Conference Talk

Navier-Stokes AI

A verification-first approach to physics-grounded AI for CFD-based medical-device design optimization

Amir Sajjadi

The Engineering Question

Can a CFD-trained surrogate screen device designs while reserving full flow solves for selected cases?

This does not claim to solve the general Navier–Stokes existence-and-smoothness problem. Navier–Stokes remains the physics foundation — A Gaussian Process estimates four scalar CFD responses within declared design bounds. Physical validation remains pending.

Navier-Stokes AI — Approach & Methodology

The trust gap

Pure AI surrogate

Fast, within tested bounds

Predicts selected metrics from numerical examples. It cannot guarantee field conservation for a new geometry.

Full CFD

Numerical reference

Solves flow fields under stated assumptions. New geometries require a new mesh and numerical solve.

The answer:

Train on CFD cases that pass numerical gates. Return selected designs to CFD for an independent check.

Navier-Stokes AI — Approach & Methodology

The methodology: a closed verification loop

1 · Geometry + BCs

Catheter/vessel geometry, flow rate, fluid properties

2 · Navier–Stokes / CFD

Axisymmetric incompressible solver

3 · Verification & QA gate

Analytical, mass, residual, mesh checks

4 · Selected CFD dataset

Only gate-passing cases proceed

5 · AI surrogate

Four scalar outputs + GP uncertainty

6 · Design-space search

Thousands of candidates, surrogate-scored

7 · Best candidates

Pressure drop + device WSS objectives

8 · CFD confirmation

Independent production-mesh CFD

CFD supplies numerical training labels. Surrogate screening uses no new CFD solves; total time savings remain unmeasured.

Scientific guardrails

Distinguish analytical, reduced-order, numerical CFD, and AI results — never blur the four

Verify the numerical solver before it ever generates AI training data

Report what the final-model split tests; reserve a prospective set before active learning next time

Report measured prediction error and solver times; benchmark end-to-end cost before claiming savings

Never present synthetic or example values as measured results

Confirm selected designs with independent production-mesh CFD; test shear on a fine mesh next

Passing numerical gates supports this model and mesh; physical or clinical validation remains open.

Milestone 1 · complete

A verified physics baseline

Concentric-annulus flow around a straight catheter — the one case with a known closed-form solution, used as ground truth before any CFD result is trusted.

Reference operating point

Reynolds number	≈ 107.1
Analytical pressure drop	736.92 Pa
Analytical max. velocity	0.2666 m/s
Catheter wall shear	4.29 Pa
Vessel wall shear	3.38 Pa

Grid convergence (numerical vs. exact)

Grid points	ΔP error	Velocity L2 error
21	0.259%	0.248%
41	0.065%	0.062%
81	0.016%	0.016%
161	0.004%	0.004%

Error decreases monotonically with mesh refinement — a convergence study, not a single lucky run

NS-0001 · Straight annulus

What the solver actually computes

Axial velocity field

Pressure field

Straight-annulus benchmark; the enlarged-body NS-0003 device solve follows.

Why we trust it

Convergence, not coincidence

Numerical vs. analytical velocity profile

Residual convergence history

28 iterations to convergence · continuity residual 9.5e-08 · no-slip and mass conservation both pass

Technical detail · the NS-0003 device solver

Body-fitted, finite-volume, SIMPLE-coupled axisymmetric Navier-Stokes

Governing equations — steady, incompressible, no swirl

Continuity, axial momentum, and radial momentum equations in cylindrical coordinates

Discretization

Body-fitted mapping r(z,eta) = R_i(z) + H(z) S(eta)

Face decomposition S = S_orth + S_nonorth and implicit orthogonal diffusion plus deferred correction

Finite-volume, cell-centered, collocated. Every face area / cell volume is the exact solid of revolution (2πr weighting). Convection: first-order upwind on conservative shared face mass fluxes.

Pressure-velocity coupling: SIMPLE

Pressure correction equation A p' = b, face flux correction, and under-relaxed update

Loop: momentum predictor (implicit, under-relaxed) → predicted face fluxes + BCs → solve for p' → correct p, u, fluxes → repeat until continuity + flux-change both fall under tolerance.

