fluid simulation fundamentals

laval

Before simulating a lava lamp, learn how a computer represents a fluid. A sustained thermal flow needs an energy source and an energy sink. The heater adds temperature while ambient cooling continuously removes it.

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\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n

\n\n

Click a cell to inspect its coordinates.

—selected cell

—array index

—cell value

0update steps

\n\n

section 1

A fluid needs an address

We divide a rectangular region into cells. Each cell has an x and y coordinate, just like a pixel.

neighbor blend per step25%

Each cell closes this percentage of the difference between itself and its neighbors.

distance per step0.75 cells

How far backward each cell looks along its velocity vector.

pressure iterations60

More iterations approximate the pressure solution more closely.

mean |divergence|: —

wall leakage

outward-pointing wall cells: —

buoyancy strength0.50

How much upward velocity one step adds in hotter cells.

heater per step0.12cooling per step2%

The heater adds temperature at the bottom; cooling removes this fraction from every cell.

temperature diffusion15%wall cooling8%

Conduction exchanges heat with neighbors; wall cooling removes heat only at the glass boundary.

neutral temperature0.50density response1.20

Wax above neutral gets an upward push; wax below neutral gets a downward push.

melting temperature0.55transition width0.10solid drag85%

Cold wax resists velocity; hot wax becomes mobile across a smooth melting range.

interface sharpening50%

Pushes mixed wax fractions toward carrier (0) or wax (1) after advection.

latent heat coupling70%

How strongly excess heat near the melt point changes melt progress instead of temperature.

surface tension0.35

Scales the curvature force applied only near the wax–carrier interface.

carrier viscosity5%hot-wax viscosity10%cold-wax viscosity45%

Each value is the fraction of the difference toward neighboring velocity closed per step.

time per frame1.00substeps4

max travel per substep: —

lamp presetsteps per second6

wax —avg temp —melt —max speed —

heater unevenness60%

Creates deterministic hot and cool patches across the bottom heater.

neck sensitivity55%pinch strength45%

detected neck cells: 0 · wax error: 0.000

Cyan outlines show narrow bridges. Removed wax is returned to nearby thicker wax.

mass correction100%

raw drift 0.000after correction 0.000

Measures wax before the substep and returns numerical gain or loss across mixed interface cells.

energy ledger this substep

heater +0.000cooling 0.000stored latent 0.000numerical drift 0.000

Thermal energy is temperature plus energy stored in melt progress. Heating and cooling are intentional transfers; drift is not.

closed material inventory

total wax 0.0last moved 0.00

Clicking an empty cell withdraws the same wax amount from existing cells, preferring the bottom reservoir.

wax motion

height 0%rising 0%falling 0%

The blue trace is the wax center of mass. Repeated peaks and valleys indicate a circulating cycle.

wax connection threshold35%

blobs 0largest 0%split events 0merge events 0

Colored outlines identify cells belonging to the same four-neighbor connected region.

persistent identity matching

next identity 1matched 0

A region keeps its outline color when most cells overlap a region from the previous frame.

identity search radius1 cell

When direct overlap fails, search this far around the new region for its previous identity.

cycle quality

height range 0%active blobs 0

A useful lamp has a changing center height and more than one connected wax region without becoming unstable.

cell width0.5 cmseconds per frame0.10 s

reported maximum speed: 0.0 cm/s

complete model

blobs 0cycle range 0%rising 0%falling 0%

This runs the closed, mass-corrected, heated, cooled, two-phase timestep with pinch-off and persistent blob tracking.

wax particle mass220base concentration75%particle cohesion70%glass cooling10%

particles 0 · blobs 0

target neighbor density3.0volume pressure90%

maximum particle density: 0.0

const field = createScalarField(28, 18);\n\nfield.at(x, y);

What to noticeA two-dimensional cell can live in a one-dimensional array.

Deliberately not includedThere is no motion, pressure, viscosity, or fluid physics yet.

\n\n

The path from this grid to a lava lamp

checkpoint 01Grid + scalar field

02Velocity vectors

03Advection

04Diffusion

05Divergence

06Pressure projection

07Boundaries

08Heat + buoyancy

09Heat transport gap

10Coupled thermal step

11Heating + cooling

12Conduction + wall cooling

13Wax material field

14Temperature-driven density

15Phase change + solid drag

16Wax interface

17Latent heat

18Cohesion + surface tension

19Phase-dependent viscosity

20Stable complete timestep

21Continuous calibrated lamp

22Uneven heating + bottom pool

23Pinch-off + recombination

24Wax conservation

25Energy budget

26Closed vessel

27Cycle diagnostics

28Blob components

29Persistent identity

30Motion matching

31Cycle calibration

32Physical units

33Final simulator

34Shaped lamp vessel

35Particle volume pressure

Current claim: this demonstrates a grid, a scalar field, and neighbor-based smoothing. It is not yet a fluid simulation.

