A Time Finite Element Method Based on the Differential Quadrature Rule and Hamilton’s Variational Principle

Xing, Yufeng; Qin, Mingbo; Guo, Jiaxing

doi:10.3390/app7020138

Cited by 11 publications

(11 citation statements)

References 45 publications

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“…2. The method of storing incremental matrix Equation (19) indicates that A(h N ) can be expressed as…”

Section: M Algorithmmentioning

confidence: 99%

“…However, excessive single‐step calculations limit the use of this type of methods. The second type is the use of the higher‐order differential quadrature time element . Nevertheless, this approach expands system size and has unavoidable numerical difficulty in calculating higher‐order interpolation polynomials.…”

Section: Introductionmentioning

confidence: 99%

“…The second type is the use of the higher-order differential quadrature time element. [18][19][20] Nevertheless, this approach expands system size and has unavoidable numerical difficulty in calculating higher-order interpolation polynomials. Higher-order method can also be developed by employing multistep or multi-substep approaches.…”

Section: Introductionmentioning

confidence: 99%

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Highly precise time integration method for linear structural dynamic analysis

Xing

Zhang

Wang

2018

Numerical Meth Engineering

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Summary To achieve high accuracy, conventional time integration methods require small time steps, especially when applied to large‐scale finite element systems. The use of small time steps calls for a large number of computations. In this paper, a strategy is proposed to construct an efficient and highly precise time integration method for linear time‐invariant systems. Adopting the multi‐substep notion, the highly precise time integration method creates an integrated amplification matrix prior to recursive calculations by multiplying the amplification matrices of N equal substeps, where a 2m algorithm and the method of storing incremental matrix are respectively employed to reduce computational cost and rounding error. On the basis of the Newmark method, two highly precise symplectic schemes and one highly precise dissipative scheme are developed. In addition, the consistent stability of the used Newmark schemes is presented. The free vibration of a three‐dimensional truss, the impact of a particle against a rod, and the forced vibrations of a rectangular thin plate and a stiff system are investigated to validate the performance of the proposed method.

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“…2. The method of storing incremental matrix Equation (19) indicates that A(h N ) can be expressed as…”

Section: M Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Highly precise time integration method for linear structural dynamic analysis

Xing

Zhang

Wang

2018

Numerical Meth Engineering

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“…Besides of the finite difference method, this is one of the main numerical methods for initial value problems (IVPs). In recent years, many alternative finite element methods (FEMs) for IVPs have been developed [16], which are as well-known as time FEM. According to the different ways of construction, the time FEM can be roughly classified into three kinds: (1) Constructed by Gurtin variational principle, which transforms the initial-boundary value problem into the equivalent boundary value problem (BVP) by means of Laplace positive and inverse transformation [17,18]; (2) Constructed by Hamilton's variational principle or Hamilton's law of variation [19][20][21], in which Bailey [22,23], Simikins [24], and Borri [25] have made important achievements successively and; (3) Constructed by the weighted residual method, and is what we use in the present paper.…”

Section: Introductionmentioning

confidence: 99%

A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics

Xing¹,

Yang²,

Wang³

2019

Applied Sciences

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This paper presents a step-by-step time integration method for transient solutions of nonlinear structural dynamic problems. Taking the second-order nonlinear dynamic equations as the model problem, this self-starting one-step algorithm is constructed using the Galerkin finite element method (FEM) and Newton–Raphson iteration, in which it is recommended to adopt time elements of degree m = 1,2,3. Based on the mathematical and numerical analysis, it is found that the method can gain a convergence order of 2m for both displacement and velocity results when an ordinary Gauss integral is implemented. Meanwhile, with reduced Gauss integration, the method achieves unconditional stability. Furthermore, a feasible integration scheme with controllable numerical damping has been established by modifying the test function and introducing a special integral rule. Representative numerical examples show that the proposed method performs well in stability with controllable numerical dissipation, and its computational efficiency is superior as well.

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“…Bathe and Wilson [39] presented a perfect and underlying process for the accuracy and stability analysis of time integration methods. By referencing to predecessors' procedure, this paper makes comprehensive researches on the improved DQTEM which is different from the Differential Quadrature Time Finite Element Method (DQTFEM) [40] proposed recently by the authors. DQTFEM solved the weak form of ordinary differential equations of motion based on the general form of Hamiltonian Variational Principle, the spatial domain was discretized by the standard FEM, and the Gauss-Lobatto-Legendre integration points were used in the calculation of the time derivatives through DQ rule.…”

Section: Introductionmentioning

confidence: 99%

An Improved Differential Quadrature Time Element Method

Qin

Xing

Guo³

2017

Applied Sciences

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A Differential Quadrature Time Element Method (DQTEM) was proposed by the author and co-worker, its drawback is the need of larger storage capacity since the dimension of the coefficients matrix for solution is the product of both spatial degrees of freedom and temporal degrees of freedom. To solve this problem, an improved DQTEM is developed in this work, in which the differential quadrature method is used to discretize both spatial and time domains, sequentially, and the dimension of the coefficients matrix is greatly reduced without losing solution accuracy. Theoretical studies demonstrate the improved DQTEM features superiorities including higher-order accuracy, adequate stability and symplectic characteristics. The improvement of DQTEM is validated by extensive comparisons of the present DQTEM with the original DQTEM.

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A Time Finite Element Method Based on the Differential Quadrature Rule and Hamilton’s Variational Principle

Cited by 11 publications

References 45 publications

Highly precise time integration method for linear structural dynamic analysis

Highly precise time integration method for linear structural dynamic analysis

A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics

An Improved Differential Quadrature Time Element Method

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