Physics-based deformable object simulations have been playing an increasingly important role in 3D computer graphics. They have been adopted for humanoid character animations as well as special effects such as fire and explosion. However, simulations of large, complex systems can consume large amounts of computation and mostly remain offline, which prohibits their use for interactive applications.

We present several highly efficient schemes for deformable object simulation using custom spatial coordinate systems. Our choices span the spectrum of subspace to full space and both Lagrangian and Eulerian viewpoints.

Subspace methods achieve massive speedups over their “full space” counterparts by drastically reducing the degrees of freedom involved in the simulation. A long standing difficulty in subspace simulation is incorporating various non-linearities. They introduce expensive computational bottlenecks and quite often cause novel deformations that are outside the span of the subspace.

We address these issues in articulated deformable body simulations from a Lagrangian viewpoint. We remove the computational bottleneck of articulated self-contact handling by deploying a pose-space cubature scheme, a generalization of the standard “cubature” approximation. To handle novel deformations caused by arbitrary external collisions, we introduce a generic approach called subspace condensation, which activates full space simulation on the fly when an out-of-basis event is encountered. Our proposed framework

efficiently incorporates various non-linearities and allows subspace methods to be used in cases where they previously would not have been considered.

Deformable solids can interact not only with each other, but also with fluids. We

design a new full space method that achieves a two-way coupling between deformable solids and an incompressible fluid where the underlying geometric representation is entirely Eulerian. No-slip boundary conditions are automatically satisfied by imposing a global divergence-free condition. We are able to simulate multiple solids undergoing complex, frictional contact while simultaneously interacting with a fluid. The complexity of the scenarios we are able to simulate surpasses those that we have seen from any previous method.

Reduced-order methods are an attractive model for physical simulation in computer graphics. They aim to reduce the computational cost of a full-rank simulation by project- ing onto a reduced set of bases. However, an improper application of such methods can be unstable and scale poorly, which prevents many methods from being used in practice.

In this work, we aim to improve the scalability and stability of reduced order methods for two such scenarios. For the Eigenfluids method of fluid simulation, we show that by carefully applying the discrete sine and cosine transforms, the prohibitive scaling of its memory usage can be reduced asymptotically. The simulation is further stabilized by using a variational approach with different basis functions. We also show that the basis functions can be made quite general by extending them to spherical and polar coordinate systems. By using an orthogonalization method, we show a wide variety of basis functions can be designed that maintain fast transformations. For stochastic structure optimization, which assesses whether a fabricated object will break under real- world conditions, we show that the computation can be made asymptotically faster by carefully exploiting certain tensor structures. We then stabilize the optimization by applying an alternate basis to the function gradients.

Fluids have inspired generations of artists and scientists throughout history. Aesthetically, the wide variety of abstract shapes they form is both surprising and pleasing. Besides visual art, which until the digital age mostly captured frozen moments in time, late 19th-century composers such as Debussy and Ravel wrote works of music inspired by the movement of fluids over time. With the framework of several basic conservation laws of physics, earlier 19th-century scientific work discovered a set of differential equations called the Navier-Stokes equations that described the time evolution of fluid velocity fields.

In recent years, the advent of higher computing power and the birth of computer graphics as a discipline has given rise to computational methods for approximating and visualizing solutions to the Navier-Stokes equations, which had previously remained intractably complex. Many artists and musicians have also embraced digital technologies, allowing for the development of algorithmically generated music as well as multimodal representations of large, complex data sets.

With this new technology, it is natural to consider the following question: is it possible to systematically generate sounds from fluid dynamics while retaining an underlying musicality? In this dissertation, we present a framework for generating correlated correlated fluid motions and musical sounds using the empirical eigenvalues of a subspace fluid simulation. Our method is multimodal in nature, allowing for the generation of musical sound as well as novel visual forms. The specific mapping from fluid velocity to sound chosen allows for control and modulation of both the visuals and the audio in an integrated, unifying fashion.

The method of subspace simulation, which our mapping framework relies on, has a known drawback of high memory consumption. As a means of overcoming this technical obstacle, we also present a data compression framework for fluid subspaces. Our proposed algorithm can achieve an order of magnitude data compression without any noticeable visual artifacts. Using this compression algorithm allows the potential for simulating greater variety of complex scenes on powerful computers as well as the ability to run previously too-complex scenes on a laptop.