10-EFT.WP.Core.Tension v1.0 | Appendix C Boundary and Connection Library (Tension Volume)

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Appendix C Boundary and Connection Library (Tension Volume)

I. Scope and Conventions

1. Objective: Provide a standard library of boundary conditions and connection (node/interface) operators for tension systems, spanning line elements, membrane boundaries, and multi-port networks, unified with S72-*, Mx-7*, and I70-*.
2. Variable conventions
   • Line-element primaries: u(ell,t), v = ∂_t u, T_fil(ell,t), q(ell,t).
   • Impedance: Z = rho_l * c = sqrt( T0 * rho_l ) = T0 / c (uniform segment with constant tension T0).
   • Interface wave amplitudes use force/pressure-type magnitudes; reflections/transmissions follow S72-9, S72-10, S72-11.
3. Fundamental conservation
   • Node equilibrium: S72-7 : ∑ T_i * t_hat_i + f_ext = 0.
   • Power consistency: lossless interfaces satisfy R_ref + T_trans = 1.

II. End Boundary Conditions (Line Elements)

1. Dirichlet (displacement constraint)
   • Form: u(ell_end,t) = u_bar(t).
   • Equivalent impedance: Z_bc -> ∞ (rigid clamp with u_bar = 0).
   • Reflection (treating the second medium as Z2 = Z_bc): r_amp = ( Z_bc - Z ) / ( Z_bc + Z ) -> +1.
2. Neumann (traction/slope constraint)
   • Form: T_fil * ∂_ell u |_{ell_end} = t_bar(t); free end uses t_bar = 0.
   • Free-end equivalent: Z_bc = 0, so r_amp = ( 0 - Z ) / ( 0 + Z ) = -1.
3. Robin (mixed boundary)
   • Form: alpha * u + beta * ( T_fil * ∂_ell u ) = g(t).
   • Equivalent impedance (frequency domain with i*omega): Z_bc(omega) = ( beta^{-1} ) * alpha / ( i * omega ) (for g = 0).

III. End Components and Equivalent Impedance

1. Linear spring support
   • Condition: T_fil * ∂_ell u = K_s * u; thus Z_bc(omega) = K_s / ( i * omega ).
   • Low-frequency absorber match: if K_s << Z * | i * omega | then |r_amp| -> 1 (soft support).
2. Viscous damper
   • Condition: T_fil * ∂_ell u = C_s * ∂_t u; hence Z_bc = C_s.
   • Matched termination: C_s = Z ⇒ r_amp = 0.
3. Tip mass
   • Condition: T_fil * ∂_ell u = M_t * ∂_tt u; Z_bc(omega) = i * omega * M_t.
   • Behavior: low frequency ≈ free end, high frequency ≈ clamped, with continuous phase in r_amp.
4. Kelvin–Voigt terminal (spring–damper–mass in parallel)
   Z_bc(omega) = K_s / ( i * omega ) + C_s + i * omega * M_t.
5. End excitation
   Equivalent end force: t_bar(t), superposed on Z_bc model: T_fil * ∂_ell u = t_bar + Z_bc * v.

IV. Line-Element Guidance / Frictional Connections

1. Ideal ring guide (frictionless)
   • Constraint: |T_fil| is continuous through the guide, the normal reaction does no work, and t_hat updates with geometry.
   • Energy: no added dissipation, only direction changes.
2. Capstan (frictional pulley)
   • Tension ratio: T_out = T_in * exp( ± mu_k * theta_wrap ) (wrap angle theta_wrap in radians).
   • Sign: take the positive sign along the direction of frictional hindrance; close the reaction in S72-7.
3. Line–line sliding knot (lock/slip hybrid)
   • Threshold: lock if |ΔT| ≤ T_lock; when |ΔT| > T_lock, use the Capstan model.
   • Equivalent: piecewise Z_joint(omega); locked segment acts as a rigid constraint, slip segment follows the above dissipative law.

V. Membrane/Interface Boundary Operators

• Clamped boundary (zero displacement/normal displacement)
  u|_{∂A} = 0 or u • n_hat = 0; surface energy closes via S72-3 together with the boundary line integral ∮ sigma_s d s.
• Free boundary (no external traction)
  t = sigma • n_hat = 0; equivalent to line-tension balance along the boundary.
• Contact angle condition (Young equation)
  S72-4 : sigma_sv = sigma_sl + sigma_lv * cos(theta_c); solve theta_c at the boundary to determine geometry.
• Boundary line damping/mass distribution
  Linear model: Z_line(omega) = K_line / ( i * omega ) + C_line + i * omega * M_line, distributed along ∂A.

VI. Two-Port Connections and Interface Transmission

1. Series junction of two uniform segments (Z1 to Z2)
   • Reflection: S72-9 : r_amp = ( Z2 - Z1 ) / ( Z2 + Z1 ).
   • Transmission: S72-10 : t_amp = 2 * Z2 / ( Z1 + Z2 ).
   • Lossless conservation: S72-11 : R_ref = |r_amp|^2 , T_trans = 1 - R_ref.
2. Insert a series element
   Z2' = Z2 + Z_series; substitute Z2' for Z2 in the above.
3. Insert a shunt dissipative branch (to ground)
   Z_eq = ( (1/Z2) + Y_shunt )^{-1}; then use Z_eq in place of Z2.

