02-EFT.WP.Core.Equations v1.1 | Chapter 6 — Boundary and Initial Conditions

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Chapter 6 — Boundary and Initial Conditions

I. Aims and Scope

• Provide unified templates for boundary and initial conditions of q(x,t) over a domain Ω (covering rho(x,t), E(x,t), T_fil(x,t), etc.), including strong forms, weak forms, compatibility notes, and implementation binding.
• Use the boundary partition ∂Ω = ∂Ω_D ∪ ∂Ω_N ∪ ∂Ω_R; the outward unit normal is nu. Fluxes are written uniformly as J_q(x,t) and sources as S_q(x,t) (see Chapter 5).

II. Boundary Decomposition and Notation (S60-1)

III. Dirichlet Conditions (S60-2)

1. S60-2 (strong form)
   q(x,t) = g_D^q(x,t) on ∂Ω_D × (0,t_end]
2. Applicability and examples
   • Fixed field value: T_fil = T_ref; density clamp: rho = rho_ref.
   • Units and dimensions must match the corresponding q (see Core.Terms §6).

IV. Neumann Conditions (S60-3)

1. S60-3 (strong form)
   J_q(x,t) • nu(x) = g_N^q(x,t) on ∂Ω_N × (0,t_end]
2. Notes
   • For diffusive flux J_q = - D_q * grad[q], one has - D_q * grad[q] • nu = g_N^q.
   • Pure Neumann problems require an integral constraint to ensure solvability (see P60-2).

V. Robin (Convective-Type) Conditions (S60-4)

• S60-4 (strong form)
  J_q • nu + h_q * ( q - q_env ) = 0 on ∂Ω_R × (0,t_end], where h_q ≥ 0.
• Implementation (aligned with I20-2)
  This can be rewritten as an equivalent Neumann form: J_q • nu = - h_q * ( q - q_env ) and incorporated as a boundary linear operator during assembly.

VI. Interface and Internal Boundary Conditions (S60-5)

1. S60-5 (jump conditions across media interfaces)
   Let an interface Γ_int split Ω into Ω^- and Ω^+, with the normal oriented from Ω^- to Ω^+:
   • Field continuity (optional): [[ q ]]_Γ def= q^+ - q^- = 0.
   • Flux conservation (mandatory): [[ J_q • nu ]]_Γ = 0.
2. Anisotropic case
   If J_q = - K(x) * grad[q], continuity must be enforced on nu • K * grad[q].

VII. Initial Conditions and Consistency (S60-6)

• S60-6 (strong initial condition)
  q(x,0) = q_init(x) on Ω
• Compatibility with boundaries
  On ∂Ω_D, q_init should satisfy q_init(x) = g_D^q(x,0); on ∂Ω_N, satisfy J_q(x,0) • nu = g_N^q(x,0) (if prescribed).

VIII. Weak Embedding and Natural Boundaries (S60-7)

IX. Compatibility and Well-Posedness (P60-1 ~ P60-3)

• P60-1 (symbol and dimension consistency)
  dim( g_N^q ) = dim( J_q • nu ), and dim( h_q * ( q - q_env ) ) = dim( J_q • nu ).
• P60-2 (pure Neumann compatibility)
  If ∂Ω = ∂Ω_N and there is no source term, then the constraint
  ( ∫_{∂Ω} g_N^q d A ) = ( ∫_{Ω} S_q d V )
  must hold; otherwise the solution does not exist or is not unique.
• P60-3 (corner and boundary-overlap priority)
  On the measure-zero set ∂Ω_D ∩ ∂Ω_N, the priority is Dirichlet > Robin > Neumann. Implementations should enforce this via projection or penalization.

X. Specialization to the Tension Field (S60-8)

1. S60-8 (boundary families for T_fil(x,t))
   • Dirichlet: T_fil = T_ref (fixed tension potential).
   • Neumann: grad[T_fil] • nu = g_N^{T} (prescribe the normal component of the tension gradient).
   • Robin: - k_Tn * grad[T_fil] • nu + h_T * ( T_fil - T_env ) = 0.
2. Mapping consistency
   If n_eff def= F_map(T_fil, ...) (see Chapter 3), boundary values of T_fil and its normal gradient influence propagation and T_arr through F_map.

