Steklov Spectral Geometry for Extrinsic Shape Analysis

We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.Comment: Additional experiments adde

Development and Optimization of Non-Hydrostatic Models for Water Waves and Fluid-Vegetation Interaction

The primary objective of this study is twofold: 1) to develop an efficient and accurate non-hydrostatic wave model for fully dispersive highly nonlinear waves, and 2) to investigate the interaction between waves and submerged flexible vegetation using a fully coupled wave-vegetation model. This research consists of three parts. Firstly, an analytical dispersion relationship is derived for waves simulated by models utilizing Keller-box scheme and central differencing for vertical discretization. The phase speed can be expressed as a rational polynomial function of the dimensionless water depth, $kh$, and the layer distribution in water column becomes an optimizable parameter in this function. For a given tolerance dispersion error, the range of $kh$ is extended and the layer thicknesses are optimally selected. The derived theoretical dispersion relationship is tested with linear and nonlinear standing waves generated by an Euler model. The optimization method is applicable to other non-hydrostatic models for water waves. Secondly, an efficient and accurate approach is developed to solve Euler equations for fully dispersive and highly nonlinear water waves. Discontinuous Galerkin, finite difference, and spectral element formulations are used for horizontal discretization, vertical discretization, and the Poisson equation, respectively. The Keller-box scheme is adopted for its capability of resolving frequency dispersion accurately with low vertical resolution (two or three layers). A three-stage optimal Strong Stability-Preserving Runge-Kutta (SSP-RK) scheme is employed for time integration. Thirdly, a fully coupled wave-vegetation model for simulating the interaction between water waves and submerged flexible plants is presented. The complete governing equation for vegetation motion is solved with a high-order finite element method and an implicit time differencing scheme. The vegetation model is fully coupled with a wave model to explore the relationship between displacement of water particle and plant stem, as well as the effect of vegetation flexibility on wave attenuation. This vegetation deformation model can be coupled with other wave models to simulate wave-vegetation interactions

Theoretical Investigation of Electroosmotic Flows and Chaotic Stirring in Rectangular Cavities

Two dimensional, time-independent and time-dependent electro-osmotic flows driven by a uniform electric field in a closed rectangular cavity with uniform and nonuniform zeta potential distributions along the cavity’s walls are investigated theoretically. First, we derive an expression for the one-dimensional velocity and pressure profiles for a flow in a slender cavity with uniform (albeit possibly different) zeta potentials at its top and bottom walls. Subsequently, using the method of superposition, we compute the flow in a finite length cavity whose upper and lower walls are subjected to non-uniform zeta potentials. Although the solutions are in the form of infinite series, with appropriate modifications, the series converge rapidly, allowing one to compute the flow fields accurately while maintaining only a few terms in the series. Finally, we demonstrate that by time-wise periodic modulation of the zeta potential, one can induce chaotic advection in the cavity. Such chaotic flows can be used to stir and mix fluids. Since devices operating on this principle do not require any moving parts, they may be particularly suitable for microfluidic devices

Phurbas: An Adaptive, Lagrangian, Meshless, Magnetohydrodynamics Code. II. Implementation and Tests

We present an algorithm for simulating the equations of ideal magnetohydrodynamics and other systems of differential equations on an unstructured set of points represented by sample particles. The particles move with the fluid, so the time step is not limited by the Eulerian Courant-Friedrichs-Lewy condition. Full spatial adaptivity is required to ensure the particles fill the computational volume, and gives the algorithm substantial flexibility and power. A target resolution is specified for each point in space, with particles being added and deleted as needed to meet this target. We have parallelized the code by adapting the framework provided by GADGET-2. A set of standard test problems, including 1e-6 amplitude linear MHD waves, magnetized shock tubes, and Kelvin-Helmholtz instabilities is presented. Finally we demonstrate good agreement with analytic predictions of linear growth rates for magnetorotational instability in a cylindrical geometry. This paper documents the Phurbas algorithm as implemented in Phurbas version 1.1.Comment: 14 pages, 14 figures, ApJS accepted, revised in accordance with changes to paper I (arXiv:1110.0835