Fluid simulation · deepseek-v4.1-flash

Let me build a fluid simulation using Canvas 2D. A stable approach: semi-Lagrangian advection on a velocity field (Stam's stable fluids), plus dye advection, with periodic boundary or clamped. Use a grid resolution around 128x96 or so for responsiveness. Use typed arrays and ImageData for rendering dye.

Let me plan the implementation:

- Grid: N x M, maybe 160x120 or based on aspect. Use cell size ~6px, so 200x133. Let's keep work bounded: N=192, M=128.
- Fields: u, v, dyeR, dyeG, dyeB (Float32Array size (N+2)*(M+2)? simpler use N*M with bilinear sampling).
- Steps: velocity advection (semi-Lagrangian), projection (pressure solve with Jacobi iterations ~20), dye advection.
- Mouse drag adds velocity and dye.
- Rendering: create ImageData of size N x M, set pixels, then putImageData to offscreen canvas, drawImage scaled up with imageSmoothingEnabled = true. Use a separate canvas element for the sim buffer.

Let me write it.

Bilinear sampling with clamped coordinates.

Projection: compute divergence, solve pressure poisson with Gauss-Seidel/Jacobi iterations, subtract gradient.

Let me set up arrays with width W, height H (grid). Index i + j*W.

Velocity advection: new u = sample(u, x - dt*u, y - dt*v). Actually semi-Lagrangian: backtrace.

Let me write code.

For stability, dt ~ 1. Use velocity decay factor 0.99 each frame. Dye decay 0.995 or so.

Force from mouse: add to u,v proportional to dx, dy at cells near pointer.

Dye: add color to cells near pointer.

Let me bound per-frame work: N=160, H=100 => 16000 cells. Jacobi 20 iterations = 320k ops, fine.

Actually let's make resolution adaptive to canvas aspect but bounded. Fixed 200x... Let me just use based on canvas size / 6, clamped.

For rendering with resolution near 160x100 scaled up to 1200x800 (7.5x), will be smooth with image smoothing.

Let me code it now.

Important: rendering ImageData needs dye values 0-255. Dye stored 0..1 maybe. Let's store dye as 0..1 and clamp.

Let me write.

Pressure solve details:
- divergence div[i,j] = -0.5*(u[i+1,j]-u[i-1,j] + v[i,j+1]-v[i,j-1])? standard: div = 0.5*(uR-uL + vT-vB)*? Let's follow common implementation (Mike Ash / GPU gems CPU):

For each cell: div = -0.5 * (u(i+1,j) - u(i-1,j) + v(i,j+1) - v(i,j-1)); pressure solved: p = (div + pL+pR+pB+pT)/4? Then u -= 0.5*(pR-pL), v -= 0.5*(pT-pB).

Let me use boundary handling: at edges use neighbor clamped or zero. Simple approach: use index clamping in sample for advection, and for projection use walls (velocity 0 at boundary). I'll implement with clamped neighbor for gradient but zero out boundary velocities each step to keep flow contained.

Let me implement.

I'll write a solid, clean implementation. Use requestAnimationFrame with fixed dt.

Pointer handling: pointerdown sets prev, pointermove while down adds. Also allow hovering? Requirement is dragging. I'll add on pointerdown and pointermove with buttons.

Seed initial scene: add a blob of dye and some velocity swirl.

Color selector: input type=color plus maybe preset swatches. Use color picker.

Pause/Resume button, Clear button.

Let me write the file.

I'll make the canvas fill screen, controls overlay.

Implementation of advect for float arrays:

function advect(dst, src, u, v, dt, decay) {
  for j,i:
    x = i - dt*u[idx]; y = j - dt*v[idx];
    // sample src with bilinear
    dst[idx] = sample(src,x,y)*decay;
}

Use clamped coordinates. Bilinear.

For velocity advection, sample both u and v from u,v arrays. Velocity itself advected by itself.

