A hybrid variationally consistent homogenization approach based on Ritz's method

Bleier, Nikolaus; Mosler, Jörn

doi:10.1002/nme.4465

Cited by 8 publications

(7 citation statements)

References 55 publications

(119 reference statements)

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“…Therefore, the computational cost is tremendously high when practical problems are to be solved. Although there are some remedies to reduce the cost such as parallel computations [17] and the decoupling [32][33][34] or mixing [26,35] schemes, the present study takes the reduced-order approach based on the proper orthogonal decomposition, which is the subject of the next section, by partially following the previous developments in the literature.…”

Section: Solution Scheme: Fementioning

confidence: 99%

“…Second, localization analyses are carried out with the N c patterns of macroscopic deformation gradients and the response data vectors are constructed as (35) where P il is the microscopic 1st PK stress in Voigt form at calculation point l in element e and have N p ( = 4 in case of plane strain problems) components. Since the calculation points commonly corresponds to Gaussian integration points in FEA, the calculation point number is identified with l := (e − 1) × N g + k where N g is the number of Gaussian integration points per element e and k stands for the element number.…”

Section: Offline Computationsmentioning

confidence: 99%

“…Here, the order of components in each of these vectors is in accordance with that of POD bases s j and Eqn. (35). Furthermore,• is a value at each microscopic Gaussian point.…”

Section: Offline Computationsmentioning

confidence: 99%

See 2 more Smart Citations

$${\text {FE}}^r$$ method with surrogate localization model for hyperelastic composite materials

Hatano

Matsubara

Moriguchi

et al. 2020

Adv. Model. and Simul. in Eng. Sci.

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This study presents a method for constructing a surrogate localization model for a periodic microstructure, or equivalently, a unit cell, to efficiently perform micro-macro coupled analyses of hyperelastic composite materials. The offline process in this approach is to make a response data matrix that stores the microscopic stress distributions in response to various patterns of macroscopic deformation gradients, which is followed by the proper orthogonal decomposition (POD) of the matrix to construct a reduced order model (ROM) of the microscopic analysis (localization) with properly extracted POD bases. Then, response surfaces of the POD coefficients are constructed so that the ROM can be continuous with respect to the input datum, namely, the macroscopic deformation gradient. The novel contributions of this study are the application of the L2 regularization to the interpolation approximations of the POD coefficients by use of radial basis functions (RBFs) to make the response surfaces continuous and the combined use of the cross-validation and the Bayesian optimization to search for the optimal set of parameters in both the RBFs and L2regularization formula. The resulting model can be an alternative to microscopic finite element (FE) analyses in the conventional $${\text {FE}}^2$$ FE 2 method and realizes $${\text {FE}}^r$$ FE r with $$1<r<<2$$ 1 < r < < 2 accordingly. Representative numerical examples of micro-macro coupled analysis with the $${\text {FE}}^r$$ FE r are presented to demonstrate the capability and promise of the surrogate localization model constructed with the proposed approach in comparison with the results with high-fidelity direct $${\text {FE}}^2$$ FE 2 .

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Section: Solution Scheme: Fementioning

confidence: 99%

Section: Offline Computationsmentioning

confidence: 99%

See 1 more Smart Citation

$${\text {FE}}^r$$ method with surrogate localization model for hyperelastic composite materials

Hatano

Matsubara

Moriguchi

et al. 2020

Adv. Model. and Simul. in Eng. Sci.

View full text Add to dashboard Cite

show abstract

“…[3]) is based on the principle of energy equivalence, in which the macroscopic energies for isothermal reversible process of a representative volume element (RVE) are given…”

Section: Atomistic-to-continuum Modelmentioning

confidence: 99%

Continuum modeling of material interfaces and surfaces based on molecular statics computations

Sievers

Mosler

2015

Proc Appl Math and Mech

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The effective macroscopic properties of a broad variety of different materials are defined by the properties of the involved material interfaces and surfaces. However, in contrast to classical bulk materials, the mechanical properties of material interfaces can usually only be determined indirectly. For that purpose, two different approaches are presented which allow to compute macroscopic properties of material interfaces based on molecular statics computations. While the first of those is based on the principle of energy equivalence, the second one relies on the principle of stress equivalence. The advantages and disadvantages of both frameworks are analyzed.

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“…The main idea of the new improved homogenization model is to employ the principle of energy equivalence, i.e., the representative volume elements associated with a molecular statics computation and that related to the continuum scale should lead to the same total energy, cf. [2]. Considering isothermal reversible processes, such energies can be defined as…”

Section: Atomistic-to-continuum Modelmentioning

confidence: 99%

Macroscopic modeling of material interfaces based on atomistic descriptions

Sievers

Clausmeyer

Mosler

2014

Proc Appl Math and Mech

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The effective macroscopic properties of a broad variety of different materials are defined by the properties of the involved material interfaces. Grain boundaries in polycrystals represent a well-known example. However, in contrast to classical bulk materials, the mechanical properties of material interfaces can oftentimes not be identified in a straightforward manner. In this paper, a multiscale framework is presented which allows to compute macroscopic properties of material interfaces based on molecular dynamics. The key idea of the elaborated multiscale framework is the coupling of an RVE associated with the atomistic scale to an RVE based on a continuum mechanics description by means of the principle of energy equivalence.

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A hybrid variationally consistent homogenization approach based on Ritz's method

Cited by 8 publications

References 55 publications

$${\text {FE}}^r$$ method with surrogate localization model for hyperelastic composite materials

$${\text {FE}}^r$$ method with surrogate localization model for hyperelastic composite materials

Continuum modeling of material interfaces and surfaces based on molecular statics computations

Macroscopic modeling of material interfaces based on atomistic descriptions

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