VII. N-Port Star Junction (Velocity-Continuity Model)

1. Assumptions
   • Ports i = 1..N, segment characteristic impedances Z_i, incident amplitudes a_i, outgoing amplitudes b_i (force-type).
   • Node velocity continuity: v_i = ( a_i - b_i ) / Z_i = v_node. Node force balance: ∑ ( a_i + b_i ) = 0.
2. Solution
   • Aggregate impedance: Z_sum = ∑_k Z_k.
   • Scattering matrix: b_i = a_i - ( 2 * Z_i / Z_sum ) * ( ∑_j a_j ).
   • Single incidence at port m: r_mm = ( Z_sum - 2 * Z_m ) / Z_sum, t_im = - 2 * Z_i / Z_sum (i ≠ m).
   • When directions are normalized uniformly, take |t_im| = 2 * Z_i / Z_sum; power weighting preserves losslessness.
3. Special case (3-port, equal impedance)
   Z_1 = Z_2 = Z_3 = Z0 ⇒ r = -1/3, |t| = 2/3, R_ref = 1/9, T_trans = 8/9.

VIII. Loop and Branch Library (Structural Patterns)

• Series ladder
  Segment k has (Z_k, L_k); compose via transfer matrices or cumulative S72-9..11.
• Closed loop
  Phase closure: ∑_k k_k * L_k = 2 * π * m (standing-wave index m ∈ N+), with tension consistency T_fil = rho_l * c^2.
• Branching (Y/T)
  Use Section VII with N = 3. If terminal elements exist, first update port impedances Z_i -> Z_i + Z_series or apply shunt corrections.

IX. Power and Energy Checks (Interface Level)

• Instantaneous power flow along a line element
  P(ell,t) = ( T_fil * ∂_ell u ) * ( ∂_t u ) (positive along +t_hat).
• Average power (harmonic, displacement amplitude U, velocity amplitude V)
  One-way traveling wave: P_avg = (1/2) * Z * |V|^2 = (1/2) * |F|^2 / Z, with F = T_fil * ∂_ell u.
• Node conservation (passive)
  ∑ P_in = ∑ P_out; for lossless two-ports, R_ref + T_trans = 1.

X. JSON Snippets (Aligned with Appendix B)

• End impedance and boundaries

{

"bc": [

{ "type": "dirichlet", "location": "path_end",

"data": { "node_id": "n0", "u_bar": [0,0,0] } },

{ "type": "robin", "location": "path_end",

"data": { "node_id": "n1",

"Z_bc": { "model": "KCM",

"K_s": 1.0e4, "C_s": 0.12, "M_t": 0.02 } } }

]

}

• Junction and scattering

{

"junctions": [

{

"node_id": "J1",

"model": "star_v_cont",

"ports": [

{ "edge_id": "E1", "Z": 0.75 },

{ "edge_id": "E2", "Z": 0.50 },

{ "edge_id": "E3", "Z": 0.50 }

],

"scattering": { "type": "auto" }

}

]

}

• Frictional guide (Capstan)

{

"guides": [

{ "id": "G1", "type": "capstan",

"theta_wrap_rad": 2.356, "mu_k": 0.20,

"in_edge": "E4", "out_edge": "E5" }

]

}

XI. Computational Workflow Summary (Interfaces for Mx-74)

• Assemble the network: load paths gamma(ell) and properties E, A, rho_m; construct graph(nodes, edges).
• Compute each segment’s Z = rho_l * c or frequency-dependent Z(omega).
• Fold terminal elements into Z_bc(omega) and update port-equivalent impedances.
• Apply the Section VII scattering operator at each node to obtain r_amp, t_amp.
• Check power conservation and R_ref + T_trans = 1; if mismatched, back-trace terminal/node parameters.
• Output network-level responses and boundary reactions; record audit quantities ΔE_rel and delta_form (if T_arr is involved).

XII. Common Terminals and Interface Quick Reference

• Rigid end (clamped): Z_bc -> ∞, r_amp = +1.
• Free end (zero traction): Z_bc = 0, r_amp = -1.
• Damped match: Z_bc = Z, r_amp = 0.
• Spring end: Z_bc = K_s / ( i * omega ), |r_amp| = | ( K_s/(i*omega) - Z ) / ( K_s/(i*omega) + Z ) |.
• Mass end: Z_bc = i * omega * M_t, |r_amp| = | ( i*omega*M_t - Z ) / ( i*omega*M_t + Z ) |.
• Three-port equal-Z junction: r = -1/3, |t| = 2/3, R_ref = 1/9.

XIII. Minimal Unit Test Set

• BC-01 Fixed–fixed string (length L, wave speed c)
  Fundamental: f_1 = c / ( 2 * L ); verify end reflections r_amp ≈ +1.
• BC-02 Free–free string
  Fundamental: f_1 = c / ( 2 * L ); ends behave as antinodes in displacement, with r_amp ≈ -1.
• JC-03 Three-port equal-impedance star
  R_ref = 1/9, T_trans = 8/9; passes power conservation.
• TM-04 Matched damping termination
  C_s = Z; reflected peak vanishes (R_ref ≈ 0).

XIV. Anchors to Main-Text Equations

• Node equilibrium and energy: S72-7, S72-8.
• Two-port reflection/transmission: S72-9, S72-10, S72-11.
• Dynamics and wave speed: S72-5, S72-6.
• Stability and time-stepping: S72-12 (to be checked as part of explicit stepping with dissipative boundaries).