Fractional Sensitivity Equation Method: Application to Fractional Model Construction

Kharazmi, Ehsan; Zayernouri, Mohsen

doi:10.1007/s10915-019-00935-0

Cited by 21 publications

(17 citation statements)

References 74 publications

(88 reference statements)

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“…Phase‐field models derived from free‐energy potentials with nonlocal effects were first discussed by Giacomin and Lebowitz 54,55 with recent contributions from Abels et al, 56 and Ainsworth and Mao 53 . Fractional‐order models for structural analysis have been developed, 57 to which corresponding fractional uncertainty/sensitivity analyses can be formulated via operator‐based uncertainty quantification 58 and fractional sensitivity equation method 59 …”

Section: Numerical Resultsmentioning

confidence: 99%

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An integrated sensitivity‐uncertainty quantification framework for stochastic phase‐field modeling of material damage

Moraes

Zayernouri

Meerschaert

2020

Int J Numer Methods Eng

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Materials accumulate energy around voids and defects under external loading, causing the formation of microcracks. With increasing or repeated loads, those microcracks eventually coalesce to form macrocracks, which in a brittle material can cause catastrophic failure without apparent permanent deformation. At the continuum level, a stochastic phase-field model is employed to simulate failure through introducing damage and fatigue variables. The damage phase-field is introduced as a continuous dynamical variable representing the volumetric portion of fractured material and fatigue is treated as a continuous internal field variable to model the effects of microcracks arising from energy accumulation. We formulate a computational-mathematical framework for quantifying the corresponding model uncertainties and sensitivities in order to unfold and mitigate the salient sources of unpredictability in the model, hence, leading to new possible modeling paradigms. Considering an isothermal isotropic linear elastic material with viscous dissipation under the hypothesis of small deformations, we employed Monte Carlo and probabilistic collocation methods to perform the forward uncertainty propagation, in addition to local-to-global sensitivity analysis. We demonstrate that the model parameters associated with free-energy potentials contribute significantly more to the total model output uncertainties, motivating further investigations for obtaining more predictable model forms, representing the damage diffusion.

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Section: Numerical Resultsmentioning

confidence: 99%

“…53 Fractional-order models for structural analysis have been developed, 57 to which corresponding fractional uncertainty/sensitivity analyses can be formulated via operator-based uncertainty quantification 58 and fractional sensitivity equation method. 59…”

Section: F I G U R E 17mentioning

confidence: 99%

An integrated sensitivity‐uncertainty quantification framework for stochastic phase‐field modeling of material damage

Moraes

Zayernouri

Meerschaert

2020

Int J Numer Methods Eng

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“…Fractional calculus introduces well-established mathematical tools for an accurate description of anomalous phenomena, ubiquitous in a wide range of applications from bio-tissues (Ionescu et al 2017;Naghibolhosseini & Long 2018) and material science (Meral, Royston & Magin 2010;Suzuki & Zayernouri 2021;Suzuki et al 2021a) to vibration (Suzuki et al 2021b), porous media (Xie & Fang 2019;Zaky, Hendy & Macías-Díaz 2020;Samiee et al 2020b). As an alternative approach to standard methods, they leverage their inherent potentials in representing long-range interactions, self-similar structures, sharp peaks and memory effects in a variety of applications (see Kharazmi & Zayernouri 2019;Burkovska, Glusa & D'Elia 2020). This potential is substantially indicated by power-law or logarithmic kernels of convolution type in the corresponding fractional operators.…”

Section: Preliminaries On Tempered Fractional Calculusmentioning

confidence: 99%

Tempered fractional LES modeling

2021

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The presence of non-local interactions and intermittent signals in the homogeneous isotropic turbulence grant multi-point statistical functions a key role in formulating a new generation of large-eddy simulation (LES) models of higher fidelity. We establish a tempered fractional-order modelling framework for developing non-local LES subgrid-scale models, starting from the kinetic transport. We employ a tempered Lévy-stable distribution to represent the source of turbulent effects at the kinetic level, and we rigorously show that the corresponding turbulence closure term emerges as the tempered fractional Laplacian, $(\varDelta +\lambda )^{\alpha } (\cdot )$ , for $\alpha \in (0,1)$ , $\alpha \neq \frac {1}{2}$ and $\lambda >0$ in the filtered Navier–Stokes equations. Moreover, we prove the frame invariant properties of the proposed model, complying with the subgrid-scale stresses. To characterize the optimum values of model parameters and infer the enhanced efficiency of the tempered fractional subgrid-scale model, we develop a robust algorithm, involving two-point structure functions and conventional correlation coefficients. In an a priori statistical study, we evaluate the capabilities of the developed model in fulfilling the closed essential requirements, obtained for a weaker sense of the ideal LES model (Meneveau, Phys. Fluids, vol. 6, issue 2, 1994, pp. 815–833). Finally, the model undergoes the a posteriori analysis to ensure the numerical stability and pragmatic efficiency of the model.

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“…They typically convert the problem into an optimization problem, and then, formulate a suitable estimator by minimizing an objective function, see e.g. 7–15 . Here, we overcome this difficulty by developing physics-informed neural networks (PINNs) both for integer-order and fractional-order models 16 .…”

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confidence: 99%

Identifiability and predictability of integer- and fractional-order epidemiological models using physics-informed neural networks

Kharazmi

Cai

Zheng

et al. 2021

Preprint

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We analyze a plurality of epidemiological models through the lens of physics-informed neural networks (PINNs) that enable us to identify multiple time-dependent parameters and to discover new data-driven fractional differential operators. In particular, we consider several variations of the classical susceptible-infectious-removed (SIR) model by introducing more compartments and delay in the dynamics described by integer-order, fractional-order, and time-delay models. We report the results for the spread of COVID-19 in New York City, Rhode Island and Michigan states, and Italy, by simultaneously inferring the unknown parameters and the unobserved dynamics. For integer-order and time-delay models, we fit the available data by identifying time-dependent parameters, which are represented by neural networks (NNs). In contrast, for fractional differential models, we fit the data by determining different time-dependent derivative orders for each compartment, which we represent by NNs. We investigate the identifiability of these unknown functions for different datasets, and quantify the uncertainty associated with NNs and with control measures in forecasting the pandemic.

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Fractional Sensitivity Equation Method: Application to Fractional Model Construction

Cited by 21 publications

References 74 publications

An integrated sensitivity‐uncertainty quantification framework for stochastic phase‐field modeling of material damage

An integrated sensitivity‐uncertainty quantification framework for stochastic phase‐field modeling of material damage

Tempered fractional LES modeling

Identifiability and predictability of integer- and fractional-order epidemiological models using physics-informed neural networks

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