A path-following technique via an asymptotic-numerical method

Cochelin, Bruno

doi:10.1016/0045-7949(94)90165-1

Cited by 230 publications

(239 citation statements)

References 9 publications

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“…Besides, the convergence toward the equilibrium solution is never assured. An alternative is to use an asymptotic numerical method [4] [5]. Such a method principle is, starting from a solution point, to seek the branch of solution as an asymptotic expansion of an arc length measure, leading to a semi-analytical solution.…”

Section: Asymptotic Numerical Methodsmentioning

confidence: 99%

A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables

Cottanceau

Thomas

Véron

et al. 2018

Finite Elements in Analysis and Design

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. Summary A flexible cable is modeled by a geometrically exact beam model with 3D rotations described using quaternion parameters. The boundary value problem is then discretized by the finite element method. The use of an asymptotic numerical method to solve the problem, requiring quadratic equations, is well suited to the quaternion parametrization. This combination of methods leads to a fast, robust and accurate algorithm very well-adapted for the simulation of the assembly process of cables. This is proved by running many examples involving complicated solutions.

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Section: Asymptotic Numerical Methodsmentioning

confidence: 99%

A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables

Cottanceau

Thomas

Véron

et al. 2018

Finite Elements in Analysis and Design

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“…A small step size leads to a slow convergence, while a large step size compromises the accuracy of the results. Our method is similar to [Cochelin 1994], which adaptively selects the step size as large as possible while retaining sufficient accuracy to ensure quadratic convergence at each step. Our key contribution is the development of asymptotic expansions for the inverse neo-Hookean model.…”

Section: Related Workmentioning

confidence: 99%

“…To this end, we need to quickly estimate the residual of X(a). As suggested by [Cochelin 1994], a simple estimation is motivated by the observation that when a − a0 is within the convergence radius of the power series, the difference between two consecutive approximation orders remains small. However, when a − a0 reaches the convergence radius, they separate rapidly (see Figure 4).…”

Section: Residual Estimation Of X(a)mentioning

confidence: 99%

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An asymptotic numerical method for inverse elastic shape design

Chen

Zheng

et al. 2014

ACM Trans. Graph.

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Inverse shape design for elastic objects greatly eases the design efforts by letting users focus on desired target shapes without thinking about elastic deformations. Solving this problem using classic iterative methods (e.g., Newton-Raphson methods), however, often suffers from slow convergence toward a desired solution. In this paper, we propose an asymptotic numerical method that exploits the underlying mathematical structure of specific nonlinear material models, and thus runs orders of magnitude faster than traditional Newton-type methods. We apply this method to compute rest shapes for elastic fabrication, where the rest shape of an elastic object is computed such that after physical fabrication the real object deforms into a desired shape. We illustrate the performance and robustness of our method through a series of elastic fabrication experiments.

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“…Let us note that these problems have the same tangent stiffness matrix and hence the terms (U i , λ i ) (1 ≤ i ≤ N ) of (2.2) are computed by inverting only one stiffness matrix. The last step is the continuation technique [1,2]. The end of the branch j is the starting point of the next branch (j + 1).…”

Section: Local Parameterizations In the Anmmentioning

confidence: 99%

Local Parameterization and the Asymptotic Numerical Method

Mottaqui

Braikat

Damil

2010

Math. Model. Nat. Phenom.

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Abstract. The Asymptotic Numerical Method (ANM) is a family of algorithms, based on computation of truncated vectorial series, for path following problems [2]. In this paper, we present and discuss some techniques to define local parameterization [4,6,7] in the ANM. We give some numerical comparisons of pseudo arc-length parameterization and local parameterization on nonlinear elastic shells problems.

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A path-following technique via an asymptotic-numerical method

Cited by 230 publications

References 9 publications

A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables

A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables

An asymptotic numerical method for inverse elastic shape design

Local Parameterization and the Asymptotic Numerical Method

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