\n\n`;\n\nexport default function(){\n return new Response(page,{headers:{\"content-type\":\"text/html; charset=utf-8\"}});\n}\n","label",[130],"278739fe-68dc-4321-99ca-3c0d23161170","README.md","d9aa0084-86fa-11f1-87e4-1607ee4eb77e",228,["D",1785250338848],43,"# laval\n\nA step-by-step fluid simulation learning project for TypeScript developers.\n\n## Current checkpoint: uneven heating and symmetry breaking\n\nThis project does **not** simulate a complete lava lamp yet. It builds the pieces of a simple Eulerian fluid solver one concept at a time, keeping the numerical details behind clearly named TypeScript operations until each idea is intuitive.\n\nThe live lessons now cover:\n\n1. **The grid** — select a cell and inspect its `(x, y)` coordinate and flat-array index.\n2. **A scalar field** — paint an abstract dye amount from `0` to `1`.\n3. **Diffusion** — observe values move toward neighboring values.\n4. **A velocity field** — store a two-dimensional vector in every cell and render it as an arrow.\n5. **Advection** — transport dye through the velocity field by tracing backward and sampling the previous scalar field.\n6. **Advection plus diffusion** — separate movement of the pattern from spreading of the pattern.\n7. **Divergence** — measure local imbalance between flow entering and leaving each cell.\n8. **Pressure projection** — solve a pressure field and use its gradient to reduce divergence.\n9. **Boundary conditions** — prevent velocity from pointing through the outer walls while preserving motion parallel to them.\n10. **Temperature and buoyancy** — store heat in another scalar field and convert it into upward velocity.\n11. **Rising heat gap** — reveal why local buoyancy followed directly by backward advection can pin bottom-row heat.\n12. **Coupled thermal step** — project and constrain velocity before using it to advect temperature.\n13. **Continuous heating and cooling** — add temperature at the heater while removing a fraction throughout the field every step.\n14. **Conduction and wall cooling** — exchange temperature with neighboring cells and remove additional heat at the glass boundary.\n15. **Wax material field** — store and advect wax fraction independently from temperature.\n16. **Temperature-driven density** — combine wax fraction and temperature into a signed buoyancy force.\n17. **Temperature-driven phase** — derive a liquid-wax fraction and use the solid fraction to resist motion.\n18. **Wax interface** — render mixed cells fractionally and counter numerical smearing with an explicit sharpening pass.\n19. **Latent heat** — store melt progress so phase can change while temperature remains near the melting point.\n20. **Surface tension** — estimate interface normals and curvature, then apply a curvature force near the wax boundary.\n21. **Phase-dependent viscosity** — diffuse velocity using a blend rate derived from carrier fraction, wax fraction, and melt progress.\n22. **Stable complete timestep** — place every operation in one authoritative order and subdivide frame time into smaller solves.\n23. **Continuous simulator** — schedule the stable timestep at a fixed rate with presets, pause/step controls, and live diagnostics.\n24. **Uneven heating** — start from a broad bottom wax pool and introduce deterministic heater variation that seeds localized buoyant regions.\n\n## Boundary model\n\nThe outer cells act as solid walls. `enforceBoundaries(velocity)` removes the component of velocity normal to each wall:\n\n- left and right walls set horizontal velocity to zero;\n- top and bottom walls set vertical velocity to zero;\n- the tangential component remains, so fluid may slide along the wall.\n\nThis is a simple **free-slip, no-through-wall** boundary condition. The lesson highlights outward-pointing wall cells in red and reports a leakage count. One boundary step should reduce that count to zero.\n\n## Current heat model\n\nThe temperature field stores a normalized heat value from `0` to `1` in every cell. `applyBuoyancy(velocity, temperature)` adds upward velocity in proportion to local temperature and the selected buoyancy strength. The isolated heat lesson intentionally changes velocity without moving temperature.\n\n## The rising-heat gap\n\n`stepRisingHeat()` composes two operations already introduced separately:\n\n1. `applyBuoyancy(velocity, temperature)` changes vertical velocity.\n2. `advect(temperature, velocity)` moves the temperature field through that velocity.\n\nThis exposes a missing link: local buoyancy may leave neighboring destination cells stationary, so backward advection cannot pull heat into them.\n\n## Current coupled step\n\n`stepCoupledThermal()` uses the fuller ordering:\n\n1. Apply buoyancy to velocity.\n2. Pressure-project velocity so the local push becomes a divergence-reduced surrounding flow.\n3. Enforce no-through-wall boundaries.\n4. Advect temperature through the corrected velocity.\n\nProjection does not move temperature itself. It prepares the velocity field that advection uses.\n\n## Current heat-cycle step\n\nEach step now performs two scalar-field updates before the coupled flow calculation:\n\n1. Multiply every temperature by `1 - coolingRate`.\n2. Add `heaterStrength` to cells above the bottom-center heater, clamped to `1`.\n\nThe result depends on a competition between energy input and loss. A stronger heater accumulates heat faster; stronger cooling shortens the lifetime of transported heat.\n\n## Current conduction step\n\nThe temperature update now distinguishes three mechanisms:\n\n1. **Ambient cooling** scales down temperature throughout the volume.\n2. **Conduction** moves each cell toward the average of its direct neighbors, even when velocity is zero.\n3. **Wall cooling** removes additional temperature only from edge cells touching the glass.\n\nAdvection remains different from conduction: advection carries heat with bulk motion; conduction spreads heat through local neighbor exchange.\n\n## Current material-field step\n\nEvery cell now contains another scalar, `waxFraction`:\n\n- `0` means carrier liquid.\n- `1` means wax.\n- A value between `0` and `1` is a numerically mixed cell produced by sampling.\n\nTemperature and wax fraction are separate fields. Both are transported through the same velocity field, but temperature represents thermal state while wax fraction represents material identity.\n\n## Current density step\n\nWax buoyancy is derived rather than stored:\n\n```ts\ndensityAnomaly = waxFraction * (neutralTemperature - temperature);\nverticalVelocity += densityAnomaly * densityResponse;\n```\n\nHot wax produces a negative canvas-y change and rises. Cold wax produces a positive canvas-y change and sinks. Carrier-only cells have `waxFraction = 0`, so this wax-specific force is zero regardless of their temperature.\n\n## Current phase step\n\nWax identity remains in `waxFraction`; phase is derived separately from temperature:\n\n```ts\nliquidFraction = smoothMelt(temperature, meltingTemperature, transitionWidth);\nsolidFraction = waxFraction * (1 - liquidFraction);\nvelocity *= 1 - solidFraction * solidDrag;\n```\n\nThe smooth transition avoids an abrupt numerical switch at one exact temperature. Cold wax resists motion, while hot wax becomes increasingly mobile without ceasing to be wax.\n\n## Current interface step\n\nSemi-Lagrangian advection and bilinear sampling produce fractional wax cells at a blob boundary. Rendering now blends the carrier and phase colors using the actual `waxFraction`. An optional sharpening pass applies a smoothstep curve and blends toward it:\n\n```ts\nsharpened = wax * wax * (3 - 2 * wax);\nwax = mix(wax, sharpened, interfaceSharpness);\n```\n\nThis makes the boundary more legible, but it is not perfectly mass-conserving.\n\n## Current latent-heat step\n\nEach wax cell now stores melt progress separately from temperature. Excess temperature above the melting point can be absorbed into melt progress; cooling wax can release that stored phase energy back into temperature. The result is a visible temperature plateau during melting and solidification.\n\nThis is an enthalpy-style teaching abstraction with normalized values, not a calibrated joule-based energy model.\n\n## Current surface-force step\n\nThe wax-fraction gradient estimates an interface normal. The divergence of that normalized direction estimates curvature. A continuum-surface-force approximation then adds velocity near the interface:\n\n```ts\nnormal = normalize(gradient(waxFraction));\ncurvature = -divergence(normal);\nforce = surfaceTension * curvature * gradient(waxFraction);\n```\n\nThis encourages coherent blobs by resisting ragged, highly curved interfaces.\n\n## Current viscosity step\n\nViscosity is represented as neighbor-based velocity diffusion. Each cell derives an effective per-step blend from its material and phase:\n\n```ts\nwaxViscosity = mix(coldWaxViscosity, hotWaxViscosity, liquidFraction);\neffectiveViscosity = mix(carrierViscosity, waxViscosity, waxFraction);\nvelocity = diffuseVelocity(velocity, effectiveViscosity);\n```\n\nThis smooths local momentum differences rather than merely scaling velocity toward zero.\n\n## Current complete timestep\n\nOne substep now follows this order:\n\n1. Add heater energy and apply ambient cooling.\n2. Conduct temperature and cool the walls.\n3. Exchange latent heat.\n4. Apply density, phase drag, and surface-tension forces.\n5. Diffuse velocity by material-dependent viscosity.\n6. Pressure-project velocity and enforce walls.\n7. Advect temperature, wax fraction, and melt progress.\n8. Sharpen the wax interface.\n\nA frame divides its requested time by the substep count. Rate-like changes and advection distance use that smaller duration.\n\n## Complete simulator runtime\n\nThe final module runs the stable timestep through a fixed-rate accumulator. Playback speed is measured in simulation frames per second and remains independent of display refresh rate. Presets set related parameters together, while diagnostics expose wax amount, average temperature, melt fraction, and maximum speed.\n\nThe simulator can be paused for cell inspection or advanced one frame at a time.\n\n## Current symmetry-breaking step\n\nA perfectly uniform pool over a perfectly uniform heater can remain artificially symmetric. The new heater scales its local input with a deterministic spatial and slowly changing perturbation. The perturbation changes only temperature; density and buoyancy amplify those small thermal differences into distinct regions of motion.\n\nAt zero unevenness, the heater remains uniform for direct comparison.\n\n## Current pinch-off step\n\nA 28 × 18 grid cannot resolve a wax neck once it becomes thinner than one cell. This lesson therefore makes that missing sub-grid decision visible:\n\n1. Detect a partially filled cell connecting thicker wax on opposite sides.\n2. Require velocity strain so a quiet narrow shape is not cut automatically.\n3. Remove a strength-scaled fraction from the neck.\n4. Return the removed wax to the thicker neighboring regions.\n5. Measure the remaining wax error after correction.\n\nCyan outlines mark detected necks. Recombination is already represented by the material field: when advection brings fractional interfaces into contact, they become one connected wax region and surface tension smooths the joined boundary.\n\nThis is a conservative topology correction for teaching, not a resolved Rayleigh–Plateau calculation.\n\n## Current conservation audit\n\nThe simulator now records the sum of every wax fraction before a substep. After advection, pinch-off, and interface sharpening, it measures the numerical drift and optionally redistributes that difference through cells that already contain wax:\n\n```ts\ntargetMass = sum(waxBefore);\nrunPhysics();\nrawDrift = sum(waxAfter) - targetMass;\nredistribute(-rawDrift, waxCells);\n```\n\nAt 0% correction, the raw numerical drift remains visible. At 100%, the corrected drift should remain near zero. This preserves normalized wax volume globally while allowing wax to move freely between individual cells.\n\n## Current energy-budget step\n\nThermal energy is approximated as the sum of temperature and melt progress. The ledger measures the system at four points in each substep:\n\n```ts\nstart = sum(temperature + meltProgress);\nheaterInput = afterHeater - start;\ncoolingLoss = afterCooling - afterHeater;\nnumericalDrift = finish - afterCooling;\n```\n\nTemperature and latent storage can exchange energy without changing their combined total. The total is not held constant because the lamp is an open system: its heater deliberately adds energy and its cooling boundaries deliberately remove it.\n\n## Current closed-vessel step\n\nEvery cell represents a full volume split between two complementary fractions:\n\n```ts\ncarrierFraction = 1 - waxFraction;\nwaxFraction + carrierFraction === 1;\n```\n\nThe earlier paint gesture created wax and therefore increased the field total before conservation began. In the closed-vessel lesson, painting an empty location withdraws the same amount from existing wax, preferring the bottom reservoir. The gesture changes the distribution without changing the inventory.\n\n## Current cycle-diagnostics step\n\nA single frame cannot establish whether the lamp is cycling. The simulator now records three wax-weighted measurements over time:\n\n- Center height: the vertical center of mass of the wax field.\n- Rising fraction: wax in cells with upward velocity.\n- Falling fraction: wax in cells with downward velocity.\n\nA repeating rise and fall in center height is evidence of circulation. A flat trace indicates a steady arrangement even when small local velocities remain.\n\n## Current blob-tracking step\n\nA blob is derived rather than stored. Cells above a configurable wax-fraction threshold form a binary mask, and a four-neighbor flood fill labels each connected component. The diagnostics report component count, the largest component’s share of all wax, and apparent split or merge events when that count changes.\n\nThe threshold matters: a weak fractional bridge may connect two regions at 20% but separate them at 50%. This makes blob identity a measurement choice layered over the continuous wax field.\n\n## Current persistent-identity step\n\nConnected components are now matched to the previous frame by cell overlap. The strongest overlap inherits the previous identity and outline color; an unmatched component receives a new identity. Multiple children sharing one prior identity indicate a split, while one region overlapping multiple prior identities indicates a merge.\n\n## What this does not claim\n\nThe pressure solution is approximate and the collocated grid is intentionally simple. The boundary operation does not yet model no-slip wall friction, internal obstacles, or boundary-aware pressure coupling. Buoyancy is the first generated force, but temperature does not yet move, spread, cool, or interact with material density. There is currently no wax/carrier-liquid phase interface.\n\n## Build order\n\n1. Grid and scalar field — complete\n2. Diffusion — complete\n3. Velocity vector field — complete\n4. Advection — complete\n5. Advection plus diffusion — complete\n6. Divergence — complete\n7. Pressure projection — complete\n8. Boundary conditions — complete\n9. Temperature and buoyancy — complete\n10. Rising heat gap — complete\n11. Coupled thermal step — complete\n12. Continuous heating and cooling — complete\n13. Temperature diffusion and wall cooling — complete\n14. Wax material field — complete\n15. Temperature-driven density — complete\n16. Temperature-driven phase change and solid drag — complete\n17. Wax interface — complete\n18. Latent heat — complete\n19. Cohesion and surface tension — complete\n20. Phase-dependent viscosity — complete\n21. Stable complete timestep — complete\n22. Continuous simulator and calibration — complete\n23. Uneven heating and bottom wax pool — complete\n24. Pinch-off and recombination — complete\n25. Global wax conservation — complete\n26. Thermal energy budget — complete\n27. Closed-vessel material inventory — complete\n28. Cycle diagnostics — complete\n29. Connected-component blob tracking — complete\n30. Persistent blob identity — complete\n31. Motion-tolerant identity matching — complete\n32. Coordinated cycle calibration — complete\n33. Physical-unit interpretation — complete\n34. Final streamlined simulator — complete\n\nEach checkpoint adds one concept, one visualization, and one falsifiable claim.\n\n\n## Final simulator\n\nThe final mode runs the same authoritative physics assembled throughout the course. Motion-tolerant identity matching searches nearby previous cells when exact overlap fails. Calibration presets adjust related coefficients together. The units lesson interprets cells-per-frame using selected length and time scales without claiming the normalized coefficients are measured material constants.\n\nThe completed simulation includes a closed wax inventory, heating and cooling, conduction, latent heat, temperature-dependent buoyancy and phase drag, surface tension, material-dependent viscosity, pressure projection, wall boundaries, advection, pinch-off, wax-volume correction, cycle diagnostics, and persistent blob tracking.\n\n\n## Shaped-vessel circulation\n\nModule 36 replaces the rectangular active domain with a tapered lamp mask. Falling wax is confined toward a narrow base, the heater is concentrated around the center, and temperature is removed along the stair-stepped glass boundary. The complete substep still performs phase exchange, buoyancy, surface tension, viscosity, pressure projection, advection, pinch-off, and wax-volume correction.\n\nThe geometry boundary is an educational cell mask rather than a cut-cell pressure solve, but it directly tests the hypothesis that a shaped base and localized heat source help fallen wax pool, re-form, and rise repeatedly.\n\n\n### Bottom-collection correction\n\nThe first shaped-vessel attempt only cancelled velocity at stair-stepped walls. That prevented leakage but did not reproduce the inward pressure response of a genuinely sloped boundary, so fallen wax still remained a horizontal layer.\n\nModule 36 now applies a visible, adjustable inward collection force to cool wax near the base. It also corrects the base-concentration mapping: higher concentration now means a narrower Gaussian heater. This is an explicit coarse-grid correction for missing cut-cell wall physics.\n\n\n## Hybrid particle module\n\nModule 36 now represents wax as a fixed set of Lagrangian particles rather than a fraction stored in Eulerian cells. Each particle stores position, velocity, and temperature. Temperature changes its buoyant acceleration, pairwise cohesion keeps nearby wax together, and analytic collisions constrain particles to the tapered vessel.\n\nParticle velocities are deposited onto the existing grid, pressure-projected there, and sampled back to the particles. A temporary wax field is rasterized only for diagnostics and rendering; it is no longer the authoritative material inventory. Conservation follows from the fixed particle count.\n\n\n### Particle mass and thermal breakup\n\nThe shaped particle module now defaults to 220 wax particles and exposes an 80–320 particle inventory control. Changing the inventory resets the example because particle count is the conserved mass.