# Moving circles on the uniform dense grid — S0 DRAFT contract

Status: **DRAFT for review, not frozen.** No implementation exists. Base: main
`b2034f87e03656c5f67ab88562a70d6c338b9918`. Scope approved by Frank (via TRON,
thread C0C78UFKGRM/1793335557.460509): uniform-grid prescribed translation and
single-cylinder free transverse VIV. Out of scope: refinement/blocked path,
pinball assembly, EsoPull, shifted storage, D3Q19, DynamisLab, manuscript.

Items marked **OPEN** are undecided; items marked **CANDIDATE** are proposals.

## 1. Working standards

- Lean, readable kernels in the existing idiom (`kernels/step/*.cu`, shared
  `apply_bouzidi_link`); Python manages state and validation. No framework,
  no compatibility layer, no new dependency.
- The existing stationary/rotating path (`CurvedBoundaryKernel`, host link
  packing, `sync_bodies`) stays byte-identical. Moving circles are a separate,
  explicitly enabled path with its own numerical contract (this file).
- One authority per state (section 4). No per-step full-domain download,
  rebuild or recompile; local swept-band device updates are in scope.
- Small slices (S1 host spec, S2 kernels without motion, S3 motion + refill,
  S4 oscillator); each reviewed SHA is frozen before targeted runs.
- Tolerances, benchmark parameters and the force composition used in the
  structural RHS are declared before the runs that judge them. Fresh vs carried
  evidence and existing baseline failures are recorded; smoke, convergence and
  physical validation are reported separately. No undeclared force filtering.
- External code (waLBerla GPLv3, FluidX3D source-available, VIVSIM) is a
  concept reference only; nothing is copied.

## 2. Current update order (main b2034f87, dense double buffer)

`Simulation.run(n)`: upload host action (if `upload_act`) → zero obs (if
`zero_obs`) → `DoubleBufferStepper.step(n)` → download obs → synchronize.

Per step (`lbm/stepper.py` `step`):

1. `CurvedBoundaryKernel` over the packed cut links: reads post-collision
   `G^{n-1}` at `k_f` (and `k_f - c` for q < 1/2), adds the moving-wall term
   (`Uw = -omega*ry, Vw = omega*rx` + optional per-link `surface_velocity`),
   writes the reflected value into slot `opp` of the solid node `k_s`, and
   accumulates `c(f_toward + f_reflected)` and torque into `obs`.
2. `OneStep` pull/collide `ddf → temp` (collides FLUID cells; obstacle
   interiors flagged `SOLID|OBSTACLE|BC_CURVED`), then buffer swap.
3. `ForceRegionKernel`, `SensorKernel`.

Raw `obs` force is the force **on the body** (link derivation in section 6;
DFG docs report `Cd > 0` from it). Translation `vx, vy` in the action slots is
currently stored but not read by any curved kernel.

## 3. Proposed update order (moving circles)

Per step n, before the existing step 1:

- **M1 `MovingCircleCells`** (one thread per swept-band cell): classify with the
  analytic predicates `in_old = inside(x, c^{n-1})`, `in_new = inside(x, c^n)`;
  write `flag = in_new ? SOLID|OBSTACLE|BC_CURVED : FLUID`; covered cells
  (`!in_old && in_new`) accumulate `node_cover`; fresh cells
  (`in_old && !in_new`) are refilled (section 5) and accumulate `node_uncover`.
- **M2 `MovingCircleLinks`** (one thread per band cell × direction): compute
  q and hit point at `c^n` (FP64, same formula as `find_circle_intersection`,
  same clamp [0.01, 1] and fallback classes as `CircleGeometry.build_cut_links`),
  skip non-cut slots, call `apply_bouzidi_link` with
  `u_w = (vx^n, vy^n) + omega × (hit - c^n)`, accumulate the link terms
  (section 6).

Then the unchanged steps 1–3. M2 must follow M1 because a q < 1/2 link may
read `k_f - c` at a fresh cell.

