Reduction of the number of material parameters by ANN approximation

Sumelka, Wojciech; Łodygowski, Tomasz

doi:10.1007/s00466-012-0812-9

Cited by 22 publications

(11 citation statements)

References 28 publications

(36 reference statements)

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“…Such formulations introduce the possibility of analysis of materials showing scale effect, but depending on direction. Let us mention herein the papers on this subject by: Kuhl et al [26], where quasi-brittle materials were considered; Stumpf et al [27], where the crack analysis based on the concept of continua with microstructure and evolving defects was discussed; Germain et al [28], where the anisotropic layered material was formulated; Abu-Al-Rub et al [29], where coupled anisotropic damage and plasticity constitutive model to predict the concrete distinct behavior in tension and compression was considered; Alastrue et al [30], where fully three-dimensional anisotropic elastic model for vascular tissue modelling was shown; and finally by Perzyna [31] and Sumelka et al [32,33], where the class of implicitly non-local (rate type [34]) anisotropic models for metallic materials was considered. Nevertheless, considering the subject of this paper, the fractional anisotropic non-local models are nowadays still under development.…”

Section: Introductionmentioning

confidence: 99%

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Fractional calculus for continuum mechanics – anisotropic non-locality

Sumelka¹

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. In this paper, a generalisation of previous author's formulation of fractional continuum mechanics for the case of anisotropic non-locality is presented. The discussion includes a review of competitive formulations available in literature. The overall concept is based on the fractional deformation gradient which is non-local due to fractional derivative definition. The main advantage of the proposed formulation is its structure, analogous to the general framework of classical continuum mechanics. In this sense, it allows to define similar physical and geometrical meaning of introduced objects. The theoretical discussion is illustrated by numerical examples assuming anisotropy limited to single direction. NotationSection 2and finallywhere] denotes the interval of non-local interaction, and Γ is the Euler gamma functionNext, based on non-standard definition of motion by Eq. (4) the authors define the deformation gradient F α aswhere e a and E A are base vectors in coordinate systems {x a } and {X A }, respectively.

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Section: Introductionmentioning

confidence: 99%

“…In this paper, the non-local elasticity relation was proposed as α 31 α 32 α 33 and finally 3 ] denotes the interval of non-local interaction, and Γ is gamma function…”

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confidence: 99%

Fractional calculus for continuum mechanics – anisotropic non-locality

Sumelka¹

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…To understand the general concept of 3D s-FCM, the reader should consult [20,30]. Based on the results presented in [20], the governing equations for a 3D s-FCM elastic body that occupies a volume Ω with boundary ∂Ω, together with the assumption of a variable length scale [31], have the form…”

Section: D Fractional Elasticitymentioning

confidence: 99%

Discrete mass-spring structure identification in nonlocal continuum space-fractional model

Szajek

Sumelka

2019

Eur. Phys. J. Plus

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This paper considers discrete mass-spring structure identification in a nonlocal continuum spacefractional model, defined as an optimization task. Dynamic (eigenvalues and eigenvectors) and static (displacement field) solutions to discrete and continuum theories are major constituents of the objective function. It is assumed that the masses in both descriptions are equal (and constant), whereas the spring stiffness distribution in a discrete system becomes a crucial unknown. The considerations include a variety of configurations of the nonlocal parameter and the order of the fractional model, which makes the study comprehensive, and for the first time provides insight into the possible properties (geometric and mechanical) of a discrete structure homogenized by a space-fractional formulation.Recently, the application of fractional calculus (FC) in mechanics for describing scale effects (in a broad sense [1]) has gained considerable attention; such concepts are called fractional mechanics (FM). FM formulations exist that describe the following materials: polymers [2], granular soils [3-7], silts [8], granites [9], biological tissues [1,10], metals [6,11], and silicon [12]. FC introduces a continuum-type description through the definition of the fractional derivative (FD) (derivative of an arbitrary order [13-16]), revealing certain hidden aspects of the discrete nature of the phenomena analysed. Furthermore, FC has an indisputable attribute, namely, that FC provides an infinite number of possible FD realizations [17,18]; thus we can identify a specific type of FD depending on the experimental evidence.An important family of FM concepts includes formulations where, in classic continuum mechanics (CCM) theory, the integer differential operators acting on a spatial variable are replaced by fractional operators; such models are called space-fractional continuum mechanics (s-FCM) [19][20][21][22][23][24][25]. The physical interpretation of s-FCM is that FD homogenizes, in a phenomenological sense, the underlying microstructure [26]; thus, the scale effect is included within FC, and the s-FCM model is nonlocal. Nevertheless, to the best of our knowledge no one has answered the following question: Is there a discrete structure that is homogenized by s-FCM?.In this paper, we provide the answer to the above question by identifying a discrete mass-spring structure, whose behaviour (static and dynamic) is mapped with high precision by an s-FCM approach. Such a concept is a kind of inverse problem to recent trends in physics/mechanics on the study of continualization techniques, such as the Taylor series-based method or the shift operator expansion method [27][28][29]. The overall problem is defined as a 1D optimization task with several variables (altogether approximately 2.5 million analyses were conducted). Furthermore, the considerations include a variety of configurations of the non-local parameter and the order of the fractional model (altogether twenty discrete spring-mass structures were identified), which makes the s...

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“…Artificial neural networks constitute an interesting direction in numerical investigations in mechanics of materials and structures and in engineering [15][16][17][18][19][20]. A properly structured network trained on experimental data can become a remarkable asset in either building or verifying mechanical models or surrogate models.…”

Section: Introductionmentioning

confidence: 99%

ANN Architecture Specifications for Modelling of Open-Cell Aluminum under Compression

Dudzik

Stręk

2020

Mathematical Problems in Engineering

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e knowledge on strength properties of porous metals in compression is essential in tailored application design, as well as in elaboration of general material models. In this article, the authors propose specification details of the ANN architecture for adequate modelling of the phenomenon of compressive behaviour of open-cell aluminum. In the presented research, an algorithm was used to build different structures of artificial neural networks (ANNs), which approximated stress-strain relations of an aluminum sponge subjected to compression. Next, the quality of the built approximations was appraised. e mean absolute relative error (MARE), coefficient of determination between outputs and targets (R 2 ), root mean square error (RMSE), and mean square error (MSE) were assumed as criterial measures for the assessment of the fitting quality. e studied neural networks (NNs) were two-layer feedforward networks with different numbers of neurons in the hidden layer. A set of experimental stress-strain data from quasistatic uniaxial compression tests of open-cell aluminum of various apparent densities was used as data for training of neural networks. Analysis was performed in two modes: in the first one, all samples were taken for training, and in the second case, one sample was left out during training in order to play the role of external data for testing the trained network later. e taken out samples were maximum and minimum density samples (for extrapolation) and one random from within the density interval. e results showed that good approximation on the engineering level (MARE < 5%) was reached for teaching networks with ≥7 neurons in the hidden layer for the first studied case and with ≥8 neurons for the second. Calculations on external data proved that 8 neurons are enough to actually obtain MARE < 10%. Moreover, it was shown that the quality of approximation can be significantly improved to MARE ≈ 7% (tested on external data) if the initial region of the stress-strain relation is modelled by an additional network.

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Reduction of the number of material parameters by ANN approximation

Cited by 22 publications

References 28 publications

Fractional calculus for continuum mechanics – anisotropic non-locality

Fractional calculus for continuum mechanics – anisotropic non-locality

Discrete mass-spring structure identification in nonlocal continuum space-fractional model

ANN Architecture Specifications for Modelling of Open-Cell Aluminum under Compression

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