\n\nBreakup is not driven by mass alone. The focused heater now moves slowly across the base and varies within the particle body, creating temperature and buoyancy differences. Cohesion is stronger for cold particles and weaker for hot particles, allowing heated lobes to stretch away while the cold reservoir remains available for later reheating.\n\n\n## Particle volume pressure\n\nModule 37 estimates each particle's local density with a compact smoothing kernel. When neighborhood density exceeds the selected rest density, symmetric outward pressure is applied before cohesion. A hard-core term also prevents extreme overlap.\n\nParticle velocities are now deposited bilinearly across four grid cells and projected velocities are sampled back bilinearly. This removes the nearest-cell quantization that previously gave an entire collapsed clump one shared velocity sample.\n",[141],"ec02ba1f-9c0f-4799-880e-882f8edb6681","src","c305d0ec-86fa-11f1-ace5-1607ee4eb77e",2,["D",1784854699691],"directory",[149],"pulls","branches",1,"forks","envVars",{"_163":62,"_165":62},"connections",{"_163":62,"_164":62},"count","unfilledCount","emptyCount",{"_43":148,"_32":149,"_30":150,"_131":-5,"_41":42,"_40":151,"_133":24,"_38":279,"_13":62,"_34":153,"_36":-5,"_23":26,"_138":-5,"_182":280,"_27":281},{"_43":274,"_32":225,"_30":221,"_131":150,"_41":42,"_40":275,"_133":26,"_38":276,"_13":62,"_34":153,"_36":-5,"_23":26,"_138":-5,"_182":277,"_27":278},{"_43":266,"_32":267,"_30":268,"_131":221,"_41":42,"_40":269,"_133":26,"_38":270,"_13":62,"_34":35,"_36":271,"_23":26,"_138":-5,"_182":272,"_27":273},{"_43":258,"_32":259,"_30":260,"_131":221,"_41":42,"_40":261,"_133":26,"_38":262,"_13":62,"_34":35,"_36":263,"_23":26,"_138":-5,"_182":264,"_27":265},{"_43":250,"_32":251,"_30":252,"_131":221,"_41":42,"_40":253,"_133":26,"_38":254,"_13":62,"_34":35,"_36":255,"_23":26,"_138":-5,"_182":256,"_27":257},{"_43":242,"_32":243,"_30":244,"_131":221,"_41":42,"_40":245,"_133":26,"_38":246,"_13":62,"_34":35,"_36":247,"_23":26,"_138":-5,"_182":248,"_27":249},{"_43":234,"_32":235,"_30":236,"_131":221,"_41":42,"_40":237,"_133":26,"_38":238,"_13":62,"_34":35,"_36":239,"_23":26,"_138":-5,"_182":240,"_27":241},{"_43":226,"_32":227,"_30":228,"_131":221,"_41":42,"_40":229,"_133":26,"_38":230,"_13":62,"_34":35,"_36":231,"_23":26,"_138":-5,"_182":232,"_27":233},{"_43":220,"_32":33,"_30":31,"_131":221,"_41":42,"_40":14,"_133":26,"_38":222,"_13":62,"_34":35,"_36":37,"_23":26,"_138":-5,"_182":223,"_27":224},{"_43":215,"_32":206,"_30":200,"_131":150,"_41":42,"_40":216,"_133":24,"_38":217,"_13":62,"_34":153,"_36":-5,"_23":26,"_138":-5,"_182":218,"_27":219},{"_43":207,"_32":208,"_30":209,"_131":200,"_41":42,"_40":210,"_133":26,"_38":211,"_13":62,"_34":35,"_36":212,"_23":26,"_138":-5,"_182":213,"_27":214},{"_43":197,"_32":198,"_30":199,"_131":200,"_41":42,"_40":201,"_133":26,"_38":202,"_13":62,"_34":35,"_36":203,"_23":26,"_138":-5,"_182":204,"_27":205},{"_43":188,"_32":189,"_30":190,"_131":150,"_41":42,"_40":191,"_133":26,"_38":192,"_13":193,"_34":35,"_36":194,"_23":26,"_138":-5,"_182":195,"_27":196},{"_43":140,"_32":141,"_30":142,"_131":-5,"_41":42,"_40":143,"_133":26,"_38":185,"_13":145,"_34":20,"_36":146,"_23":26,"_138":-5,"_182":186,"_27":187},{"_43":129,"_32":130,"_30":57,"_131":-5,"_41":42,"_40":132,"_133":26,"_38":181,"_13":135,"_34":136,"_36":137,"_23":26,"_138":45,"_182":183,"_27":184},["D",1785250298615],"children",[],[130],["D",1785250338848],[],[141],"b048c151-ff04-4baa-843a-72fdd4180858","contracts.ts","db9d981a-86fa-11f1-aed7-1607ee4eb77e",229,["D",1785250348184],40,"/**\n * Named contracts for the fluid-simulation learning checkpoints.\n *\n * Each interface exposes the operation a lesson teaches while allowing the\n * numerical implementation to remain hidden until the learner is ready for it.\n */\n\nexport type CellCoordinate = Readonly<{\n x: number;\n y: number;\n}>;\n\nexport interface ScalarField {\n readonly width: number;\n readonly height: number;\n at(cell: CellCoordinate): number;\n set(cell: CellCoordinate, value: number): void;\n clear(): void;\n}\n\nexport type Vector2 = Readonly<{\n x: number;\n y: number;\n}>;\n\nexport interface VectorField {\n readonly width: number;\n readonly height: number;\n at(cell: CellCoordinate): Vector2;\n set(cell: CellCoordinate, value: Vector2): void;\n clear(): void;\n}\n\nexport interface ScalarFieldRenderer {\n render(field: ScalarField): void;\n}\n\nexport interface VectorFieldRenderer {\n render(field: VectorField): void;\n}\n\nexport interface FieldUpdate {\n step(current: ScalarField): ScalarField;\n}\n\nexport interface DiffusionUpdate extends FieldUpdate {\n readonly strength: number;\n}\n\nexport interface AdvectionUpdate {\n readonly distancePerStep: number;\n step(quantity: ScalarField, velocity: VectorField): ScalarField;\n}\n\nexport interface TransportAndDiffusionStep {\n step(quantity: ScalarField, velocity: VectorField): ScalarField;\n}\n\nexport interface DivergenceMeasurement {\n measure(velocity: VectorField): ScalarField;\n}\n\nexport interface PressureProjection {\n solvePressure(divergence: ScalarField): ScalarField;\n subtractPressureGradient(\n velocity: VectorField,\n pressure: ScalarField,\n ): VectorField;\n}\n\n/**\n * A free-slip, no-through-wall boundary operation.\n *\n * It removes velocity normal to the outer walls while preserving velocity\n * tangent to them. A future no-slip condition can additionally remove the\n * tangential component.\n */\nexport interface BoundaryCondition {\n enforce(velocity: VectorField): VectorField;\n countOutwardLeaks(velocity: VectorField): number;\n}\n\n/** Converts a temperature scalar field into a vertical velocity change. */\nexport interface BuoyancyForce {\n readonly strength: number;\n apply(velocity: VectorField, temperature: ScalarField): VectorField;\n}\n\n/** Applies buoyancy, then transports temperature through the new velocity. */\nexport interface ThermalFlowStep {\n step(velocity: VectorField, temperature: ScalarField): Readonly<{\n velocity: VectorField;\n temperature: ScalarField;\n }>;\n}\n\n/** Orders force, projection, walls, and temperature transport into one step. */\nexport interface CoupledThermalStep {\n step(\n velocity: VectorField,\n temperature: ScalarField,\n projection: PressureProjection,\n boundaries: BoundaryCondition,\n ): Readonly<{ velocity: VectorField; temperature: ScalarField }>;\n}\n\n/** Adds heat at the lamp base and removes a fraction of heat everywhere. */\nexport interface ThermalSourceAndSink {\n readonly heaterStrength: number;\n readonly coolingRate: number;\n apply(temperature: ScalarField): ScalarField;\n}\n\n/** Separates neighbor-based conduction from additional boundary heat loss. */\nexport interface ThermalConduction {\n readonly diffusionStrength: number;\n readonly wallCoolingRate: number;\n diffuse(temperature: ScalarField): ScalarField;\n coolWalls(temperature: ScalarField): ScalarField;\n}\n\n/** A scalar fraction identifying wax independently from thermal state. */\nexport interface MaterialField extends ScalarField {\n /** 0 is carrier liquid, 1 is wax, intermediate values are mixed cells. */\n at(cell: CellCoordinate): number;\n}\n\nexport interface MaterialTransport {\n advect(material: MaterialField, velocity: VectorField): MaterialField;\n}\n\n/** Derives a signed wax-density force relative to the carrier liquid. */\nexport interface TemperatureDrivenDensity {\n readonly neutralTemperature: number;\n readonly response: number;\n apply(\n velocity: VectorField,\n temperature: ScalarField,\n waxFraction: MaterialField,\n ): VectorField;\n}\n\n/** Derives phase state from temperature without replacing material identity. */\nexport interface TemperatureDrivenPhase {\n readonly meltingTemperature: number;\n readonly transitionWidth: number;\n readonly solidDrag: number;\n liquidFraction(temperature: number): number;\n applyDrag(\n velocity: VectorField,\n temperature: ScalarField,\n waxFraction: MaterialField,\n ): VectorField;\n}\n\n/** Controls fractional-cell sharpening after material advection. */\nexport interface MaterialInterface {\n readonly sharpness: number;\n sharpen(waxFraction: MaterialField): MaterialField;\n}\n\n/** Stores phase energy separately so melting can occur at near-constant temperature. */\nexport interface LatentHeatExchange {\n readonly coupling: number;\n exchange(\n temperature: ScalarField,\n waxFraction: MaterialField,\n meltProgress: ScalarField,\n ): Readonly<{ temperature: ScalarField; meltProgress: ScalarField }>;\n}\n\n/** Applies a curvature force near the wax–carrier interface. */\nexport interface SurfaceTensionForce {\n readonly strength: number;\n apply(velocity: VectorField, waxFraction: MaterialField): VectorField;\n}\n\n/** Diffuses velocity at a rate derived from material identity and phase. */\nexport interface PhaseDependentViscosity {\n readonly carrierViscosity: number;\n readonly hotWaxViscosity: number;\n readonly coldWaxViscosity: number;\n diffuse(\n velocity: VectorField,\n waxFraction: MaterialField,\n meltProgress: ScalarField,\n ): VectorField;\n}\n\n/** Runs the authoritative coupled order using one or more stable substeps. */\nexport interface CompleteTimestep {\n readonly frameTime: number;\n readonly substeps: number;\n step(): void;\n}\n\nexport interface SimulatorRuntime {\n readonly playing: boolean;\n readonly stepsPerSecond: number;\n play(): void;\n pause(): void;\n step(): void;\n reset(): void;\n}\n\n/** Adds deterministic spatial variation to a distributed bottom heater. */\nexport interface UnevenHeatSource {\n readonly unevenness: number;\n apply(temperature: ScalarField, simulationStep: number): ScalarField;\n}\n\n/** Makes a coarse-grid topology correction while preserving total wax. */\nexport interface PinchOffCorrection {\n readonly neckSensitivity: number;\n readonly pinchStrength: number;\n detect(\n waxFraction: MaterialField,\n velocity: VectorField,\n ): ReadonlyArray;\n apply(waxFraction: MaterialField, velocity: VectorField): Readonly<{\n waxFraction: MaterialField;\n detectedNecks: number;\n conservationError: number;\n }>;\n}\n\n/** Audits and corrects global wax-volume drift across a complete substep. */\nexport interface WaxConservationAudit {\n readonly correctionStrength: number;\n total(waxFraction: MaterialField): number;\n correct(waxFraction: MaterialField, targetTotal: number): Readonly<{\n waxFraction: MaterialField;\n rawDrift: number;\n correctedDrift: number;\n }>;\n}\n\n/** Separates intentional thermal transfers from unexplained numerical drift. */\nexport interface ThermalEnergyBudget {\n total(temperature: ScalarField, meltProgress: ScalarField): number;\n step(): Readonly<{\n heaterInput: number;\n coolingLoss: number;\n latentStorage: number;\n numericalDrift: number;\n }>;\n}\n\n/** Relocates a fixed material inventory instead of creating wax while editing. */\nexport interface ClosedVesselInventory {\n totalWax(waxFraction: MaterialField): number;\n carrierFractionAt(cell: CellCoordinate): number;\n relocate(\n waxFraction: MaterialField,\n target: CellCoordinate,\n ): Readonly<{ waxFraction: MaterialField; moved: number }>;\n}\n\n/** Reduces a sequence of full fields to signals that reveal circulation. */\nexport interface CycleDiagnostics {\n readonly centerHeightHistory: ReadonlyArray;\n record(waxFraction: MaterialField, velocity: VectorField): Readonly<{\n centerHeight: number;\n risingFraction: number;\n fallingFraction: number;\n }>;\n}\n\nexport interface TrackedBlob {\n readonly label: number;\n readonly waxAmount: number;\n readonly cells: ReadonlyArray;\n}\n\n/** Derives connected wax regions from a thresholded material field. */\nexport interface BlobTracker {\n readonly connectionThreshold: number;\n identify(waxFraction: MaterialField): ReadonlyArray;\n}\n\n/** Matches derived regions across adjacent frames using spatial overlap. */\nexport interface PersistentBlobTracker {\n match(previous: ReadonlyArray, current: ReadonlyArray): Readonly<{\n blobs: ReadonlyArray;\n splits: number;\n merges: number;\n }>;\n}\n\nexport interface WaxParticle {\n position: Readonly<{ x: number; y: number }>;\n velocity: Readonly<{ x: number; y: number }>;\n temperature: number;\n}\n\n/** Couples conserved Lagrangian wax particles to an Eulerian pressure grid. */\nexport interface HybridParticleLamp {\n readonly particles: ReadonlyArray;\n depositVelocity(): VectorField;\n projectCarrierFlow(): VectorField;\n sampleProjectedVelocity(velocity: VectorField): void;\n step(): void;\n}\n\n/** Preserves finite particle volume with neighbor-density pressure. */\nexport interface ParticleVolumeConstraint {\n readonly restDensity: number;\n readonly pressureStrength: number;\n estimateDensity(particles: ReadonlyArray): ReadonlyArray;\n applyPressure(particles: ReadonlyArray): void;\n}\n\nexport const roadmap = [\n \"grid-and-scalar-field\",\n \"diffusion\",\n \"velocity-field\",\n \"advection\",\n \"advection-and-diffusion\",\n \"divergence\",\n \"pressure-projection\",\n \"boundaries\",\n \"temperature-and-buoyancy\",\n \"rising-heat-gap\",\n \"coupled-thermal-step\",\n \"continuous-heating-and-cooling\",\n \"temperature-diffusion-and-wall-cooling\",\n \"wax-material-field\",\n \"temperature-driven-density\",\n \"phase-change-and-viscosity\",\n \"two-phase-flow\",\n \"stable-complete-timestep\",\n \"continuous-simulator\",\n \"uneven-heating\",\n \"pinch-off-and-recombination\",\n \"global-wax-conservation\",\n \"thermal-energy-budget\",\n \"closed-vessel-inventory\",\n \"cycle-diagnostics\",\n \"connected-component-blob-tracking\",\n \"persistent-blob-identity\",\n \"motion-tolerant-matching\",\n \"cycle-calibration\",\n \"physical-unit-interpretation\",\n \"final-lava-lamp-simulator\",\n \"shaped-vessel-circulation\",\n \"particle-volume-pressure\",\n] as const;\n",[],[149,189],"d779e42e-99ec-4a68-86e3-d67878e23ba3","styles.ts","4cb17186-8a96-11f1-a9d3-1607ee4eb77e","34766f7c-8a96-11f1-8334-1607ee4eb77e",232,["D",1785251356153],"export const styles = \" :root { color-scheme:dark; --bg:#0d0f12; --panel:#15181d; --line:#303640; --text:#edf0f4; --muted:#929aa6; --blue:#50a7ff; --yellow:#ffd166; }\\n * { box-sizing:border-box; }\\n body { margin:0; background:var(--bg); color:var(--text); font:15px/1.5 ui-monospace,SFMono-Regular,Menlo,monospace; }\\n button,input,select { font:inherit; }\\n select { width:100%; color:var(--text); background:#242a32; border:1px solid #414955; border-radius:7px; padding:7px; }\\n .final-only { display:none; } .final-only.visible { display:inline-block; }\\n .shell { width:min(1050px,calc(100% - 28px)); margin:auto; padding:32px 0 60px; }\\n header { display:flex; justify-content:space-between; gap:30px; align-items:flex-end; margin-bottom:26px; }\\n h1 { margin:0; font:750 52px/.9 system-ui,sans-serif; letter-spacing:-.06em; }\\n h1 span { color:var(--blue); }\\n header p { max-width:560px; margin:0; color:var(--muted); }\\n .eyebrow { color:var(--yellow); text-transform:uppercase; letter-spacing:.14em; font-size:11px; margin-bottom:8px; }\\n .lesson-nav { display:grid; grid-template-columns:repeat(auto-fit,minmax(135px,1fr)); border:1px solid var(--line); border-radius:12px; overflow:hidden; margin-bottom:16px; }\\n .lesson-button { padding:13px; color:var(--muted); background:#111419; border:0; cursor:pointer; text-align:left; }\\n .lesson-button+.lesson-button { border-left:1px solid var(--line); }\\n .lesson-button b { display:block; color:inherit; margin-bottom:2px; }\\n .lesson-button small { font-size:10px; }\\n .lesson-button.active { color:var(--text); background:#20252c; box-shadow:inset 0 -3px var(--blue); }\\n .workspace { display:grid; grid-template-columns:minmax(0,1fr) 290px; gap:16px; align-items:start; }\\n .card { background:var(--panel); border:1px solid var(--line); border-radius:14px; overflow:hidden; }\\n .canvas-head { display:flex; justify-content:space-between; align-items:center; gap:14px; padding:12px 14px; border-bottom:1px solid var(--line); }\\n .canvas-head span { color:var(--muted); font-size:11px; }\\n .actions { display:flex; gap:7px; }\\n button.control { border:1px solid #414955; color:var(--text); background:#242a32; border-radius:7px; padding:6px 10px; cursor:pointer; }\\n button.control.primary { background:var(--blue); color:#07111b; border-color:var(--blue); font-weight:700; }\\n button:disabled { opacity:.35; cursor:not-allowed; }\\n .canvas-wrap { padding:14px; }\\n canvas { display:block; width:100%; aspect-ratio:14/9; background:#0b1017; border:1px solid #343b45; image-rendering:pixelated; cursor:crosshair; }\\n canvas.mini-chart { width:100%; height:92px; aspect-ratio:auto; cursor:default; margin-top:8px; }\\n .readout { display:grid; grid-template-columns:repeat(4,1fr); border-top:1px solid var(--line); }\\n .readout div { padding:10px 12px; min-width:0; }\\n .readout div+div { border-left:1px solid var(--line); }\\n .readout b { display:block; color:var(--yellow); white-space:nowrap; }\\n .readout small { color:var(--muted); font-size:9px; text-transform:uppercase; letter-spacing:.08em; }\\n aside { padding:17px; }\\n aside h2 { font:700 17px system-ui,sans-serif; margin:0 0 8px; }\\n aside p { color:var(--muted); margin:0 0 18px; }\\n .code { padding:13px; background:#0d1117; border:1px solid #2b323b; border-radius:9px; color:#c9d1d9; white-space:pre-wrap; font-size:12px; margin-bottom:18px; }\\n .code em { color:var(--blue); font-style:normal; }\\n .concept { border-top:1px solid var(--line); padding-top:14px; margin-top:14px; }\\n .concept b { display:block; margin-bottom:5px; }\\n .concept span { color:var(--muted); font-size:12px; }\\n .slider { display:none; margin-bottom:18px; }\\n .slider.visible { display:block; }\\n .slider label { display:flex; justify-content:space-between; margin-bottom:6px; font-size:12px; }\\n input[type=range] { width:100%; accent-color:var(--blue); }\\n .roadmap { margin-top:16px; padding:18px; }\\n .roadmap h2 { font:700 18px system-ui,sans-serif; margin:0 0 14px; }\\n .steps { display:grid; grid-template-columns:repeat(4,1fr); gap:8px; }\\n .step { border:1px solid var(--line); border-radius:9px; padding:11px; color:var(--muted); min-height:82px; }\\n .step.current { border-color:#397fbd; background:#142131; color:var(--text); }\\n .step small { color:var(--blue); }\\n .step b { display:block; font:650 13px system-ui,sans-serif; margin-top:6px; }\\n .truth { margin-top:14px; padding:12px; border-left:3px solid var(--yellow); background:#1d1b15; color:#bdb7a7; font-size:12px; }\\n @media(max-width:780px){ header{align-items:flex-start;flex-direction:column}.workspace{grid-template-columns:1fr}.steps{grid-template-columns:repeat(2,1fr)} }\\n @media(max-width:500px){.lesson-nav{grid-template-columns:1fr}.lesson-button+.lesson-button{border-left:0;border-top:1px solid var(--line)}.readout{grid-template-columns:repeat(2,1fr)}.readout div:nth-child(3){border-left:0;border-top:1px solid var(--line)}.readout div:nth-child(4){border-top:1px solid var(--line)}}\";\n",[],[149,206,198],"ui","81c27834-228f-4220-9787-a4a40e4dbb8e","document.ts","52b29326-8a96-11f1-bb6b-1607ee4eb77e",234,["D",1785251366224],"export const documentHtml = \"\\n\\n\\n \\n \\n laval — fluid simulation fundamentals\\n \\n\\n\\n

