Method of separation of variables and Hamiltonian system

Zhong, Wanxie; Zhong, Xiang-Xiang

doi:10.1002/num.1690090107

Cited by 36 publications

(6 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Fluid mechanics has a larger collection of references related to Hamiltonain 21),22),23), 24) , which employ advanced numerical analysis techniques. As for computational mechanics, we can find some relatively old works 25),26),27),28), 29) . In structural mechanics, a Hamiltonian is used to introduce a phase space that consists of location and velocity 30) used in structural dynamics.…”

Section: Literature Surveymentioning

confidence: 99%

See 1 more Smart Citation

Rigorous Derivation of Hamiltonian From Lagrangian for Solid Continuum

et al. 2018

View full text Add to dashboard Cite

This paper provides rigorous derivation of a Hamiltonian from a given Lagrangian of solid continuum, for a possible improvement of dynamic analysis. The derived Hamiltonian, unlike an ordinary one, has momentum and strain as argument, and the associated canonical equation includes spatial derivative in it. Characteristics of the Hamiltonian of this form are studied. The appearance of strain rate in the canonical equation, which has been overlooked in the past researches, is pointed out. For numerical computation, a Hamiltonian for discretized functions is derived. It is shown that discretized strain rate appears in the canonical equation when suitable discretization scheme is applied.

show abstract

Section: Literature Surveymentioning

confidence: 99%

“…5 for the procedures of deriving Eqs. (24) and (25). The temporal derivative of p α and ϵ β are com- puted in terms of the derivative of H * multiplied by B βα that serves as a role of spatial derivative.…”

Section: (1) Modal Analysismentioning

confidence: 99%

Rigorous Derivation of Hamiltonian From Lagrangian for Solid Continuum

et al. 2018

View full text Add to dashboard Cite

show abstract

“…In the 1990s, a novel symplectic methodology for elasticity was proposed by Zhong et al [22][23][24]. The methodology was rapidly developed in applied mechanics for its capability of going beyond the limitation of the semi-inverse method and extending the scope of analytic solutions.…”

Section: Introductionmentioning

confidence: 99%

Analytic bending solutions of free rectangular thin plates resting on elastic foundations by a new symplectic superposition method

Zhong

2013

Proc. R. Soc. A.

View full text Add to dashboard Cite

Analytic bending solutions of free rectangular thin plates resting on elastic foundations, based on the Winkler model, are obtained by a new symplectic superposition method. The proposed method offers a rational elegant approach to solve the problem analytically, which was believed to be difficult to attain. By way of a rigorous but simple derivation, the governing differential equations for rectangular thin plates on elastic foundations are transferred into Hamilton canonical equations. The symplectic geometry method is then introduced to obtain analytic solutions of the plates with all edges slidingly supported, followed by the application of superposition, which yields the resultant solutions of the plates with all edges free on elastic foundations. The proposed method is capable of solving plates on elastic foundations with any other combinations of boundary conditions. Comprehensive numerical results validate the solutions by comparison with those obtained by the finite element method.

show abstract

“…The traditional semi-inverse method belongs to the Lagrangian system of one kind of variables and lacks rational analysis so that the solving range is limited very largely. A new systematic methodology for theory of elasticity has been established and some achievements have been acquired [3,4] . And the systematic methodology can be applied to many branches of applied mechanics [5] .…”

Section: Introductionmentioning

confidence: 99%

Theoretic solution of rectangular thin plate on foundation with four edges free by symplectic geometry method

钟阳

张永山

2006

Appl Math Mech

View full text Add to dashboard Cite

The theoretic solution for rectangular thin plate on foundation with four edges free is derived by symplectic geometry method. In the analysis proceeding, the elastic foundation is presented by the Winkler model. Firstly, the basic equations for elastic thin plate are transferred into Hamilton canonical equations. The symplectic geometry method is used to separate the whole variables and eigenvalues are obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees are developed. Since the basic elasticity equations of thin plate are only used and it is not need to select the deformation function arbitrarily. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example is given to demonstrate the accuracy and convergence of the current solution.

show abstract

Method of separation of variables and Hamiltonian system

Cited by 36 publications

References 7 publications

Rigorous Derivation of Hamiltonian From Lagrangian for Solid Continuum

Rigorous Derivation of Hamiltonian From Lagrangian for Solid Continuum

Analytic bending solutions of free rectangular thin plates resting on elastic foundations by a new symplectic superposition method

Theoretic solution of rectangular thin plate on foundation with four edges free by symplectic geometry method

Contact Info

Product

Resources

About