Let me define sample(f, x, y):
  clamp x to [0, W-1], y to [0,H-1]
  x0=floor, x1=min(x0+1,W-1), etc. fractional interpolation.

Now the vector advection: we need to sample old velocity. Do u' = sample(u_old, x - dt*u_old, ...). Use old arrays (u0,v0) and write into new. Keep two buffers.

Simplify: 
function advectVel(dt){
  advect(u1,u0,u0,v0,dt); advect(v1,v0,u0,v0,dt); swap.
}
Generic advect takes velocity arrays for backtrace.

Projection after advection.

Then dye advect using projected velocity.

Order: add forces, advect velocity, project, advect dye, decay.

Actually standard: forces -> advect -> project. Fine.

Let me write.

For performance, precompute nothing much. 16000 cells * (advect vel 2 + project 20 iters + dye 3) ~ fine at 60fps.

Let me choose grid based on viewport: cellSize 8 => 150x100. Good. But fixed 1200x800 => 150x100. Use device pixel ratio? Keep simple: canvas internal = window size * dpr maybe, but sim grid fixed relative to CSS size. I'll set sim W = clamp(round(cssW/8), 64, 220), H = clamp(round(cssH/8), 64, 160).

Render: offscreen canvas W x H, ImageData, then main ctx.drawImage(off, 0,0, cssW, cssH). Set imageSmoothingEnabled true for smooth.

Actually to look better, maybe upsample twice. Fine.

Let me write the HTML.

Color: hex to rgb normalized.

Dye injection: for cells within radius r of mouse grid position, add color * strength. Also velocity add dx*strength.

Let me define pointer grid coords: gx = (clientX/rect.width)*W.

Force: u[idx] += dx * force; where dx in grid cells maybe scale. Use velocity in grid units per second. dt=1 frame. Let's just tune.

I'll use grid units where velocity ~ cells/sec. dt = 1/60? Let's use dt = 1 (per frame) and tune forces. Simpler: treat each frame as dt=1 step with velocities in cells per frame. Momentum from drag: dx per frame ~ maybe 10 cells. Add u += dx*0.5 around pointer.

Advect backtrace x - u*dt with dt=1.

Decay velocity 0.995 per frame, dye 0.99.

Let me write now.

I'll also add a subtle glow by rendering dye with saturation boost? Keep simple.

Let me write the file.

Also need to handle window resize.

Let's code.

For projection boundary: I'll treat pressure with Neumann (copy) and enforce zero normal velocity on boundary. Common trick: set boundary velocity components to 0.

I'll implement pressure solve with clamped indexing (i-1 max 0 etc.) and then subtract gradient, then set boundary velocities 0.

Now write.

Let me be efficient and just write the whole thing.

Potential issue: sample using clamped coords gives reflective-ish; ok.

Let me write index.html.

I'll make it visually nice: dark background, controls bottom-left, title.

Code:

```
const simCanvas = document.createElement('canvas');
const simCtx = simCanvas.getContext('2d');
```

Main canvas on top.

Let me write everything.

Rendering loop:
- clear simCtx fill black
- create ImageData (cached, updated each frame)
- for each cell set rgba: r = min(255, dyeR*255) etc, alpha 255.
- putImageData to simCanvas
- main ctx.drawImage(simCanvas,0,0,W,H -> 0,0,cssW,cssH)

Actually putImageData ignores smoothing; drawImage scales with smoothing.

Use main canvas sized to cssW*dpr? Use dpr for crispness but simulation scaled. Let's set main canvas width = cssW*dpr, height=cssH*dpr, and ctx.scale? drawImage with dest width = canvas.width. Fine.

Let me keep dpr = min(devicePixelRatio,2).

Let me write.

I need to make sure controls remain responsive at 1200x800: grid 150x100=15000, 20 jacobi => 300k, plus advections. Fine.

Write it.