Exclusivity: moving bodies are never packed into the `CurvedBoundaryKernel`
link arrays (`n_curved` covers stationary/rotating bodies only), so M2 is the
sole writer of their solid slots and the sole source of their link force; no
double write and no double `obs`. The stationary path's packing, launch and
arithmetic are unchanged.

Completeness: after M1, every (fluid cell, direction) pair in the swept band
whose neighbour is `in_new` is visited by M2. This holds because the band is
the circle AABB at the declared extreme offsets plus 2 cells, and the host
rejects per-step moves of 1 cell or more and bands touching the domain edge,
other bodies, sensors or force regions. Each M2 write lands at
`index_f(k_s, opp)`, the exact slot the adjacent fluid cell pulls, so every
fluid pull from a solid slot of a moving body is BFL-written.

## 4. State, masks and authority

- Host `ObjectState` is the single authority for `offset = (dx, dy)`,
  `velocity = (vx, vy)` and `omega`. Velocity is supplied, never differenced
  from offsets.
- Per body a small device **motion record** `{c^{n-1}, c^n, u^n}` is uploaded
  with the action buffer; action-slot meaning is not overloaded. (CANDIDATE
  layout; OPEN.)
- Membership is never read from a flag that M1 is overwriting. The host
  verifies at `add_body` that every swept-band cell has base flag FLUID, so the
  band's base state is known without reading `flag_gpu`.
- `flag_gpu` is the device authority for band cells while a moving body
  exists; the host flag mirror is stale for those cells and is not used.

## 5. Refill of fresh cells (CANDIDATE)

- Stored time level: everything in `ddf` at M1 is post-collision `G^{n-1}`; the
  refill writes all 9 slots at that same level.
- `G = f_eq(rho_bar, u_w(x)) + G_neq(x_d)` where `rho_bar` = mean density of
  valid neighbours and `x_d` = the valid neighbour best aligned with the outward
  normal `(x - c^n)/|x - c^n|`; `G_neq(x_d) = G(x_d) - f_eq(rho(x_d), u(x_d))`.
- Valid donor: inside the domain, base FLUID, `!in_old && !in_new`.
- No valid donor for the NEQ part → equilibrium only (counted). No valid
  neighbour at all → cannot happen when the per-step move is < 1 cell
  (validated on host); counted anyway.
- Fresh cells only write, old-fluid cells only read: no race.

## 6. Discrete momentum ledger (to review; not a selection)

Notation per step n: `F^n` = fluid set after M1; links L = (k_f, dir) at
`c^n`; `f_t = G_dir(k_f)`; `f_r` = value M2 writes into `(k_s, opp)`.
Fluid momentum before streaming:

    P_pre = P(G^{n-1} on F^{n-1}) - sum_cover sum_q c_q G_q + sum_fresh sum_q c_q G_q^refill

During the pull, every population entering a fluid cell comes from a fluid
cell except the BFL-written slots; `f_t` leaves into solid and is never pulled
by a fluid cell. Collision conserves momentum for SRT/MRT without forcing.
Hence, in exact arithmetic,

    P_fluid^n - P_fluid^{n-1} = - sum_L c (f_t + f_r) - I_cover + I_uncover + B^n + S^n

with `B^n` external-boundary momentum (inlet/outlet/walls) and `S^n` forcing.
So **`F_CME + node_cover - node_uncover`** is, by construction, the body
momentum that balances the fluid ledger of *this* storage/update order.
Branch details: halfway fallback links (`f_r = f_t` plus the moving-wall
addend) and q < 1/2 links enter the link accumulators exactly like Bouzidi
links, with `c(f_t + f_r)`. In the q < 1/2 branch `G_dir(k_f - c)` is an
interpolation input only, not a fluid-momentum sink. Pull-slot completeness
is the section 3 contract. The scope excludes `ddf_shifting` (rejected), so
the unshifted force form applies. The balance is a fluid-side identity; it
does not close body+fluid globally in a domain with inlet/outlet/forcing
(`B^n`, `S^n`), and the solid-slot writes are boundary plumbing, not physical
streaming packets.

