Eva Kanso

Discrete differential forms for computational modeling

The emergence of computers as an essential tool in scientific research has shaken the very foundations of differential modeling. Indeed, the deeply-rooted abstraction of smoothness, or differentiability, seems to inherently clash with a computers ability of storing only finite sets of numbers. While there has been a series of computational techniques that proposed discretizations of differential equations, the geometric structures they are supposed to simulate are often lost in the process.

Locomotion of Articulated Bodies in a Perfect Fluid

AbstractThis paper is concerned with modeling the dynamics of N articulated solid bodies submerged in an ideal fluid. The model is used to analyze the locomotion of aquatic animals due to the coupling between their shape changes and the fluid dynamics in their environment. The equations of motion are obtained by making use of a two-stage reduction process which leads to significant mathematical and computational simplifications. The first reduction exploits particle relabeling symmetry: that is, the symmetry associated with the conservation of circulation for ideal, incompressible fluids. As a result, the equations of motion for the submerged solid bodies can be formulated without explicitly incorporating the fluid variables. This reduction by the fluid variables is a key difference with earlier methods, and it is appropriate since one is mainly interested in the location of the bodies, not the fluid particles. The second reduction is associated with the invariance of the dynamics under superimposed rigid motions. This invariance corresponds to the conservation of total momentum of the solid-fluid system. Due to this symmetry, the net locomotion of the solid system is realized as the sum of geometric and dynamic phases over the shape space consisting of allowable relative motions, or deformations, of the solids. In particular, reconstruction equations that govern the net locomotion at zero momentum, that is, the geometric phases, are obtained. As an illustrative example, a planar three-link mechanism is shown to propel and steer itself at zero momentum by periodically changing its shape. Two solutions are presented: one corresponds to a hydrodynamically decoupled mechanism and one is based on accurately computing the added inertias using a boundary element method. The hydrodynamically decoupled model produces smaller net motion than the more accurate model, indicating that it is important to consider the hydrodynamic interaction of the links.

Stable, circulation-preserving, simplicial fluids

Visual quality, low computational cost, and numerical stability are foremost goals in computer animation. An important ingredient in achieving these goals is the conservation of fundamental motion invariants. For example, rigid and deformable body simulation benefits greatly from the conservation of linear and angular momenta. In the case of fluids, however, none of the current techniques focuses on conserving invariants, and consequently, often introduce a visually disturbing numerical diffusion of vorticity. Just as important visually is the resolution of complex simulation domains. Doing so with regular (even if adaptive) grid techniques can be computationally delicate. In this article, we propose a novel technique for the simulation of fluid flows. It is designed to respect the defining differential properties, that is, the conservation of circulation along arbitrary loops as they are transported by the flow. Consequently, our method offers several new and desirable properties: Arbitrary simplicial meshes (triangles in 2D, tetrahedra in 3D) can be used to define the fluid domain; the computations involved in the update procedure are efficient due to discrete operators with small support; and it preserves discrete circulation, avoiding numerical diffusion of vorticity.

Geometric, variational integrators for computer animation

We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems---an important computational tool at the core of most physics-based animation techniques. Several features make this particular time integrator highly desirable for computer animation: it numerically preserves important invariants, such as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energy behavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quite simply; finally, our simple methodology allows for the design of high-order accurate schemes if needed. Two key properties set the method apart from earlier approaches. First, the nonlinear equations that must be solved during an update step are replaced by a minimization of a novel functional, speeding up time stepping by more than a factor of two in practice. Second, the formulation introduces additional variables that provide key flexibility in the implementation of the method. These properties are achieved using a discrete form of a general variational principle called the Pontryagin-Hamilton principle, expressing time integration in a geometric manner. We demonstrate the applicability of our integrators to the simulation of non-linear elasticity with implementation details.

Source Seeking for Two Nonholonomic Models of Fish Locomotion

In this paper, we present a method of locomotion control for underwater vehicles that are propelled by a periodic deformation of the vehicle body, which is similar to the way a fish moves. We have developed control laws employing ldquoextremum seekingrdquo for two different ldquofishrdquo models. The first model consists of three rigid body links and relies on a 2-degree-of-freedom (DOF) movement that propels the fish without relying on vortices. The second fish model uses a Joukowski airfoil that has only 1 DOF in its movement and, thus, relies on vortex shedding for propulsion. We achieve model-free and position-free ldquosource seeking,rdquo and, if position is available, navigation along a predetermined path.

Swimming due to transverse shape deformations

Balance laws are derived for the swimming of a deformable body due to prescribed shape changes and the effect of the wake vorticity. The underlying balances of momenta, though classical in nature, provide a unifying framework for the swimming of three-dimensional and planar bodies and they hold even in the presence of viscosity. The derived equations are consistent with Lighthills reactive force theory for the swimming of slender bodies and, when neglecting vorticity, reduce to the model developed in Kanso et al . ( J. Nonlinear Sci ., vol. 15, 2005, p. 255) for swimming in potential flow. The locomotion of a deformable body is examined through two sets of examples: the first set studies the effect of cyclic shape deformations, both flapping and undulatory, on the locomotion in potential flow while the second examines the effect of the wake vorticity on the net locomotion. In the latter, the vortex wake is modelled using pairs of point vortices shed periodically from the tail of the deformable body.

Stability of a coupled body-vortex system

This paper considers the dynamics of a rigid body interacting with point vortices in a perfect uid. The uid velocity is obtained using the classical complex variables theory and conformal transformations. The equations of motion of the solid-uid system are formulated in terms of the solid variables and the position of the point vortices only. These equations are applied to study the dynamic interaction of an elliptic cylinder with vortex pairs for its relevance to understanding the swimming of sh in an ambient vorticity eld. One

Hydrodynamically coupled rigid bodies

This paper considers a finite number of rigid bodies moving in potential flow. The dynamics of the solid--fluid system is described in terms of the solid variables only using Kirchhoff potentials. The equations of motion are first derived for the problem of two submerged bodies where one is forced into periodic oscillations. The hydrodynamic coupling causes the free body to drift away from or towards the oscillating body. The method of multiple scales is used to separate the slow drift from the fast response. Interestingly, the free body, when attracted towards the forced one, starts to drift away after it reaches certain separation distance. This suggests that the hydrodynamic coupling helps in preventing collisions. The fluids role in collision avoidance and motion coordination is examined further through examples. In particular, we show that a free body can coordinate its motion with that of its neighbours, which may be relevant to understanding the coordinated motion in fish schooling.

Optimal Motion of an Articulated Body in a Perfect Fluid

An articulated body can propel and steer itself in a perfect fluid by changing its shape only. Our strategy for motion planning for the submerged body is based on finding the optimal shape changes that produce a desired net locomotion; that is, motion planning is formulated as a nonlinear optimization problem.

Passive locomotion via normal-mode coupling in a submerged spring–mass system

The oscillations of a class of submerged mass–spring systems are examined. An inviscid fluid model is employed to show that the hydrodynamic effects couple the normal modes of these systems. This coupling of normal modes can excite the displacement mode – yielding passive locomotion of the system – even when starting with zero displacement velocity. This is in contrast with the fact that under similar initial conditions but without the hydrodynamic coupling, such systems cannot achieve a net displacement. These ideas are illustrated via two examples of a two-mass and a three-mass system.