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fluid simulation fundamentals

laval

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\\n\\n

\\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n \\n

\\n\\n

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Click a cell to inspect its coordinates.

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—selected cell

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—array index

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—cell value

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0update steps

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\\n\\n

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section 1

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A fluid needs an address

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We divide a rectangular region into cells. Each cell has an x and y coordinate, just like a pixel.

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neighbor blend per step25%

Each cell closes this percentage of the difference between itself and its neighbors.

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distance per step0.75 cells

How far backward each cell looks along its velocity vector.

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pressure iterations60

More iterations approximate the pressure solution more closely.

mean |divergence|: —

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wall leakage

outward-pointing wall cells: —

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buoyancy strength0.50

How much upward velocity one step adds in hotter cells.

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heater per step0.12cooling per step2%

The heater adds temperature at the bottom; cooling removes this fraction from every cell.

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temperature diffusion15%wall cooling8%

Conduction exchanges heat with neighbors; wall cooling removes heat only at the glass boundary.

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neutral temperature0.50density response1.20

Wax above neutral gets an upward push; wax below neutral gets a downward push.

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melting temperature0.55transition width0.10solid drag85%

Cold wax resists velocity; hot wax becomes mobile across a smooth melting range.

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interface sharpening50%

Pushes mixed wax fractions toward carrier (0) or wax (1) after advection.

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latent heat coupling70%

How strongly excess heat near the melt point changes melt progress instead of temperature.

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surface tension0.35

Scales the curvature force applied only near the wax–carrier interface.

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carrier viscosity5%hot-wax viscosity10%cold-wax viscosity45%

Each value is the fraction of the difference toward neighboring velocity closed per step.

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time per frame1.00substeps4

max travel per substep: —

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lamp presetsteps per second6

wax —avg temp —melt —max speed —

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heater unevenness60%

Creates deterministic hot and cool patches across the bottom heater.

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neck sensitivity55%pinch strength45%

detected neck cells: 0 · wax error: 0.000

Cyan outlines show narrow bridges. Removed wax is returned to nearby thicker wax.

\\n

mass correction100%

raw drift 0.000after correction 0.000

Measures wax before the substep and returns numerical gain or loss across mixed interface cells.

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energy ledger this substep

heater +0.000cooling 0.000stored latent 0.000numerical drift 0.000

Thermal energy is temperature plus energy stored in melt progress. Heating and cooling are intentional transfers; drift is not.

\\n

closed material inventory

total wax 0.0last moved 0.00

Clicking an empty cell withdraws the same wax amount from existing cells, preferring the bottom reservoir.

\\n

wax motion

height 0%rising 0%falling 0%

The blue trace is the wax center of mass. Repeated peaks and valleys indicate a circulating cycle.

\\n

wax connection threshold35%

blobs 0largest 0%split events 0merge events 0

Colored outlines identify cells belonging to the same four-neighbor connected region.

\\n

persistent identity matching

next identity 1matched 0

A region keeps its outline color when most cells overlap a region from the previous frame.

\\n

identity search radius1 cell

When direct overlap fails, search this far around the new region for its previous identity.

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cycle quality

height range 0%active blobs 0

A useful lamp has a changing center height and more than one connected wax region without becoming unstable.

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cell width0.5 cmseconds per frame0.10 s

reported maximum speed: 0.0 cm/s

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complete model

blobs 0cycle range 0%rising 0%falling 0%

This runs the closed, mass-corrected, heated, cooled, two-phase timestep with pinch-off and persistent blob tracking.

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wax particle mass220base concentration75%particle cohesion70%glass cooling10%

particles 0 · blobs 0

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target neighbor density3.0volume pressure90%

maximum particle density: 0.0

\\n

const field = createScalarField(28, 18);\\n\\nfield.at(x, y);

\\n

What to noticeA two-dimensional cell can live in a one-dimensional array.

\\n

Deliberately not includedThere is no motion, pressure, viscosity, or fluid physics yet.

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\\n\\n

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The path from this grid to a lava lamp

\\n

checkpoint 01Grid + scalar field

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02Velocity vectors

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03Advection

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04Diffusion

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05Divergence

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06Pressure projection

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07Boundaries

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08Heat + buoyancy

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09Heat transport gap

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10Coupled thermal step

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11Heating + cooling

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12Conduction + wall cooling

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13Wax material field

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14Temperature-driven density

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15Phase change + solid drag

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16Wax interface

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17Latent heat

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18Cohesion + surface tension

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19Phase-dependent viscosity

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20Stable complete timestep

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21Continuous calibrated lamp

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22Uneven heating + bottom pool

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23Pinch-off + recombination

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24Wax conservation

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25Energy budget

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26Closed vessel

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27Cycle diagnostics

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28Blob components

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29Persistent identity

\\n

30Motion matching

\\n

31Cycle calibration

\\n

32Physical units

\\n

33Final simulator

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34Shaped lamp vessel

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35Particle volume pressure

\\n

Current claim: this demonstrates a grid, a scalar field, and neighbor-based smoothing. It is not yet a fluid simulation.

