Generalizations of Aitken's process for accelerating the convergence of sequences

Brezinski, Claude; Redivo–Zaglia, Michela

doi:10.1590/s0101-82052007000200001

Cited by 17 publications

(10 citation statements)

References 14 publications

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“…When the brute-force sub-iterations lead to severe numerical instabilities during strong added-mass effects, the NIFC-based correction provides a stability to the partitioned FSI coupling [38]. The present force correction scheme can be interpreted as a generalization of Aitken's ∆ 2 extrapolation scheme [45,46] to provide convergent behavior to the interface force sequence generated through the nonlinear iterations between the fluid and the structure. The geometric extrapolations with the aid of dynamic weighting parameter allow to transform a divergent fixedpoint iteration to a stable and convergent iteration [38,25].…”

Section: Interface Force Correction Schemementioning

confidence: 97%

A 3D common-refinement method for non-matching meshes in partitioned variational fluid–structure analysis

Law

Joshi

et al. 2018

Journal of Computational Physics

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We present a three-dimensional (3D) common-refinement method for non-matching meshes between discrete non-overlapping subdomains of incompressible fluid and nonlinear hyperelastic structure. The fluid flow is discretized using a stabilized Petrov-Galerkin method, and the large deformation structural formulation relies on a continuous Galerkin finite element method. An arbitrary Lagrangian-Eulerian formulation with a nonlinear iterative force correction (NIFC) coupling is achieved in a staggered partitioned manner by means of fully decoupled implicit procedures for the fluid and solid discretizations. To begin, we first investigate the accuracy of common-refinement method (CRM) to satisfy traction equilibrium condition along the fluid-elastic interface with non-matching meshes. We systematically assess the accuracy of CRM against the matching grid solution by varying grid mismatch between the fluid and solid meshes over a cylindrical tubular elastic body. We demonstrate second-order accuracy of CRM through uniform refinements of fluid and solid meshes along the interface. We then extend the error analysis to transient data transfer across non-matching meshes between fluid and solid solvers. We show that the common-refinement discretization across non-matching fluid-structure grids yields accurate transfer of the physical quantities across the fluid-solid interface. We next solve a 3D benchmark problem of a cantilevered hyperelastic plate behind a circular bluff body and verify the accuracy of coupled solutions with respect to the available solution in the literature. By varying the solid interface resolution, we generate various non-matching grid ratios and quantify the accuracy of CRM for the nonlinear structure interacting with a laminar flow. We illustrate that the CRM with the partitioned NIFC treatment is stable for low solid-to-fluid density ratio and non-matching meshes. Finally, we demonstrate the 3D parallel implementation of common-refinement with NIFC scheme for a realistic engineering problem of drilling riser undergoing complex vortex-induced vibration with strong added mass effects.

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Section: Interface Force Correction Schemementioning

confidence: 97%

A 3D common-refinement method for non-matching meshes in partitioned variational fluid–structure analysis

Law

Joshi

et al. 2018

Journal of Computational Physics

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“…According to Brezinski and Zaglia [16] it is recommended to calculate the second line in (42) in the following form. It is mathematically equivalent to the second line in (42) but less susceptible to round-off errors according to [16]:…”

Section: The Algorithmmentioning

confidence: 99%

“…Our method is not based on polynomial fits. It was noted in [14,15] that the results of two Secant steps can be combined into a better approximant of the root in a way reminiscent of Aitken's delta-squared method [16] or Shank's transformation [17]. We take this idea further.…”

Section: Introductionmentioning

confidence: 99%

A Method to Accelerate the Convergence of the Secant Algorithm

Nijmeijer¹

2014

Advances in Numerical Analysis

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We present an acceleration technique for the Secant method. The Secant method is a root-searching algorithm for a general function f. We exploit the fact that the combination of two Secant steps leads to an improved, so-called first-order approximant of the root. The original Secant algorithm can be modified to a first-order accelerated algorithm which generates a sequence of first-order approximants. This process can be repeated: two nth order approximants can be combined in a (n+1)th order approximant and the algorithm can be modified to an (n+1)th order accelerated algorithm which generates a sequence of such approximants. We show that the sequence of nth order approximants converges to the root with the same order as methods using polynomial fits of f of degree n.

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“…Calude Breziniski and Michela Redivo Zaglia [11] proposed extension of Aitken's extrapolation into a general form involving transforming the sequence of iteration into a different form using known sequences which can lead to stabilization and convergence of the original iteration. The transformation, however, is not simple and straight forward and required further refinement.…”

Section: ( )mentioning

confidence: 99%

Higher Order Aitken Extrapolation with Application to Converging and Diverging Gauss-Seidel Iterations

Tiruneh¹

2013

JAMP

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In this paper Aitken's extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduced that enables successive decomposition of high Eigen values of the iteration matrix to enable convergence. While extrapolation of a convergent fixed point iteration using a geometric series sum is a known form of Aitken acceleration, it is shown in this paper that the same formula can be used to estimate the solution of sets of linear equations from diverging Gauss -Seidel iterations. In both convergent and divergent iterations, the ratios of differences among the consecutive values of iteration eventually form a convergent (divergent) series with a factor equal to the largest Eigen value of the iteration matrix. Higher order Aitken extrapolation is shown to eliminate the influence of dominant Eigen values of the iteration matrix in successive order until the iteration is determined by the lowest possible Eigen values. For the convergent part of the Gauss-Seidel iteration, further acceleration is made possible by coupling of the extrapolation technique with the successive over relaxation (SOR) method. Application examples from both convergent and divergent iterations have been provided. Coupling of the extrapolation with the SOR technique is also illustrated for a steady state two dimensional heat flow problem which was solved using MATLAB programming.

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Generalizations of Aitken's process for accelerating the convergence of sequences

Cited by 17 publications

References 14 publications

A 3D common-refinement method for non-matching meshes in partitioned variational fluid–structure analysis

A 3D common-refinement method for non-matching meshes in partitioned variational fluid–structure analysis

A Method to Accelerate the Convergence of the Secant Algorithm

Higher Order Aitken Extrapolation with Application to Converging and Diverging Gauss-Seidel Iterations

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