Dynamic response of mechanical systems by a weak Hamiltonian formulation

Borri, M.; Ghiringhelli, G.L.; Lanz, M.; Mantegazza, Paolo; Merlini, T.

doi:10.1016/0045-7949(85)90098-7

Cited by 95 publications

(5 citation statements)

References 10 publications

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“…To solve the linear IVP of ODEs in Equation 5, a time-domain FEM based on the Galerkin weak form [25,34,42] was used in the present paper. Let v ∈ H 1 v and u h ∈ H 1 E denote the test function and the trial function, respectively.…”

Section: Galerkin Finite Element Methodsmentioning

confidence: 99%

“…In this case, the present algorithm automatically degrades into the so-called 'GW (Galerkin Weak form) method' which has been already proposed [34] for linear elastodynamic problems, and turns out to be an unconditionally stable time integration method with the spectral radii ρ(A) = 1, similar to quadratic and cubic elements. Borri [25] discovered that, when reduced Gauss integration was used in calculation of the stiffness matrix, the GW method for linear elastodynamic equations is unconditionally stable, otherwise, it is conditionally stable. This conclusion might be understood as the loss of accuracy earns the improvement of stability.…”

Section: Stabilitymentioning

confidence: 99%

“…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%

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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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Section: Galerkin Finite Element Methodsmentioning

confidence: 99%

Section: Stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Xing¹,

Yang²,

Wang³

2019

Applied Sciences

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“…[30][31][32][33][34][35][36][37][38] Recently, authors of this article have developed a single-field, velocity based, tDG-ST/FEM (hereafter, read as v-ST/FEM) to solve the problems related to soil dynamics and earthquake engineering. [39][40][41][42][43][44] In v-ST/FEM, velocity is the independent variable which is discontinuous in time. Displacement field is evaluated by the time integration of velocity field.…”

Section: Introductionmentioning

confidence: 99%

A methodology to control numerical dissipation characteristics of velocity based time discontinuous Galerkin space‐time finite element method

Sharma

Fujisawa

Murakami

et al. 2022

Numerical Meth Engineering

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Direct time integration schemes are an integral part of the FEM simulation of structural dynamics problems. Such schemes should be at least second-order accurate, unconditionally stable, and numerically dissipates the high-frequency components. To this end, this article develops a time integration scheme, called modified v-ST/FEM, which is based on the time-discontinuous Galerkin method. The proposed method employs an unsymmetric triangular bubble function for relating the displacement field to the velocity field. The modified v-ST/FEM contains two-parameter 𝛼 ∈ (0, 0.5) and 𝛽 ∈ (−1, 𝛽 c ) for controlling the dissipation of high-frequency components. A comprehensive study of the influence of 𝛼 and 𝛽 on the numerical performance of the proposed method is conducted. It is found that the error in the solution increases when the value of 𝛼 increases. However, for all practical purposes, 𝛽 has a negligible influence on the accuracy of the proposed method. The modified v-ST/FEM is second-order accurate for 𝛼 ≠ 0.0, and third-order accurate for 𝛼 = 0.0. The numerical efficacy of the modified v-ST/FEM is demonstrated by solving some benchmark problems and comparing its result to those obtained by other popular methods such as Trapezoidal rule, HHT-𝛼, and Bathe's scheme.

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“…Argyris and Scharpf , Fried , Bailey , Simkins , and Borri et al. . This law can be regarded as a generalization of Hamilton's principle: It accounts for any initial and final velocities by considering non‐zero variations in the displacement at the boundaries of the time domain.…”

Section: Introductionmentioning

confidence: 99%

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

2016

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We develop a class of C1‐continuous time integration methods that are applicable to conservative problems in elastodynamics. These methods are based on Hamilton's law of varying action. From the action of the continuous system we derive a spatially and temporally weak form of the governing equilibrium equations. This expression is first discretized in space, considering standard finite elements. The resulting system is then discretized in time, approximating the displacement by piecewise cubic Hermite shape functions. Within the time domain we thus achieve C1‐continuity for the displacement field and C0‐continuity for the velocity field. From the discrete virtual action we finally construct a class of one‐step schemes. These methods are examined both analytically and numerically. Here, we study both linear and nonlinear systems as well as inherently continuous and discrete structures. In the numerical examples we focus on one‐dimensional applications. The provided theory, however, is general and valid also for problems in 2D or 3D. We show that the most favorable candidate — denoted as p2‐scheme — converges with order four. Thus, especially if high accuracy of the numerical solution is required, this scheme can be more efficient than methods of lower order. It further exhibits, for linear simple problems, properties similar to variational integrators, such as symplecticity. While it remains to be investigated whether symplecticity holds for arbitrary systems, all our numerical results show an excellent long‐term energy behavior.

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Dynamic response of mechanical systems by a weak Hamiltonian formulation

Cited by 95 publications

References 10 publications

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

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

A methodology to control numerical dissipation characteristics of velocity based time discontinuous Galerkin space‐time finite element method

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

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Dynamic response of mechanical systems by a weak Hamiltonian formulation

Cited by 95 publications

References 10 publications

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

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

A methodology to control numerical dissipation characteristics of velocity based time discontinuous Galerkin space‐time finite element method

C1‐continuous space‐time discretization based on Hamilton's law of varying action

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C¹‐continuous space‐time discretization based on Hamilton's law of varying action