\\n

\\n\\n\";\n",[],[149,206,208],"f1340c10-1007-45af-9382-df0fe0c3c541",230,["D",1785251315458],[176,177],[149,206],"36a0cc68-90da-4a57-8266-64bd10b798cf","37516f62-8a96-11f1-b54f-1607ee4eb77e",["D",1785251400936],[],[149,225,33],"client","4a754bbb-7b29-46f4-8203-b3a4f16b68d0","06-lesson-content.ts","646d5cd6-8a96-11f1-a4d0-1607ee4eb77e",239,["D",1785251395970],"export const lessonContent = \" var copy={\\n grid:{number:\\\"section 1\\\",title:\\\"A fluid needs an address\\\",text:\\\"We divide a rectangular region into cells. Each cell has an x and y coordinate, just like a pixel.\\\",instruction:\\\"Click a cell to inspect its coordinates.\\\",code:\\\"const field = createScalarField(28, 18);\\\\n\\\\nfield.at(x, y);\\\",notice:\\\"A two-dimensional cell can live in a one-dimensional array.\\\",notYet:\\\"There is no stored fluid quantity and no motion yet.\\\"},\\n scalar:{number:\\\"section 2\\\",title:\\\"One number in every cell\\\",text:\\\"A scalar field stores one value at each location. Here the value represents an abstract amount of blue dye.\\\",instruction:\\\"Click and drag to paint dye into cells.\\\",code:\\\"field.set(x, y, dyeAmount);\\\\n\\\\nrenderScalarField(field);\\\",notice:\\\"Color is only a picture of the numbers. The Float32Array is the model.\\\",notYet:\\\"The dye does not move or spread by itself.\\\"},\\n diffusion:{number:\\\"section 3\\\",title:\\\"State changes in steps\\\",text:\\\"A simulation repeatedly derives a new field from the current field. The slider controls how far each cell moves toward its neighbor average during the next step.\\\",instruction:\\\"For a fair comparison: choose a percentage, press example, then press step.\\\",code:\\\"while (running) {\\\\n field = diffuseScalarField(field);\\\\n renderScalarField(field);\\\\n}\\\",notice:\\\"The slider affects future steps; it cannot recalculate steps already taken. Compare 5% and 100% from the same example state.\\\",notYet:\\\"This is diffusion only—not velocity, advection, pressure, or a complete fluid solver.\\\"},\\n velocity:{number:\\\"checkpoint 2\\\",title:\\\"Two numbers in every cell\\\",text:\\\"A velocity field stores a horizontal component vx and a vertical component vy at each location. An arrow visualizes that pair.\\\",instruction:\\\"Drag from one cell toward another to assign a direction, or press example.\\\",code:\\\"velocity.set(x, y, { vx, vy });\\\\n\\\\nrenderVectorField(velocity);\\\",notice:\\\"Arrow direction shows the vector direction. Arrow length represents its magnitude.\\\",notYet:\\\"The arrows describe possible motion, but nothing follows them yet. Transport comes next with advection.\\\"},\\n advection:{number:\\\"checkpoint 3\\\",title:\\\"Use velocity to transport dye\\\",text:\\\"Advection connects the fields. For every destination cell, we trace backward along its velocity and sample the dye value found there.\\\",instruction:\\\"Press example, then advance the dye one step at a time.\\\",code:\\\"dye = advectScalarField(\\\\n dye,\\\\n velocity\\\\n);\\\\n\\\\nrenderScalarField(dye);\\\",notice:\\\"The velocity field is not the dye. It tells the update where each new dye value should come from.\\\",notYet:\\\"This uses semi-Lagrangian sampling. It is stable and visual, but it can blur dye and does not conserve its total amount exactly.\\\"},\\n transportDiffusion:{number:\\\"checkpoint 4\\\",title:\\\"Transport, then spread\\\",text:\\\"A fluid step can apply multiple updates in sequence. First advection transports the dye. Then diffusion exchanges dye with neighboring cells.\\\",instruction:\\\"Press example, choose both strengths, and advance one combined step at a time.\\\",code:\\\"dye = advect(dye, velocity);\\\\ndye = diffuse(dye, diffusion);\\\\n\\\\nrenderScalarField(dye);\\\",notice:\\\"Advection changes where the pattern goes. Diffusion changes how sharp or spread-out the pattern is.\\\",notYet:\\\"This diffuses the dye quantity, not the velocity field. We also have not enforced incompressibility or solved pressure.\\\"},\\n divergence:{number:\\\"checkpoint 5\\\",title:\\\"Where flow expands or compresses\\\",text:\\\"Divergence measures the net flow leaving each cell. Positive values behave like sources; negative values behave like sinks; values near zero have balanced inflow and outflow.\\\",instruction:\\\"Press example. Red cells are sources, blue cells are sinks. Click a cell to inspect its divergence.\\\",code:\\\"divergence = measureDivergence(velocity);\\\\n\\\\n// target for incompressible flow\\\\ndivergence ≈ 0;\\\",notice:\\\"A velocity field can look smooth while still creating or destroying volume. Divergence gives us a number that exposes that problem.\\\",notYet:\\\"We only measure the error here. The next lesson will solve pressure and subtract its gradient to reduce divergence.\\\"},\\n projection:{number:\\\"checkpoint 6\\\",title:\\\"Use pressure to correct velocity\\\",text:\\\"Projection converts divergence into a pressure field, then subtracts the pressure gradient from velocity. The correction pushes away from compression and resists expansion.\\\",instruction:\\\"Press example, note the red and blue error, then press step to project the velocity.\\\",code:\\\"divergence = measureDivergence(velocity);\\\\npressure = solvePressure(divergence);\\\\nvelocity -= gradient(pressure);\\\",notice:\\\"Pressure is not painted manually. It is solved from the divergence error, used to correct velocity, and recalculated on the next step.\\\",notYet:\\\"The iterative pressure solve is approximate, and wall boundary conditions are still simplified. The next lesson will make boundaries explicit.\\\"},\\n boundaries:{number:\\\"checkpoint 7\\\",title:\\\"Fluid cannot cross a solid wall\\\",text:\\\"A wall removes the velocity component pointing through it. Motion parallel to the wall can remain, so the fluid slides along the surface instead of escaping.\\\",instruction:\\\"Press example. Red wall cells leak outward; press step to enforce no-through boundaries.\\\",code:\\\"velocity = enforceNoThroughWalls(velocity);\\\\n\\\\nnormalVelocityAtWall = 0;\\\",notice:\\\"At left and right walls we remove horizontal velocity. At top and bottom walls we remove vertical velocity.\\\",notYet:\\\"This is a free-slip wall: tangential motion remains. We have not yet implemented no-slip friction, obstacles, or coupled projection at the boundary.\\\"},\\n temperature:{number:\\\"checkpoint 8\\\",title:\\\"Heat creates upward acceleration\\\",text:\\\"Temperature is another scalar field. Hotter fluid is less dense, so buoyancy adds upward velocity where temperature is high.\\\",instruction:\\\"Press example, choose a buoyancy strength, then press step. Watch the arrows grow upward inside the hot region.\\\",code:\\\"velocity = applyBuoyancy(\\\\n velocity,\\\\n temperature\\\\n);\\\",notice:\\\"Temperature does not move anything directly. It changes velocity; velocity can move dye later through advection.\\\",notYet:\\\"Temperature does not diffuse or advect yet. We also omit ambient cooling, gravity calibration, density, and wax phase changes.\\\"},\\n thermalFlow:{number:\\\"checkpoint 9\\\",title:\\\"The local-force gap\\\",text:\\\"Buoyancy changes velocity only where heat already exists. Backward advection in neighboring stationary cells cannot discover that heat, so a bottom row can remain pinned.\\\",instruction:\\\"Paint the bottom row and step. This deliberately preserves the gap you noticed.\\\",code:\\\"velocity = applyBuoyancy(velocity, temperature);\\\\ntemperature = advect(temperature, velocity);\\\",notice:\\\"Composing two correct operations is not enough if the velocity field has not been made into a coherent fluid-wide flow.\\\",notYet:\\\"This version skips pressure projection and boundary enforcement between force and transport. The next module inserts them.\\\"},\\n coupledThermal:{number:\\\"checkpoint 10\\\",title:\\\"Spread the push before transport\\\",text:\\\"Pressure projection turns the local buoyant push into a divergence-reduced velocity field. Wall enforcement keeps that corrected flow inside before it transports temperature.\\\",instruction:\\\"Paint a hot bottom row or press example, then step. Compare its motion with Module 11.\\\",code:\\\"velocity = applyBuoyancy(velocity, temperature);\\\\nvelocity = project(velocity);\\\\nvelocity = enforceWalls(velocity);\\\\ntemperature = advect(temperature, velocity);\\\",notice:\\\"Projection does not move heat. It reshapes velocity so neighboring cells participate; advection then moves the temperature.\\\",notYet:\\\"The heater is not continuous and temperature does not diffuse or cool. Those processes come next.\\\"},\\n heatedFlow:{number:\\\"checkpoint 11\\\",title:\\\"Temperature needs a source and a sink\\\",text:\\\"A real lamp keeps receiving energy from its bulb while losing energy to its surroundings. Each step adds heat near the bottom and removes a percentage everywhere.\\\",instruction:\\\"Press clear, then step repeatedly. Adjust heater and cooling to see whether heat accumulates or fades.\\\",code:\\\"temperature = cool(temperature);\\\\ntemperature = addBottomHeat(temperature);\\\\nvelocity = applyBuoyancyAndProject(velocity, temperature);\\\\ntemperature = advect(temperature, velocity);\\\",notice:\\\"The temperature field is no longer just initial data. It is driven over time by competing input and loss rates.\\\",notYet:\\\"Cooling is uniform rather than stronger at the glass, and heat does not diffuse by conduction. We still have only one material.\\\"},\\n conductiveFlow:{number:\\\"checkpoint 12\\\",title:\\\"Heat also travels without bulk motion\\\",text:\\\"Conduction exchanges temperature between neighboring cells. The glass boundary then removes extra heat only from edge cells, creating a spatially meaningful heat sink.\\\",instruction:\\\"Press clear and step repeatedly. Compare low and high conduction, then change wall cooling.\\\",code:\\\"temperature = addBottomHeat(temperature);\\\\ntemperature = diffuseTemperature(temperature);\\\\ntemperature = coolAtWalls(temperature);\\\\nvelocity = solveFlow(velocity, temperature);\\\\ntemperature = advect(temperature, velocity);\\\",notice:\\\"Advection carries heat with moving fluid. Conduction spreads heat between neighbors even when the local velocity is zero.\\\",notYet:\\\"The walls all use one cooling rate and the heater has an abstract strength. We still have one continuous material rather than wax in a carrier liquid.\\\"},\\n waxField:{number:\\\"checkpoint 13\\\",title:\\\"Material identity is another scalar field\\\",text:\\\"Wax fraction stores how much of each cell is wax: 0 is carrier liquid, 1 is wax, and values between represent a numerically mixed cell.\\\",instruction:\\\"Press example or paint wax, then step. Pink shows wax while the underlying warmth still affects the shared velocity.\\\",code:\\\"wax = advect(wax, velocity);\\\\ntemperature = advect(temperature, velocity);\\\",notice:\\\"Wax fraction and temperature coexist at every cell. Advection can transport both fields through the same velocity without confusing what they mean.\\\",notYet:\\\"Wax does not yet have different density or viscosity, and it does not melt or solidify. This is material tracking, not phase-change physics.\\\"},\\n densityWax:{number:\\\"checkpoint 14\\\",title:\\\"Wax buoyancy depends on relative density\\\",text:\\\"Temperature changes wax density relative to the carrier. Wax hotter than neutral gets an upward force; colder wax gets a downward force.\\\",instruction:\\\"Press example: the left blob is hot and low, the right blob is cold and high. Step and compare their directions.\\\",code:\\\"densityError = wax * (neutralTemperature - temperature);\\\\nvelocity.y += densityError * response;\\\",notice:\\\"The force combines temperature and material identity. Equally hot carrier liquid does not receive this wax-specific density force.\\\",notYet:\\\"Values are normalized rather than kg/m³, and the relationship is linear. Wax still has no temperature-dependent viscosity or solid state.\\\"},\\n phaseWax:{number:\\\"checkpoint 15\\\",title:\\\"Temperature changes how freely wax moves\\\",text:\\\"Wax identity remains constant, but a smooth phase fraction is derived from temperature. Cold wax is mostly solid and damps local velocity; hot wax is mostly liquid and follows the flow.\\\",instruction:\\\"Press example, then step. Watch the initially cold wax resist motion until heating moves it through the melting range.\\\",code:\\\"liquid = smoothMelt(temperature, meltPoint, width);\\\\nvelocity *= 1 - wax * (1 - liquid) * solidDrag;\\\",notice:\\\"Material identity and phase state are different: wax remains wax while its liquid fraction changes continuously from 0 to 1.\\\",notYet:\\\"Solid drag is a teaching approximation, not a full viscosity or viscoelastic stress solve. Latent heat and surface tension are still absent.\\\"},\\n interfaceWax:{number:\\\"checkpoint 16\\\",title:\\\"A wax boundary is made of fractional cells\\\",text:\\\"Bilinear advection creates cells containing partial wax. The renderer now respects that fraction, and an optional sharpening pass nudges mixed values toward 0 or 1.