Byungmoon Kim

An Unconditionally Stable MacCormack Method

Abstract The back and forth error compensation and correction (BFECC) method advects the solution forward and then backward in time. The result is compared to the original data to estimate the error. Although inappropriate for parabolic and other non-reversible partial differential equations, it is useful for often troublesome advection terms. The error estimate is used to correct the data before advection raising the method to second order accuracy, even though each individual step is only first order accurate. In this paper, we rewrite the MacCormack method to illustrate that it estimates the error in the same exact fashion as BFECC. The difference is that the MacCormack method uses this error estimate to correct the already computed forward advected data. Thus, it does not require the third advection step in BFECC reducing the cost of the method while still obtaining second order accuracy in space and time. Recent work replaced each of the three BFECC advection steps with a simple first order accurate unconditionally stable semi-Lagrangian method yielding a second order accurate unconditionally stable BFECC scheme. We use a similar approach to create a second order accurate unconditionally stable MacCormack method.

FlowFixer: using BFECC for fluid simulation

Back and Forth Error Compensation and Correction (BFECC) was recently developed for interface computation by using the level set method. We show that it can be applied to reduce dissipation and diffusion encountered in various advection steps in uid simulation such as velocity, smoke density and image advections. BFECC can be implemented easily on top of the r st order upwinding or semi-Lagrangian integration of advection equations, while providing second order accuracy both in space and time. When applied to level set evolution, BFECC reduces volume loss signi cantly . We combine these techniques with variable density projection and show that they yield a realistic animations of two-phase ows. We demonstrate the bene ts of this approach on the image advection and on the simulation of smoke, of bubbles in water, and of a highly dynamic interaction between water, a solid, and air.

Advections with Significantly Reduced Dissipation and Diffusion

Back and forth error compensation and correction (BFECC) was recently developed for interface computation using a level set method. We show that BFECC can be applied to reduce dissipation and diffusion encountered in a variety of advection steps, such as velocity, smoke density, and image advections on uniform and adaptive grids and on a triangulated surface. BFECC can be implemented trivially as a small modification of the first-order upwind or semi-Lagrangian integration of advection equations. It provides second-order accuracy in both space and time. When applied to level set evolution, BFECC reduces volume loss significantly. We demonstrate the benefits of this approach on image advection and on the simulation of smoke, bubbles in water, and the highly dynamic interaction between water, a solid, and air. We also apply BFECC to dye advection to visualize vector fields

Simulation of bubbles in foam with the volume control method

Liquid and gas interactions often produce bubbles that stay for a long time without bursting on the surface, making a dry foam structure. Such long lasting bubbles simulated by the level set method can suffer from a small but steady volume error that accumulates to a visible amount of volume change. We propose to address this problem by using the volume control method. We track the volume change of each connected region, and apply a carefully computed divergence that compensates undesired volume changes. To compute the divergence, we construct a mathematical model of the volume change, choose control strategies that regulate the modeled volume error, and establish methods to compute the control gains that provide robust and fast reduction of the volume error, and (if desired) the control of how the volume changes over time.

Controllers for unicycle-type wheeled robots: Theoretical results and experimental validation

Mobile robots offer a typical example of systems with nonholonomic constraints. Several controllers have been proposed in the literature for stabilizing these systems. However, few experimental studies have been reported comparing the characteristics and the performance of these controllers with respect to neglected dynamics, quantization, noise, delays, etc. In this paper, we use a Khepera mobile robot to perform experimental comparison of several control laws. Khepera has two dc motor-powered wheels and introduces many realistic difficulties, such as different motor dynamics for the two wheels, time delay, quantization, sensor noise, and saturation. We emphasize the implementation difficulties of two discontinuous controllers proposed herein, and we compare their performance with several other controllers reported in the literature. Ways to improve the performance of each controller are also discussed.

Collision prediction for polyhedra under screw motions

The prediction of collisions amongst N rigid objects may be reduced to a series of computations of the time to first contact for all pairs of objects. Simple enclosing bounds and hierarchical partitions of the space-time domain are often used to avoid testing object-pairs that clearly will not collide. When the remaining pairs involve only polyhedra under straight-line translation, the exact computation of the collision time and of the contacts requires only solving for intersections between linear geometries. When a pair is subject to a more general relative motion, such a direct collision prediction calculation may be intractable. The popular brute force collision detection strategy of executing the motion for a series of small time steps and of checking for static interferences after each step is often computationally prohibitive. We propose instead a less expensive collision prediction strategy, where we approximate the relative motion between pairs of objects by a sequence of screw motion segments, each defined by the relative position and orientation of the two objects at the beginning and at the end of the segment. We reduce the computation of the exact collision time and of the corresponding face/vertex and edge/edge collision points to the numeric extraction of the roots of simple univariate analytic functions. Furthermore, we propose a series of simple rejection tests, which exploit the particularity of the screw motion to immediately decide that some objects do not collide or to speed-up the prediction of collisions by about 30%, avoiding on average 3/4 of the root-finding queries even when the object actually collide.

GeoFilter: Geometric Selection of Mesh Filter Parameters

When designing a lowpass filter to eliminate noise in a triangle mesh, the cutoff frequency is typically chosen by a cumbersome trial-and-error process. Therefore, it is important to provide a guideline in selectingfilter frequencies. Here, we explore the relation between the frequencies in a mesh filter and the geometric measures of user-selected features. In addition, by combining previously proposed implicit and explicit formulations, we develop a second order filter that can act as lowpass, bandpass, highpass, notch, and band exaggeration/reduction filters. The proposed GeoFilter framework allows the user to choose the frequencies for that filter based on the physical size of a blob (ellipsoid) automatically fit to a user-selected feature in the mesh. For example, the size of a bump in a noisy pattern can be used as a cutoff frequency in a lowpass filter, while the size of a nose may be used to smoothen a face or to exaggerate its features as in a caricature.

Multi-phase fluid simulations using regional level sets

We address the problem of Multi-Phase (or Many-Phase) Fluid simulations. We propose to use the regional level set (RLS) that can handle a large number of regions and materials, and hence, is appropriate for simulations of many immiscible materials. Towards this goal, we improve the interpolation of the RLS, and develop the regional level set graph (RLSG), which registers connected components and their contacts, and tracks their properties such as region volumes, film life times, and film material types, as regions evolve, merge, split, or are squeezed into films. Using RLSGs tracking feature, we generate particles from tiny regions or rupturing films.

Finger sculpting with Digital Clay: 3D shape input and output through a computer-controlled real surface

The NSF Digital Clay project is focused on the design, prototyping, integration, and validation of a computer-controlled physical device capable of taking any of a wide range of possible shapes in response to changes in a digital 3D model or to changes in the pressure exercised upon it by human hands. Although it clearly is a natural and unavoidable evolution of 3D graphical user interfaces, its unprecedented capabilities constitute a major leap in technologies and paradigms for 3D display, for 3D input, and for collaborative 3D design. In this paper, we provide an overview of the Digital Clay project and discuss the challenges, design choices, and initial solutions for a new finger sculpting interface designed for the Digital Clay and prototyped using conventional 3D I/O hardware.

Designing a low-cost spacecraft simulator

The greatest difficulty in implementing spacecraft control laws is that ground-based experiments must take place in a 1g environment, whereas the actual spacecraft will operate under 0g conditions. This article describes a spacecraft simulator facility used to educate undergraduates in spacecraft attitude dynamics and control.