The Wen 2014 relative-velocity link sum adds `-u_w (f_t - f_r) = u_w dm` per
link (`dm = f_r - f_t`, the BFL mass defect). That term has no counterpart in
the fluid ledger above. Peng et al. (PRE 2016 submitted, §II.C, Eqs. 20–21)
show for a straight wall that the bounce-back error accumulated before a node
switch cancels with the switching momentum. Whether `sum u_w dm` approximates
`I_cover - I_uncover` here (implying double counting if both are used) is
**NOT demonstrated** for this implementation: it needs a bounded experiment
(prescribed slow translation; cumulative `sum u_w dm` vs cumulative node
impulse, per crossing cycle). Momentum conservation is also distinct from
Galilean invariance, which G4 tests.

Accumulators (separate, per body, per step window; all reported raw):
`link_cme`, `link_wen_correction` (= `sum u_w dm`), `node_cover`,
`node_uncover`, plus counters of crossings and refill fallbacks, so force
spikes can be correlated with crossing events.

**OPEN — structural RHS candidates (not selected):**
(a) `link_cme + node_cover - node_uncover`; (b) `link_cme + link_wen_correction`
(no node impulse). Selection follows the ledger review and the bounded
experiment above.

## 7. Rejection semantics

- Before any mutation (`ValueError`, no device write, host state unchanged):
  moving body with EsoPull, `ddf_shifting`, refinement/blocked stepper, D3Q19,
  Jets or surface actuation on that body, `sync_bodies`/checkpoint while a
  moving body exists; swept band touching domain boundaries, other bodies,
  sensors or force regions; `set_body` offset outside the declared box,
  per-step move ≥ 1 cell, speed above the declared bound.
- Host state is committed only after a successful upload (same contract as Jet
  amplitude updates); an uncertain device-write failure makes the simulation
  unusable.
- Device-side soft diagnostics (refill fallbacks) are counters, not errors.

## 8. Coupling and units (CANDIDATE)

Lattice units, rho0 = 1, per unit span. Step n uses `c^n`, `u^n` and yields
`F^n`. Weak staggered update
`v^{n+1} = v^n + (F^n - c v^n - k y^n)/m`, `y^{n+1} = y^n + v^{n+1}`
(Newmark is an alternative; OPEN). Mass, natural frequency, damping and `U_r`
follow the chosen benchmark's own definitions. Leontini, Thompson & Hourigan,
JFS 22:857–864 (2006), DOI 10.1016/j.jfluidstructs.2006.04.003, as read by
Grok from the full text: Re = 200, `m* = m/m_f = 10` (m_f displaced fluid
mass), `zeta = c/c_crit = 0.01`, `U* = U/(f_N D)` with `f_N` the natural
frequency in fluid assuming an inviscid added-mass term; the paper also
reports frequencies normalised by the in-vacuo value. Our runner derives `k`
from that in-fluid `f_N` (added-mass coefficient as stated in the paper) and
never from the vacuum formula; `c_crit` definition to be confirmed (OPEN). Stability/convergence is gated by a halved-time-step run, not
assumed from the mass ratio.

## 9. Gates (map; tolerances OPEN, declared before runs)

G0 zero-motion regression (stationary path unchanged; moving path at rest
reproduces stationary link set and DDF) · G1 raw force sign (`Cd > 0`) ·
G2/G3 prescribed translation incl. ledger accumulators · G4 matched
relative-flow (moving body in quiescent fluid vs fixed body in uniform
stream), optional boost, co-moving sanity · G5 Dütsch et al. 1998 Re 100,
KC 5 in-line (needs 2-component offset) · G6 Leontini free transverse VIV
(m* = 10, ζ = 0.01, Re = 200, definitions pending) · G7 time-level contract
frozen before the oscillator.

## 10. Remaining decisions

Structural RHS (section 6); motion-record layout; refill details; coupling
scheme; benchmark definitions and digitized reference data; tolerances;
execution/resource feasibility (Astra, pending); whether a quiescent domain
(zero inlet velocity) runs cleanly in the current config.