02-EFT.WP.Core.Equations v1.1 | Appendix A — Equation List

Home ／ Docs-Technical WhitePaper ／ 02-EFT.WP.Core.Equations v1.1

Appendix A — Equation List

I. Fields & Ordering

II. Inventory (sorted by code)

1. S10-0 Mother Form & Writing Template
   • eqn: Form: A[q] = B; weak template weak= inner_V[w, A[q]] = inner_V[w, B]
   • anchors: Chapter 1 I–III
   • kind: definition
2. S20-1 Arrival Time (General Form)
   • eqn: T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell )
   • anchors: Chapter 2 I–III
   • kind: identity
3. S20-2 Segment Additivity of Arrival Time
   • eqn: If gamma = ⋃_k gamma_k, then T_arr(gamma) = ( ∑_k T_arr(gamma_k) )
   • anchors: Chapter 2 IV
   • kind: identity
4. S20-3 Constant-Factoring Convention
   • eqn: If c_ref is constant along gamma(ell), then T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell )
   • anchors: Chapter 2 II
   • kind: identity
5. S30-1 Constitutive Mapping for Effective Index
   • eqn: n_eff def= F_map( T_fil, TensionGrad, rho, ... )
   • anchors: Chapter 3 I–II
   • kind: definition
6. S30-2 First-Order Approximation Family (Linearized Example)
   • eqn: n_eff approx a0 + a1 * T_fil + a2 * div[TensionGrad] (coefficients constant or slowly varying within the domain)
   • anchors: Chapter 3 III–IV
   • kind: guideline
7. S30-3 Differentiability & Applicability of the Mapping
   • eqn: grad_{args} F_map exists and is bounded; failure regions are marked by the constraint set D_forbid
   • anchors: Chapter 3 V
   • kind: criterion
8. S40-1 Steady Minimal Equation for the Tension Field
   • eqn: - div( K_T * grad[T_fil] ) + alpha * T_fil = S_src on Ω
   • anchors: Chapter 4 II
   • kind: strong
9. S40-2 Unsteady Minimal Equation for the Tension Field
   • eqn: ∂_t T_fil - div( K_T * grad[T_fil] ) + alpha * T_fil = S_src on Ω × (0, T]
   • anchors: Chapter 4 II–III
   • kind: strong
10. S40-3 Weak Form for the Tension Field
    • eqn: weak= inner_Ω[w, ∂_t T_fil] + ( ∫_Ω grad[w] • ( K_T * grad[T_fil] ) d V ) + inner_Ω[w, alpha * T_fil] = inner_Ω[w, S_src] + B[w]
    • anchors: Chapter 4 IV
    • kind: weak
11. S50-1 Continuity Equation
    • eqn: ∂_t rho + div( J ) = S_src
    • anchors: Chapter 5 I–II
    • kind: strong
12. S50-2 Flux Definition (Advection–Diffusion Paradigm)
    • eqn: J def= - D * grad[rho] + U * rho
    • anchors: Chapter 5 II–III
    • kind: definition
13. S60-1 Boundary Condition Template
    • eqn: T_fil|_{∂Ω_D} = g_D; nu • ( K_T * grad[T_fil] )|_{∂Ω_N} = g_N
    • anchors: Chapter 6 II–III
    • kind: procedure
14. S60-2 Initial Condition Template
    • eqn: T_fil(x, 0) = T_init(x); or, in general, q(x, 0) = q_init(x)
    • anchors: Chapter 6 II
    • kind: procedure
15. S70-1 Variational Functional
    • eqn: Lagr[q] = ( ∫_Ω ( 1/2 * grad[q] • K_T • grad[q] + 1/2 * alpha * q^2 - S_src * q ) d V ) + BC_terms
