Abstracts

| A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |

A

A discrete mechanics viewpoint of finite elasticity

Marino Arroyo

Abstract:
We identify a nonlinear differential complex for nonlinear continuum mechanics, which collects a number of classical results of differential geometry and generalizes the linear Kroner complex proposed by Hauret, Kuhl and Ortiz IJNME 2007. This complex has a discrete version on simplicial complexes. We also interpret a classical result by Moser in terms of the nonlinear version of the isotropic complex relevant to incompressibility. We prove that this complex is also exact in the discrete setting for diamond elements and other schemes, showing that the linearization of the Jacobian into the divergence and the discretization of the theory commute.

G

Discrete Elastic Rods

Eitan Grinspun

Abstract:
We present a discrete treatment of adapted framed curves and discrete parallel transport, which provides the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach is distinguished from existing simulation techniques both in the kinematic description—we represent the material frame by its angular deviation from the natural Bishop frame—as well as in the dynamical treatment—we treat the centerline as dynamic but the material frame as quasistatic. We validate the proposed model via quantitative buckling, stability, and coupled-mode experiments as well as via qualitative knot-tying comparisons.

This is joint work with Basile Audoly, Miklós Bergou, Stephen Robinson, and Max Wardetzky.

L

Global Symplectic Uncertainty Propagation on SO(3)

Melvin Leok

Abstract:
We introduces a global uncertainty propagation scheme for rigid body dynamics, through a combination of numerical parametric uncertainty techniques, noncommutative harmonic analysis, and geometric numerical integration. This method is distinguished from prior approaches, as it allows one to consider probability densities that are global, and which are not restricted to being supported on only a single coordinate chart on the manifold.

The use of Lie group variational integrators, that are symplectic and stay on the Lie group, as the underlying numerical propagator ensures that the advected probability densities respect the geometric properties of uncertainty propagation in Hamiltonian systems, which arise as consequence of the Gromov nonsqueezing theorem from symplectic geometry. We also describe how the global uncertainty propagation scheme can be applied to the problem of global attitude estimation.

Dynamic optimisation of a three-dimensional compass gait biped

Sigrid Leyendecker

Abstract:
In this work, a three-dimensional compass biped is modelled as a spherical kinematic pair in which the legs are combined at the hip by a spherical joint. The contact between a foot and the ground is modelled as a perfectly plastic impact, constraining the foot to stay fixed on the ground during the other leg's swing phase. This contact condition is represented by another spherical joint connection between the foot and the point of contact on the ground. During each swing phase, a variational integrator in combination with the discrete null space method for the treatment of the constraints yields the discrete equations of motion. The contact is transferred instantaneously when the second foot hits the ground and the first one is released. Here, the discrete null space method is used again to facilitate the formulation of the switching contact conditions.Then, DMOC is applied to determine actuating torques in the hip joint during half a gait cycle.

Uncertainty Quanitifcation using Concentration-of-Measure

Lenny Lucas

Abstract:
Typically one uses a sampling procedure (e.g., Monte Carlo) to gain insight into the probability density of an output based on the probability densities of inputs into a system, which can be integrated over a "failure" region to obtain a probability of failure. The CM phenomenon, which predicts that a system responds more predictably when more randomness is injected, provides an analytical upper bound for the probability of failure of a system. The result of this method of estimating failure probability has applications in reliability analysis, and certification of engineering systems. This method requires an optimization of random variables in the design space to compute characteristics of the system's extreme responses to describe familiar terms in UQ such as margin and uncertainty. Previous trends are shown for numerical examples such as an imploding ring, a linear system, and current development for a system with random controls.

O

Near Boltzmann-Gibbs Measure Preserving Stochastic Variational Integrator

Houman Owhadi

Abstract:
In this joint work with Nawaf Bou-Rabee we present a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. Analogously to discrete mechanics and variational integrators, this theory leads to structure preserving numerical integrators for noisy mechanical systems by extremizing a discrete stochastic action. In this talk, we will focus on a particular one, designed to approximate the solutions of Langevin equations with uniform friction and noise. This new integrator is characterized by the following two properties:

• It exactly preserves the exponential rate of decay of the symplectic form.
• It inherits a near Boltzmann-Gibbs measure preservation property from the near energy preservation property of its associated noise/friction-free symplectic Euler integrator.

V

The geometry behind rigid body-fluid interactions

Joris Vankerschaver

Abstract:
In this talk, we present some of our results on the geometry behind rigid bodies in perfect flows interacting with point vortices. We use symplectic reduction by stages to re-derive the equations of motion, and introduce a number of geometric structures along the way, most notably a special connection which encapsulates the response of the fluid to motions of the rigid body.

As an interesting aside, we show that a number of classical results, such as the expression for the Kutta-Joukowski force on a rigid body with circulation, or the form of the interaction between the vortices and the body, are consequences of the fact that the curvature of this connection is non-zero.

This is joint work with E. Kanso and J. E. Marsden.