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https://github.com/gszep/fluid-structure-interactive

Evolving differentiable digital creatures in complex fluids
https://github.com/gszep/fluid-structure-interactive

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Evolving differentiable digital creatures in complex fluids

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# Interactive Fluid-Structure Simulation



## Bleeding-edge WebGPU



As WebGPU is still in active development, it is likely not yet available in release builds of recent browsers. In particular, chrome users on linux systems must use the following startup flags



```bash

--enable-unsafe-webgpu --enable-features=Vulkan,VulkanFromANGLE,DefaultANGLEVulkan

```



Other methods can be found [here](https://developer.chrome.com/en/docs/web-platform/webgpu/#use).



## References



### Fluid Simulation References



-   Jos Stam Paper : https://www.dgp.toronto.edu/public_user/stam/reality/Research/pdf/GDC03.pdf

-   Nvidia GPUGem's Chapter 38 : https://developer.nvidia.com/gpugems/gpugems/part-vi-beyond-triangles/chapter-38-fast-fluid-dynamics-simulation-gpu

-   Jamie Wong's Fluid simulation : https://jamie-wong.com/2016/08/05/webgl-fluid-simulation/

-   PavelDoGreat's Fluid simulation : https://github.com/PavelDoGreat/WebGL-Fluid-Simulation

-   Stam's Stable Fluids : https://pages.cs.wisc.edu/~chaol/data/cs777/stam-stable_fluids.pdf



### WebGPU References



-   WebGPU Official Reference : https://www.w3.org/TR/webgpu/

-   WGSL Official Reference : https://www.w3.org/TR/WGSL/

-   Get started with GPU Compute on the web : https://web.dev/gpu-compute/

-   Raw WebGPU Tutorial : https://alain.xyz/blog/raw-webgpu



## Theoretical Background



Let's begin with the Navier–Stokes and continuity equations for a two-dimensional velocity field $\mathbf{u}=\hat{\mathbf{x}}u+\hat{\mathbf{y}}v$ of an incompressible homogenous non-Newtonian fluid with viscosity law $\eta(\dot\gamma)$ that is only a function of a scalar shear rate $\dot\gamma$



$$

\rho\left(\frac{\partial\mathbf{u}}{\partial t}+\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}\right)

-\nabla\cdot\left(2\eta\mathbf{D}\right)+\nabla p=0\\

\mathrm{where}\quad

\nabla\cdot\mathbf{u}=0\\

\quad\\

\mathbf{D}:=

\frac{1}{2}\begin{pmatrix}

2\frac{\partial u}{\partial x} &

\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y} \\

\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y} &

2\frac{\partial v}{\partial y}

\end{pmatrix}\\

\quad\\

\dot\gamma:=\sqrt{

 \left(\frac{\partial u}{\partial x}\right)^2

+\frac{1}{2}\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)^2

+\left(\frac{\partial v}{\partial y}\right)^2

}

$$



Taking the curl of this vector equation to obtain a scalar equation for the vorticity $\omega$ allows us to eliminate the pressure term $\nabla p$ and vortex stretching term $\left(\boldsymbol{\omega}\cdot\nabla\right)\mathbf{u}$ since partial derivatives with respect to $z$ vanish



$$

\rho\left(\frac{\partial}{\partial t}+u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}\right)\omega=\nabla\times\nabla\cdot\left(2\eta\mathbf{D}\right)

\quad\mathrm{where}\quad \omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}

$$



Without loss of generality we can let the divergence-free velocity field be defined by a stream function $\psi$



$$

u=\frac{\partial\psi}{\partial y}

\qquad

v=-\frac{\partial\psi}{\partial x}

$$



yielding scalar non-linear equations in the vorticity $\omega$ and stream function $\psi$



$$

\omega=-\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)\psi\\

\rho\left(\frac{\partial}{\partial t}+\frac{\partial\psi}{\partial y}\frac{\partial}{\partial x}-\frac{\partial\psi}{\partial x}\frac{\partial}{\partial y}\right)\omega=\nabla\times\nabla\cdot\left(2\eta\mathbf{D}\right)