    • anchors: Chapter 7 II
    • kind: definition
16. S70-2 Euler–Lagrange / Weak-Form Correspondence
    • eqn: delta[Lagr] = 0 weak= inner_Ω[w, ∂_t q] + ( ∫_Ω grad[w] • ( K_T * grad[q] ) d V ) + inner_Ω[w, alpha * q] - inner_Ω[w, S_src] = 0
    • anchors: Chapter 7 III–IV
    • kind: weak
17. S80-1 Coarse-Graining Closure (Subgrid Flux)
    • eqn: bar_q def= avg_V[q; V] or avg_t[q; Δt]; J_sgs^q def= - K_sgs^q * grad[ bar_q ]
    • anchors: Chapter 8 II–III
    • kind: guideline
18. S90-1 Discrete Conservation & Discrete Inner Products
    • eqn: inner_h[u,v] def= ( ∑_{i} u_i * v_i * |V_i| ); ∑_{i} ( div_h F )_i * |V_i| = ∑_{faces} ( F_f • nu_f ) * A_f
    • anchors: Chapter 9 II
    • kind: metric
19. S90-2 Discrete Arrival Time
    • eqn: T_arr_h = ( ∑_i ( n_eff_i / c_ref ) * ds_i ); segment additivity T_arr_h = ( ∑_k T_arr_h^{(k)} )
    • anchors: Chapter 9 III
    • kind: identity
20. S90-3 Assembly Consistency (Strong → Weak → Algebraic)
    • eqn: M * dq/dt + K(q) = f + b; Newton linearization K(q) ≈ K(q^n) + J(q^n) * ( q^{n+1} - q^n )
    • anchors: Chapter 9 IV
    • kind: procedure
21. S90-4 Time-Stepping Stability Bounds
    • eqn: Convective bound Δt ≤ CFL * min_i( Δx_i / |u|_{max,i} ); diffusive bound Δt ≤ σ * min_i( Δx_i^2 / ( 2 * d * κ_{max,i} ) )
    • anchors: Chapter 9 V
    • kind: guideline
22. S90-5 Boundary Priority Rule
    • eqn: When value and flux are both given at the same location, prefer Dirichlet; Neumann serves only as a diagnostic term
    • anchors: Chapter 9 VI
    • kind: criterion
23. S90-6 Error and Conservation Metrics
    • eqn: E_L2 def= sqrt( ∑_i |x_i - y_i|^2 * |V_i| ); E_Linf def= max_i |x_i - y_i|; E_T def= | T_arr(x) - T_arr(y) |
    • anchors: Chapter 9 VII
    • kind: metric
24. S90-7 Order-of-Accuracy Estimation
    • eqn: p_est def= log2( E_k / E_{k+1} ), where E_k = L2_h[ q_{h_k} - q_{h_{k+1}} ]
    • anchors: Chapter 9 VIII
    • kind: metric
25. S90-8 Numerical Realization of Statistics
    • eqn: avg_t[q; Δt] ≈ ( 1 / N_t ) * ( ∑_{k=0}^{N_t-1} q^{n-k} ); avg_V[q; V] ≈ ( 1 / |V| ) * ( ∑_{c ∈ V} q_c * |V_c| )
    • anchors: Chapter 9 IX
    • kind: procedure
26. S90-9 Regression Pass Criterion
    • eqn: pass if ( L2 ≤ τ_L2 and L_inf ≤ τ_Linf ) and | T_arr - T_arr_ref | ≤ τ_T
    • anchors: Chapter 9 XI
    • kind: criterion

III. Quick Jump Index (by topic)

• Arrival time & paths: S20-1, S20-2, S20-3, S90-2
• Constitutive & mappings: S30-1, S30-2, S30-3
• Tension-field control equations: S40-1, S40-2, S40-3
• Continuity & flux: S50-1, S50-2
• Boundary & initial data: S60-1, S60-2
• Variational & weak forms: S70-1, S70-2
• Statistics & coarse-graining: S80-1, S90-8
• Numerics & consistency: S90-1, S90-3, S90-4, S90-5, S90-6, S90-7, S90-9