Related papers: Geometric Numerical Integration for Complex Dynami…
In this work, we present a symplectic integration scheme to numerically compute space debris motion. Such an integrator is particularly suitable to obtain reliable trajectories of objects lying on high orbits, especially geostationary ones.…
We introduce new Langevin-type equations describing the rotational and translational motion of rigid bodies interacting through conservative and non-conservative forces, and hydrodynamic coupling. In the absence of non-conservative forces…
This work introduces a new approach for accelerating the numerical analysis of time-domain partial differential equations (PDEs) governing complex physical systems. The methodology is based on a combination of a classical reduced-order…
Symplectic integration algorithms have become popular in recent years in long-term orbital integrations because these algorithms enforce certain conservation laws that are intrinsic to Hamiltonian systems. For problems with large variations…
In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian defined on QxQ, where Q is the configuration manifold, and a…
The PID controller is an elegant and versatile controller for set point tracking in double integrator systems of which mechanical systems evolving on Euclidean space constitute a large class. But since mechanical systems are typically…
We consider mechanics of composite materials in which thin inclusions are modeled by lower-dimensional manifolds. By successively applying the dimensional reduction to junctions and intersections within the material, a geometry of…
An explicit second-order numerical method to integrate the isokinetic equations of motion is derived by fitting circular arcs through every three consecutive points of the discretized trajectory, so that the tangent and the curvature…
We consider geometric numerical integration algorithms for differential equations evolving on symmetric spaces. The integrators are constructed from canonical operations on the symmetric space, its Lie triple system (LTS), and the…
Foliate systems are those which preserve some (possibly singular) foliation of phase space, such as systems with integrals, systems with continuous symmetries, and skew product systems. We study numerical integrators which also preserve the…
This article reviews some integrators particularly suitable for the numerical resolution of differential equations on a large time interval. Symplectic integrators are presented. Their stability on exponentially large time is shown through…
In this paper, a flexible tether-net system is applied to capture the space debris and a numerical framework is established to explore its nonlinear dynamic behaviors, which comprises four principal phases: folding, spreading, contacting,…
"Stiff" differential equations are commonplace in engineering and dynamical systems. To solve them we need flexible integrators that can deal with rapidly-changing righthand sides. This tutorial describes the application of "adaptive" […
A geometric derivation of numerical integrators for nonholonomic systems and optimal control problems is obtained. It is based in the classical technique of generating functions adapted to the special features of nonholonomic systems and…
Learning workable representations of dynamical systems is becoming an increasingly important problem in a number of application areas. By leveraging recent work connecting deep neural networks to systems of differential equations, we…
We develop a unified geometric framework for mechanical systems that combine conservative and dissipative dynamics by formulating them on contact manifolds. Within this setting, we identify the Reeb vector field as the intrinsic generator…
In this paper we use retraction and discretization maps (see [Barbero Li\~n\'an and Mart\'in de Diego, 2022]) as a tool for deriving in a systematic way numerical integrators preserving geometric structures (such as symplecticity or…
A numerical integration method for guiding-center orbits of charged particles in toroidal fusion devices with three-dimensional field geometry is described. Here, high order interpolation of electromagnetic fields in space is replaced by a…
Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete…
A geometric derivation of nonholonomic integrators is developed. It is based in the classical technique of generating functions adapted to the special features of nonholonomic systems. The theoretical methodology and the integrators…