A generic and efficient Taylor series–based continuation method using a quadratic recast of smooth nonlinear systems

Guillot, Louis; Cochelin, Bruno; Vergez, Christophe

doi:10.1002/nme.6049

Cited by 53 publications

(64 citation statements)

References 33 publications

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“…The systems can be solved with the help of an efficient condensation algorithm which finally result in the Taylor series (2) sought for. The interested reader is referred to [22] for all the details on the practical computation of the series. It shall be emphasized here that once the quadratic recast is available, all the operations necessary to compute the Taylor series developments (2) can be automatized.…”

Section: The Quadratic Framework and The Asymptotic Numerical Methodsmentioning

confidence: 99%

“…The key is to use a differentiated form of these variables. In the book [21], these recasts are discussed in more details and a table of recasts of common functions is given in the article [22]. Hereafter, the example of sine function's recast is given as an highlight of the general approach.…”

Section: Transcendental Nonlinearitymentioning

confidence: 99%

“…This works easily as the differentiated form is quadratic with respect to the variables (x, y, z, dx, dy, dz). This form is very generic and can be computed for all the elementary transcendental functions [21,22]. It gives the possibility to recast a very wide range of nonlinearities quadratically.…”

Section: Transcendental Nonlinearitymentioning

confidence: 99%

“…The third one relies on the quadratic recast of the equations. This latest approach is the one developed by the authors for its genericity and its performance [22]. It has first been described in [9,10,11] in the case of equilibrium branches of structures and then extended to periodic solutions of ODE using the Harmonic Balance Method [12].…”

Section: Introductionmentioning

confidence: 99%

“…Periodic solutions of fractional order systems can be continued in this framework [42]. Some very recent work [22] has shown that the periodic solutions of delay differential equations (possibly of neutral type) could also be investigated with this method.…”

Section: Introductionmentioning

confidence: 99%

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A Taylor series-based continuation method for solutions of dynamical systems

2019

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This paper describes a generic Taylor series based continuation method, the so-called Asymptotic Numerical Method, to compute the bifurcation diagrams of nonlinear systems. The key point of this approach is the quadratic recast of the equations as it allows to treat in the same way a wide range of dynamical systems and their solutions. Implicit Differential-Algebraic Equations, forced or autonomous, possibly with time-delay or fractional order derivatives are handled in the same framework. The static, periodic and quasi-periodic solutions can be continued as well as transient solutions.

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Section: The Quadratic Framework and The Asymptotic Numerical Methodsmentioning

confidence: 99%

Section: Transcendental Nonlinearitymentioning

confidence: 99%

Section: Transcendental Nonlinearitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Taylor series-based continuation method for solutions of dynamical systems

2019

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Stationary solutions in applied dynamics: A unified framework for the numerical calculation and stability assessment of periodic and quasi‐periodic solutions based on invariant manifolds

Hetzler

Bäuerle

2023

GAMM-Mitteilungen

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The determination of stationary solutions of dynamical systems as well as analyzing their stability is of high relevance in science and engineering. For static and periodic solutions a lot of methods are available to find stationary motions and analyze their stability. In contrast, there are only few approaches to find stationary solutions to the important class of quasi-periodic motions-which represent solutions of generalized periodicity-available so far. Furthermore, no generally applicable approach to determine their stability is readily available. This contribution presents a unified framework for the analysis of equilibria, periodic as well as quasi-periodic motions alike. To this end, the dynamical problem is changed from a formulation in terms of the trajectory to an alternative formulation based on the invariant manifold as geometrical object in the state space. Using a so-called hypertime parametrization offers a direct relation between the frequency base of the solution and the parametrization of the invariant manifold. Over the domain of hypertimes, the invariant manifold is given as solution to a PDE, which can be solved using standard methods as Finite Differences (FD), Fourier-Galerkin-methods (FGM) or quasi-periodic shooting (QPS). As a particular advantage, the invariant manifold represents the entire stationary dynamics on a finite domain even for quasi-periodic motions -whereas obtaining the same information from trajectories would require knowing them over an infinite time interval. Based on the invariant manifold, a method for stability assessment of quasi-periodic solutions by means of efficient calculation of Lyapunov-exponents is devised. Here, the basic idea is to introduce a Generalized Monodromy Mapping, which may be determined in a pre-processing step: using this mapping, the Lyapunov-exponents may efficiently be calculated by iterating this mapping.

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Stretch‐induced wrinkling analysis of thin sheets with splines

Tian

Chen

2023

Numerical Meth Engineering

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Numerical simulation of stretch-induced wrinkling in thin elastic sheets is a challenging problem due to a vanishing bending stiffness and the coexistence of superabundant equilibrium solutions. In this work, we present a computational framework to capture the morphological evolution of stretch-induced wrinkles. The application of modified Föppl-von Kármán plate model results in a fourth-order partial differential equation. The convergence of finite-element solutions necessitates C 1 -continuous approximations. Herein, Powell-Sabin B-splines, which are based on triangles, are utilized for both the approximation of the field variables and the description of the geometry. To trace the wrinkling behavior in thin sheets, a path-following technique using asymptotic numerical method is considered. The advantage of this method is an adaptive step length, which works incredibly well near the bifurcation points and allows for the computation of the post-bifurcation diagrams with a quite small perturbation. The versatility and accuracy of the developed computational approach are assessed in three case studies, featuring wrinkling in highly stretched rectangular and annular thin sheets.

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A generic and efficient Taylor series–based continuation method using a quadratic recast of smooth nonlinear systems

Cited by 53 publications

References 33 publications

A Taylor series-based continuation method for solutions of dynamical systems

A Taylor series-based continuation method for solutions of dynamical systems

Stationary solutions in applied dynamics: A unified framework for the numerical calculation and stability assessment of periodic and quasi‐periodic solutions based on invariant manifolds

Stretch‐induced wrinkling analysis of thin sheets with splines

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