A note on the equivalence of two recent time‐integration schemes for <i>N</i>‐body problems

Graham, Edward; Jelenić, Gordan; Crisfield, M. A.

doi:10.1002/cnm.520

Cited by 17 publications

(9 citation statements)

References 9 publications

(22 reference statements)

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“…In this regard, let us note that Betsch and Steinmann [5] have proposed a different energy-momentum scheme with the introduction of the so-called assumed distance method in the context of the time-finite element method. Nevertheless, as it was later revealed by Graham, Jelenic, Crisfield [17], this assumed distance scheme is only 'almost' energymomentum conserving; i.e. a nearly (but not exactly) energy-momentum conserving scheme.…”

Section: Remarksmentioning

confidence: 94%

Energy conserving and dissipative time finite element schemes for N‐body stiff problems

Bui

2004

Numerical Meth Engineering

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SUMMARYPetrov-Galerkin finite element method is adopted to develop a family of temporal integrators, which preserves the feature of energy conservation or numerical dissipation for non-linear N -body dynamical systems. This leads to an enhancement of numerical stability and the integrators may therefore offer some advantage for the numerical solution of stiff systems in long-term simulations. Dynamically tuneable numerical integration is exploited to improve the accuracy of the time-stepping schemes. Representative simulations for simple non-linear systems show the performance of the schemes in controlling over or damping out unresolved high frequencies.

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Section: Remarksmentioning

confidence: 94%

Energy conserving and dissipative time finite element schemes for N‐body stiff problems

Bui

2004

Numerical Meth Engineering

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“…It is the exact energymomentum conserving algorithm for a special case of the potential energy which polynomials of degree two or lower; see [44]. 2.…”

Section: Gssss Framework Encompassing Lms Methods: Conservative Systementioning

confidence: 99%

“…Algorithm 1 is identical to the so-called assumed distance method proposed by Betsch and Steinmann [13]. It is the exact energy-momentum conserving algorithm when the internal potential energy is given as a polynomial function of degree two or less; see [44]. The assumed distance method is:…”

Section: Energy-momentum Conserving Algorithms For N -Body Systemsmentioning

confidence: 99%

An Overview and Recent Advances in Vector and Scalar Formalisms: Space/Time Discretizations in Computational Dynamics—A Unified Approach

Tamma

Har

Zhou

et al. 2011

Arch Computat Methods Eng

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An overview of recent advances in computational dynamics for modeling and simulation is described. The targeted objectives are towards a wide variety of science and engineering applications in particle and continuum dynamics of structures and materials which fall in this class. Starting with the supposition that in the beginning, the well known Newton's law of motion for N-body systems is given, and is a statement of the principle of balance of linear momentum, subsequently, using this as a landmark, firstly, the principal relations to various other distinctly different fundamental principles which are of primary interest here are established. Likewise, for continuum dynamics of structures, via the principle of balance of linear momentum, analogous developments are also established. Consequently, these distinctly different fundamental principles are shown to serve as the starting point for the various developments. Stemming from three distinctly different fundamental principles, we present recent advances in N-body dynamical systems, and also continuous-body dynamical systems with focus on numerical aspects in space/time discretization. The fundamental principles are the following:the Principle of Virtual Work in Dynamics, Hamilton's Principle and as an alternate (due to inconsistencies associated with Hamilton's principle), Hamilton's Law of Varying Action, and the Principle of Balance of Mechanical Energy. Both vector and scalar formalisms are described in detail with particular focus towards general numerical discretizations in space and/or time for N-body and continuumelastodynamics applications which are encountered in a wide class of holonomic-scleronomic problems. The formulations include the classical Newtonian mechanics framework with vector formalism, and new scalar formalisms with descriptive functions such as the Lagrangian, the Hamiltonian, and the Total Mechanical Energy to readily enable numerical discretizations. The concepts emanating from the present developments and distinctly different fundamental principles inherently: (1) can independently be shown to yield the strong form of the governing mathematical model equations of motion that are continuous in space and/or time together with the natural boundary conditions; the various frameworks for the case of holonomic-scleronomic systems with the mentioned limitations are indeed all equivalent, (2) can explain naturally how the weak statement of the Bubnov-Galerkin weighted-residual form that is customarily employed for discretization with vector formalism arises for both space and time, and (3) can circumvent relying upon traditional practices of conducting numerical discretizations starting either from balance of linear momentum (Newton's second law) involving Cauchy's equations of motion (governing equations) arising from continuum mechanics or via (1) and (2) above if one chooses this option. The concepts instead provide new avenues for numerical space/time discretization for continuum-dynamical systems or time discretization for N-body system...

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“…The eG(1) method for many-particle dynamics is identical with the energy and momentum conserving second order accurate scheme in the References [28,32] as well as in Part I (cf. Reference [55]). This time stepping scheme can be written in a more explicit form as the following equations:…”

Section: The Enhanced Galerkin (Eg) Methodsmentioning

confidence: 99%

Conservation properties of a time FE method. Part IV: Higher order energy and momentum conserving schemes

Groß

Betsch

Steinmann

2005

Int. J. Numer. Meth. Engng

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SUMMARYIn the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite-dimensional Hamiltonian systems with symmetry is shown. The result of this development is the enhanced Galerkin (eG) finite element method in time. The conservation of the eG method is generally related to its collocation property. Total energy conservation, in particular, is obtained by a new projection technique. The eG method is, moreover, based on objective time discretization of the used strain measure. This paper is concerned with particle dynamics and semi-discrete non-linear elastodynamics. The related numerical examples show good performance in presence of stiffness as well as for calculating large-strain motions.

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A note on the equivalence of two recent time‐integration schemes for N‐body problems

Cited by 17 publications

References 9 publications

Energy conserving and dissipative time finite element schemes for N‐body stiff problems

Energy conserving and dissipative time finite element schemes for N‐body stiff problems

An Overview and Recent Advances in Vector and Scalar Formalisms: Space/Time Discretizations in Computational Dynamics—A Unified Approach

Conservation properties of a time FE method. Part IV: Higher order energy and momentum conserving schemes

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