An optimized implementation of the Newmark/Newton‐Raphson algorithm for the time integration of non‐linear problems

Jacob, Breno Pinheiro; Ebecken, Nelson F. F.

doi:10.1002/cnm.1640101204

Cited by 41 publications

(19 citation statements)

References 6 publications

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“…(14), " A is the effective nonlinear stiffness matrix and " B ðkÞ is the effective residual vector. Further details on an optimized implementation of the Newmark/NewtonRaphson algorithm can be found, for instance, in Jacob and Ebecken [18]. Equations (13) and (14) enable the computation of the FEM response at time t n .…”

Section: Nonlinear Dynamic Fem Analysismentioning

confidence: 99%

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Iterative coupling of BEM and FEM for nonlinear dynamic analyses

Soares

Estorff

Mansur

2004

Computational Mechanics

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The present work deals with the iterative coupling of boundary element and finite element methods. First, the domain of the original problem is subdivided into two subdomains, which are separately modeled by the FEM and BEM. Thus the special features and advantages of the two methodologies can be taken into account. Then, prescribing arbitrary transient boundary conditions, a successive renewal of the variables on the interface between the two subdomains is performed through an iterative procedure until the final convergence is achieved. In the case of local nonlinearities within the finite element subdomain, it is straightforward to perform the iterative coupling together with the iterations needed to solve the nonlinear system. The procedure turns out to be very efficient. Moreover, a special formulation allows taking into account different durations of the time steps in each subdomain.The numerical simulation of arbitrarily shaped continuous bodies subjected to harmonic or transient loads remains, despite much effort and progress over the last decades, a challenging area of research. In most cases, discrete techniques, such as the finite element method (FEM) and the boundary element method (BEM) have been employed and continuously further developed with respect to accuracy and efficiency. Both methodologies can be formulated in the time domain or in the frequency domain, and each approach has relative benefits and limitations. The finite element method, for instance, is well suited for inhomogeneous and anisotropic materials as well as for dealing with the nonlinear behavior of a body. For systems with infinite extension and regions of high stress concentration, however, the use of the boundary element method is by far more advantageous. More details are given, e.g., by Hughes [17] or Bathe [2] for the FEM and by Becker [3] or Dominguez [10] in the case of the BEM.In fact, it did not take long until some researchers started to combine the FEM and the BEM in order to profit from their respective advantages by trying to evade their disadvantages. Up to now, quite a few publications concentrate on such coupling approaches. Many details are given, e.g., by Zienkiewicz et al. [25,26], who were among the first suggesting a ''mariage à la mode -the best of two worlds'', by von Estorff and Prabucki [13], von Estorff and Antes [14], Belytschko and Lu [4], Yu et al. [23], or Rizos and Wang [21]. A rather complete overview is provided by Beskos [ [5][6][7]. It should be mentioned, that in most cases the BEM has been used to model those parts of the investigated bodies which are of semi-infinite extension, while finite parts were represented with the FEM. Moreover, it has been assumed that the considered systems behave linearly, which means that only elastic material and small displacements are considered.The FEM/BEM coupling in the time domain has also been successfully used to take into account nonlinear effects. Thus Pavlatos and Beskos [20] as well as Yazdchi et al. [24], for instance, modeled an inelastic structure ...

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Section: Nonlinear Dynamic Fem Analysismentioning

confidence: 99%

“…It turns out to be very efficient with respect not only to the first but also to the second and third aspect mentioned above. The FEM and the BEM parts are based on usual formulations, as suggested, e.g., by Crisfield [9], Jacob and Ebecken [18], Bathe [2] and Mansur [19], Dominguez [10], Soares Jr et al [22], respectively.…”

Section: Introductionmentioning

confidence: 99%

Iterative coupling of BEM and FEM for nonlinear dynamic analyses

Soares

Estorff

Mansur

2004

Computational Mechanics

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“…(A) Initial computations for time instant n + 1 (A.1) For each subdomain/processor node, Initialize total displacements, incremental displacements, and elastic forces: Calculate constant term of effective residualsb * / effective residuals for first iterationb (0) [37].…”

