The asymptotic-numerical method: an efficient perturbation technique for nonlinear structural mechanics

Cochelin, Bruno; Damil, Noureddine; Potier‐Ferry, Michel

doi:10.1080/12506559.1994.10511124

Cited by 75 publications

(69 citation statements)

References 15 publications

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“…The equilibrium equation stated in the reference configuration looks like ∇P+B = 0 in 0 (18) in which B is the body force. The boundary conditions of the body are defined by…”

Section: Problem Formulation For Kirchhoff-saint Venant Modelsmentioning

confidence: 99%

“…The technique is best viewed by considering the Saint-Venant Kirchhoff model, as done in [16,18], since only geometric non-linearities are present. Later on, in Section 3, we will generalize the method for other hyperelastic models.…”

Section: Problem Formulation For Kirchhoff-saint Venant Modelsmentioning

confidence: 99%

“…Following [18], we consider a linear and a non-linear terms for the Green-Lagrange strain tensor, E, in the form…”

Section: Problem Formulation For Kirchhoff-saint Venant Modelsmentioning

confidence: 99%

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Model order reduction for hyperelastic materials

Niroomandi

Alfaro

Cueto

et al. 2009

Numerical Meth Engineering

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International audienceIn this paper, we develop a novel algorithm for the dimensional reduction of the models of hyperelastic solids undergoing large strains. Unlike standard proper orthogonal decomposition methods, the proposed algorithm minimizes the use of the Newton algorithms in the search of non-linear equilibrium paths of elastic bodies.The proposed technique is based upon two main ingredients. On one side, the use of classic proper orthogonal decomposition techniques, that extract the most valuable information from pre-computed, complete models. This information is used to build global shape functions in a Ritz-like framework.On the other hand, to reduce the use of Newton procedures, an asymptotic expansion is made for some variables of interest. This expansion shows the interesting feature of possessing one unique tangent operator for all the terms of the expansion, thus minimizing the updating of the tangent stiffness matrix of the problem.The paper is completed with some numerical examples in order to show the performance of the technique in the framework of hyperelastic (Kirchhoff-Saint Venant and neo-Hookean) solids

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“…The equilibrium equation stated in the reference configuration looks like ∇P+B = 0 in 0 (18) in which B is the body force. The boundary conditions of the body are defined by…”

Section: Problem Formulation For Kirchhoff-saint Venant Modelsmentioning

confidence: 99%

Section: Problem Formulation For Kirchhoff-saint Venant Modelsmentioning

confidence: 99%

See 1 more Smart Citation

Model order reduction for hyperelastic materials

Niroomandi

Alfaro

Cueto

et al. 2009

Numerical Meth Engineering

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show abstract

“…It is now clear that the use of model reduction seems to be an appealing alternative for reaching such performances. However, techniques based on the use of POD, POD with interpolation (PODI), even combined with asymptotic numerical methods to avoid the computation of the tangent stiffness matrix [28,54], exhibit serious difficulties to fulfil such requirements as discussed in [60][61][62][63].…”

Section: Surgery Simulatorsmentioning

confidence: 99%

PGD-Based Computational Vademecum for Efficient Design, Optimization and Control

Chinesta

Leygue²,

Bordeu³

et al. 2013

Arch Computat Methods Eng

280

291

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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. to face the challenges posed by current ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, some challenging problems remain today intractable. These problems, that are common to many branches of science and engineering, are of different nature. Among them, we can cite those related to highdimensional problems, which do not admit mesh-based approaches due to the exponential increase of degrees of freedom. We developed in recent years a novel technique, called Proper Generalized Decomposition (PGD). It is based on the assumption of a separated form of the unknown field and it has demonstrated its capabilities in dealing with highdimensional problems overcoming the strong limitations of classical approaches. But the main opportunity given by this technique is that it allows for a completely new approach for classic problems, not necessarily high dimensional. Many challenging problems can be efficiently cast into a multidimensional framework and this opens new possibilities to solve old and new problems with strategies not envisioned until now. For instance, parameters in a model can be set as additional extra-coordinates of the model. In a PGD framework, the resulting model is solved once for life, in order to obtain a general solution that includes all the solutions for every possible value of the parameters, that is, a sort of computational vademecum. Under this rationale, optimization of complex problems, uncertainty quantification, simulation-based control and real-time simulation are now at hand, even in highly complex scenarios, by combining an off-line stage in which the general PGD solution, the vademecum, is computed, and an on-line phase in which, even on deployed, handheld, platforms such as smartphones or tablets, real-time response is obtained as a result of our queries.

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“…Further developments in the real-time simulation based upon POD techniques include the employ of asymptotic numerical methods (ANM) [61][62][63][64] in combination with POD [65][66][67]. By means of an asymptotic expansion of the variables of interest, POD-ANM techniques allow to solve non-linear problems without the burden associated to the update and inversion of tangent stiffness matrices.…”

Section: Proper Orthogonal Decompositionmentioning

confidence: 99%

Real time simulation for computational surgery: a review

Cueto

Chinesta

2014

Advanced Modeling and Simulation in Engineering Sciences

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In this paper a non-exhaustive review is made on the existing literature for real-time simulation in the field of computational surgery. Many methods have been proposed so far to deal with the very astringent assumption of real-time response in the field, specially for simulators equipped with haptic peripherals. A special emphasis is made on techniques that respond to the so-called second generation of surgery simulators, that able to adequately model the mechanics of the problem. Techniques employing supercomputing facilities, notably those base upon parallel implementations on GPUs will be covered, while special attention will be paid to techniques based upon model order reduction, a promising technique in the field. Finally, some review is made on techniques able to give some insight in the so-called third generation of surgery simulators, i.e., that able to include physiological details into the simulation.

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The asymptotic-numerical method: an efficient perturbation technique for nonlinear structural mechanics

Cited by 75 publications

References 15 publications

Model order reduction for hyperelastic materials

Model order reduction for hyperelastic materials

PGD-Based Computational Vademecum for Efficient Design, Optimization and Control

Real time simulation for computational surgery: a review

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