Boundary conditions

Walls: no-slip, from actual cell-to-face geometry | Inlet/outlet: specified mass flow rate | Scope: axisymmetric device study; separate free-tip work appears in the research appendix

ρ = 1060 kg/m³, μ = 0.0035 Pa·s (blood-like)

Typical case: 100 mL/min

NS-0003 · Physics Engine v1

The mid-shaft device, meshed and frozen

Verified 2D solve · merged to main

Real body-fitted 161 by 41 mesh conforming to the device geometry and vessel wall, with wall-clustered radial spacing

Actual solver output: 161×41 body-fitted production mesh

Frozen baseline parameters

Vessel ID	4.0 mm
Shaft OD	2.0 mm
Device body OD	2.4 mm
Body length	15 mm
Transition length	3 mm
Flow rate	100 mL/min
Production mesh	161 × 41

Shaft continues on both sides of the bulge — a different geometry from the free-ended circular-tip candidate shown later in this deck

NS-0003 · CFD solution

A real 2D flow solve around the mid-shaft device

Velocity magnitude contour: flow accelerates through the device constriction from about 0.27 to 0.31 metres per second

Velocity accelerates through the constriction, exactly as continuity predicts

Pressure contour decreasing smoothly and monotonically along the vessel from inlet to outlet

Pressure drops smoothly and monotonically along the vessel

Pressure drop

846 Pa

Peak velocity

0.314 m/s

Max device WSS

6.53 Pa

Converged in

57 iters

Negligible mass imbalance at convergence — independently re-executed and reproduced to 11 significant figures

NS-0003 · Mesh sensitivity

What's solid, and what's explicitly flagged

Wall shear stress along the device and vessel walls, with visible numerical oscillation near the transitions and along the body

Wall shear stress: the sawtooth noise near the transitions is real and undisguised

Medium → fine mesh change (121×31 → 161×41 → 201×51)

Pressure drop	0.60%
Peak velocity	0.07%
Device max WSS	0.93%
Device P95 WSS	3.37%
Vessel max WSS	1.63%

Documented, not hidden

WSS mesh sensitivity is explicitly carried as a qualified, open numerical-quality flag in the project's own decision log — pressure and velocity are not.

161×41 frozen as the production mesh for dataset generation; 201×51 reserved for selective fine-mesh verification

NS0003-DATASET-v1 · After active-learning round 1

The AI surrogate, on real CFD data

Predicted versus actual pressure drop for seven cases excluded from the final evaluation model fit

Evaluation GP: 18 train / 7 test

Pressure drop	0.999
Peak velocity	0.9997
Vessel max WSS	0.994
Device max WSS	0.953

Pressure RMSE 2.80 Pa on this single split.

How the five additions were chosen

A pressure-drop GP ranked 4,000 candidates by predictive uncertainty. Five separated inputs were solved with CFD. Device WSS R² is 0.953 on the later split; the earlier 0.890 used different test cases.

25 cases (20 initial + 5 additions). Five of seven test cases informed the earlier acquisition GP. A prospective test is next.

AI proposes, CFD confirms · mid-shaft device geometry

20,000 surrogate evaluations; 3 independent CFD checks

Scatter plot of 20,000 surrogate-screened candidate designs in pressure-drop vs device-WSS space, showing a strong positive correlation, with the 3-design Pareto front and the CFD-confirmed actual points overlaid almost exactly on top of the predictions

The honest finding

Both modeled objectives improve near the 2.10 mm lower body-OD bound. The saved screen has 3 nondominated designs. Device function constraints and clinical outcomes were outside this search.

CFD confirmation, 3/3 candidates

Design	ΔP error	WSS error
OPT1-001	0.05%	3.43%
OPT1-002	0.31%	3.38%
OPT1-003	0.39%	3.39%

At fixed 100 mL/min, 20,000 geometries were scored without additional CFD solves. Twenty-five dataset solves and three confirmations are counted separately.

Real CFD, both designs, same solver · mid-shaft device geometry

What the optimization actually changed

Side-by-side real CFD post-processing: velocity magnitude, static pressure, and device wall shear stress for the baseline 2.4mm device versus the surrogate-optimized 2.1mm device. The optimized design shows a visibly smaller velocity peak and roughly half the wall shear stress bump.

Pressure drop

846 → 753 Pa (−11.0%)

Device max WSS

6.52 → 4.55 Pa (−30.3%)

Peak velocity

0.314 → 0.277 m/s

Both fields come from independent CFD at 161×41. The selected design still needs fine-mesh shear verification.

