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Algorithms

This page summarizes current assembly/solve algorithms as implemented.

1. Unknown Layouts

Monophasic (StokesLayout)

x = [uomega_1; ugamma_1; ...; uomega_N; ugamma_N; pomega]

Two-phase (StokesLayoutTwoPhase)

x = [uomega1_1; ...; uomega1_N; uomega2_1; ...; uomega2_N; ugamma_1; ...; ugamma_N; pomega1; pomega2]

omega denotes cell/face unknowns, gamma denotes cut/interface trace unknowns.

2. Discrete Operators

At high level:

  • velocity diffusion-style blocks use cut-cell operator products (G' * Winv * G, G' * Winv * H),
  • pressure coupling is split into gradient/divergence blocks,
  • interface/cut equations are imposed by row overwrite/insertion,
  • boundary side laws (traction/symmetry/outlet) overwrite boundary-adjacent momentum rows,
  • active/inactive row masks are computed from capacity support.

3. Steady Monophasic Assembly

assemble_steady!(sys, model::StokesModelMono, t):

  1. Build momentum, trace-tie, and continuity blocks.
  2. Insert forcing and cut Dirichlet trace RHS.
  3. Apply side-level traction/symmetry overwrite (if present).
  4. Apply component-wise velocity BC row modifications.
  5. Apply optional pressure wall BC rows.
  6. Apply pressure gauge row replacement.
  7. Apply identity-row regularization on inactive/halo rows.

4. Unsteady Monophasic Assembly

assemble_unsteady!(..., scheme):

  • theta scheme (:BE, :CN, numeric theta),
  • adds (rho/dt) * V mass term on momentum omega rows,
  • adds history contribution from previous step (x_prev),
  • keeps continuity/gauge algebraic,
  • applies boundary/gauge/row-activity logic at t_{n+1}.

5. Moving Monophasic Assembly

assemble_unsteady_moving!:

  1. Build slab and end-time capacities/operators (_build_moving_slab!).
  2. Assemble slab-integrated momentum terms and end-time pressure coupling.
  3. Enforce cut trace at end-time via bc_cut_u.
  4. Apply end-time box BCs and gauge.
  5. Mask inactive rows from end-time supports and regularize.

Interpretation:

  • geometry/mass/diffusion terms come from slab integration,
  • cut trace and outer BCs are strongly imposed at end-time rows.

6. Two-Phase Fixed-Interface Assembly

assemble_steady! / assemble_unsteady! for StokesModelTwoPhase:

  • separate phase-1 and phase-2 momentum/continuity blocks,
  • shared ugamma rows used for traction/interface-force equations,
  • pressure coupling from both phase pressure blocks into traction rows,
  • gauge applied on phase-1 pressure block row replacement,
  • same row-activity/identity regularization strategy.

7. FSI Stepping Algorithms

Split (loose) coupling

step_fsi!:

  1. Predict rigid state over slab.
  2. Solve moving-boundary fluid step.
  3. Build end-time static model and compute integrated hydrodynamic force/torque.
  4. Advance rigid-body ODE state (:symplectic_euler or :forward_euler).

Strong partitioned coupling

step_fsi_strong!:

  1. Initialize state guess at t_{n+1}.
  2. Alternate fluid solve and rigid ODE update.
  3. Relax updates (:none, :constant, :aitken).
  4. Stop by absolute/relative residual tolerance or maxiter.

8. Regularization Strategy

Rows with no physical support (inactive/halo/uncoupled rows after masking) are set to identity with zero RHS.

Why this is used:

  • keeps linear systems nonsingular,
  • preserves fixed global indexing/layout,
  • simplifies coupling between assembly paths and postprocessing.

9. Practical Notes

  • First-order loss can appear near cut/wall closure depending on collocation and BC type.
  • Traction/symmetry/outlet rows are side-level constraints and must be declared consistently across velocity components on a side.
  • bc_p cannot coexist with traction/symmetry side laws on the same side.
  • For pressure errors, compare shift-invariant quantities when gauge differs.