\\\",instruction:\\\"Press example and step. Compare 0% and 100% sharpening, and inspect the wax value at the edge.\\\",code:\\\"wax = advect(wax, velocity);\\\\nwax = sharpenInterface(wax, strength);\\\",notice:\\\"Faint edge cells are numerical mixtures, not fully solid wax. Wax fraction must affect both the simulation and its rendering.\\\",notYet:\\\"This smoothstep sharpening is not mass-conserving and is not a full VOF, level-set, or surface-tension method.\\\"},\\n latentWax:{number:\\\"checkpoint 17\\\",title:\\\"Melting consumes energy\\\",text:\\\"Wax now stores melt progress separately from temperature. Near the melting point, added energy advances the phase instead of immediately making the wax hotter.\\\",instruction:\\\"Press example, select a wax cell, and step. Watch temperature linger near the melt point while melt progress rises.\\\",code:\\\"energy = exchangeLatentHeat(temperature, meltProgress);\\\\n// temperature plateaus while phase changes\\\",notice:\\\"A phase transition can store energy without a matching temperature increase. Solidifying releases that stored energy back into temperature.\\\",notYet:\\\"The latent capacity is normalized and approximate; we do not yet use joules, specific heat, or a fully conservative enthalpy formulation.\\\"},\\n surfaceWax:{number:\\\"checkpoint 18\\\",title:\\\"The wax boundary creates a force\\\",text:\\\"A gradient estimates the interface normal. Changes in that normal estimate curvature. Surface tension applies a force proportional to curvature near the wax boundary.\\\",instruction:\\\"Press example and step. Compare surface tension at 0 and 1; cyan cells mark the mixed interface.\\\",code:\\\"normal = normalize(gradient(wax));\\\\ncurvature = -divergence(normal);\\\\nforce = surfaceTension * curvature * gradient(wax);\\\",notice:\\\"Surface tension acts at the interface, not throughout the wax. It resists highly curved, ragged boundaries and encourages coherent blobs.\\\",notYet:\\\"This coarse continuum-surface-force approximation can create numerical currents and is not quantitatively calibrated.\\\"},\\n viscosityWax:{number:\\\"checkpoint 19\\\",title:\\\"Viscosity diffuses momentum\\\",text:\\\"Each cell moves its velocity toward the average of neighboring velocities. The blend strength depends on whether the cell is carrier, hot liquid wax, or cold wax.\\\",instruction:\\\"Press example and step. Compare low and high cold-wax viscosity, then inspect viscosity at hot and cold wax cells.\\\",code:\\\"viscosity = mixByMaterialAndPhase(carrier, hotWax, coldWax);\\\\nvelocity = diffuseVelocity(velocity, viscosity);\\\",notice:\\\"Diffusing dye spreads a scalar quantity. Viscosity uses similar neighbor exchange on velocity, smoothing differences in momentum.\\\",notYet:\\\"These are normalized per-step blend rates, not Pa·s. Variable-viscosity diffusion is approximate and solid drag remains a separate teaching force.\\\"},\\n stableLamp:{number:\\\"checkpoint 20\\\",title:\\\"One frame can contain several smaller solves\\\",text:\\\"The complete physics order is fixed. A frame time is divided by the selected substep count, and rate-like updates are scaled by that smaller duration.\\\",instruction:\\\"Press example. Compare one substep with eight at the same time per frame, and watch maximum travel per substep.\\\",code:\\\"for (const dt of subdivide(frameTime)) {\\\\n addHeatAndCool(dt);\\\\n updatePhaseAndForces(dt);\\\\n diffuseAndProjectVelocity(dt);\\\\n enforceWalls();\\\\n advectFields(dt);\\\\n}\\\",notice:\\\"Substeps do not add new physics. They reduce how far information travels before fields are recomputed, improving numerical stability.\\\",notYet:\\\"This still uses normalized units and explicit approximations. The final lesson adds continuous time, playback controls, and lamp-oriented presets.\\\"},\\n simulator:{number:\\\"checkpoint 21\\\",title:\\\"The complete lamp runs continuously\\\",text:\\\"A fixed-rate animation loop repeatedly executes the stable timestep. Presets choose coherent groups of parameters rather than changing one coefficient in isolation.\\\",instruction:\\\"Choose a preset, press example, then play. Pause at any time to inspect a cell or advance one frame manually.\\\",code:\\\"while (playing) {\\\\n accumulator += elapsedTime;\\\\n while (accumulator >= frameInterval)\\\\n stepCompleteLamp();\\\\n}\\\",notice:\\\"The simulator runtime does not add physics. It schedules the validated timestep repeatedly and exposes diagnostics that reveal drift or instability.\\\",notYet:\\\"Parameters remain normalized and visually calibrated. This is a complete educational browser simulator, not a quantitatively validated industrial multiphase CFD model.\\\"},\\n unevenHeating:{number:\\\"checkpoint 22\\\",title:\\\"Perfect symmetry prevents separate blobs\\\",text:\\\"A broad wax pool now rests above a heater whose strength varies across space and slowly over time. Small thermal differences seed separate buoyant regions.\\\",instruction:\\\"Press example, compare unevenness at 0% and 100%, then play. Watch for localized hot spots instead of uniform lifting.\\\",code:\\\"localHeat = heater * (1 + spatialPerturbation);\\\\ntemperature.bottom += localHeat;\\\",notice:\\\"The perturbation does not create motion directly. It creates temperature differences that density and buoyancy amplify into different rising regions.\\\",notYet:\\\"Pinch-off and merging still depend on the coarse interface method. The next step focuses on necking, separation, and recombination.\\\"},\\n pinchOff:{number:\\\"checkpoint 23\\\",title:\\\"Thin wax bridges can become separate blobs\\\",text:\\\"The coarse grid cannot resolve a neck thinner than one cell, so this lesson adds an explicit topology rule: detect a strained, partially filled bridge, remove wax from its neck, and return that wax to its thicker neighbors.\\\",instruction:\\\"Press example, then play. Cyan cells are neck candidates. Compare pinch strength at 0% and 100%, and watch separated blobs merge naturally when advection brings their interfaces together.\\\",code:\\\"neck = detectThinBridge(wax, velocity);\\\\nremoved = pinch(neck);\\\\nredistributeConservatively(removed);\\\",notice:\\\"Pinch-off is a change in connectivity. The algorithm severs only narrow, strained bridges and conserves the wax it moves. Recombination needs no opposite rule: overlapping interfaces already become one connected field.\\\",notYet:\\\"This is a transparent teaching approximation, not a resolved Rayleigh–Plateau instability. A finer adaptive grid or particle/interface method would produce breakup from the governing forces themselves.\\\"},\\n conservation:{number:\\\"checkpoint 24\\\",title:\\\"Wax may move, but its total amount should not change\\\",text:\\\"Advection and interface sharpening operate cell by cell, so rounding and nonlinear sharpening can create or destroy wax numerically. This lesson records total wax before each substep and audits it afterward.\\\",instruction:\\\"Press example and play. Set mass correction to 0% to expose raw drift, then 100% and compare the two diagnostics.\\\",code:\\\"targetMass = sum(waxBefore);\\\\nrunPhysics();\\\\nerror = targetMass - sum(waxAfter);\\\\nredistribute(error, interfaceCells);\\\",notice:\\\"Conservation is a constraint on the whole field, not a property of one cell. Individual cell values may change while their sum remains constant.\\\",notYet:\\\"The correction preserves normalized wax volume, not physical mass derived from density and cell volume. Energy and momentum also deserve separate conservation audits.\\\"},\\n energyBudget:{number:\\\"checkpoint 25\\\",title:\\\"Heat has sources, sinks, and storage\\\",text:\\\"Unlike wax volume, thermal energy is not expected to remain constant. The heater adds it, ambient and wall cooling remove it, and melting transfers it between temperature and latent storage.\\\",instruction:\\\"Press example and play. Change heater and cooling strength, then watch which ledger entries respond. Stored latent energy can rise even while temperature pauses.\\\",code:\\\"energy = sum(temperature + meltProgress);\\\\nchange = heaterInput + coolingLoss + numericalDrift;\\\",notice:\\\"A conservation law is an accounting identity: system change equals energy crossing the boundary plus numerical error. Constant energy is expected only in a closed system.\\\",notYet:\\\"The ledger uses normalized energy units and omits kinetic and gravitational potential energy. The next refinement would audit the complete mechanical-plus-thermal budget.\\\"},\\n closedVessel:{number:\\\"checkpoint 26\\\",title:\\\"A closed lamp has a fixed material inventory\\\",text:\\\"Previously, painting set a cell to wax without removing wax anywhere else. Conservation then faithfully preserved that newly enlarged total. In this lesson, painting relocates wax from the existing field instead.\\\",instruction:\\\"Press example. Note the total wax, then click or drag through empty cells. Wax leaves other locations—starting with the bottom reservoir—as the selected cells fill.\\\",code:\\\"amount = withdraw(waxInventory);\\\\ntarget.wax += amount;\\\\n// carrier fraction is always 1 - wax fraction\\\",notice:\\\"Every cell is already full: wax fraction plus carrier-liquid fraction equals one. Adding wax to a cell means displacing carrier there and taking wax from somewhere else in the closed lamp.\\\",notYet:\\\"The relocation gesture is an educational editing constraint, not fluid motion. During play, advection and pressure remain responsible for physical transport.\\\"},\\n cycleDiagnostics:{number:\\\"checkpoint 27\\\",title:\\\"A cycle is a pattern measured over time\\\",text:\\\"One frame cannot tell us whether wax is circulating. We track the wax-weighted center height and the fractions currently moving upward or downward, then retain a short history.\\\",instruction:\\\"Press example and play. Look for the height trace to rise and fall repeatedly. A flat trace means the lamp has approached a steady arrangement, even if local cells still move.\\\",code:\\\"height = weightedCenter(wax);\\\\nrising = sum(wax where velocity.y < 0);\\\\nfalling = sum(wax where velocity.y > 0);\\\",notice:\\\"Oscillation is a property of a sequence of states, not one state. Center of mass compresses the whole wax field into one useful—but incomplete—signal.\\\",notYet:\\\"The trace does not identify individual blobs. Object tracking would be needed to measure each blob’s lifetime, speed, split, and merge events.\\\"},\\n particleVolume:{number:\\\"checkpoint 35\\\",title:\\\"Wax particles need pressure as well as cohesion\\\",text:\\\"A smoothing kernel estimates crowding. Density above the target creates outward pressure, preventing many particles from occupying one point.\\\",instruction:\\\"Press example and play. Raise volume pressure until wax spans several cells, then tune cohesion to keep that finite volume connected.\\\",code:\\\"density = kernelSum(neighbors);\\\\npressure = max(0, density - restDensity);\\\",notice:\\\"Pressure preserves particle volume; cohesion shapes it. Bilinear transfer couples each particle to four grid cells.\\\",notYet:\\\"This is an SPH-style teaching approximation, not an iterative PBF solve.\\\"},\\n shapedLamp:{number:\\\"checkpoint 34\\\",title:\\\"The vessel shape completes the circulation path\\\",text:\\\"Wax is a configurable fixed particle mass moving through the projected carrier grid. Uneven heating creates internal density differences, while hot particles weaken their cohesion enough for a large cluster to neck and separate.\\\",instruction:\\\"Press example and play. Compare low and high base concentration. Watch fallen wax gather centrally, reheat locally, and launch again.\\\",code:\\\"inside = lampShape(cell);\\\\nheat = focusedBaseHeater(x);\\\\nenforceCurvedWalls(velocity);\\\",notice:\\\"More particles create a deformable wax body rather than a tiny point cluster. Cold particles cohere strongly at the base; unevenly heated particles rise at different rates and can pull the body apart.\\\",notYet:\\\"The particle-to-grid transfer is deliberately simple and the rendered surface is an overlap visualization rather than a calibrated SPH kernel.\\\"},\\n finalLamp:{number:\\\"checkpoint 33\\\",title:\\\"The complete educational lava lamp\\\",text:\\\"The final mode runs one authoritative timestep: heat, cooling, phase exchange, buoyancy, surface force, viscosity, pressure projection, walls, transport, pinch-off, and conservation.\\\",instruction:\\\"Choose a preset, press example, then play. Use the compact diagnostics to judge whether wax is forming blobs and completing a rise-and-fall cycle.\\\",code:\\\"while (playing) stepLavaLamp();\\\",notice:\\\"The final picture is the consequence of interacting fields and constraints, not a scripted blob animation.\\\",notYet:\\\"This remains a coarse normalized teaching model, not a quantitatively validated industrial multiphase CFD solver.