Galerkin variational integrators and modified symplectic Runge–Kutta methods

Ober‐Blöbaum, Sina

doi:10.1093/imanum/drv062

Cited by 38 publications

(54 citation statements)

References 23 publications

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“…3 where we applied an order four spRK method. Further discussions and details with regard to higher order accurate VIs and spRK methods are given in [3].…”

Section: Results and Outlookmentioning

confidence: 99%

Variational integration in endochronic theory for small strain elastoplastodynamics

Bär¹,

Groß²

2016

Proc Appl Math and Mech

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This paper presents a basic endochronic plasticity model with isotropic hardening for small strain theory according to Valanis. The key point of this model is a convolution integral over an intrinsic time scale involving past values of the strain measure and a so-called memory kernel which leads to a smooth evolution equation for the internal variable. For the temporal discretization of the underlying constitutive equations and the resulting evolution equation we use higher order accurate variational integrators (VI). The remarkable feature is the fact that we approximate the position vector in terms of velocities according to partitioned Runge-Kutta methods (pRK). As a representative model problem serves a quasistatic, uniaxial tensile testing as well as a dynamic, elastoplastic cantilever beam with a smooth plasticity model according to the Valanis framework.

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“…3 where we applied an order four spRK method. Further discussions and details with regard to higher order accurate VIs and spRK methods are given in [3].…”

Section: Results and Outlookmentioning

confidence: 99%

Variational integration in endochronic theory for small strain elastoplastodynamics

Bär¹,

Groß²

2016

Proc Appl Math and Mech

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“…As discussed in the previous subsection, the optimal control problem consists of finding trajectories of the state variables and control inputs, satisfying the equations (69a), (69b), (69c) and (70), subject to boundary conditions and minimizing the cost to high-order numerical methods (where here, high-order refers to the local truncation error). This problem, in the context of principal bundles and integration of Lagrange-Poincare equations, is a promising line of investigation, in particular how to relate higher-order constrained variational problems on principal bundles with higher-order integrators, such as Galerkin variational integrators and modified symplectic Runge-Kutta methods, using the results for first-order systems given in [17] and [55].…”

Section: Energy Minimum Control Of Two Coupled Rigid Bodiesmentioning

confidence: 99%

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

Bloch¹,

Colombo²,

Jiménez³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincaré equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincaré equations.Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.2010 Mathematics Subject Classification. Primary: 37K05 ; Secondary: 37J15; 37N05; 49M25; 49S05; 49J15.

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“…Indeed, well known methods, like Störmer-Verlet [1], Newmark [2] and symplectic partitioned Runge-Kutta [1,3], fit in this framework. This approach can be used also for the study of fairly new methods, like the Galerkin variational integrators [4][5][6][7], that will be analyzed in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…Some of the cited articles already contains partial results for some particular Galerkin variational integrators: in all these papers the space of polynomials of degree at most N is used to approximate the space of trajectories and different quadrature rules are employed (Gauss-Legendre quadrature with N nodes [1] or Gauss-Legendre-Lobatto quadrature with N + 1 nodes [5,6]). In particular in [6] the equivalence between some Galerkin variational integrators and a particular class of Runge-Kutta methods has been proved. This equivalence determines the convergence properties of these methods.…”

Section: Introductionmentioning

confidence: 99%

High-Order Variational Time Integrators for Particle Dynamics

Miglio

Parolini

Penati

et al. 2018

Communications in Applied and Industrial Mathematics

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The general family of Galerkin variational integrators are analyzed and a complete classification of such methods is proposed. This classification is based upon the type of basis function chosen to approximate the trajectories of material points and the numerical quadrature formula used in time. This approach leads to the definition of arbitrarily high order method in time. The proposed methodology is applied to the simulation of brownout phenomena occurring in helicopter take-off and landing.

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Galerkin variational integrators and modified symplectic Runge–Kutta methods

Cited by 38 publications

References 23 publications

Variational integration in endochronic theory for small strain elastoplastodynamics

Variational integration in endochronic theory for small strain elastoplastodynamics

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

High-Order Variational Time Integrators for Particle Dynamics

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