cleanup of forw ns2
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NT
@@ -8,6 +8,11 @@
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"source": [
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"# Differentiable Fluid Simulations\n",
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"\n",
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"\n",
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"**TODO , NEXT update diffphys example, then check burgers examples**\n",
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"\n",
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"\n",
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"\n",
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"Next, we'll target a more complex example with the Navier-Stokes equations as model. We'll target a 2D case with velocity $\\mathbf{u}$, no explicit viscosity term, and a marker density $d$ that drives a simple Boussinesq buoyancy term $\\eta d$ adding a force along the y dimension:\n",
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"\n",
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"$\\begin{aligned}\n",
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@@ -8,8 +8,6 @@
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"source": [
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"# Navier-Stokes Forward Simulation\n",
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"\n",
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"**TODO , double check here - sync with latest local-colab-notebook version, then update diffphys example**\n",
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"\n",
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"Now let's target a somewhat more complex example: a fluid simulations based on the Navier-Stokes equations. This is still very simple with Φ<sub>Flow</sub>, as differentiable operators for all steps exist. The Navier-Stokes equations (in their incompressible form) introduce an additional pressure field $p$, and a constraint for conservation of mass:\n",
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"\n",
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"$\\begin{align}\n",
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@@ -69,14 +67,14 @@
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"execution_count": 2,
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"metadata": {
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"id": "WrA3IXDxv31P"
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},
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"outputs": [],
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"source": [
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"DOMAIN = Domain(x=40, y=32, boundaries=CLOSED, bounds=Box[0:100, 0:80])\n",
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"INFLOW = domain.grid(Sphere(center=(30, 15), radius=10)) * 0.2\n",
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"INFLOW = DOMAIN.grid(Sphere(center=(30, 15), radius=10)) * 0.2\n",
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"\n",
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"DT = 1.5\n",
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"NU = 0.01"
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@@ -98,7 +96,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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@@ -118,7 +116,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"execution_count": 4,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/",
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@@ -138,10 +136,10 @@
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{
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"data": {
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"text/plain": [
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"<matplotlib.image.AxesImage at 0x7f7f88ebd4c0>"
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"<matplotlib.image.AxesImage at 0x7fb2c07a7c40>"
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]
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},
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"execution_count": 12,
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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},
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@@ -159,15 +157,15 @@
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}
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],
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"source": [
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"def step(velocity, smoke, pressure, buoyancy_factor=1.0):\n",
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" smoke = advect.semi_lagrangian(smoke, velocity, dt) + inflow\n",
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"def step(velocity, smoke, pressure, dt=1.0, buoyancy_factor=1.0):\n",
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" smoke = advect.semi_lagrangian(smoke, velocity, dt) + INFLOW\n",
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" buoyancy_force = smoke * (0, buoyancy_factor) >> velocity # resamples smoke to velocity sample points\n",
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" velocity = advect.semi_lagrangian(velocity, velocity, 1) + dt * buoyancy_force\n",
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" velocity = diffuse.explicit(velocity, nu, dt)\n",
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" velocity, pressure, iterations, divergence = fluid.make_incompressible(velocity, domain, pressure_guess=pressure)\n",
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" velocity = diffuse.explicit(velocity, NU, dt)\n",
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" velocity, pressure, iterations, divergence = fluid.make_incompressible(velocity, DOMAIN, pressure_guess=pressure)\n",
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" return velocity, smoke, pressure\n",
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"\n",
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"velocity, smoke, pressure = step(velocity, smoke, None)\n",
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"velocity, smoke, pressure = step(velocity, smoke, None, dt=DT)\n",
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"\n",
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"print(\"Max. velocity and mean density: \" + format( [ math.max(velocity.values) , math.mean(smoke.values) ] ))\n",
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"\n",
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@@ -184,36 +182,88 @@
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"\n",
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"The pressure projection step in `make_incompressible` is typically the computationally most expensive step in the sequence above. It solves a Poisson equation for the boundary conditions of the domain, and updates the velocity field with the gradient of the computed pressure.\n",
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"\n",
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"Just for testing, we've also printed the mean value of the velocities, and the max density after the update. As you can see in the resulting image, we have a first round region of smoke, with a slight upwards motion (which does not show here yet). "
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"Just for testing, we've also printed the mean value of the velocities, and the max density after the update. As you can see in the resulting image, we have a first round region of smoke, with a slight upwards motion (which does not show here yet). \n",