\\\"},\\n physicalUnits:{number:\\\"checkpoint 32\\\",title:\\\"Normalized values need a scale to become units\\\",text:\\\"A velocity of one means one cell per simulation frame. Choose a cell width and frame duration to interpret that same value as centimeters per second.\\\",instruction:\\\"Play, then change cell width and seconds per frame. The motion is unchanged; only its physical interpretation changes.\\\",code:\\\"speedCmPerSecond = velocityCellsPerFrame * cellWidthCm / secondsPerFrame;\\\",notice:\\\"Units do not appear automatically. A nondimensional simulation becomes physical only after selecting reference scales.\\\",notYet:\\\"The coefficients are visually calibrated and cannot yet be claimed as measured wax viscosity, density, or surface tension.\\\"},\\n calibration:{number:\\\"checkpoint 31\\\",title:\\\"A lamp is calibrated as a system\\\",text:\\\"Heating, cooling, viscosity, density response, and surface tension must produce a bounded repeating cycle together. No single best slider value exists in isolation.\\\",instruction:\\\"Compare the balanced, slow, and active presets. Watch height range and blob count instead of judging only instantaneous speed.\\\",code:\\\"preset = calibrateTogether({heat,cooling,viscosity,density,surface});\\\",notice:\\\"Calibration chooses a useful regime of behavior; it does not add new governing physics.\\\",notYet:\\\"These presets are visual targets rather than fits to measurements from a specific manufactured lamp.\\\"},\\n motionMatch:{number:\\\"checkpoint 30\\\",title:\\\"Identity matching needs motion tolerance\\\",text:\\\"A fast blob may move into entirely new cells between frames. When direct overlap fails, the tracker searches nearby previous cells before declaring a new identity.\\\",instruction:\\\"Play and compare a search radius of zero with one or two cells. Larger radii preserve identity through faster motion but can confuse nearby blobs.\\\",code:\\\"match = overlap || nearbyPreviousIdentity(searchRadius);\\\",notice:\\\"Tracking is an inference problem layered over the simulation. Greater tolerance reduces lost identities but increases ambiguous matches.\\\",notYet:\\\"The matcher expands spatially rather than predicting a velocity-directed future centroid.\\\"},\\n blobIdentity:{number:\\\"checkpoint 29\\\",title:\\\"Identity comes from overlap across frames\\\",text:\\\"Each connected region compares its cells with the previous frame. The strongest overlap inherits the old identity; unmatched regions receive a new one.\\\",instruction:\\\"Press example and play. Outline colors now persist as blobs travel. Splits share a parent identity and merges overlap multiple previous identities.\\\",code:\\\"identity = largestOverlap(currentBlob, previousBlobs);\\\\nif (!identity) identity = nextId++;\\\",notice:\\\"Connectivity answers what is one blob now. Temporal matching answers whether it is the same blob as before.\\\",notYet:\\\"Fast motion can defeat pure overlap matching. A production tracker would also compare centroid, volume, and velocity.\\\"},\\n blobTracking:{number:\\\"checkpoint 28\\\",title:\\\"A blob is a connected region, not a special particle\\\",text:\\\"We threshold the wax fraction, then flood-fill neighboring wax cells. Every connected component receives a label, size, and colored outline.\\\",instruction:\\\"Press example and play. Change the connection threshold and watch weak fractional bridges either join regions or separate them. Split and merge counts respond when the component count changes.\\\",code:\\\"mask = wax > threshold;\\\\nblobs = connectedComponents(mask, fourNeighbors);\\\",notice:\\\"Blob identity is derived from the wax field. The simulation never stores a blob object; connected shapes emerge from neighboring cell values.\\\",notYet:\\\"A changing component count is only an apparent split or merge. Robust event tracking must match component overlap across frames so threshold flicker is not mistaken for topology change.\\\"}\\n };\\n\";\n",[],[149,225,227],"66bbe82a-d375-49a5-be76-d5e3f08b4dd2","05-rendering-and-input.ts","61265abe-8a96-11f1-8774-1607ee4eb77e",238,["D",1785251390439],"export const renderingAndInput = \" function colorForLatentWax(tempValue,waxValue,progress){\\n var liquid=waxValue>0?Math.max(0,Math.min(1,progress/waxValue)):0;\\n var phaseR=150+90*liquid,phaseG=95-20*liquid,phaseB=205-55*liquid;\\n var baseR=11,baseG=16,baseB=23;\\n if(tempValue>0){baseR=40+215*tempValue;baseG=35+95*tempValue;baseB=50-25*tempValue;}\\n return \\\"rgb(\\\"+Math.round(baseR*(1-waxValue)+phaseR*waxValue)+\\\",\\\"+Math.round(baseG*(1-waxValue)+phaseG*waxValue)+\\\",\\\"+Math.round(baseB*(1-waxValue)+phaseB*waxValue)+\\\")\\\";\\n }\\n\\n function colorForPhaseWax(tempValue,waxValue){\\n var liquid=liquidWaxFraction(tempValue);\\n var phaseR=150+90*liquid,phaseG=95-20*liquid,phaseB=205-55*liquid;\\n var baseR=11,baseG=16,baseB=23;\\n if(tempValue>0){baseR=40+215*tempValue;baseG=35+95*tempValue;baseB=50-25*tempValue;}\\n return \\\"rgb(\\\"+Math.round(baseR*(1-waxValue)+phaseR*waxValue)+\\\",\\\"+Math.round(baseG*(1-waxValue)+phaseG*waxValue)+\\\",\\\"+Math.round(baseB*(1-waxValue)+phaseB*waxValue)+\\\")\\\";\\n }\\n\\n function colorForWax(tempValue,waxValue){\\n var baseR=11,baseG=16,baseB=23;\\n if(tempValue>0){baseR=40+215*tempValue;baseG=35+95*tempValue;baseB=50-25*tempValue;}\\n var waxR=224,waxG=80,waxB=170;\\n return \\\"rgb(\\\"+Math.round(baseR*(1-waxValue)+waxR*waxValue)+\\\",\\\"+Math.round(baseG*(1-waxValue)+waxG*waxValue)+\\\",\\\"+Math.round(baseB*(1-waxValue)+waxB*waxValue)+\\\")\\\";\\n }\\n\\n function colorForTemperature(value){\\n if(value<=0)return \\\"#0b1017\\\";\\n var r=Math.round(40+215*value),g=Math.round(35+95*value),b=Math.round(50-25*value);\\n return \\\"rgb(\\\"+r+\\\",\\\"+g+\\\",\\\"+b+\\\")\\\";\\n }\\n\\n function seedDivergenceExample(){\\n var source={x:8,y:9},sink={x:20,y:9};\\n for(var y=0;y1){vx/=length;vy/=length;}\\n velocityX[indexOf(x,y)]=vx;velocityY[indexOf(x,y)]=vy;\\n }\\n computeDivergence();\\n }\\n function velocityAt(array,x,y){\\n x=Math.max(0,Math.min(WIDTH-1,x));y=Math.max(0,Math.min(HEIGHT-1,y));return array[indexOf(x,y)];\\n }\\n function computeDivergence(){\\n for(var y=0;y0){sum+=at(x-1,y);count++;} if(x0){sum+=at(x,y-1);count++;} if(y0)return \\\"rgb(\\\"+Math.round(22+145*amount)+\\\",\\\"+Math.round(25+45*amount)+\\\",\\\"+Math.round(34+25*amount)+\\\")\\\";\\n if(value<0)return \\\"rgb(\\\"+Math.round(22+45*amount)+\\\",\\\"+Math.round(25+35*amount)+\\\",\\\"+Math.round(34+145*amount)+\\\")\\\";\\n return \\\"#111821\\\";\\n }\\n function colorFor(value){\\n if(value<=0) return \\\"#0b1017\\\";\\n var r=Math.round(40+value*65),g=Math.round(105+value*110),b=Math.round(170+value*80);\\n return \\\"rgb(\\\"+r+\\\",\\\"+g+\\\",\\\"+b+\\\")\\\";\\n }\\n function draw(){\\n ctx.clearRect(0,0,canvas.width,canvas.height);\\n for(var y=0;y=0?[\\\"#50a7ff\\\",\\\"#ffd166\\\",\\\"#ef6f9a\\\",\\\"#75d39b\\\",\\\"#b28cff\\\",\\\"#ff9f5a\\\"][blobLabels[indexOf(x,y)]%6]:mode===\\\"pinchOff\\\"&&neckMask[indexOf(x,y)]?\\\"#7de3ff\\\":mode===\\\"surfaceWax\\\"&&wax[indexOf(x,y)]>.05&&wax[indexOf(x,y)]<.95?\\\"#7de3ff\\\":\\\"#28313c\\\";ctx.strokeRect(x*CELL+.5,y*CELL+.5,CELL-1,CELL-1);\\n }\\n if(mode===\\\"shapedLamp\\\"||mode===\\\"particleVolume\\\")drawWaxParticles();\\n if(mode===\\\"velocity\\\"||mode===\\\"advection\\\"||mode===\\\"transportDiffusion\\\"||mode===\\\"divergence\\\"||mode===\\\"projection\\\"||mode===\\\"boundaries\\\"||mode===\\\"temperature\\\"||mode===\\\"thermalFlow\\\"||mode===\\\"coupledThermal\\\"||mode===\\\"heatedFlow\\\"||mode===\\\"conductiveFlow\\\"||mode===\\\"waxField\\\"||mode===\\\"densityWax\\\"||mode===\\\"phaseWax\\\"||mode===\\\"interfaceWax\\\"||mode===\\\"latentWax\\\"||mode===\\\"surfaceWax\\\"||mode===\\\"viscosityWax\\\"||mode===\\\"stableLamp\\\"||mode===\\\"simulator\\\"||mode===\\\"unevenHeating\\\"||mode===\\\"pinchOff\\\"||mode===\\\"conservation\\\"||mode===\\\"energyBudget\\\"||mode===\\\"closedVessel\\\"||mode===\\\"cycleDiagnostics\\\"||mode===\\\"blobTracking\\\"||mode===\\\"blobIdentity\\\"||mode===\\\"motionMatch\\\"||mode===\\\"calibration\\\"||mode===\\\"physicalUnits\\\"||mode===\\\"finalLamp\\\"||mode===\\\"shapedLamp\\\"||mode===\\\"particleVolume\\\"){\\n for(var y=0;y.01){ctx.beginPath();ctx.moveTo(endX,endY);ctx.lineTo(endX-head*Math.cos(angle-.55),endY-head*Math.sin(angle-.55));ctx.lineTo(endX-head*Math.cos(angle+.55),endY-head*Math.sin(angle+.55));ctx.closePath();ctx.fill();}\\n ctx.lineWidth=1;\\n }\\n function cellFromEvent(event){\\n var rect=canvas.getBoundingClientRect();\\n return {x:Math.max(0,Math.min(WIDTH-1,Math.floor((event.clientX-rect.left)*canvas.width/rect.width/CELL))),y:Math.max(0,Math.min(HEIGHT-1,Math.floor((event.clientY-rect.top)*canvas.height/rect.height/CELL)))};\\n }\\n function relocateWax(cell){\\n if(mode===\\\"shapedLamp\\\"&&!vesselMask[indexOf(cell.x,cell.y)])return 0;\\n var target=indexOf(cell.x,cell.y),needed=1-wax[target],moved=0;\\n if(needed<=0)return 0;\\n for(var y=HEIGHT-2;y>=1&&needed>1e-6;y--)for(var x=1;x1e-6;x++){\\n var donor=indexOf(x,y);if(donor===target||wax[donor]<=0)continue;\\n var take=Math.min(needed,wax[donor]);var ratio=wax[donor]>0?take/wax[donor]:0;\\n wax[donor]-=take;meltProgress[donor]*=1-ratio;wax[target]+=take;needed-=take;moved+=take;\\n }\\n document.getElementById(\\\"inventoryMetric\\\").textContent=totalWax().toFixed(1);\\n document.getElementById(\\\"movedWaxMetric\\\").textContent=moved.toFixed(2);\\n return moved;\\n }\\n\\n function interact(event){\\n var cell=cellFromEvent(event);\\n if(mode===\\\"closedVessel\\\"||mode===\\\"cycleDiagnostics\\\"||mode===\\\"blobTracking\\\"||mode===\\\"blobIdentity\\\"||mode===\\\"motionMatch\\\"||mode===\\\"calibration\\\"||mode===\\\"physicalUnits\\\"||mode===\\\"finalLamp\\\"||mode===\\\"shapedLamp\\\"||mode===\\\"particleVolume\\\"){\\n selected=cell;if(painting||event.type===\\\"pointerdown\\\")relocateWax(cell);\\n }else if(mode===\\\"waxField\\\"||mode===\\\"densityWax\\\"||mode===\\\"phaseWax\\\"||mode===\\\"interfaceWax\\\"||mode===\\\"latentWax\\\"||mode===\\\"surfaceWax\\\"||mode===\\\"viscosityWax\\\"||mode===\\\"stableLamp\\\"||mode===\\\"simulator\\\"||mode===\\\"unevenHeating\\\"||mode===\\\"pinchOff\\\"||mode===\\\"conservation\\\"||mode===\\\"energyBudget\\\"||mode===\\\"closedVessel\\\"||mode===\\\"cycleDiagnostics\\\"||mode===\\\"blobTracking\\\"||mode===\\\"blobIdentity\\\"||mode===\\\"motionMatch\\\"||mode===\\\"calibration\\\"||mode===\\\"physicalUnits\\\"||mode===\\\"finalLamp\\\"||mode===\\\"shapedLamp\\\"||mode===\\\"particleVolume\\\"){\\n selected=cell;if(painting||event.type===\\\"pointerdown\\\")wax[indexOf(cell.x,cell.y)]=1;\\n }else if(mode===\\\"temperature\\\"||mode===\\\"thermalFlow\\\"||mode===\\\"coupledThermal\\\"||mode===\\\"heatedFlow\\\"||mode===\\\"conductiveFlow\\\"){\\n selected=cell;if(painting||event.type===\\\"pointerdown\\\")temperature[indexOf(cell.x,cell.y)]=1;\\n }else if(mode===\\\"velocity\\\"||mode===\\\"divergence\\\"||mode===\\\"projection\\\"||mode===\\\"boundaries\\\"){\\n if(!vectorOrigin) vectorOrigin=cell;\\n selected=vectorOrigin;\\n var dx=cell.x-vectorOrigin.x,dy=cell.y-vectorOrigin.y,length=Math.sqrt(dx*dx+dy*dy);\\n if(length>0){velocityX[indexOf(selected.x,selected.y)]=dx/length;velocityY[indexOf(selected.x,selected.y)]=dy/length;if(mode===\\\"divergence\\\"||mode===\\\"projection\\\"){computeDivergence();if(mode===\\\"projection\\\")updateProjectionMetric(null,meanAbsoluteDivergence());}if(mode===\\\"boundaries\\\")updateBoundaryMetric(null,countBoundaryLeaks());}\\n }else{\\n selected=cell;\\n if(mode!