\quad\\\quad\\

\mathbf{D}:=

\frac{1}{2}\begin{pmatrix}

2\frac{\partial^2}{\partial x \partial y} &

\frac{\partial^2}{\partial y^2}-\frac{\partial^2}{\partial x^2} \\

\frac{\partial^2}{\partial y^2}-\frac{\partial^2}{\partial x^2} &

-2\frac{\partial^2}{\partial x \partial y}

\end{pmatrix}\psi\\

\quad\\

\dot\gamma:=\sqrt{

 2\left(\frac{\partial^2\psi}{\partial x \partial y}\right)^2

+\frac{1}{2}\left(\frac{\partial^2\psi}{\partial y^2}-\frac{\partial^2\psi}{\partial x^2}\right)^2

}

$$



Let's try to simplify the ghastly viscosity term



$$

\nabla\times\nabla\cdot\left(2\eta\mathbf{D}\right)=

\frac{\partial}{\partial x}\left(

 \frac{\partial(2\eta[\mathbf{D}]_{xy})}{\partial x}

+\frac{\partial(2\eta[\mathbf{D}]_{yy})}{\partial y}

\right)-

\frac{\partial}{\partial y}\left(

 \frac{\partial(2\eta[\mathbf{D}]_{xx})}{\partial x}

+\frac{\partial(2\eta[\mathbf{D}]_{yx})}{\partial y}

\right)\\

=

\left(

 \frac{\partial^2}{\partial x^2}

-\frac{\partial^2}{\partial y^2}\right)\left(2\eta[\mathbf{D}]_{xy}\right)

+

2\frac{\partial^2}{\partial x \partial y}\left(

\eta[\mathbf{D}]_{yy}

-\eta[\mathbf{D}]_{xx}

\right)\\

\quad\qquad=

-\left(

 \frac{\partial^2}{\partial x^2}

-\frac{\partial^2}{\partial y^2}\right)\left(

 \eta\left(

 \frac{\partial^2}{\partial x^2}

-\frac{\partial^2}{\partial y^2}\right)\psi

\right)

-4\frac{\partial^2}{\partial x \partial y}\left(

 \eta\frac{\partial^2 \psi}{\partial x \partial y }

\right)\\

\qquad\qquad\quad

=\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)\left(\eta\omega\right)

+2\left(

  \frac{\partial^2\psi}{\partial^2 x}\frac{\partial^2\eta}{\partial^2 y}

-2\frac{\partial^2\psi}{\partial x\partial y}\frac{\partial^2\eta}{\partial x\partial y}

+ \frac{\partial^2\psi}{\partial^2 y}\frac{\partial^2\eta}{\partial^2 x}

\right)

$$



## Numerical Integration



We begin with the two dimensional velocity field $(u,v)$ at the start time $t$ and grid points $(x,y)$. Lets assume a uniform discretization over the grid with spacing $\Delta$



#### 1. Compute the vorticity $\omega$ from velocity field



This can be done with a central finite-difference stencil on the grid



$$

\omega=\frac{v_{x+\Delta}-v_{x-\Delta}-u_{y+\Delta}+u_{y-\Delta}}{2\Delta}

$$



#### 2. Solve poisson equation for $\psi$ starting with initial guess $\psi'$



This can be done with iterative methods



$$

\nabla^2\psi=-\omega

$$



#### 3. Perform time step for vorticity $\omega$ and store previous $\psi$



$$

\omega\leftarrow\omega+

\left(\nabla^2\left(\eta\omega\right)

+

2\frac{\psi_x\eta_y-2\psi_{xy}\eta_{xy}+\eta_x\psi_y}{\Delta^4}

-\rho\frac{\omega_{xy}}{2\Delta }\right)\frac{\Delta t}{\rho}

\qquad

\psi'\leftarrow\psi

\\\quad\\

\mathrm{where}\quad \omega_{xy}:=\omega_{x+\Delta}u-\omega_{x-\Delta}u+\omega_{y+\Delta}v-\omega_{y-\Delta}v

$$



#### 4. Repeat step 2.



#### 5. Semi-Lagrangian Advection



The semi-Lagrangian advection method traces the characteristics of the flow backward in time and interpolates the values from the previous time step. This ensures stability by tracing the flow backward in time to find the departure point and then interpolating the values from the previous time step at that point.