What the main device study establishes

1 · Verified physics baseline

2 · Surrogate architecture prototype

3 · Real 2D coupled solve: NS-0002 + NS-0003 (merged to main)

4 · Production CFD dataset, mid-shaft device: 25/25 cases accepted (20 DOE + 5 active-learning)

5 · AI surrogate + active learning round 1, mid-shaft device (GP, held-out R² 0.95-0.9997)

6 · Surrogate optimization, mid-shaft device (20k screened) + CFD confirmation (3/3 matched)

7 · Public case study live; validation next

Core principle

Numerically check CFD labels. Use the surrogate to decide where new solves are most useful.

The 25-case model screened 20,000 geometries and three choices received independent CFD solutions. Next: prospective testing, selected-design fine-mesh shear checks, physical validation, and a fair timed cost comparison.

Amir Sajjadi

Public case study: github.com/asajjadi/navier-stokes-ai-case-study

Not connected to the NS-0003 dataset, surrogate, or optimization results shown earlier — a distinct geometry module, on its own solver

Research appendix · separate free-tip geometry

A free-ended catheter tip, meshed for real

Boundary condition verified against exact solution

Two panels: the full 100mm domain mesh where the tip region looks compressed because the axes are not equal scale, and a zoomed equal-aspect-ratio view of the tip showing it is a true semicircle exactly matching a reference circle outline

Top: full-domain mesh (axes not equal scale). Bottom: zoomed, true-to-scale view — the tip is an exact semicircle, verified against a reference circle outline

Verified: axis boundary condition, mesh convergence

Tested against exact Hagen-Poiseuille pipe flow (translation-invariant, no entrance-length ambiguity). L2 profile error decreases monotonically with refinement: 0.87% → 0.26% → 0.17% across 3 grid levels — a real release-gate test now, not a one-off run.

Rounded tip still open; a conical tip is now solved

The rounded tip diverged under every mesh fix tried (Entries 021–030). The conical tip converges once pressure-velocity coupling is fixed, and matches OpenFOAM to 0.1% in pressure drop (035).

Research appendix · conical tip · what field inspection caught

Stable is not the same as correct

Velocity field of the earlier conical-tip solve: speeds near 1.25 m/s around the shaft and a fast streak along the axis behind the tip, several times the physical value

Before: residuals fell smoothly, yet peak speed was 4.3 m/s (physical ~0.26), with a 300 Pa pressure checkerboard

After: converged in 876 iterations; peak 0.264 m/s, mass conserved, no checkerboard

Root cause: pressure and velocity were decoupled on the collocated grid, so a checkerboard could hide from the residuals. Fix: Rhie-Chow momentum interpolation.

Dev log Entry 035 · 481×61 mesh · same geometry, same flow rate, same solver family

Research appendix · conical tip · converged solution, 481×61

The converged pressure and velocity fields

Pressure field: a smooth, nearly linear fall along the annulus, a gentler slope in the open pipe beyond the tip, and a small low-pressure dip at the shaft-to-cone shoulder; zoom panel of the tip region below

Pressure falls 276.5 Pa inlet to outlet: 4,545 Pa/m along the annulus (exact: 4,543), gentler once the vessel opens

Velocity: annulus peak 0.239 m/s, a slow wake behind the apex filling in over ~5 mm, pipe centreline 0.264 m/s at the outlet

100 mL/min, vessel 4 mm, shaft 1.6 mm, 0.8 mm cone · converged to 1e-7 in 876 iterations · dev log Entry 035

Research appendix · conical tip · independent cross-check

Converged, and it matches OpenFOAM

Axial velocity profiles at eight stations from z = 25 to 99 mm: the custom solver's points lie on the OpenFOAM curves upstream of the tip and from 55 mm downstream; just behind the tip at 51 to 52 mm the custom wake fills in somewhat faster

Axial-velocity profiles at 8 stations: custom solver (dots) vs OpenFOAM (lines)

	This solver	OpenFOAM
Pressure drop	276.5 Pa	276.7 Pa
Peak speed	0.264 m/s	0.264 m/s
Flow at 25 / 80 mm	100 mL/min	100 mL/min
Iterations	876	~300
Mesh	481×61	22,640 cells

OpenFOAM itself checks out: upstream gradient within 0.05% of the exact solution, mesh-independent to 0.2%.

Remaining gap: just behind the apex the wake fills in faster. With the same first-order scheme at the same resolution, OpenFOAM matches it: this is convection order plus tip resolution, not a bug.

Dev log Entry 035 · OpenFOAM v2606 simpleFoam, 5° axisymmetric wedge, developed-annulus inlet · not connected to the NS-0003 dataset