n_eff and clarify meanings inline.
• Do not omit parentheses or the explicit path in integrals (e.g., avoid ∫ n_eff d ell / c_ref); write ( ∫ ( n_eff / c_ref ) d ell ) and attach gamma(ell).
• Disallow unscaled mixed terms (e.g., kappa_T * grad[T_fil] + T_fil); bring terms to consistent dimensions or non-dimensionalize first.

XIII. Pre-Release Checklist

• Are Ω, ∂Ω_D, ∂Ω_N, and nu explicitly declared and not confused with n(x,t)?
• Do S40-1 / S40-2 with the chosen J_T closure pass check_dim_equation?
• Do boundary and initial conditions match the implementation (bc_dirichlet / bc_neumann)?
• If coupled to arrival time, is ∂ n_eff / ∂ T_fil (or an equivalent sensitivity) provided for adjoint_sensitivity?
• Are non-dimensional parameters (e.g., Pe_T^{-1}) fixed within a problem family and their provenance recorded?

XIV. Minimal Working Example (Strong → Weak → Assembly)

• Strong-form selection
  ∂_t T_fil - div( kappa_T * grad[T_fil] ) = S_src on Ω;
  T_fil = 0 on ∂Ω_D; ( - kappa_T * grad[T_fil] ) • nu = g_N on ∂Ω_N.
• Weak form
  weak= ( ∫_{Ω} w * ∂_t T_fil d V ) + ( ∫_{Ω} kappa_T * grad[w] • grad[T_fil] d V ) = ( ∫_{Ω} w * S_src d V ) + ( ∫_{∂Ω_N} w * g_N d A ).
• Assembly and solve sequence
  assemble_operator("mass", grid, ...) -> M_T; assemble_operator("stiffness", grid, {kappa_T}) -> K_T;
  form M_T * d/dt T_vec + K_T * T_vec = f_T, evolve with solve_linear or solve_nonlinear;
  evaluate n_eff from the computed T_fil → check T_arr via propagate_time for consistency.
• 。