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This article presents an ultraweak discontinuous Petrov-Galerkin (DPG) formulation of the time-harmonic Maxwell equations for the vectorial envelope of the electromagnetic field in a weakly-guiding multi-mode fiber waveguide. This formulation is derived using an envelope ansatz for the vector-valued electric and magnetic field components, factoring out an oscillatory term of exp(-ikz) with a user-defined wavenumber k, where z is the longitudinal fiber axis and field propagation direction. The resulting formulation is a modified system of the time-harmonic Maxwell equations for the vectorial envelope of the propagating field. This envelope is less oscillatory in the z-direction than the original field, so that it can be more efficiently discretized and computed, enabling solutions to the vectorial DPG Maxwell system in fibers that are 1000x longer than previously possible. Different approaches for incorporating a perfectly matched layer for absorbing the outgoing wave modes at the fiber end are derived and compared numerically. The resulting formulation is used to solve a 3D Maxwell model of an ytterbium-doped active gain fiber amplifier, coupled with the heat equation for including thermal effects. The nonlinear model is then used to simulate thermally-induced transverse mode instability (TMI). The numerical experiments demonstrate that it is computationally feasible to perform simulations and analysis of real-length optical fiber laser amplifiers using discretizations of the full vectorial time-harmonic Maxwell equations. The approach promises a new high-fidelity methodology for analyzing TMI in high-power fiber laser systems and is extendable to including other nonlinearities.
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This content will become publicly available on September 1, 2026
Finding mechanisms of exceptional mobility using numerical algebraic geometry
Mechanisms of exceptional mobility, including both overconstrained mechanisms and robots with self-motion, move with more degrees of freedom than predicted by the Grübler–Kutzbach formulas. Although a number of such cases are known, it is difficult to find new examples. This article explains a geometric formulation, called a fiber product, that facilitates finding exceptional mechanisms using tools from numerical algebraic geometry. The purpose of this article is to specialize the mathematical theory developed in A.J. Sommese and C.W. Wampler (2008) to the realm of kinematics and to present simple planar, spherical, and spatial examples that illustrate basic concepts. Although the formulation is general, its application to more complicated mechanisms will require the development of more refined solution techniques that exploit the symmetry inherent in fiber products.
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- Award ID(s):
- 2144732
- PAR ID:
- 10656471
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Mechanism and Machine Theory
- Volume:
- 211
- Issue:
- C
- ISSN:
- 0094-114X
- Page Range / eLocation ID:
- 106033
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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