446 research outputs found
Multisymplectic Lie group variational integrator for a geometrically exact beam in R3
In this paper we develop, study, and test a Lie group multisymplectic
integra- tor for geometrically exact beams based on the covariant Lagrangian
formulation. We exploit the multisymplectic character of the integrator to
analyze the energy and momentum map conservations associated to the temporal
and spatial discrete evolutions.Comment: Article in press. 22 pages, 18 figures. Received 20 November 2013,
Received in revised form 26 February 2014, Accepted 27 February 2014.
Communications in Nonlinear Science and Numerical Simulation. 201
Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms
This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a
discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators
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
Adaptive Geometric Numerical Integration of Mechanical Systems
This thesis is about structure preserving numerical integration of initial value problems, i.e., so called geometric numerical integrators. In particular, we are interested in how time-step adaptivity can be achieved in conjunction with structure preserving properties without destroying the good long time integration properties which are typical for geometric integration methods. As a specific application we consider dynamic simulations of rolling bearings and rotor dynamical problems. The work is part of a research collaboration between SKF (www.skf.com) and the Centre of Mathematical Sciences at Lund University
C1-continuous space-time discretization based on Hamilton's law of varying action
We develop a class of C1-continuous time integration methods that are
applicable to conservative problems in elastodynamics. These methods are based
on Hamilton's law of varying action. From the action of the continuous system
we derive a spatially and temporally weak form of the governing equilibrium
equations. This expression is first discretized in space, considering standard
finite elements. The resulting system is then discretized in time,
approximating the displacement by piecewise cubic Hermite shape functions.
Within the time domain we thus achieve C1-continuity for the displacement field
and C0-continuity for the velocity field. From the discrete virtual action we
finally construct a class of one-step schemes. These methods are examined both
analytically and numerically. Here, we study both linear and nonlinear systems
as well as inherently continuous and discrete structures. In the numerical
examples we focus on one-dimensional applications. The provided theory,
however, is general and valid also for problems in 2D or 3D. We show that the
most favorable candidate -- denoted as p2-scheme -- converges with order four.
Thus, especially if high accuracy of the numerical solution is required, this
scheme can be more efficient than methods of lower order. It further exhibits,
for linear simple problems, properties similar to variational integrators, such
as symplecticity. While it remains to be investigated whether symplecticity
holds for arbitrary systems, all our numerical results show an excellent
long-term energy behavior.Comment: slightly condensed the manuscript, added references, numerical
results unchange
Computational methods and software systems for dynamics and control of large space structures
Two key areas of crucial importance to the computer-based simulation of large space structures are discussed. The first area involves multibody dynamics (MBD) of flexible space structures, with applications directed to deployment, construction, and maneuvering. The second area deals with advanced software systems, with emphasis on parallel processing. The latest research thrust in the second area involves massively parallel computers
Discrete mechanics and optimal control for constrained systems
The equations of motion of a controlled mechanical system subject to holonomic constraints may be formulated in terms
of the states and controls by applying a constrained version of the Lagrange-d’Alembert principle. This paper derives a
structure-preserving scheme for the optimal control of such systems using, as one of the key ingredients, a discrete analogue
of that principle. This property is inherited when the system is reduced to its minimal dimension by the discrete null
space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced discrete
equations serve as nonlinear equality constraints for the minimization of a given objective functional. The algorithm yields
a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the
initial to the desired final state. In particular, for the optimal control of multibody systems, a force formulation consistent
with the joint constraints is introduced. This enables one to prove the consistency of the evolution of momentum maps.
Using a two-link pendulum, the method is compared with existing methods. Further, it is applied to a satellite reorientation
maneuver and a biomotion problem
Formulation and performance of variational integrators for rotating bodies
Variational integrators are obtained for two mechanical systems whose configuration spaces are, respectively, the rotation group and the unit sphere. In the first case, an integration algorithm is presented for Euler’s equations of the free rigid body, following the ideas of Marsden et al. (Nonlinearity 12:1647–1662, 1999). In the second example, a variational time integrator is formulated for the rigid dumbbell. Both methods are formulated directly on their nonlinear configuration spaces, without using Lagrange multipliers. They are one-step, second order methods which show exact conservation of a discrete angular momentum which is identified in each case. Numerical examples illustrate their properties and compare them with existing integrators of the literature
- …