5,276 research outputs found
An Overview of Variational Integrators
The purpose of this paper is to survey some recent advances in variational
integrators for both finite dimensional mechanical systems as well as continuum
mechanics. These advances include the general development of discrete
mechanics, applications to dissipative systems, collisions, spacetime integration algorithms,
AVI’s (Asynchronous Variational Integrators), as well as reduction for
discrete mechanical systems. To keep the article within the set limits, we will only
treat each topic briefly and will not attempt to develop any particular topic in
any depth. We hope, nonetheless, that this paper serves as a useful guide to the
literature as well as to future directions and open problems in the subject
Symplectic Model Reduction of Hamiltonian Systems
In this paper, a symplectic model reduction technique, proper symplectic
decomposition (PSD) with symplectic Galerkin projection, is proposed to save
the computational cost for the simplification of large-scale Hamiltonian
systems while preserving the symplectic structure. As an analogy to the
classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is
designed to build a symplectic subspace to fit empirical data, while the
symplectic Galerkin projection constructs a reduced Hamiltonian system on the
symplectic subspace. For practical use, we introduce three algorithms for PSD,
which are based upon: the cotangent lift, complex singular value decomposition,
and nonlinear programming. The proposed technique has been proven to preserve
system energy and stability. Moreover, PSD can be combined with the discrete
empirical interpolation method to reduce the computational cost for nonlinear
Hamiltonian systems. Owing to these properties, the proposed technique is
better suited than the classical POD-Galerkin approach for model reduction of
Hamiltonian systems, especially when long-time integration is required. The
stability, accuracy, and efficiency of the proposed technique are illustrated
through numerical simulations of linear and nonlinear wave equations.Comment: 25 pages, 13 figure
Lagrangian Data-Driven Reduced Order Modeling of Finite Time Lyapunov Exponents
There are two main strategies for improving the projection-based reduced
order model (ROM) accuracy: (i) improving the ROM, i.e., adding new terms to
the standard ROM; and (ii) improving the ROM basis, i.e., constructing ROM
bases that yield more accurate ROMs. In this paper, we use the latter. We
propose new Lagrangian inner products that we use together with Eulerian and
Lagrangian data to construct new Lagrangian ROMs. We show that the new
Lagrangian ROMs are orders of magnitude more accurate than the standard
Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to
construct the ROM basis. Specifically, for the quasi-geostrophic equations, we
show that the new Lagrangian ROMs are more accurate than the standard Eulerian
ROMs in approximating not only Lagrangian fields (e.g., the finite time
Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction).
We emphasize that the new Lagrangian ROMs do not employ any closure modeling to
model the effect of discarded modes (which is standard procedure for
low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase
in the new Lagrangian ROMs' accuracy is entirely due to the novel Lagrangian
inner products used to build the Lagrangian ROM basis
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