Asymptotic Numerical Method for strong nonlinearities

Zahrouni, Hamid; Aggoune, Woihida; Brunelot, Juliette; Potier‐Ferry, Michel

doi:10.3166/reef.13.97-118

Cited by 14 publications

(15 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This enables a simple computation of the Taylor series coefficients, which is essential for the effectiveness of the approach. This requirement of representing the nonlinearities in a quadratic form is the main drawback of the ANM, but fortunately expressions for many common nonlinearities can be found in the literature: contact [5,6,7], friction [6,7], vibro-impact [6], cubic [23]. Using these expressions, the system of equations can be written in the following form:…”

Section: Deterministic Nonlinear Problemmentioning

confidence: 99%

“…The major restrain of the ANM is that the nonlinearities must be expressed in a quadratic form, somewhat restraining its wide applicability, especially for complex industrial problems where the non-linearities cannot be accurately represented by a polynomial quadratic form. However, several kinds of nonlinearities have been successfully processed with the ANM [5,6,7] leading to a significant reduction in the computational time.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Uncertainty propagation for nonlinear vibrations: A non-intrusive approach

Panunzio

Salles

Schwingshackl

2017

Journal of Sound and Vibration

View full text Add to dashboard Cite

The propagation of uncertain input parameters in a linear dynamic analysis is reasonably well established today, but with the focus of the dynamic analysis shifting towards nonlinear systems, new approaches will be required to compute the uncertain nonlinear responses. A combination of stochastic methods (Polynomial Chaos Expansion, PCE) with an Asymptotic Numerical Method (ANM) for the solution of the nonlinear dynamic systems will be presented to predict the propagation of random input uncertainties and assess their influence on the nonlinear vibrational behaviour of a system. The proposed method allows the computation of stochastic resonance frequencies and peak amplitudes based on multiple input uncertainties, leading to a series of uncertain nonlinear dynamic responses. One of the main challenges when using the PCE is thereby the Gibbs phenomenon, which can heavily impact the resulting stochastic nonlinear response by introducing spurious oscillations. A novel technique to avoid the Gibbs phenomenon will be presented in this paper, leading to high quality frequency response predictions. A comparison of the proposed stochastic nonlinear analysis technique to traditional Monte Carlo simulations, demonstrated comparable accuracy at a significantly reduced computational cost, thereby validating the proposed approach.

show abstract

Section: Deterministic Nonlinear Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Uncertainty propagation for nonlinear vibrations: A non-intrusive approach

Panunzio

Salles

Schwingshackl

2017

Journal of Sound and Vibration

View full text Add to dashboard Cite

show abstract

“…It can also be applied to more complicated problems involving singularities like contact conditions and plasticity. In these problems, the non regular relations are replaced by regular ones for using the perturbation technique; [4,5,18,53,54]. The validity range of the power series expansion can be largely improved by replacing the truncated power series by rational fractions called Padé approximants (see [6,7,14,19,34]).…”

Section: Introductionmentioning

confidence: 99%

Solving hyperelastic material problems by asymptotic numerical method

2010

Self Cite

View full text Add to dashboard Cite

This paper presents a numerical algorithm based on a perturbation technique named asymptotic numerical method (ANM) to solve nonlinear problems with hyperelastic constitutive behaviors. The main advantages of this technique compared to Newton-Raphson are: (a) a large reduction of the number of tangent matrix decompositions; (b) in presence of instabilities or limit points no special treatment such as arc-length algorithms is necessary. The ANM uses high order series approximation with auto-adaptive step length and without need of any iteration. Introduction of this expansion into the set of nonlinear equations results into a sequence of linear problems with the same linear operator. The present work aims at providing algorithms for applying the ANM to the special case of compressible and incompressible hyperelastic materials. The efficiency and accuracy of the method are examined by comparing this algorithm with Newton-Raphson method for problems involving hyperelastic structures with large strains and instabilities.

show abstract

“…The same idea of a condensation at order n has been applied to contact variables as h n or n n (Equations (23) and (24)). See References [19,31,32] for general discussions of this technique within ANM.…”

Section: Remarksmentioning

confidence: 99%

Asymptotic numerical methods for unilateral contact

Aggoune

Zahrouni

Potier‐Ferry

2006

Int. J. Numer. Meth. Engng

View full text Add to dashboard Cite

SUMMARYNew algorithms based upon the asymptotic numerical method (ANM) are proposed to solve unilateral contact problems. ANM leads to a representation of a solution path in terms of series or Padé approximants. To get a smooth solution path, a hyperbolic relation between contact forces and clearance is introduced. Three key points are discussed: the influence of the regularization of the contact law, the discretization of the contact force by Lagrange multipliers and prediction-correction algorithms. Simple benchmarks are considered to evaluate the relevance of the proposed algorithms.

show abstract

Asymptotic Numerical Method for strong nonlinearities

Cited by 14 publications

References 17 publications

Uncertainty propagation for nonlinear vibrations: A non-intrusive approach

Uncertainty propagation for nonlinear vibrations: A non-intrusive approach

Solving hyperelastic material problems by asymptotic numerical method

Asymptotic numerical methods for unilateral contact

Contact Info

Product

Resources

About