XI. Path Endpoints and Arrival-Time Hook (S60-9)

1. Endpoint conditions
   For a path gamma(ell) with x_src = gamma(0) and x_rec = gamma(L_gamma):
   • Source-end prescription: q( gamma(0), t ) = q_src(t) or J_q( gamma(0), t ) • tau = g_src(t), where tau is the tangent.
   • Receiver-end prescription: q( gamma(L_gamma), t ) = q_rec(t) or J_q( gamma(L_gamma), t ) • tau = g_rec(t).
2. Consistency with T_arr
   When T_arr = ( ∫_{gamma} ( n_eff / c_ref ) d ell ), if endpoint conditions alter n_eff, recompute ( n_eff_path, ds ) and call propagate_time.

XII. Non-Dimensional Boundary and Initial Conditions (S60-10)

1. Scale mapping (aligned with Core.Terms §6)
   bar_x := ( x / L0 ), bar_t := ( t / t0 ), bar_q := ( q / q0 ).
2. Non-dimensional BC/IC
   • Dirichlet: bar_q = ( g_D^q / q0 ).
   • Neumann: ( J_q • nu ) / J0 = ( g_N^q / J0 ), with J0 def= q0 * L0 / t0 (for diffusion-dominated cases, one may take J0 := D_char * q0 / L0).
   • Robin: ( J_q • nu ) / J0 + Bi_q * ( bar_q - bar_q_env ) = 0, where Bi_q def= ( h_q * L0 / J0 ).
   • Initial value: bar_q(x,0) = ( q_init(x) / q0 ).

XIII. Lint Rules and Forbidden Patterns

• Do not omit boundary sets or normals (e.g., avoid J_q = g_N); write J_q • nu = g_N^q on ∂Ω_N.
• Do not omit parentheses or the path declaration in line integrals (e.g., avoid ∫ n_eff d ell / c_ref); write ( ∫ ( n_eff / c_ref ) d ell ) and annotate gamma(ell) with d ell.
• Do not mix T_fil with T_trans; n must not replace n_eff.

XIV. Implementation Binding and Assembly (I20-2 Alignment)

1. Boundary interfaces
   • Dirichlet: bc_dirichlet(mask:any, value:any) -> BC.
   • Neumann: bc_neumann(mask:any, flux:any) -> BC.
   • Robin via equivalent Neumann: set flux := - h_q * ( q - q_env ) on the mask, or add the h_q term in a discrete boundary operator.
2. Registration examples
   register_equation("S60-2", "q = g_D^q on ∂Ω_D", "bc", anchors=["§VI"], depends=["S50-1"])
   register_equation("S60-3", "J_q • nu = g_N^q on ∂Ω_N", "bc", anchors=["§IV"], depends=["S50-1"])
3. Dimensional check
   check_dim_equation("J_q • nu = g_N^q") -> "[J_q]", which must match g_N^q (see P60-1).

XV. Minimal Working Example (BC/IC Package for Density Transport)

1. Strong equation (from Chapter 5)
   ∂_t rho - div( D_rho * grad[rho] ) = S_rho on Ω
2. Boundary and initial data
   • rho = rho_ref on ∂Ω_D
   • - D_rho * grad[rho] • nu = g_N^{rho} on ∂Ω_N
   • - D_rho * grad[rho] • nu + h_rho * ( rho - rho_env ) = 0 on ∂Ω_R
   • rho(x,0) = rho_init(x) on Ω
3. Weak-form assembly
   weak= ( ∫_{Ω} w * ∂_t rho d V ) + ( ∫_{Ω} D_rho * grad[w] • grad[rho] d V ) = ( ∫_{Ω} w * S_rho d V ) + ( ∫_{∂Ω_N} w * g_N^{rho} d A ) + ( ∫_{∂Ω_R} w * ( - h_rho * ( rho - rho_env ) ) d A ).
4. Implementation sequence
   • Apply bc_dirichlet on ∂Ω_D; apply bc_neumann on ∂Ω_N.
   • Convert Robin to equivalent flux and include via bc_neumann or assemble a boundary operator.
   • Advance with solve_*. If arrival time is required, evaluate updated n_eff(x,t), generate ( n_eff_path, ds ) along gamma(ell), and call propagate_time(n_eff_path, ds, c_ref).