Mathematical and numerical connections between polynomial extrapolation and Padé approximants: applications in structural mechanics

Damil, N.; Cadou, Jean-Marc; Potier‐Ferry, Michel

doi:10.1002/cnm.705

Cited by 11 publications

(17 citation statements)

References 17 publications

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“…Second remark: the proposed method converges more rapidly than the preconditioned conjugate method, with the same preconditioning matrix. Third, the proposed method and the HPP solver of the Section 2.3 lead to the same convergence curve: this is in accordance with the results of Reference [17], where one has established the mathematical equivalence between a sequence accelerated by MMPE and the 'homotopy-perturbation-Padé' method. Consequently, the HPP will be no longer considered in the following numerical tests.…”

Section: Evaluation Of the Proposed Solversupporting

confidence: 85%

“…In the practice, the results from MMPE and Padé approximants are very close [18]. It has been established that, in some specific cases, the Padé approximants introduced in [15] lead to the same mathematical results as MMPE [17].…”

Section: Convergence Accelerationmentioning

confidence: 75%

“…The method to deduce the Padé approximants is exactly the same as in many previous works (see, for example, [15]) and this permits to accelerate the convergence of the polynomial approximation (17). In this form, the algorithm is not iterative because the series (17) and the Padé approximants are explicit functions of .…”

Section: A Variant Based On Homotopy and Perturbationmentioning

confidence: 94%

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A solver combining reduced basis and convergence acceleration with applications to non‐linear elasticity

Cadou

Potier‐Ferry²

2009

Numer Methods Biomed Eng

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SUMMARYAn iterative solver is proposed to solve the family of linear equations arising from the numerical computation of non-linear problems. This solver relies on two quantities coming from previous steps of the computations: the preconditioning matrix is a matrix that has been factorized at an earlier step and previously computed vectors yield a reduced basis. The principle is to define an increment in two sub-steps. In the first sub-step, only the projection of the unknown on a reduced subspace is incremented and the projection of the equation on the reduced subspace is satisfied exactly. In the second sub-step, the full equation is solved approximately with the help of the preconditioner. Last, the convergence of the sequences is accelerated by a well-known method, the modified minimal polynomial extrapolation. This algorithm assessed by classical benchmarks coming from shell buckling analysis. Finally, its insertion in path following techniques is discussed. This leads to non-linear solvers with few matrix factorizations and few iterations.

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Section: Evaluation Of the Proposed Solversupporting

confidence: 85%

Section: Convergence Accelerationmentioning

confidence: 75%

Section: A Variant Based On Homotopy and Perturbationmentioning

confidence: 94%

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A solver combining reduced basis and convergence acceleration with applications to non‐linear elasticity

Cadou

Potier‐Ferry²

2009

Numer Methods Biomed Eng

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“…Furthermore it has been proved [17] that, with a proper choice of the projection vectors Y i , they become mathematically identical. With the MMPE algorithm, the orthonormalization procedure could be avoided and the possibility of choosing vectors Y i leads to more freedom to optimize the procedure.…”

Section: Two Convergence Acceleration Techniquesmentioning

confidence: 99%

“…3 and 4 that when these two accelerating techniques are used with a low number of iterations (lower than 10), the evolution of the residual versus the number of iterations are nearly the same for the two techniques. In a recent work [17], the authors have established the mathematical equivalence between these two techniques when a specific choice of vectors Y i is made but some differences are observed in the numerical practice, especially when a large number of vectors is used.…”

Section: A First Comparisonmentioning

confidence: 99%

Convergence acceleration of iterative algorithms. Applications to thin shell analysis and Navier–Stokes equations

et al. 2008

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This work deals with the convergence acceleration of iterative nonlinear methods. Two convergence accelerating techniques are evaluated: the Modified Mininal Polynomial Extrapolation Method (MMPE) and the Padé approximants. The algorithms studied in this work are iterative correctors: Newton's modified method, a high-order iterative corrector presented in Damil et al. (Commun Numer Methods Eng 15:701-708, 1999) and an original algorithm for vibration of viscoelastic structures. We first describe the iterative algorithms for the considered nonlinear problems. Secondly, the two accelerating techniques are presented. Finally, through several numerical tests from the thin shell theory, Navier-Stokes equations and vibration of viscoelastic shells, the advantages and drawbacks of each accelerating technique is discussed.

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An algorithm for the computation of multiple Hopf bifurcation points based on Pade approximants

Girault

Guevel

Cadou

2011

Numerical Methods in Fluids

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SUMMARYRecently, a numerical method was proposed to compute a Hopf bifurcation point in fluid mechanics. This numerical method associates a bifurcation indicator and a Newton method. The former gives initial guesses to the iterative method. These initial values are the minima of the bifurcation indicator. However, sometimes, these minima do not lead to the convergence of the Newton method. Moreover, as only a single initial guess is obtained for each computation of the indicator, the computational time to obtain a Hopf bifurcation point can be quite long. The present algorithm is an enhancement of the previous one. It consists in automatically computing several initial guesses for each indicator curve. The majority of these initial values leads to the convergence of the Newton method. This method is evaluated through the problem of the lid-driven cavity with several aspect ratios in the framework of the finite element analysis of the 2D Navier-Stokes equations. The results prove the efficiency and the robustness of the proposed algorithm.

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Mathematical and numerical connections between polynomial extrapolation and Padé approximants: applications in structural mechanics

Cited by 11 publications

References 17 publications

A solver combining reduced basis and convergence acceleration with applications to non‐linear elasticity

A solver combining reduced basis and convergence acceleration with applications to non‐linear elasticity

Convergence acceleration of iterative algorithms. Applications to thin shell analysis and Navier–Stokes equations

An algorithm for the computation of multiple Hopf bifurcation points based on Pade approximants

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