02-EFT.WP.Core.Equations v1.1 | Chapter 7 — Variational and Weak Forms

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Chapter 7 — Variational and Weak Forms

I. Aims and Scope

• Standardize the volume’s variational notation, weak-form templates, and function-space conventions for primary fields such as T_fil(x,t) and rho(x,t).
• Establish a generalized workflow from strong form → weak form; clarify the roles of ∂Ω_D / ∂Ω_N / ∂Ω_R in weak statements, consistent with Chapter 6.
• Provide registrable minimal equation labels S70-* and postulates P70-* to enable implementation binding I20-* and cross-volume citations.

II. Variational Notation and Function Spaces (S70-1)

• Trial space V def= { w | w = 0 on ∂Ω_D }; unknown space U def= { q | q = g_D^q on ∂Ω_D }.
• Volume inner product: inner_Ω[u,v] def= ( ∫_{Ω} u * v d V ).
• Boundary inner product: inner_∂Ω[u,v] def= ( ∫_{∂Ω} u * v d A ).
• Functional and variation: Lagr[q]; deltaLagr; q def= ( d/dε |_{ε=0} Lagr[q + ε w] ).

III. Generic Template from Strong to Weak (S70-2)

1. Strong mother form (placeholder)
   ∂_t q + div( J_q ) = S_q on Ω，其中 J_q 与 S_q 可依具体物理定义（见第4/5章）。
2. S70-2 (weak template)
   • For any w ∈ V,
   • weak= inner_Ω[w, ∂t q] + ( ∫{Ω} grad[w] • J_q d V ) - ( ∫_{∂Ω} w * ( J_q • nu ) d A ) = inner_Ω[w, S_q]。
   • On ∂Ω_N or ∂Ω_R, use the flux expressions from Chapter 6 in place of J_q • nu; on ∂Ω_D we have w = 0.

IV. Postulates and Symbol Isolation (P70-1 ~ P70-2)

• P70-1 (symbol isolation)
  T_fil exclusively denotes the tension field; T_trans the transmission coefficient; c_ref the reference propagation ceiling; n the number density; n_eff the effective refractive index. These symbols must not be interchanged or overloaded in the same context.
• P70-2 (weak-form notation)
  weak= denotes equality in the distributional sense; inner_Ω[•,•] and inner_∂Ω[•,•] denote the L2 inner products on Ω and ∂Ω, respectively; the definition of V must entail w = 0 on ∂Ω_D.

V. Weak Form for the Tension Field (building on Chapter 4, S70-3)

1. Strong example (see Chapter 4)
   ∂_t T_fil + div( J_T ) = S_T on Ω，其中 J_T def= - K_T * grad[T_fil] + U_T * T_fil。
2. S70-3 (weak form)
   • For w ∈ V:
   • weak= inner_Ω[w, ∂t T_fil] + ( ∫{Ω} grad[w] • ( - K_T * grad[T_fil] + U_T * T_fil ) d V ) = inner_Ω[w, S_T] + ( ∫{∂Ω_N} w * g_N^{T} d A ) + ( ∫{∂Ω_R} w * ( - h_T * ( T_fil - T_env ) ) d A )。
   • Here g_N^{T} def= ( J_T • nu ) on ∂Ω_N, with h_T ≥ 0 (see Chapter 6).

VI. Weak Form for Continuity and Transport (building on Chapter 5, S70-4)

1. Strong example (see Chapter 5)
   ∂_t rho + div( J_rho ) = S_rho，J_rho def= - D_rho * grad[rho] + V_rho * rho。
2. S70-4 (weak form)
   • For w ∈ V:
   • weak= inner_Ω[w, ∂t rho] + ( ∫{Ω} grad[w] • ( - D_rho * grad[rho] + V_rho * rho ) d V ) = inner_Ω[w, S_rho] + ( ∫{∂Ω_N} w * g_N^{rho} d A ) + ( ∫{∂Ω_R} w * ( - h_rho * ( rho - rho_env ) ) d A )。

VII. Obtaining Weak Forms from a Stationary Principle (S70-5)

1. Energy-type functional (self-adjoint case; coefficients may vary in time but remain fixed over a time slab):
   Lagr[q] def= ( 1/2 ) * ( ∫{Ω} ( grad[q] • K * grad[q] + c_q * q * q ) d V ) - ( ∫{Ω} f_q * q d V ) - ( ∫_{∂Ω_N} g_N^q * q d A )。
2. S70-5 (Euler–Lagrange weak form)
   • deltaLagr; q = 0 for all w ∈ V is equivalent to
   • weak= ( ∫{Ω} grad[w] • ( K * grad[q] ) d V ) + inner_Ω[w, c_q * q] = inner_Ω[w, f_q] + ( ∫{∂Ω_N} w * g_N^q d A )。
   • If a time term ∂_t q is needed, approximate on a time slab via inner_Ω[w, ( q^{n+1} - q^{n} ) / Δt ] and incorporate it in the discretized Lagr.