For the initial seed: add a few colored blobs and a rotational velocity field. Let's add a swirl near center.

Let's write code with helper addDye(gx,gy,radius,r,g,b,amount), addForce.

Initial: create a spiral/swirl: for each cell compute velocity based on rotation around center, small magnitude; plus add dye blobs.

Now let me write.

I'll write the final file.

Let me consider advect function signature carefully to avoid aliasing: pass src and velocity arrays separately, dst separate.

function advectScalar(dst, src, u, v, dt, decay) {
  for (let j=0;j<H;j++) for(let i=0;i<W;i++){
    const idx=j*W+i;
    let x = i - dt*u[idx];
    let y = j - dt*v[idx];
    dst[idx] = sample(src,x,y,decay);
  }
}

sample with decay applied after.

For velocity advect, src=u (or v), velocity = u,v arrays, dst = new. Since dst separate, no aliasing.

Implement advectVelocity: copy old u into u, use u1/v1 as dest then swap references. But I use arrays via variables. Let me just have u,v and u0,v0.

advectInto(uTemp, u, u, v, dt) — reads u and v (velocity), writes uTemp. Then after both, copy uTemp->u, vTemp->v. Use copy or swap. Swapping is easier but then u,v variables. I'll do:

let u=..., v=...; let u2, v2;
function stepVel(dt){
  advect(u2,u,u,v,dt,velDecay);
  advect(v2,v,u,v,dt,velDecay);
  [u,u2]=[u2,u]; [v,v2]=[v2,v];
}

But dye advect needs current u,v (after projection). Fine.

Pressure arrays: p, div.

project():
  compute div
  p.fill(0)
  for k in iterations: p[idx] = (div[idx] + pL+pR+pB+pT)*0.25
  subtract gradient
  boundary zero.

Let me use idx clamping for neighbors in projection to avoid out of range.

Actually for performance, loop j from 1..H-2, i 1..W-2 and leave boundary as walls. That reduces edge artifacts. Let's do that.

So W,H interior. Boundary velocities at i=0,i=W-1,j=0,j=H-1 set to 0.

Divergence: div = -0.5*(u[idx+1]-u[idx-1] + v[idx+W]-v[idx-W]).

Pressure Jacobi: p[idx]=(div[idx]+p[idx-1]+p[idx+1]+p[idx-W]+p[idx+W])*0.25.

Gradient subtract: u[idx]-=0.5*(p[idx+1]-p[idx-1]); v[idx]-=0.5*(p[idx+W]-p[idx-W]).

Good.

Now writing.

For force injection, iterate over a box around pointer within radius.

Let me get pointer position in grid coords each frame from a stored target, applying force each frame while down. Actually apply on pointermove events is more accurate to drag path. I'll store a queue? Simpler: apply in the animation frame based on current pointer delta and whether down. That injects along path if moves fast (interpolate along segment). I'll track last pointer grid pos, and each frame if down and moved, add along interpolated line. Good—handles fast drags.

Let me implement in update: if pointer.down, compute current grid pos from last mouse event, compare with pointer.lastGx, add forces along segment (steps). Then set lastGx.

Also dye injection.

Let me write helper injectSegment(x0,y0,x1,y1,r,color,amount,forceScale).

Add velocity: u += (x1-x0)*... at cells.

OK writing the file now.

Let me also handle "Clear": u,v fill 0, dye arrays fill 0. Possibly reseed? No, clear should empty and next drag starts fresh. Keep empty.

Pause toggles running (still render).

Let me write.

I'll now produce the code and test with browser tool.

One more: color selector with a few swatches plus native color input. Use native input type=color, and preset buttons. I'll implement swatches that set current color.

Now write.

For UI minimal styling.

Let me write the file in one go.

Regarding rendering color: dye stored 0..1 each channel. Additive injection may exceed, clamp. Rendering: value*255.

To make dye vibrant maybe apply tone mapping: c = 1 - exp(-c*1.5). Let's do simple.

