Multiphase-Flow Solver Improvement Meeting – Detailed Notes

Background & Meeting Context

• Internal technical meeting between ASML engineers (Deepanjay, Francisco, et al.) and Siemens/STAR-CCM+ team (Eduardo, Dimitris, David) about multiphase-flow issues in immersion-lithography simulations.
• Main goal: revisit long-standing instabilities in Volume-of-Fluid (VOF) simulations (meniscus rupture, spurious currents, contact-line singularities) and agree on short-term “low-hanging-fruit” improvements.
• Slide deck shown was an updated version; latest file and related SIM files will be circulated after the call.

Key Pain Points Observed in Current STAR-CCM+ Models

• Geometric & numerical complexity → solver instability.
– Triple-phase contact line under no-slip ⇒ stress singularity.
• Frequent meniscus rupture during scans → solution divergence.
• Results highly mesh-dependent; first-order schemes often more stable than second-order.
• Parasitic (spurious) currents dominate in static-bubble tests; size increases with mesh refinement.
• Difficulty switching to second-order convective scheme in some cases.
• Doubts about adequacy of improvised Cox–Voinov contact-angle implementation (via field functions).

Published Benchmark Evidence

• Paper by ex-colleague (screenshots shown) compared OpenFOAM, Fluent, STAR-CCM+ for:
  • Static air bubble in water.
  • Toluene droplet in water.
• Parameters varied: λ, n (OpenFOAM); μ (STAR-CCM+).
• Metrics examined: interface shape & RMS spurious velocity.
• Findings:
  • Fluent → best match to reference, lowest spurious currents.
  • OpenFOAM → mixed but acceptable.
  • STAR-CCM+ → poorest surface preservation, highest parasitic currents.
  • Conclusion aligns with ASML’s internal experience of meniscus rupture.

Numerical Root Causes Discussed

• Standard Continuum-Surface-Force (CSF) model uses $\mathbf{F}<em>\sigma = \sigma \kappa \nabla \alpha$ where $\kappa = -\nabla \cdot \left( \frac{\nabla \alpha}{|\nabla \alpha|} \right)$. – $\alpha$ = volume fraction field (noisy, discontinuous). – Curvature based on ∇α → numerical noise → spurious currents $\mathbf{u}</em>s$.
• Mesh refinement with unstructured cells raises curvature frequency → larger $|\mathbf{u}_s|$ (counter-intuitive degradation).
• Adaptive Mesh Refinement (AMR) adds time-dependent perturbations.

Alternative VOF Improvements Reviewed

1 Isoadvector (InterIsoFoam) Advection

• OpenFOAM variant uses geometrically reconstructed fluxes at interface; markedly reduces spurious currents vs. vanilla InterFoam.

2 Tangent-Capillary-Stress (TCS) Surface-Tension Tensor

• Converts scalar CSF term to tensor form: $\mathbf{T}<em>\sigma = \sigma \left( \mathbf{I} - \mathbf{n}\mathbf{n} \right) \delta</em>s$.
  • Removes normal component; forces strictly along interface.
  • Implemented in Basilisk & Fluent; yields lower parasitic currents.

3 Distance-Function (Level-Set-Like) Curvature in STAR-CCM+

• Siemens hidden feature (will ship as beta in 2025.2 (25.06)).
  • Solves PDE for signed interface distance $\phi$ (similar to wall-distance solver).
  • Curvature uses smoother field: $\kappa = \nabla \cdot \left( \frac{\nabla \phi}{|\nabla \phi|} \right)$.
  • Available via Java macro; adds choice “Volume-Fraction-Based” vs. “Distance-Function-Based” under Surface-Tension model.
  • Early tests (static bubble, rising bubble) show mesh-independent convergence; matches analytical Laplace pressure $\Delta P = 2\sigma/R$.

4 Fused-Poly Mesh Trick (internal demo)

• Dual-region ALE mesh: droplet meshed with prisms/OTYPE grid, outer fluid trimmed; interface cells fused → accurate curvature at low cost.
• Academic illustration only; impractical for industrial moving droplets.

Contact-Line Modeling Discussion

• STAR-CCM+ currently displaces contact line implicitly via solved VOF field & Navier slip from wall-velocity gradient.
• Cox–Voinov not built-in; ASML implemented via field functions but performance degrades at high scan speeds (≥ U \approx 0.8$–$1\,\text{m s}^{-1}, Ca > 0.2).
• Molecular-Kinetic/Generalized Navier Boundary Condition (GNBC)
– Adds uncompensated Young stress term $\sigma(\cos\theta<em>s - \cos\theta</em>d)$ to regularize singularity for all $Ca$.
– Literature supplied; candidate for user-defined model via existing UDF/field-function framework.
• Siemens stance: provide flexible UDF hooks; not planning native Cox–Voinov (single-client request, many competing models).

Practical / Implementation Notes

• New distance-function option compatible only with release 25.06 onward; macro required to expose GUI.
• GPU roadmap: VOF on GPU (26.02) demands local-cell stencils → favors PDE distance-function, not large geometric stencils.
• Geometric reconstruction (parabolic fit using 3×3×3 stencil) is accurate but too expensive & GPU-unfriendly.

Numerical / Physical Parameters Cited

• Capillary number: $Ca = \mu U / \sigma$.
• Example Laplace pressure target in demo: $\Delta P \approx 1.4\,\text{Pa}$.
• Mesh sizes in benchmark paper: N \approx 10^4$–$1.6\times10^5 cells (same count across OpenFOAM, Fluent, STAR-CCM+).

Action Items & Next Steps

1. Siemens
   • Provide Java macro + simple test cases (static droplet, rising bubble) to ASML once 25.06 is available.
   • Raise documentation ticket to clarify present contact-line implementation & distance-function option.
2. ASML
   • Install STAR-CCM+ 25.06 (IT rollout ~2 weeks after July release).
   • Supply latest SIM files exhibiting meniscus rupture & high-speed scan instabilities.
   • Evaluate distance-function surface tension; report stability & accuracy impact.
   • Circulate benchmark slide deck + two cited papers (isoadvector & TCS/GNBC).
3. Joint
   • Compare results on shared cases; iterate on parameter settings.
   • Decide whether GNBC or other user-defined contact-angle models are needed beyond Cox–Voinov.

Real-World Relevance & Broader Implications

• Immersion-lithography scanners operate at high scan speeds where $Ca\gg0.1$ — classical wetting theory (Cox-Voinov) insufficient; robust multiphase solver essential for yield.
• Reduction of parasitic currents will improve prediction of wafer table wetting, bubble entrapment, and thereby photolithography defect control.
• Lessons transferable to automotive water-management & ink-jet industries; unified solver enhancements benefit multiple Siemens customers.

Ethical / Pragmatic Considerations

• Accurate but expensive geometric reconstruction may misallocate HPC resources; need balance between fidelity & throughput.
• Hidden/beta features should be validated before production use; clear documentation crucial to avoid misuse.
• User-defined flexibility empowers clients yet shifts validation burden; partnership feedback loop is mandatory.