==\\\"grid\\\" && (painting||event.type===\\\"pointerdown\\\")) set(cell.x,cell.y,1);\\n }\\n draw();updateReadout();\\n }\\n function updateReadout(){\\n document.getElementById(\\\"coordinate\\\").textContent=selected ? \\\"(\\\"+selected.x+\\\", \\\"+selected.y+\\\")\\\" : \\\"—\\\";\\n document.getElementById(\\\"index\\\").textContent=selected ? indexOf(selected.x,selected.y) : \\\"—\\\";\\n document.getElementById(\\\"value\\\").textContent=selected ? ((mode===\\\"divergence\\\"||mode===\\\"projection\\\") ? divergence[indexOf(selected.x,selected.y)].toFixed(3) : (mode===\\\"velocity\\\"||mode===\\\"boundaries\\\") ? \\\"(\\\"+velocityX[indexOf(selected.x,selected.y)].toFixed(2)+\\\", \\\"+velocityY[indexOf(selected.x,selected.y)].toFixed(2)+\\\")\\\" : (mode===\\\"stableLamp\\\"||mode===\\\"simulator\\\"||mode===\\\"unevenHeating\\\"||mode===\\\"pinchOff\\\"||mode===\\\"conservation\\\"||mode===\\\"energyBudget\\\"||mode===\\\"closedVessel\\\"||mode===\\\"cycleDiagnostics\\\"||mode===\\\"blobTracking\\\"||mode===\\\"blobIdentity\\\"||mode===\\\"motionMatch\\\"||mode===\\\"calibration\\\"||mode===\\\"physicalUnits\\\"||mode===\\\"finalLamp\\\"||mode===\\\"shapedLamp\\\"||mode===\\\"particleVolume\\\") ? \\\"wax \\\"+wax[indexOf(selected.x,selected.y)].toFixed(2)+\\\" / speed \\\"+Math.sqrt(velocityX[indexOf(selected.x,selected.y)]**2+velocityY[indexOf(selected.x,selected.y)]**2).toFixed(2) : mode===\\\"viscosityWax\\\" ? \\\"wax \\\"+wax[indexOf(selected.x,selected.y)].toFixed(2)+\\\" / visc \\\"+viscosityAtIndex(indexOf(selected.x,selected.y)).toFixed(2) : mode===\\\"surfaceWax\\\" ? \\\"wax \\\"+wax[indexOf(selected.x,selected.y)].toFixed(2)+\\\" / curve \\\"+surfaceCurvature[indexOf(selected.x,selected.y)].toFixed(2) : mode===\\\"latentWax\\\" ? \\\"melt \\\"+(wax[indexOf(selected.x,selected.y)]>0?meltProgress[indexOf(selected.x,selected.y)]/wax[indexOf(selected.x,selected.y)]:0).toFixed(2)+\\\" / temp \\\"+temperature[indexOf(selected.x,selected.y)].toFixed(2) : (mode===\\\"phaseWax\\\"||mode===\\\"interfaceWax\\\") ? \\\"wax \\\"+wax[indexOf(selected.x,selected.y)].toFixed(2)+\\\" / liquid \\\"+liquidWaxFraction(temperature[indexOf(selected.x,selected.y)]).toFixed(2)+\\\" / temp \\\"+temperature[indexOf(selected.x,selected.y)].toFixed(2) : (mode===\\\"waxField\\\"||mode===\\\"densityWax\\\") ? \\\"wax \\\"+wax[indexOf(selected.x,selected.y)].toFixed(2)+\\\" / temp \\\"+temperature[indexOf(selected.x,selected.y)].toFixed(2) : (mode===\\\"temperature\\\"||mode===\\\"thermalFlow\\\"||mode===\\\"coupledThermal\\\"||mode===\\\"heatedFlow\\\"||mode===\\\"conductiveFlow\\\") ? temperature[indexOf(selected.x,selected.y)].toFixed(3)+\\\" / vy \\\"+velocityY[indexOf(selected.x,selected.y)].toFixed(2) : at(selected.x,selected.y).toFixed(3)) : \\\"—\\\";\\n document.getElementById(\\\"ticks\\\").textContent=tickCount;\\n }\\n\\n\";\n",[],[149,225,235],"1d67f4cb-4338-4dfc-b34c-7df0a2554100","04-diagnostics-and-runtime.ts","5d88d04e-8a96-11f1-b3cb-1607ee4eb77e",237,["D",1785251384397],"export const diagnosticsAndRuntime = \" function recordCycleDiagnostics(){\\n var total=0,heightSum=0,rising=0,falling=0;\\n for(var y=0;y.03)falling+=w;\\n }\\n var height=total>0?heightSum/total:0;\\n cycleHistory.push(height);if(cycleHistory.length>100)cycleHistory.shift();\\n document.getElementById(\\\"waxHeightMetric\\\").textContent=Math.round(height*100)+\\\"%\\\";\\n document.getElementById(\\\"risingMetric\\\").textContent=Math.round((total?rising/total:0)*100)+\\\"%\\\";\\n document.getElementById(\\\"fallingMetric\\\").textContent=Math.round((total?falling/total:0)*100)+\\\"%\\\";\\n var chart=document.getElementById(\\\"cycleChart\\\"),c=chart.getContext(\\\"2d\\\"),w=chart.width,h=chart.height;\\n c.clearRect(0,0,w,h);c.fillStyle=\\\"#0b1017\\\";c.fillRect(0,0,w,h);\\n c.strokeStyle=\\\"#303640\\\";c.lineWidth=1;for(var g=1;g<4;g++){var gy=g*h/4;c.beginPath();c.moveTo(0,gy);c.lineTo(w,gy);c.stroke();}\\n if(cycleHistory.length>1){c.strokeStyle=\\\"#50a7ff\\\";c.lineWidth=2;c.beginPath();for(var n=0;n=0?\\\"+\\\":\\\"\\\")+drift.toFixed(3);\\n }\\n\\n function stepEnergyBudgetLamp(){\\n var count=+substeps.value;stepScale=(+timeStep.value)/count;\\n for(var n=0;n0?meltTotal/waxTotal:0).toFixed(2);document.getElementById(\\\"speedMetric\\\").textContent=maxSpeed.toFixed(2);\\n }\\n\\n function stepFinalLamp(){stepStableLamp();updateSimulatorDiagnostics();}\\n\\n function setRunning(value){running=value;document.getElementById(\\\"play\\\").textContent=running?\\\"pause\\\":\\\"play\\\";if(!running)animationAccumulator=0;}\\n\\n function applyLampPreset(){\\n var preset=lampPreset.value;\\n if(preset===\\\"slow\\\"){heater.value=.08;cooling.value=.03;densityResponse.value=.8;surfaceTension.value=.5;hotViscosity.value=.2;coldViscosity.value=.5;timeStep.value=.75;substeps.value=6;}\\n else if(preset===\\\"active\\\"){heater.value=.2;cooling.value=.02;densityResponse.value=1.8;surfaceTension.value=.25;hotViscosity.value=.05;coldViscosity.value=.35;timeStep.value=1.25;substeps.value=6;}\\n else{heater.value=.12;cooling.value=.02;densityResponse.value=1.2;surfaceTension.value=.35;hotViscosity.value=.1;coldViscosity.value=.45;timeStep.value=1;substeps.value=4;}\\n document.getElementById(\\\"heaterOut\\\").textContent=(+heater.value).toFixed(2);document.getElementById(\\\"coolingOut\\\").textContent=Math.round(+cooling.value*100)+\\\"%\\\";document.getElementById(\\\"densityOut\\\").textContent=(+densityResponse.value).toFixed(2);document.getElementById(\\\"surfaceOut\\\").textContent=(+surfaceTension.value).toFixed(2);document.getElementById(\\\"hotViscosityOut\\\").textContent=Math.round(+hotViscosity.value*100)+\\\"%\\\";document.getElementById(\\\"coldViscosityOut\\\").textContent=Math.round(+coldViscosity.value*100)+\\\"%\\\";document.getElementById(\\\"timeStepOut\\\").textContent=(+timeStep.value).toFixed(2);document.getElementById(\\\"substepsOut\\\").textContent=substeps.value;\\n }\\n\\n function animationLoop(now){\\n if(!lastAnimationTime)lastAnimationTime=now;\\n var elapsed=Math.min(.25,(now-lastAnimationTime)/1000);lastAnimationTime=now;\\n if(running&&(mode===\\\"simulator\\\"||mode===\\\"unevenHeating\\\"||mode===\\\"pinchOff\\\"||mode===\\\"conservation\\\"||mode===\\\"energyBudget\\\"||mode===\\\"closedVessel\\\"||mode===\\\"cycleDiagnostics\\\"||mode===\\\"blobTracking\\\"||mode===\\\"blobIdentity\\\"||mode===\\\"motionMatch\\\"||mode===\\\"calibration\\\"||mode===\\\"physicalUnits\\\"||mode===\\\"finalLamp\\\"||mode===\\\"shapedLamp\\\"||mode===\\\"particleVolume\\\")){\\n animationAccumulator+=elapsed*(+playbackRate.value);var safety=0;\\n while(animationAccumulator>=1&&safety<4){if(mode===\\\"particleVolume\\\")stepParticleVolume();else if(mode===\\\"shapedLamp\\\")stepShapedLamp();else if(mode===\\\"motionMatch\\\"||mode===\\\"calibration\\\"||mode===\\\"physicalUnits\\\"||mode===\\\"finalLamp\\\")stepRemainingLesson();else if(mode===\\\"blobIdentity\\\")stepBlobIdentityLamp();else if(mode===\\\"blobTracking\\\")stepBlobTrackingLamp();else if(mode===\\\"cycleDiagnostics\\\")stepCycleDiagnosticsLamp();else if(mode===\\\"closedVessel\\\")stepClosedVesselLamp();else if(mode===\\\"energyBudget\\\")stepEnergyBudgetLamp();else if(mode===\\\"conservation\\\")stepConservationLamp();else if(mode===\\\"pinchOff\\\")stepPinchLamp();else if(mode===\\\"unevenHeating\\\")stepUnevenLamp();else stepFinalLamp();animationAccumulator-=1;safety++;}\\n }\\n requestAnimationFrame(animationLoop);\\n }\\n\\n\";\n",[],[149,225,243],"4ec7a4f9-b5fc-4f4d-aaf9-538341e80a50","03-particle-lamp.ts","5a17043a-8a96-11f1-a7b4-1607ee4eb77e",236,["D",1785251378628],"export const particleLamp = \" function buildLampMask(){var c=(WIDTH-1)/2;for(var y=0;y0&&y0)p.vy*=-.12;var limit=lampHalfWidth(p.y)-.48;if(p.xc+limit){p.x=c+limit;p.vx=-Math.abs(p.vx)*.2;}}\\n function updateParticleThermal(){var c=(WIDTH-1)/2,width=4-3*(+baseFocus.value),dt=stepScale,hotspot=1.8*Math.sin(tickCount*.045);for(var n=0;nHEIGHT-5.5)p.temp+=+heater.value*dt*Math.exp(-d*d)*uneven;p.temp*=1-(+cooling.value*dt);var wallGap=lampHalfWidth(p.y)-Math.abs(p.x-c);if(p.y<4||wallGap<1.2)p.temp*=1-(+glassCooling.value*dt);p.temp=Math.max(0,Math.min(1,p.temp));}}\\n function applyParticleForces(){var dt=stepScale,c=+collectionStrength.value,neutral=+neutralTemp.value,response=+densityResponse.value;for(var a=0;a2.25)continue;var d=Math.sqrt(d2),phaseCohesion=.22+.78*(1-(p.temp+o.temp)*.5),force=(d-.58)*.032*c*phaseCohesion*dt,fx=force*dx/d,fy=force*dy/d;p.vx+=fx;p.vy+=fy;o.vx-=fx;o.vy-=fy;}}\\n function applyParticleVolumePressure(){var count=waxParticles.length,h=1.3,density=new Float32Array(count),maxD=0,target=+restDensity.value,strength=+volumePressure.value*stepScale;for(var a=0;a=h)continue;var excess=Math.max(0,(density[a]+density[q])*.5-target),hard=Math.max(0,.48-d)*2,force=(excess*(1-d/h)*.055+hard*.12)*strength,fx=force*dx/d,fy=force*dy/d;p.vx-=fx;p.vy-=fy;o.vx+=fx;o.vy+=fy;}document.getElementById(\\\"particleDensityMetric\\\").textContent=maxD.toFixed(1);}\\n function depositParticleVelocityBilinear(){velocityX.fill(0);velocityY.fill(0);var weights=new Float32Array(WIDTH*HEIGHT);for(var n=0;n0){sum+=temperature[indexOf(x-1,y)];count++;}if(x0){sum+=temperature[indexOf(x,y-1)];count++;}if(ymelt&&meltProgress[i]0){\\n var released=Math.min((melt-temperature[i])*coupling,meltProgress[i]);\\n meltProgress[i]-=released;temperature[i]+=released;\\n }\\n }\\n }\\n\\n function applyLatentPhaseDrag(){\\n var drag=+solidDrag.value*stepScale;\\n for(var i=0;i0?Math.max(0,Math.min(1,meltProgress[i]/wax[i])):0;\\n var keep=1-wax[i]*(1-liquid)*drag;\\n velocityX[i]*=keep;velocityY[i]*=keep;\\n }\\n }\\n\\n function sampleMeltProgress(x,y){\\n x=Math.max(0,Math.min(WIDTH-1,x));y=Math.max(0,Math.min(HEIGHT-1,y));\\n var x0=Math.floor(x),y0=Math.floor(y),x1=Math.min(WIDTH-1,x0+1),y1=Math.min(HEIGHT-1,y0+1),tx=x-x0,ty=y-y0;\\n var top=meltProgress[indexOf(x0,y0)]*(1-tx)+meltProgress[indexOf(x1,y0)]*tx;\\n var bottom=meltProgress[indexOf(x0,y1)]*(1-tx)+meltProgress[indexOf(x1,y1)]*tx;\\n return top*(1-ty)+bottom*ty;\\n }\\n\\n function advectMeltProgress(){\\n var stepDistance=+distance.value*stepScale;\\n for(var y=0;y.0001?gx/length:0;surfaceNormalY[i]=length>.0001?gy/length:0;\\n }\\n for(var y=0;y0?Math.max(0,Math.min(1,meltProgress[i]/w)):0;\\n var waxViscosity=(+coldViscosity.value)*(1-liquid)+(+hotViscosity.value)*liquid;\\n return (+carrierViscosity.value)*(1-w)+waxViscosity*w;\\n }\\n\\n function diffuseVelocityByViscosity(){\\n for(var y=0;y0){sumX+=velocityX[indexOf(x-1,y)];sumY+=velocityY[indexOf(x-1,y)];count++;}\\n if(x0){sumX+=velocityX[indexOf(x,y-1)];sumY+=velocityY[indexOf(x,y-1)];count++;}\\n if(y.92)continue;\\n var left=wax[indexOf(x-1,y)],right=wax[indexOf(x+1,y)],up=wax[indexOf(x,y-1)],down=wax[indexOf(x,y+1)];\\n var vertical=up+down, horizontal=left+right;\\n var verticalBridge=vertical>1.05&&horizontal<(1.05-.7*sensitivity);\\n var horizontalBridge=horizontal>1.05&&vertical<(1.05-.7*sensitivity);\\n var strain=Math.abs(velocityY[indexOf(x,y-1)]-velocityY[indexOf(x,y+1)])+Math.abs(velocityX[indexOf(x-1,y)]-velocityX[indexOf(x+1,y)]);\\n if(!(verticalBridge||horizontalBridge)||strain<.04*(1-sensitivity))continue;\\n var cut=Math.min(w,.18*strength*stepScale*(.6+Math.min(2,strain)));\\n if(cut<=0)continue;neckMask[i]=1;waxNext[i]-=cut;removed+=cut;count++;\\n var a=verticalBridge?indexOf(x,y-1):indexOf(x-1,y),b=verticalBridge?indexOf(x,y+1):indexOf(x+1,y);\\n var capacityA=1-waxNext[a],capacityB=1-waxNext[b],capacity=capacityA+capacityB;\\n if(capacity>0){var putA=Math.min(capacityA,cut*capacityA/capacity),putB=Math.min(capacityB,cut-putA);waxNext[a]+=putA;waxNext[b]+=putB;}\\n }\\n wax.set(waxNext);\\n var after=totalWax(),error=before-after;\\n if(Math.abs(error)>1e-7){\\n for(var pass=0;pass<3&&Math.abs(error)>1e-6;pass++){\\n var capacity=0;\\n for(var i=0;i0?(wax[i]>.05?1-wax[i]:0):wax[i];\\n if(capacity<=0)break;\\n for(var i=0;i0?(wax[i]>.05?1-wax[i]:0):wax[i];wax[i]=Math.max(0,Math.min(1,wax[i]+error*room/capacity));}\\n error=before-totalWax();\\n }\\n }\\n document.getElementById(\\\"neckMetric\\\").textContent=count;\\n document.getElementById(\\\"waxErrorMetric\\\").textContent=Math.abs(before-totalWax()).toFixed(3);\\n }\\n\\n function runPinchSubstep(){\\n addUnevenBottomHeat();\\n var ambientKeep=1-(+cooling.value*stepScale);\\n for(var i=0;i1e-7;pass++){\\n var capacity=0;\\n for(var i=0;i0?(wax[i]>.02?1-wax[i]:0):wax[i];\\n if(capacity<=0)break;\\n for(var i=0;i0?(wax[i]>.02?1-wax[i]:0):wax[i];\\n wax[i]=Math.max(0,Math.min(1,wax[i]+remaining*room/capacity));\\n }\\n remaining=target-totalWax();\\n }\\n document.getElementById(\\\"rawDriftMetric\\\").textContent=(raw>=0?\\\"+\\\":\\\"\\\")+raw.toFixed(3);\\n var corrected=totalWax()-target;\\n document.getElementById(\\\"correctedDriftMetric\\\").textContent=(corrected>=0?\\\"+\\\":\\\"\\\")+corrected.toFixed(3);\\n }\\n\\n function totalThermalEnergy(){var total=0;for(var i=0;i=0)continue;var cells=[],ci=components.length;blobLabels[st]=ci;queue=[st];for(var q=0;q=WIDTH||ny<0||ny>=HEIGHT)continue;var ni=indexOf(nx,ny);if(wax[ni]>=threshold&&blobLabels[ni]<0){blobLabels[ni]=ci;queue.push(ni);}}}components.push(cells);}\\n var ids=[],matched=0,parents={};\\n for(var c=0;c=0)votes[old]=(votes[old]||0)+1;else if(persistent){var rr=+matchRadius.value,cx=components[c][k]%WIDTH,cy=Math.floor(components[c][k]/WIDTH);for(var sy=Math.max(0,cy-rr);sy<=Math.min(HEIGHT-1,cy+rr);sy++)for(var sx=Math.max(0,cx-rr);sx<=Math.min(WIDTH-1,cx+rr);sx++){var nearby=previousBlobLabels[indexOf(sx,sy)];if(nearby>=0)votes[nearby]=(votes[nearby]||0)+.15;}}}var best=-1,bestN=0,keys=Object.keys(votes);for(var j=0;jbestN){best=id;bestN=votes[id];}}if(persistent&&best>=0){ids[c]=best;matched++;parents[best]=(parents[best]||0)+1;if(countEvents&&keys.length>1)mergeEvents++;}else ids[c]=nextBlobId++;}\\n if(countEvents)for(var id in parents)if(parents[id]>1)splitEvents+=parents[id]-1;\\n var total=totalWax(),largest=0;for(var c=0;c1){vx/=speed;vy/=speed;}\\n velocityX[indexOf(x,y)]=vx;velocityY[indexOf(x,y)]=vy;\\n }\\n }\\n function isBoundaryLeak(x,y){\\n var i=indexOf(x,y);\\n return (x===0&&velocityX[i]<0)||(x===WIDTH-1&&velocityX[i]>0)||(y===0&&velocityY[i]<0)||(y===HEIGHT-1&&velocityY[i]>0);\\n }\\n function countBoundaryLeaks(){\\n var count=0;for(var y=0;y

Diff · v240