1,113 research outputs found
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
The motion of solid bodies in potential flow with circulation: a geometric outlook
The motion of a circular body in 2D potential flow is studied using symplectic reduction. The equations of motion are obtained starting front a kinetic-energy type system on a space of embeddings and reducing by the particle relabelling symmetry group and the special Euclidian group. In the process, we give a geometric interpretation for the Kutta-Joukowski lift force in terms of the curvature of a connection on the original phase space
Geometric, Variational Integrators for Computer Animation
We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systemsâan 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
Structure-Preserving Discretization of Incompressible Fluids
The geometric nature of Euler fluids has been clearly identified and
extensively studied over the years, culminating with Lagrangian and Hamiltonian
descriptions of fluid dynamics where the configuration space is defined as the
volume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewed
as a consequence of Noether's theorem associated with the particle relabeling
symmetry of fluid mechanics. However computational approaches to fluid
mechanics have been largely derived from a numerical-analytic point of view,
and are rarely designed with structure preservation in mind, and often suffer
from spurious numerical artifacts such as energy and circulation drift. In
contrast, this paper geometrically derives discrete equations of motion for
fluid dynamics from first principles in a purely Eulerian form. Our approach
approximates the group of volume-preserving diffeomorphisms using a finite
dimensional Lie group, and associated discrete Euler equations are derived from
a variational principle with non-holonomic constraints. The resulting discrete
equations of motion yield a structure-preserving time integrator with good
long-term energy behavior and for which an exact discrete Kelvin's circulation
theorem holds
Discrete Lie Advection of Differential Forms
In this paper, we present a numerical technique for performing Lie advection
of arbitrary differential forms. Leveraging advances in high-resolution finite
volume methods for scalar hyperbolic conservation laws, we first discretize the
interior product (also called contraction) through integrals over Eulerian
approximations of extrusions. This, along with Cartan's homotopy formula and a
discrete exterior derivative, can then be used to derive a discrete Lie
derivative. The usefulness of this operator is demonstrated through the
numerical advection of scalar fields and 1-forms on regular grids.Comment: Accepted version; to be published in J. FoC
The Dynamics of a Rigid Body in Potential Flow with Circulation
We consider the motion of a two-dimensional body of arbitrary shape in a
planar irrotational, incompressible fluid with a given amount of circulation
around the body. We derive the equations of motion for this system by
performing symplectic reduction with respect to the group of volume-preserving
diffeomorphisms and obtain the relevant Poisson structures after a further
Poisson reduction with respect to the group of translations and rotations. In
this way, we recover the equations of motion given for this system by Chaplygin
and Lamb, and we give a geometric interpretation for the Kutta-Zhukowski force
as a curvature-related effect. In addition, we show that the motion of a rigid
body with circulation can be understood as a geodesic flow on a central
extension of the special Euclidian group SE(2), and we relate the cocycle in
the description of this central extension to a certain curvature tensor.Comment: 28 pages, 2 figures; v2: typos correcte
The Geometry and Dynamics of Interacting Rigid Bodies and Point Vortices
We derive the equations of motion for a planar rigid body of circular shape
moving in a 2D perfect fluid with point vortices using symplectic reduction by
stages. After formulating the theory as a mechanical system on a configuration
space which is the product of a space of embeddings and the special Euclidian
group in two dimensions, we divide out by the particle relabelling symmetry and
then by the residual rotational and translational symmetry. The result of the
first stage reduction is that the system is described by a non-standard
magnetic symplectic form encoding the effects of the fluid, while at the second
stage, a careful analysis of the momentum map shows the existence of two
equivalent Poisson structures for this problem. For the solid-fluid system, we
hence recover the ad hoc Poisson structures calculated by Shashikanth, Marsden,
Burdick and Kelly on the one hand, and Borisov, Mamaev, and Ramodanov on the
other hand. As a side result, we obtain a convenient expression for the
symplectic leaves of the reduced system and we shed further light on the
interplay between curvatures and cocycles in the description of the dynamics.Comment: 52 pages, 2 figure
A Geometric Approach Towards Momentum Conservation
In this work, a geometric discretization of the Navier-Stokes equations is
sought by treating momentum as a covector-valued volume-form. The novelty of
this approach is that we treat conservation of momentum as a tensor equation
and describe a higher order approximation to this tensor equation. The
resulting scheme satisfies mass and momentum conservation laws exactly, and
resembles a staggered-mesh finite-volume method. Numerical test-cases to which
the discretization scheme is applied are the Kovasznay flow, and lid-driven
cavity flow
Coronavirus peplomer charge heterogeneity
Recent advancements in viral hydrodynamics afford the calculation of the transport properties of particle suspensions from first principles, namely, from the detailed particle shapes. For coronavirus suspensions, for example, the shape can be approximated by beading (i) the spherical capsid and (ii) the radially protruding peplomers. The general rigid bead-rod theory allows us to assign Stokesian hydrodynamics to each bead. Thus, viral hydrodynamics yields the suspension rotational diffusivity, but not without first arriving at a configuration for the cationic peplomers. Prior work considered identical peplomers charged identically. However, a recent pioneering experiment uncovers remarkable peplomer size and charge heterogeneities. In this work, we use energy minimization to arrange the spikes, charged heterogeneously to obtain the coronavirus spike configuration required for its viral hydrodynamics. For this, we use the measured charge heterogeneity. We consider 20â000 randomly generated possibilities for cationic peplomers with formal charges ranging from 30 to 55. We find the configurations from energy minimization of all of these possibilities to be nearly spherically symmetric, all slightly oblate, and we report the corresponding breadth of the dimensionless rotational diffusivity, the transport property around which coronavirus cell attachment revolves.journal articl
On the geometric character of stress in continuum mechanics
This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other fundamental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented
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