Pseudospectral time-domain (PSTD) methods for the wave equation: Realising boundary conditions with discrete sine and cosine transforms

Pseudospectral time domain (PSTD) methods are widely used in many branches of acoustics for the numerical solution of the wave equation, including biomedical ultrasound and seismology. The use of the Fourier collocation spectral method in particular has many computational advantages. However, the use of a discrete Fourier basis is also inherently restricted to solving problems with periodic boundary conditions. Here, a family of spectral collocation methods based on the use of a sine or cosine basis is described. These retain the computational advantages of the Fourier collocation method but instead allow homogeneous Dirichlet (sound-soft) and Neumann (sound-hard) boundary conditions to be imposed. The basis function weights are computed numerically using the discrete sine and cosine transforms, which can be implemented using O(N log N) operations analogous to the fast Fourier transform. Practical details of how to implement spectral methods using discrete sine and cosine transforms are provided. The technique is then illustrated through the solution of the wave equation in a rectangular domain subject to different combinations of boundary conditions. The extension to boundaries with arbitrary real reflection coefficients or boundaries that are non-reflecting is also demonstrated using the weighted summation of the solutions with Dirichlet and Neumann boundary conditions.Comment: 21 pages, 10 figure

Model-reduced variational fluid simulation

We present a model-reduced variational Eulerian integrator for incompressible fluids, which combines the efficiency gains of dimension reduction, the qualitative robustness of coarse spatial and temporal resolutions of geometric integrators, and the simplicity of sub-grid accurate boundary conditions on regular grids to deal with arbitrarily-shaped domains. At the core of our contributions is a functional map approach to fluid simulation for which scalar- and vector-valued eigenfunctions of the Laplacian operator can be easily used as reduced bases. Using a variational integrator in time to preserve liveliness and a simple, yet accurate embedding of the fluid domain onto a Cartesian grid, our model-reduced fluid simulator can achieve realistic animations in significantly less computational time than full-scale non-dissipative methods but without the numerical viscosity from which current reduced methods suffer. We also demonstrate the versatility of our approach by showing how it easily extends to magnetohydrodynamics and turbulence modeling in 2D, 3D and curved domains

Doctor of Philosophy

dissertationPhysical simulation has become an essential tool in computer animation. As the use of visual effects increases, the need for simulating real-world materials increases. In this dissertation, we consider three problems in physics-based animation: large-scale splashing liquids, elastoplastic material simulation, and dimensionality reduction techniques for fluid simulation. Fluid simulation has been one of the greatest successes of physics-based animation, generating hundreds of research papers and a great many special effects over the last fifteen years. However, the animation of large-scale, splashing liquids remains challenging. We show that a novel combination of unilateral incompressibility, mass-full FLIP, and blurred boundaries is extremely well-suited to the animation of large-scale, violent, splashing liquids. Materials that incorporate both plastic and elastic deformations, also referred to as elastioplastic materials, are frequently encountered in everyday life. Methods for animating such common real-world materials are useful for effects practitioners and have been successfully employed in films. We describe a point-based method for animating elastoplastic materials. Our primary contribution is a simple method for computing the deformation gradient for each particle in the simulation. Given the deformation gradient, we can apply arbitrary constitutive models and compute the resulting elastic forces. Our method has two primary advantages: we do not store or compare to an initial rest configuration and we work directly with the deformation gradient. The first advantage avoids poor numerical conditioning and the second naturally leads to a multiplicative model of deformation appropriate for finite deformations. One of the most significant drawbacks of physics-based animation is that ever-higher fidelity leads to an explosion in the number of degrees of freedom

Seeing and Hearing Fluid Subspaces

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