Final Project Milestone

CS184, Spring 2020

Daniel Geng and Nico Deshler

Progress Overview

We have made steady progress in creating both a spherical fluid simulator and a spectral ray tracer. Moreover, we have successfully established a pipeline for interfacing both of these elements, bringing us closer to a physics-based bubble simulation. So far we have adapted our planar fluid simulator to project properly on the sphere by wrapping the fluid over the edges of the fluid grid. We have also created a specialized BSDF in the project 3 pathtracer codebase for computing and visualizing thin film interferce (TFI) effects. Initially our thin film interference depicts We attempted to also build a physically accurate fluid simulator that uses the Navier-Stokes equations posed in spherical coordinates. While there are papers detailing implementations of such simulators, we have encountered instabilities arrising from the fact that the velocity vector fields are not defined at the poles. Thus for purposes of our initial bubble visualization, we stick to using the planar fluid simulator as a spherical texture map that describes the thickness of the thin film at each point on a spherical surface.

Spherical Fluid Simulator

Projective Simulator

• Wrote a 2D planar fluid simulator in python.
• Adapted the 2D planar fluid simulator to wrap flow accross the vertical boundaries. This allows us to maintain uninterrupted flow along the azimuthal direction of the sphere.
• Simulator does not wrap accross the horizontal boundaries. For numerical stability we treat the horizontal boundaries as sinks. Therfore, fluid does not flow over the north and south poles.
• Projective method preserves stability and simplicity a the expense of unphysical geometric distortions.

Planar fluid simulator projected onto the sphere

Physically-Accurate Simulator

• Working to build a physically-accurate simulator for fluid over the sphere by solving the Navier-Stokes equations formulated in spherical coordinates.
• We have written the simulator in MatLab as it provides access to efficient PDE solvers (Chebfun packagae) for spherical manifolds.
• Simulator still has a few bugs arrising from numerical instability at the poles that we intend to address.

Fluid simulator with Navier-Stokes equations formulated in spherical coordinates

Thin Film BSDF

The spectral ray tracing capability developed herein relies on a thin-film BSDF subclass that we have implemented for our project. Upon ray intersection with a thin film object, this BSDF queries the thickness of the bubble at the intersection point from a thickness map computed by the fluid simulator. After conducting bilinear interpolation over the nearest tabulated thickness values, the methods in this subclass compute the effects of thin-film interference and supply proper weights to each RGB channel in the backtraced ray. Additionally, to assign this specialized BSDF to objects in the scene we modified the COLLADA parser to handle creating instances of this subclass using the supplied thin film data.

Physics-based TFI:

• To associate color with the thin film interference effect, we sparsely sample the visible spectrum using three wavelengths $$ \lambda = 614nm, 549nm, 466nm $$. The thin film reflection coefficients computed for each of these wavelengths are taken to be the proportion of red, green, and blue respectively.
• Sparse sampling of the visible spectrum (3 wavelengths) turned out to be insufficient under the physics-based thin film interference scheme that we set out to implement at the beginning of the project.
• Response curves for the reflections coefficients are sharply peaked so we see only narrow bands of red, green, and blue in the initial renders (see below).

Pseudo-Physical TFI:

• We forgo the need for denser spectral sampling by implementing a pseudo-physical TFI approach originally developed for shader programs.
• In this alternative method the reflection coefficient linked to each color channel reweights the Schlick approximation of the Fresnel Factor for each color channel. Formally, $$R_\lambda = w_\lambda \cdot R_s$$ where $$w_\lambda = \frac{ 1 + \cos(\phi_\lambda)}{2} \hspace{2cm} \text{ and } \hspace{2cm} \phi_\lambda = 2\pi \frac{n \cdot t}{\lambda \cdot \cos(\theta_i)}$$

Interfacing Fluid Simulator with Spectral Ray Tracer

• Created a specialized .DAE file template for linking scene objects with the thin film BSDF class. This .DAE file format contains a spherical thickness map data as well as the refractive index information for the thin film medium and the surrounding media.
• Wrote scripts for automatically writing a .DAE file for each frame of the fluid simulaiton. Using these we can render each frame of a video for temporally-variant thin film interference patterns on the bubble.
• Modified the COLLADA parser to read in thin film data and link thin film BSDFs to objects in the scene for the pathtracer.

Next Steps

1. Address energy loss issues apparent in pseudo-physical thin film interference approach.
2. Address instabilities and noise in our physically-accurate spherical fluid simulator.