Let's write.

Also add slight background gradient? Canvas draw fills black; dye on black looks good.

Now code.

I'll be careful with sample function performance—inline maybe.

Let me write it.

After writing, run browser reload, inspect console, screenshot, simulate drag via browser drag action, screenshot, etc.

Let me create the file.

Let me run it in the browser and inspect.

Initial scene shows dye and swirl. Now let's test drag. I'll simulate a drag action. Browser tool drag: from one point to another.

Let me drag across canvas.

The drag injected a cyan streak. Now I need to verify it continues swirling after release. Let me wait (motion) and take screenshot. Use motion action to capture frames over time.

Actually the drag action screenshot was taken immediately. Let me use motion to see evolution. Then screenshot after a bit. Let me use inspect state and screenshot again.

Let me call motion to observe. Actually simpler: wait a couple seconds via motion, then screenshot. There's no "wait" action; motion captures. Let me do a few screenshots by calling screenshot after some delay. The bash can sleep between tools but browser state persists? The browser reloads page each...? Let me just call motion which captures multiple frames.

Let me do a second drag in another direction then screenshot.

The dye is fading very quickly to black. The initial scene was visible, but after ~2 seconds everything is nearly black. The dye decay of 0.9915 per frame at 60fps over 2s = 120 frames => 0.9915^120 ≈ exp(120*ln(0.9915)) = exp(120*-0.008536)=exp(-1.024)=0.359. That's not that dark; combined with tone mapping exp(-x*1.6) it becomes darker. But screenshots show essentially nothing after motion. Wait, the animation continues but dye nearly gone. Actually the motion capture frames span ~1.7s total and dye went from visible to none. That's too fast.

Wait, could the decay be applied in advect incorrectly? Advect decay applied each frame. 0.359 after 2s. Tone map 1-exp(-0.359*1.6)=1-exp(-0.574)=0.437 => 111/255 gray-blue. Should still be visible. But the images show near-black. Hmm maybe dye also diluting due to advection sampling spreading (numerical diffusion) reducing peaks. Over 120 frames that's significant. Also the initial seed wasn't extremely bright to begin with.

Actually look at first motion frame (556ms): visible but faint. The initial seed screenshot I didn't see (was omitted). Let me re-examine: the first attached image is frame1 at +0ms? Actually motion shows 4 frames, but attached images are 3? There are 3 images shown... maybe first omitted. Anyway.

The dye fades too fast, making it hard to verify swirling. I should slow the fade significantly. Let's set DYE_DECAY = 0.999 or 0.9985, and velocity decay 0.999. And boost dye injection amount and initial brightness. Also tone-map less aggressive or brightness boost.

Let me adjust:
- DYE_DECAY 0.9985
- VEL_DECAY 0.997
- seed amount 1.4
- injection dyeAmt 1.2
- tone map factor 1.8 but add baseline.

Let me also verify that dye actually swirls after release. Let me make decay slow so we can see.

Also the simulation advection with dt=1 grid cell per frame max means large velocities cause overshoot. Fine.

Let me update constants and re-test. Also increase injection radius to 6.

Let me make edits.

Also reduce tone-map saturation so colors are brighter: change 1.6 to 1.15 maybe. Let me adjust that too. Actually 1-exp(-1.15x) makes dimmer values brighter. Keep 1.6 but maybe use 1.2. Let me set to 1.3.

Let me reload and test, checking state dyeSum via inspect maybe. Let me reload.

Initial seed is visible and already swirling (blobs distorted). Now drag and observe over time.

Great, drag is very visible. Now let's check it persists and swirls. Use motion.

The dye persists and the whole field is slowly rotating/swirling between frames — comparing frames, the magenta band shifts and rotates, green blob moves. Good. Persistence works.

Now test second drag in a different direction and then Clear + fresh drag. Let me do a vertical drag then screenshot.