Section: Subcycling Technique Associated To the (Hull/lines) Decomposmentioning

confidence: 99%

“…A detailed formulation of the implementation of implicit Newmark time integration algorithms in association with the N-R technique, including the particular form of the effective matrix A and the effective load vector b (now referred as the 'effective residual vector') may be found in [37].…”

Section: Extension To Nonlinear Problemsmentioning

confidence: 99%

“…Therefore, the internal domain decomposition method presented here [49] may be seen as an extension of that methods, now specialized for nonlinear problems and tailored for the class of applications considered here. In this method, the domain decomposition scheme is associated to the combination of an implicit Newmark-type operator with the iterative N-R technique (as described in [37]). The steps of this implicit internal domain decomposition algorithm, for a given time-step of the solution procedure, can be summarized in Table I.…”

Section: Subcycling Technique Associated To the (Hull/lines) Decomposmentioning

confidence: 99%

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Implicit domain decomposition methods for coupled analysis of offshore platforms

Rodrigues

Corrêa

Jacob

2006

Commun. Numer. Meth. Engng.

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SUMMARYThis work presents the implementation of optimized numerical tools for the coupled analysis of floating platforms for offshore oil exploitation. The focus is on time-domain, nonlinear dynamic analysis, considering the coupling between the hydrodynamic behaviour of the hull and the structural behaviour of the mooring lines and risers modelled by finite elements (FEs). Some aspects of the formulation and solution of the large-amplitude equations of motion of the hull of the platform are presented, including a brief description of the hydrodynamic models and calculation of the environmental forces. The main aspects of the formulation for the spatial and time discretization of the structural model for the lines are also discussed. Since coupled analyses may require excessive computational costs, the objective of this work is to present the implementation and application of domain decomposition methods, adapted and specialized for the problem at hand, in order to optimize the efficiency of the computational tool. Two groups of domain decomposition methods are considered: the first is a subcycling technique that takes into account the natural partition that exists between the hull and the lines; the second considers the internal decomposition of the mesh of FEs to represent the mooring lines and risers. The methods are devised having in mind their implementation in computers with parallel architecture. Results of a numerical application are presented in order to assess the performance of the methods.

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Generalized shape optimization of transient vibroacoustic problems using cut elements

Dilgen

Aage

2021

Numerical Meth Engineering

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This article propose a generalized shape optimization method for transient coupled acoustic-mechanical interaction problems. The transient problem formulation allows to optimize for a broadband frequency content and the possibility to investigate transient phenomena such as pulse shaping. Throughout the work, the geometry is defined with the help of a nodal-based design field, for which its zero-level contour describes the interface between acoustic and structural domains. The approach utilizes a fixed background mesh to represent the geometry and a cut element immersed boundary method for the physical modeling. This modeling approach provides accurate solutions to the strongly coupled governing equations based on a special integration scheme. The optimization problem is solved using a gradient-based optimizer and employs a fully discrete adjoint approach for calculating the sensitivity information. A numerical examination on the accuracy of the sensitivity analysis compares the fully discrete gradients to the commonly used semidiscrete adjoint approach for transient optimization revealing the importance of consistent sensitivities. The transient design formulation is validated on a 2D benchmark problem concerning the design of a time-harmonic acoustic partitioner. The developed framework is then applied for the design of vibroacoustic pulse shaping devices to demonstrate control of a transient phenomena.

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An optimized implementation of the Newmark/Newton‐Raphson algorithm for the time integration of non‐linear problems

Cited by 41 publications

References 6 publications

Iterative coupling of BEM and FEM for nonlinear dynamic analyses

Iterative coupling of BEM and FEM for nonlinear dynamic analyses

Implicit domain decomposition methods for coupled analysis of offshore platforms

Generalized shape optimization of transient vibroacoustic problems using cut elements

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