A proper orthogonal decomposition approach to microstructure model reduction in Rodrigues space with applications to optimal control of microstructure-sensitive properties

Acharjee, Swagato; Zabaras, Nicholas

doi:10.1016/s1359-6454(03)00427-0

Cited by 38 publications

(29 citation statements)

References 14 publications

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“…Such modes for o (1) = I are depicted on the DPFs for the basic crystallographic directions. RME ) with respect to the elasticity tensors obtained for the polycrystalline copper aggregates with optimal orientation modes are summarized in Table 4 It should be pointed out that, as shown in [79], in the case of even ≥ 4 M , for crystallites with cubic elastic symmetry there always exists a set of M orientations, over which averaging of the form (26) results in isotropy of macroscopic elastic properties.…”

Section: Discussionmentioning

confidence: 99%

“…Control of the fragmentation mechanisms and rotational modes of inelastic deformation offers considerable scope for producing metals and alloys with a grain structure exhibiting the best properties for the preset operating conditions. Different issues related to the problems of designing functionally graded materials are discussed, e.g., in [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. General methodology of designing such materials based on multi-level modeling [15][16][17][18][19] is described in detail in [8].…”

Section: Introductionmentioning

confidence: 99%

“…An alternative way to represent ODF in a reduced form is considered in [1,4,5,[11][12][13][14]23]. To control the texture-dependent parameters, a finite element representation of the ODF conservation equation [3,24] within a fundamental region of the Rodrigues space [24][25][26] is used.…”

Section: Introductionmentioning

confidence: 99%

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On Elastic Symmetry Identification for Polycrystalline Materials

Трусов¹,

Ostapovich²

2017

Symmetry

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Abstract:The products made by the forming of polycrystalline metals and alloys, which are in high demand in modern industries, have pronounced inhomogeneous distribution of grain orientations. The presence of specific orientation modes in such materials, i.e., crystallographic texture, is responsible for anisotropy of their physical and mechanical properties, e.g., elasticity. A type of anisotropy is usually unknown a priori, and possible ways of its determination is of considerable interest both from theoretical and practical viewpoints. In this work, emphasis is placed on the identification of elasticity classes of polycrystalline materials. By the newly introduced concept of "elasticity class" the union of congruent tensor subspaces of a special form is understood. In particular, it makes it possible to consider the so-called symmetry classification, which is widely spread in solid mechanics. The problem of identification of linear elasticity class for anisotropic material with elastic moduli given in an arbitrary orthonormal basis is formulated. To solve this problem, a general procedure based on constructing the hierarchy of approximations of elasticity tensor in different classes is formulated. This approach is then applied to analyze changes in the elastic symmetry of a representative volume element of polycrystalline copper during numerical experiments on severe plastic deformation. The microstructure evolution is described using a two-level crystal elasto-visco-plasticity model. The well-defined structures, which are indicative of the existence of essentially inhomogeneous distribution of crystallite orientations, were obtained in each experiment. However, the texture obtained in the quasi-axial upsetting experiment demonstrates the absence of significant macroscopic elastic anisotropy. Using the identification framework, it has been shown that the elasticity tensor corresponding to the resultant microstructure proves to be almost isotropic.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Elastic Symmetry Identification for Polycrystalline Materials

Трусов¹,

Ostapovich²

2017

Symmetry

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“…There are also different approaches to process these given fields to produce a basis. For example, proper orthogonal decomposition (POD) has been used in several studies [7,29,1] to extract the basis from a given dataset that is optimal in the average L 2 -error sense for representing the given fields. Alternatively, other works, including the work by Boyaval et al relating to SPDEs, have simply used Gram Schmidt orthogonalization to directly convert the given fields into an orthogonal basis [12].…”

Section: Introductionmentioning

confidence: 99%

Adaptive reduced-basis generation for reduced-order modeling for the solution of stochastic nondestructive evaluation problems

Notghi

Ahmadpoor

Brigham

2016

Computer Methods in Applied Mechanics and Engineering

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(2016) 'Adaptive reduced-basis generation for reduced-order modeling for the solution of stochastic nondestructive evaluation problems.', Computer methods in applied mechanics and engineering., 310. pp. 172-188. Further information on publisher's website: http://dx. Additional information: Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Abstract A novel algorithm for creating a computationally efficient approximation of a system response that is defined by a boundary value problem is presented. More specifically, the approach presented is focused on substantially reducing the computational expense required to approximate the solution of a stochas-tic partial differential equation, particularly for the purpose of estimating the solution to an associated nondestructive evaluation problem with significant system uncertainty. In order to achieve this computational efficiency, the approach combines reduced-basis reduced-order modeling with a sparse grid col-location surrogate modeling technique to estimate the response of the system of interest with respect to any designated unknown parameters, provided the distributions are known. The reduced-order modeling component includes a novel algorithm for adaptive generation of a data ensemble based on a nested grid technique, to then create the reduced-order basis. The capabilities and potential applicability of the approach presented are displayed through two simulated case studies regarding inverse characterization of material properties for two different physical systems involving some amount of significant uncertainty. The first case study considered characterization of an unknown localized reduction in stiffness of a structure from simulated frequency response function based nondestructive testing. Then, the second case study considered characterization of an unknown temperature-dependent thermal conductivity of a solid from simulated thermal testing. Overall, the surrogate modeling approach was shown through both simulated examples to provide accurate solution estimates to inverse problems for systems represented by stochastic partial differential equations with a fraction of the typical computational cost.

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“…The reduced-order problem is then solved in the subspace spanned by this optimal basis. Some interesting recent applications which demonstrate the use of POD can be found in References [23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs

Acharjee

Zabaras

2006

Int. J. Numer. Meth. Engng

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SUMMARYA methodology is introduced for rapid reduced-order solution of stochastic partial differential equations. On the random domain, a generalized polynomial chaos expansion (GPCE) is used to generate a reduced subspace. GPCE involves expansion of the random variable as a linear combination of basis functions defined using orthogonal polynomials from the Askey series. A proper orthogonal decomposition (POD) approach coupled with the method of snapshots is used to generate a reduced solution space from the space spanned by the finite element basis functions on the spatial domain. POD methods have been extremely popular in fluid mechanics applications and have subsequently been applied to other interesting areas. They have been shown to be capable of representing complicated phenomena with a handful of degrees of freedom. This concurrent model reduction on the random and spatial domains is applied to stochastic partial differential equations (PDEs) in natural convection processes involving randomness in the porosity of the medium and the Rayleigh number. The results indicate that owing to the multiplicative nature of the concurrent model reduction, extremely large computational gains are realized without significant loss of accuracy.

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A proper orthogonal decomposition approach to microstructure model reduction in Rodrigues space with applications to optimal control of microstructure-sensitive properties

Cited by 38 publications

References 14 publications

On Elastic Symmetry Identification for Polycrystalline Materials

On Elastic Symmetry Identification for Polycrystalline Materials

Adaptive reduced-basis generation for reduced-order modeling for the solution of stochastic nondestructive evaluation problems

A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs

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