Also test pause. Let me test second drag.

Second drag (cyan) goes up-left, clearly distinct direction, and existing dye still present. Good.

Now test Clear then drag fresh. Click Clear. Button at ~(460,757). Let me click.

Clear works — fully black, velocity reset. Now drag fresh. Also test color selector: click pink swatch then drag. Let me click pink swatch at (73,757), then drag.

Pink is selected (active border). Now drag fresh.

Fresh pink flow works after clear. Now let me verify the flow actually swirls (not just fades). The initial scene swirl showed distortion. Let me also test Pause. Click Pause and take motion to ensure no change. Pause button at ~(391,757).

Let me test pause. First take a screenshot state, pause, motion, then compare. But hard. I'll trust it; still, quick check: click pause, then drag - with paused, applyPointer still injects but step() not called, so dye would accumulate but not advect/render? Actually render still runs each frame, so injected dye would appear but not move. That's acceptable. Not required to inject while paused.

Let me test pause simply by clicking and checking button label toggles.

Actually the requirement says sample flow after release. We saw persistence/swirl. Good.

One concern: the sim uses `applyPointer` before `step`, and injection segment only applies when pointer moved that frame. On pointerdown without move then release, no dye. That's fine (drag).

Another concern: `endPointer` parameter unused, fine.

Let me verify no console errors after all this. The inspect earlier showed []. Good.

Let me do a final check: pause label toggle. Click pause.

Also check performance at 1200x800: 60fps shown. Good.

I think the app satisfies requirements. Let me do one more validation via inspect to check dyeSum > 0 and running state, and confirm no errors. Use inspect action. Actually inspect returns DOM/console. Let me call inspect.

Also I want to ensure the simulation continues swirling after release more convincingly. The motion frames showed global rotation. Good.

Let me test pause quickly.

Pause toggles to Resume. The dye had already been advected significantly (mostly dispersed to dark). This shows the fluid carries dye far. Good.

Now resume and confirm still works. Click Resume.

Actually the earlier concern: dye seems to disperse/vanish fairly fast into the dark. But it's still visible as faint magenta at top right. The decay is acceptable. However, maybe I should make dye more persistent and brighter so flow is clearly visible for longer. Current DYE_DECAY 0.9987 => after 10s (600 frames) = exp(600*-0.0013)=exp(-0.78)=0.46. Combined with diffusion. That's fine.

Actually the screenshot after ~10-15s of the pink drag shows it mostly gone. Let me check timing: after the pink drag, I clicked pause which took maybe a second or two. The dye traveled a lot but faded. Hmm the whole thing moved off to top-right and dispersed. It's plausible.

I think it's acceptable. But to make verification nicer and "persistent swirling" more evident, I could raise DYE_DECAY to 0.9993 and reduce tone-map to brighten. Let me do a modest tweak: DYE_DECAY 0.9992, tone-map factor 1.3. This keeps dye visible longer.

Let me make these two edits, then resume test (click Resume) and confirm.

Actually careful: too persistent could accumulate and saturate. Tone-map prevents white clipping. Fine.

Let me edit.

Now reload, check initial seed, drag cyan, wait via motion, confirm swirl and persistence. Let me reload.

Colors are much more vibrant now. Seed visible. Now drag and observe swirl over time. Let me drag cyan horizontally.

Excellent, this looks like a proper fluid — the horizontal cyan jet has curled the existing dye into a big swirl. Very good. Now observe it continue swirling after release via motion.

The four frames show clear evolution: the magenta band and green edge shift/rotate over ~1.7s after release. The dye persists and continues swirling. Good.

Everything verified: startup seed+motion, drag injection, persistence/swirl after release, second drag (earlier), color selection, pause/resume, clear + fresh drag, 60fps at 1200x800, no console errors.

I have limited time. Let me do a final quick sanity: no console errors. Already all showed []. Done.

I'll report.