LA MEMBRANA MÁGICA

So far, we have illustrated how one can take the horizontal Lagrange-Poincar´e equa- tion to derive an ODE on T Qpart, which can be used as a candidate particle method.

Naturally, one is reluctant to simply drop the vertical equations. From simply judg- ing by the dimensions, the vertical equations contain “more of the action”. In this section, we present some possible ideas for further investigation into producing higher- accuracy meshless methods.

Higher-order isotropy groups The analysis presented in this chapter addresses methods in which the particles carry 0th-order data about the spatial velocity field, u ∈ Xdiv(M), of the fluid. That is, the only datum a particle carries is the velocity

of the fluid at a single point. The particles do not carry 1st order data, i.e., data obtained from ∇u, such as the vorticity at a point or the local stretching. This is unfortunate considering how important vorticity is for turbulence modeling [Cho94]. To address this, one could consider the isotropy group

This is the set of special diffeomorphisms which are equal to the identity on to 1st order. We can calculate the quotient Dµ(M)/G

(1)

to be the frame bundle of Qpart.

The fibers of the frame bundle contain the vorticity and stretch data around each particle. For example, if M is a flat Riemannian manifold then the quotient space is the set of particle configurations with a (1,1) tensor attached to them. The anti- symmetric part of the time derivative of the (1,1) is the vorticity, and the symmetric part is the stretch. It would be interesting to see if and how vortex methods can be approached in some limiting case of this perspective.

More generally one can consider the kth-order isotropy group

G(k) :={φ∈ Dµ(M) :φ() =, Tiφ = 0, i= 1, . . . , k}.

The quotient space Dµ(M)/G

(k)

is the kth-order frame bundle (i.e., the frame of the frame of the frame ... bundle). If M is a flat Riemannian manifold, then the quotient space is the set of particle configurations, each carrying tensors of rank (1,1),(1,2), . . . ,(1, k) above them. Thus, we can create new methods which carry higher-order data above u using roughly the same constructions presented in this chapter.

Navier-Stokes and complex fluids The Navier-Stokes equations can be viewed as a dissipative version of the Euler equations (see§1.12 of [AK92]). Moreover, there are a number of fluids on slightly more complex spaces where Euler-Poincar´e reduction has been performed, such as in magnetohydrodynamics and liquid crystals [Hol02, GBR09]. The particle relabeling symmetry of these systems makes the procedure presented in this chapter applicable to them as well. In the case of complex fluids, the unreduced configuration is a semidirect product with the special diffeomorphism group as the first component. Particle methods for complex fluids would attach data to the particles in addition to the instantaneous velocity.

Practical considerations for implementations When obtaining a method in any context, there are a number of things to keep in mind. Besides accuracy, we

have not addressed some of the basic issues that one comes across when evaluating the performance of a new method. In particular, consistency (i.e., convergence to the exact solutions as the number of particles goes to infinity) is something which needs to be investigated. This will likely depend on the choice of principal connection used to estimate the spatial velocity fields. For example, consistency of the SPH method relies heavily on the smoothness of Gaussian kernel functions. One would expect the smoothness of the image of the reconstruction mapping to play a similar role.

Additionally, the practical performance of a method depends heavily on the cou- pling between the particles. In the example provided in§3.5 all of the particles were coupled to each other; for high-accuracy computation, this scales very badly. Princi- pal connections which yield a large amount of coupling should be avoided when one is planning on using a large number of particles. However, this constraint will likely have some trade-off in accuracy. This also should be investigated.

Finally, the boundary conditions have not been sufficiently addressed in this chap- ter. It is certainly possible to satisfy the boundary conditions by construction simply by requiring that the range of the reconstruction mapping satisfy them.

3.7

Conclusion

In this chapter we have demonstrated that error analysis for particle methods is possible. In particular, it is possible to create remeshing procedures which do not sacrifice accuracy, and to define the error in a rigorous manner. The key insight is that a large family of particle methods can be obtained by taking the horizontal component of the Lagrange-Poincar´e equations. Additionally, these particle methods can be modified to include the case of Navier-Stokes fluids and complex fluids, since one can apply Lagrange-Poincar´e reduction to these systems as well. In summary, we have a new playground for creating new particle-based methods for fluids which can be both easily generalized and rigorously analyzed.

Chapter 4

Interpreting Swimming as a Limit

Cycle

Figure 4.1 – The periodic motion of a jellyfish. Every other snapshot appears to be identical modulo a rigid transformation. Photo taken from [KCDC11], courtesy of Kakani Katija Young.

It has long been suspected that swimming via undulatory motion has a passive component to it [AS05,LBLT03a]. This is of interest to control theorists, roboticists, and biomechanicians because passivity would reduce demands on active controllers and provide robustness to a variety of perturbations. In particular, we pose the conjecture, “Is swimming is a limit cycle?”. Upon first listen, this statement may sound like a plausible hypothesis. A basic example used in introductory control courses is that of the damped harmonic oscillator with external forcing u,

¨

x=−kx−νx˙ +u.

We think of a signal, u(t), as an input and the state, x(t), as an output. The step response is characterized by a transient oscillatory phase which settles to steady-state behavior at a fixed value of x. More importantly, if one inputs a sinusoidal signal, the output is also sinusoidal and of the same period. The goal of this chapter is to interpret the coherent motion of an undulating body immersed in a Navier-Stokes fluid

in a similar manner. For a shape-changing body in a viscous fluid, the contraction of muscles will induce a change in shape and also move the surrounding fluid. After a while, dissipation will bring the body and the fluid to rest at a new location in space, perhaps with a different shape as well. We could view this as the step response of the system with respect to muscle contraction. Building upon the analogy with the damped harmonic oscillator, one could hope that time-periodic muscle contractions could produce limit cycles of the same period in an appropriate phase space.

Moreover, we know that a dissipative Hamiltonian system with an asymptotically stable equilibrium will produce a limit cycle when a sufficiently small periodic force is applied (see “The Averaging Theorem” in [GH83]). A viscous fluid is a dissipative system with an asymptotically stable fixed point (i.e., still motion). An undulating body may exert a periodic force. It may appear that we need only write down the Hamiltonian, the viscous friction, an oscillating shape potential, and then QED. Right? Wrong! As we begin to probe the idea, we come across ambiguities:

(Q1) What is the configuration manifold?

(Q2) If a body moves through each cycle of undulation to a new location, then it is not returning to its previous position. Therefore, the state of the system is not cyclic unless the animal produces 0 net motion. Does this argument negate the hypothesis that swimming is a limit cycle?

(Q3) Conversely, if we had a limit cycle, then the system returns to the same state it began in. Would this imply no motion is produced?

The answers we have provided are:

(A1) The configuration manifold is a Lie groupoid. The particle relabeling symme- try of the fluid allows us to describe the system on the the corresponding Lie algebroid, (A, ρ,[,]).

(A2) No. The hypothesis is not negated. The Lie algebroid, A, exhibits an SE(3) invariance. Upon reducing by this invariance we can view swimming as a limit cycle in a reduced algebroid, [A].

(A3) If [γ](t) is a closed orbit in [A] then it must be reduced from a path, γ(t), in A such that γ(T) = z·γ(0) for some z ∈SE(3).

These answers may be “Greek” upon first reading. We will spend the remain- der of the paper explaining them. Additionally, while we are not able to definitively conclude that periodic forces on the shape of a body limits to swimming, the contrac- tion of phase space for finite dimensional dissipative Hamiltonian systems is strongly suggestive that such limiting behavior is likely.