92,612 research outputs found
Null cone evolution of axisymmetric vacuum spacetimes
We present the details of an algorithm for the global evolution of
asymptotically flat, axisymmetric spacetimes, based upon a characteristic
initial value formulation using null cones as evolution hypersurfaces. We
identify a new static solution of the vacuum field equations which provides an
important test bed for characteristic evolution codes. We also show how
linearized solutions of the Bondi equations can be generated by solutions of
the scalar wave equation, thus providing a complete set of test beds in the
weak field regime. These tools are used to establish that the algorithm is
second order accurate and stable, subject to a Courant-Friedrichs-Lewy
condition. In addition, the numerical versions of the Bondi mass and news
function, calculated at scri on a compactified grid, are shown to satisfy the
Bondi mass loss equation to second order accuracy. This verifies that numerical
evolution preserves the Bianchi identities. Results of numerical evolution
confirm the theorem of Christodoulou and Klainerman that in vacuum, weak
initial data evolve to a flat spacetime. For the class of asymptotically flat,
axisymmetric vacuum spacetimes, for which no nonsingular analytic solutions are
known, the algorithm provides highly accurate solutions throughout the regime
in which neither caustics nor horizons form.Comment: 25 pages, 6 figure
Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography
The aim of electrical impedance tomography is to reconstruct the admittivity
distribution inside a physical body from boundary measurements of current and
voltage. Due to the severe ill-posedness of the underlying inverse problem, the
functionality of impedance tomography relies heavily on accurate modelling of
the measurement geometry. In particular, almost all reconstruction algorithms
require the precise shape of the imaged body as an input. In this work, the
need for prior geometric information is relaxed by introducing a Newton-type
output least squares algorithm that reconstructs the admittivity distribution
and the object shape simultaneously. The method is built in the framework of
the complete electrode model and it is based on the Fr\'echet derivative of the
corresponding current-to-voltage map with respect to the object boundary shape.
The functionality of the technique is demonstrated via numerical experiments
with simulated measurement data.Comment: 3 figure
Exponential stabilization of driftless nonlinear control systems using homogeneous feedback
This paper focuses on the problem of exponential stabilization of controllable, driftless systems using time-varying, homogeneous feedback. The analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control Lie algebra. It can be shown that any continuous, time-varying controller that achieves exponential stability relative to the Euclidean norm is necessarily non-Lipschitz. Despite these restrictions, we provide a set of constructive, sufficient conditions for extending smooth, asymptotic stabilizers to homogeneous, exponential stabilizers. The modified feedbacks are everywhere continuous, smooth away from the origin, and can be extended to a large class of systems with torque inputs. The feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers
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