A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics

Kerfriden, Pierre; Goury, Olivier; Rabczuk, Timon; Bordas, Stéphane Pierre Alain

doi:10.1016/j.cma.2012.12.004

Cited by 119 publications

(65 citation statements)

References 63 publications

(157 reference statements)

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“…The latter has been widely used, e.g. in homogenisation, fluid mechanics and many other fields [6,17,18,23,26,41]. For a deeper discussion and a broader overview the reader is referred to [2,33] and references therein.…”

Section: Introductionmentioning

confidence: 99%

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

et al. 2017

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Computing the macroscopic material response of a continuum body commonly involves the formulation of a phenomenological constitutive model. However, the response is mainly influenced by the heterogeneous microstructure. Computational homogenisation can be used to determine the constitutive behaviour on the macro-scale by solving a boundary value problem at the micro-scale for every so-called macroscopic material point within a nested solution scheme. Hence, this procedure requires the repeated solution of similar microscopic boundary value problems. To reduce the computational cost, model order reduction techniques can be applied. An important aspect thereby is the robustness of the obtained reduced model. Within this study reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied for the boundary value problem on the micro-scale. This involves the Proper Orthogonal Decomposition (POD) for the primary unknown and hyper-reduction methods for the arising nonlinearity. Therein three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness. Introducing interpolation or Gappy-POD based approximations may not preserve the symmetry of the system tangent, rendering the widely used Galerkin projection sub-optimal. Hence, a different projection related to a Gauss-Newton scheme (Gauss-Newton with Approximated Tensors-GNAT) is favoured to obtain an optimal projection and a robust reduced model.

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Section: Introductionmentioning

confidence: 99%

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

et al. 2017

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“…Such approaches are common in electrical engineering, molecular simulation, or mechanics [1], and recent applications include real-time simulation [2]. The term nonlinear model reduction generally refers to the reduced modeling of a nonlinear system, yet relying on a linear (a ne) reduced description [3,4]. Nevertheless, it is apparent that many models evolve on essentially nonlinear manifolds, in particular those involving large amplitude rotations or steric constraints.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear manifold learning for model reduction in finite elastodynamics

Millán

Arroyo

2013

Computer Methods in Applied Mechanics and Engineering

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Model reduction in computational mechanics is generally addressed with linear dimensionality reduction methods such as Principal Components Analysis (PCA). Hypothesizing that in many applications of interest the essential dynamics evolve on a nonlinear manifold, we explore here reduced order modeling based on nonlinear dimensionality reduction methods. Such methods are gaining popularity in diverse fields of science and technology, such as machine perception or molecular simulation. We consider finite deformation elastodynamics as a model problem, and identify the manifold where the dynamics essentially take place -the slow manifold-by nonlinear dimensionality reduction methods applied to a database of snapshots. Contrary to linear dimensionality reduction, the smooth parametrization of the slow manifold needs special techniques, and we use local maximum entropy approximants. We then formulate the Lagrangian mechanics on these data-based generalized coordinates, and develop variational time-integrators. Our proof-of-concept example shows that a few nonlinear collective variables provide similar accuracy to tens of PCA modes, suggesting that the proposed method may be very attractive in control or optimization applications. Furthermore, the reduced number of variables brings insight into the mechanics of the system under scrutiny. Our simulations also highlight the need of modeling the net e↵ect of the disregarded degrees of freedom on the reduced dynamics at long times.

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“…Multiscale approaches can be employed to increase the computational efficiency of large-scale lattice computations (e.g. Gonella and Ruzzene (2008); ; Kerfriden et al (2011Kerfriden et al ( , 2012Kerfriden et al ( , 2013a). One of those multiscale approaches is the quasicontinuum (QC) method (Tadmor et al, 1996a,b), which has the following advantages that are not all shared by other multiscale approaches.…”

Section: Introductionmentioning

confidence: 99%

Quasicontinuum-based multiscale approaches for plate-like beam lattices experiencing in-plane and out-of-plane deformation

Beex

Kerfriden

Rabczuk

et al. 2014

Computer Methods in Applied Mechanics and Engineering

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The quasicontinuum (QC) method is a multiscale approach that aims to reduce the computational cost of discrete lattice computations. The method incorporates smallscale local lattice phenomena (e.g. a single lattice defect) in macroscale simulations. Since the method works directly and only on the beam lattice, QC frameworks do not require the construction and calibration of an accompanying continuum model (e.g. a cosserat/micropolar description). Furthermore, no coupling procedures are required between the regions of interest in which the beam lattice is fully resolved and coarse domains in which the lattice is effectively homogenized. Hence, the method is relatively straightforward to implement and calibrate. In this contribution, four variants of the QC method are investigated for their use for planar beam lattices which can also experience out-of-plane deformation. The different frameworks are compared to the direct lattice computations for three truly multiscale test cases in which a single lattice defect is present in an otherwise perfectly regular beam lattice.

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A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics

Cited by 119 publications

References 63 publications

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

Nonlinear manifold learning for model reduction in finite elastodynamics

Quasicontinuum-based multiscale approaches for plate-like beam lattices experiencing in-plane and out-of-plane deformation

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