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"\n",
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"## Datatypes and Dimensions\n",
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"\n",
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"The created grids are instances of the class `Grid`.\n",
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"Like tensors, grids also have the `shape` attribute which lists all batch, spatial and channel dimensions.\n",
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"[Shapes in Φ<sub>Flow</sub>](https://tum-pbs.github.io/PhiFlow/Math.html#shapes) store not only the sizes of the dimensions but also their names and types."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Smoke: (x=40, y=32)\n",
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"Velocity: (x=40, y=32, vector=2)\n",
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"Inflow: (x=40, y=32)\n",
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"Inflow, spatial only: (x=40, y=32)\n"
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]
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}
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],
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"source": [
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"# ? print(\"Fluid state: \" + format(fluid.state))\n",
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"# ? print(\"\\nVelocity content:\")\n",
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"# ? [print(grid.data.shape) for grid in fluid.velocity.unstack()];"
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"print(f\"Smoke: {smoke.shape}\")\n",
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"print(f\"Velocity: {velocity.shape}\")\n",
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"print(f\"Inflow: {INFLOW.shape}\")\n",
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"print(f\"Inflow, spatial only: {INFLOW.shape.spatial}\")\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# TODO\n",
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"\n",
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"Note that we actually created two variables, one for each velocity component. If you're interested in how this works, have a look at the [Struct documentation](https://github.com/tum-pbs/PhiFlow/blob/master/documentation/Structs.ipynb).\n",
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"\n",
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"If you look closely at the output from the last `print` command, you'll notice that the shapes of the variables differ. This is because the velocity is sampled in [staggered form](https://github.com/tum-pbs/PhiFlow/blob/master/documentation/Staggered_Grids.md).\n",
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"The grid values can be accessed using the `values` property."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"(x=40, y=32) float32 0.0 < ... < 0.2\n",
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"(x=(41, 40) along vector, y=(32, 33) along vector, vector=2) float32 -0.10927753 < ... < 0.135354\n",
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"(x=40, y=32) float32 0.0 < ... < 0.2\n"
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]
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}
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],
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"source": [
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"print(smoke.values)\n",
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"print(velocity.values)\n",
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"print(INFLOW.values)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Grids have many more properties which are documented [here](https://tum-pbs.github.io/PhiFlow/phi/field/#phi.field.Grid).\n",
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"Also note that the staggered grid has a [non-uniform shape](https://tum-pbs.github.io/PhiFlow/Math.html#non-uniform-tensors) because the number of faces is not equal to the number of cells. The `INFLOW` grid naturally has the same dimensions as the `smoke` grid.\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Time Evolution\n",
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"\n",
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"With this setup, we can easily advance the simulation forward in time a bit more by repeatedly calling the `step` function."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"execution_count": 7,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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@@ -241,7 +291,7 @@
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],
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"source": [
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"for time_step in range(10):\n",
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" velocity, smoke, pressure = step(velocity, smoke, pressure)\n",
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" velocity, smoke, pressure = step(velocity, smoke, pressure, dt=DT)\n",
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" print('Computed frame {}, max velocity {}'.format(time_step , np.asarray(math.max(velocity.values)) ))\n"
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]
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},
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@@ -256,7 +306,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"execution_count": 8,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/",
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@@ -269,10 +319,10 @@
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{
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"data": {
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"text/plain": [
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"<matplotlib.image.AxesImage at 0x7f7f8390fdf0>"
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"<matplotlib.image.AxesImage at 0x7fb2bc1153d0>"
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]
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},
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"execution_count": 6,
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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},
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@@ -304,7 +354,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"execution_count": 9,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/",
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@@ -342,7 +392,7 @@
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"for time_step in range(20):\n",
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" if time_step<3 or time_step%10==0: \n",
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" print('Computing time step %d' % time_step)\n",
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" velocity, smoke, pressure = step(velocity, smoke, pressure)\n",
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" velocity, smoke, pressure = step(velocity, smoke, pressure, dt=DT)\n",
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" if time_step%5==0:\n",
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" steps.append(smoke.values.numpy('y,x'))\n",
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"\n",
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