P6-b design 2: the Robin wall on the patch's Inner side — RobinWall { alpha, datum } with the wall velocity u_s + (t_f − datum)/α (explicit, damped by the wall's own viscous gain μ/(α d)) and the compliant-wall pressure term |S| p'/α implicit in the projection (the added-mass operator in the fluid's own response); Dirichlet bit for bit when off; MMS pin on the skewed annulus (orders 2.46/2.13 at α = 10 and 100 μ/h, the Dirichlet values; a 1e12 wall reproduces Dirichlet to 2e-6); OversetPisoSolver::patch_mut; harness knob RTX_FSI2O_ROBIN_ALPHA with the previous pass's tractions as the datum, printed in the header

Co-Authored-By: Claude Fable 5.1 <[email protected]>
Claude-Session: https://claude.ai/code/session_01YJPeT6WA2e7YvAnS875AHL

crates/specialized/rtx-fsi/tests/fsi2_harness/overset_march.rs

@@ -100,7 +100,7 @@ pub fn run_march_overset(case: BenchmarkCase, config: &OversetMarchConfig) -> Ov
     // Every setting the acceptance rule reads, printed once: P5-3 lost a
     // day to a floor of 2e-4 against the overnight marches' 1e-6.
     println!(
-        "  coupling: {} (reuse {}, ω0 {}, c1 {}), floor {:.1e}, rtol {:.1e}, stall accept {:.1e}, max subit {}, predictor {}, s = {}, patch offset {} h × {} rows, patch convection {:?}, bg convection {:?}, patch stretch {}, fillet {} m, tip corner {} m, fict mass α {}",
+        "  coupling: {} (reuse {}, ω0 {}, c1 {}), floor {:.1e}, rtol {:.1e}, stall accept {:.1e}, max subit {}, predictor {}, s = {}, patch offset {} h × {} rows, patch convection {:?}, bg convection {:?}, patch stretch {}, fillet {} m, tip corner {} m, fict mass α {}, robin α {}",
         cfg.coupler,
         cfg.reuse,
         cfg.initial_relaxation,
@@ -119,6 +119,7 @@ pub fn run_march_overset(case: BenchmarkCase, config: &OversetMarchConfig) -> Ov
         super::overset::fillet(),
         super::overset::tip_corner(),
         std::env::var("RTX_FSI2O_FICT_MASS").unwrap_or_else(|_| "0".into()),
+        std::env::var("RTX_FSI2O_ROBIN_ALPHA").unwrap_or_else(|_| "0".into()),
     );
 
     // Phase 1: rigid flag to t_release (`RTX_FSI2O_LOAD=dir` replaces the
@@ -295,6 +296,22 @@ pub fn run_march_overset(case: BenchmarkCase, config: &OversetMarchConfig) -> Ov
         }
         a
     };
+    // P6-b design 2 (`docs/overset_metal_campaign.md` §5.19): the Robin wall
+    // on the patch — `RTX_FSI2O_ROBIN_ALPHA` (Pa·s/m; 0 = the Dirichlet wall
+    // bit for bit; the plate's impedance is ρ_s h_s / Δt). Each pass gives
+    // the fluid the tractions of the previous pass as the datum, so at the
+    // coupled fixed point the wall is the Dirichlet one.
+    let robin_alpha: f64 = std::env::var("RTX_FSI2O_ROBIN_ALPHA")
+        .ok()
+        .and_then(|v| v.parse().ok())
+        .unwrap_or(0.0);
+    if robin_alpha > 0.0 {
+        println!(
+            "  Robin wall: α = {robin_alpha:.3e} Pa·s/m (ρ_s h_s / Δt = {:.3e})",
+            case.rho_s * 0.02 / dt
+        );
+    }
+    let robin_datum: RefCell<Vec<[f64; 2]>> = RefCell::new(Vec::new());
     // The load with the compensating term for a given previous-subiterate acceleration.
     let with_fict =
         |nodal: &[(NodeId, Vector3<f64>)], accel: &[f64]| -> Vec<(NodeId, Vector3<f64>)> {
@@ -383,6 +400,9 @@ pub fn run_march_overset(case: BenchmarkCase, config: &OversetMarchConfig) -> Ov
             let (predicted, _) = flag.borrow_mut().step(&flag_state).unwrap();
             extract(&predicted)
         };
+        if robin_alpha > 0.0 {
+            robin_datum.replace(fluid.borrow().inner_tractions());
+        }
         let saved = fluid.borrow().snapshot();
         type PassResult = (DynamicState, Vec<(NodeId, Vector3<f64>)>, f64, usize);
         let latest: RefCell<Option<PassResult>> = RefCell::new(None);
@@ -390,9 +410,15 @@ pub fn run_march_overset(case: BenchmarkCase, config: &OversetMarchConfig) -> Ov
             let fs = std::time::Instant::now();
             let mut fl = fluid.borrow_mut();
             fl.restore(&saved);
+            if robin_alpha > 0.0 {
+                fl.set_robin(robin_alpha, Some(robin_datum.borrow().clone()));
+            }
             fl.advance_subcycled(&d_n, d_candidate, cfg.subcycle, v_n.as_deref())
                 .expect("fluid pass");
             let (nodal, conservation, faces) = fl.sample_load(d_candidate);
+            if robin_alpha > 0.0 {
+                robin_datum.replace(fl.inner_tractions());
+            }
             t_fluid.set(t_fluid.get() + fs.elapsed().as_secs_f64());
             let ss = std::time::Instant::now();
             let mut flag_ref = flag.borrow_mut();