708 research outputs found
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Nonsmooth Lagrangian mechanics and variational collision integrators
Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem.
Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated
Discontinuous collocation methods and gravitational self-force applications
Numerical simulations of extereme mass ratio inspirals, the mostimportant
sources for the LISA detector, face several computational challenges. We
present a new approach to evolving partial differential equations occurring in
black hole perturbation theory and calculations of the self-force acting on
point particles orbiting supermassive black holes. Such equations are
distributionally sourced, and standard numerical methods, such as
finite-difference or spectral methods, face difficulties associated with
approximating discontinuous functions. However, in the self-force problem we
typically have access to full a-priori information about the local structure of
the discontinuity at the particle. Using this information, we show that
high-order accuracy can be recovered by adding to the Lagrange interpolation
formula a linear combination of certain jump amplitudes. We construct
discontinuous spatial and temporal discretizations by operating on the
corrected Lagrange formula. In a method-of-lines framework, this provides a
simple and efficient method of solving time-dependent partial differential
equations, without loss of accuracy near moving singularities or
discontinuities. This method is well-suited for the problem of time-domain
reconstruction of the metric perturbation via the Teukolsky or
Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and
GPU architectures are discussed.Comment: 29 pages, 5 figure
Pseudospectral Knotting Methods for Solving Optimal Control Problems
The article of record as published may be found at https://doi.org/10.2514/1.3426A class of computational methods for solving a wide variety of optimal control problems is presented; these
problems include nonsmooth, nonlinear, switched optimal control problems, as well as standard multiphase prob lems. Methods are based on pseudospectral approximations of the differential constraints that are assumed to be
given in the form of controlled differential inclusions including the usual vector field and differential-algebraic
forms. Discontinuities and switches in states, controls, cost functional, dynamic constraints, and various other
mappings associated with the generalized Bolza problem are allowed by the concept of pseudospectral (PS) knots.
Information across switches and corners is passed in the form of discrete event conditions localized at the PS
knots. The optimal control problem is approximated to a structured sparse mathematical programming problem.
The discretized problem is solved using off-the-shelf solvers that include sequential quadratic programming and
interior point methods. Two examples that demonstrate the concept of hard and soft knots are presented.Charles Stark Draper Laboratory, Inc. (Draper)Jet Propulsion Laboratory (JPL)Naval Postgraduate SchoolSecretary of the U.S. Air Forc
A mathematical framework for inverse wave problems in heterogeneous media
This paper provides a theoretical foundation for some common formulations of
inverse problems in wave propagation, based on hyperbolic systems of linear
integro-differential equations with bounded and measurable coefficients. The
coefficients of these time-dependent partial differential equations respresent
parametrically the spatially varying mechanical properties of materials. Rocks,
manufactured materials, and other wave propagation environments often exhibit
spatial heterogeneity in mechanical properties at a wide variety of scales, and
coefficient functions representing these properties must mimic this
heterogeneity. We show how to choose domains (classes of nonsmooth coefficient
functions) and data definitions (traces of weak solutions) so that optimization
formulations of inverse wave problems satisfy some of the prerequisites for
application of Newton's method and its relatives. These results follow from the
properties of a class of abstract first-order evolution systems, of which
various physical wave systems appear as concrete instances. Finite speed of
propagation for linear waves with bounded, measurable mechanical parameter
fields is one of the by-products of this theory
Continuous approximations of a class of piece-wise continuous systems
In this paper we provide a rigorous mathematical foundation for continuous
approximations of a class of systems with piece-wise continuous functions. By
using techniques from the theory of differential inclusions, the underlying
piece-wise functions can be locally or globally approximated. The approximation
results can be used to model piece-wise continuous-time dynamical systems of
integer or fractional-order. In this way, by overcoming the lack of numerical
methods for diffrential equations of fractional-order with discontinuous
right-hand side, unattainable procedures for systems modeled by this kind of
equations, such as chaos control, synchronization, anticontrol and many others,
can be easily implemented. Several examples are presented and three comparative
applications are studied.Comment: IJBC, accepted (examples revised
Numerical computation of nonlinear normal modes in mechanical engineering
This paper reviews the recent advances in computational methods for nonlinear normal modes (NNMs). Different algorithms for the computation of undamped and damped NNMs are presented, and their respective advantages and limitations are discussed. The methods are illustrated using various applications ranging from low-dimensional weakly nonlinear systems to strongly nonlinear industrial structures. © 2015 Elsevier Ltd
Geometric Numerical Integration
The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods
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