VIII. Path Constraint and Arrival-Time Coupling (S70-6)

1. Path-constraint functional
   • For a path gamma(ell) and a target arrival time T_arr^*, introduce a multiplier λ:
   • Lagr_path[q, λ] def= Lagr[q] + λ * ( ( ∫_{gamma(ell)} ( n_eff(q) / c_ref ) d ell ) - T_arr^* )。
2. S70-6 (constrained stationarity)
   • delta[Lagr_path; q, λ](w, μ) = 0 yields
   • deltaLagr; q + λ * ( ∫_{gamma} ( ( ∂ n_eff / ∂ q ) * w / c_ref ) d ell ) = 0；
   • ( ∫_{gamma} ( n_eff(q) / c_ref ) d ell ) - T_arr^* = 0。
   • Here ( ∂ n_eff / ∂ q ) follows from the differentiability of n_eff def= F_map(T_fil, ...) (see Chapter 3).

IX. Dimensional and Non-Dimensional Weak Forms (S70-7)

• S70-7 (dimensional closure)
  dim( inner_Ω[w, ∂t q] ) = dim( inner_Ω[w, S_q] )；dim( ∫{Ω} grad[w] • J_q d V ) = dim( inner_Ω[w, S_q] )。
• Non-dimensional mapping (consistent with Core.Terms Chapter 6)
  bar_x := ( x / L0 ), bar_t := ( t / t0 ), bar_q := ( q / q0 )，so every weak-form term transforms under the same scaling; boundary terms introduce Bi_q (Biot-type group; see Chapter 6).

X. Numerical Discretization and Assembly Binding (I20-2)

1. Mass and stiffness-type operators:
   M[w,q] def= inner_Ω[w, q]；A[w,q] def= ( ∫_{Ω} grad[w] • K * grad[q] d V )；C[w,q] def= inner_Ω[w, c_q * q]。
2. Implementation mapping:
   • assemble_operator("mass", grid, coeffs={"ρ":1}) -> LinOp for M；
   • assemble_operator("stiffness", grid, coeffs={"K":K}) -> LinOp for A；
   • bc_dirichlet / bc_neumann match Chapter 6 boundary terms; Robin enters via an equivalent flux.
3. Sequence suggestion:
   Discretize the path gamma(ell) to obtain ( n_eff_path, ds ) → propagate_time(n_eff_path, ds, c_ref), for constraints or a posteriori evaluation of T_arr.

XI. Lint Rules and Forbidden Items

• Do not omit the domain and boundaries: Ω and ∂Ω_* must be explicit.
• Do not drop parentheses: e.g., ( ∫ ( n_eff / c_ref ) d ell ) must include paired parentheses and declare gamma(ell) and d ell.
• Do not mix symbols: T_fil versus T_trans, n versus n_eff must not be interchanged (see P70-1).
• Do not leave the trial space unconstrained: V must include w = 0 on ∂Ω_D (see P70-2).

XII. Registrable Entries and Implementation Binding (I20-aligned)*

1. Typical registrations:
   • register_equation("S70-2", "weak= inner_Ω[w, ∂_t q] + ∫Ω grad[w] • J_q d V - ∫∂Ω w*(J_q•nu) d A = inner_Ω[w, S_q]", "weak", anchors=["§III"], depends=["S60-1"])
   • register_equation("S70-3", "weak form of T_fil", "weak", anchors=["§V"], depends=["S40-","S60-"])
   • register_equation("S70-6", "delta[Lagr_path]=0 with arrival-time constraint", "weak", anchors=["§VIII"], depends=["S20-","S30-"])
2. Dimensional check:
   check_dim_equation("inner_Ω[w, ∂_t q] = inner_Ω[w, S_q] - ∫_Ω grad[w] • J_q d V") -> "[w]*[q]/[t]"（与右端一致）。

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

1. Strong form: ∂_t q - div( K * grad[q] ) = f_q on Ω，q = g_D^q on ∂Ω_D，- K * grad[q] • nu = g_N^q on ∂Ω_N。
2. Weak form:
   weak= inner_Ω[w, ∂t q] + ( ∫{Ω} grad[w] • K * grad[q] d V ) = inner_Ω[w, f_q] + ( ∫_{∂Ω_N} w * g_N^q d A )。
3. Assembly recipe:
   • Mass operator M and stiffness operator A; load vector b built from f_q and boundary fluxes.
   • Apply bc_dirichlet on ∂Ω_D; time stepping with solve_linear / solve_nonlinear.
4. Arrival-time a posteriori (e.g., n_eff = F_map(T_fil, ...)):
   Update n_eff(x,t) → sample along gamma(ell) → compute T_arr = propagate_time(n_eff_path, ds, c_ref), compare with targets or feed into the S70-6 constraint loop.