Fabio Semperlotti

Purdue University West Lafayette

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Sansit Patnaik

Purdue University West Lafayette

Timothy Walsh

Kansas State University

Wei Ding

Mayo Clinic in Arizona

Greg Pickrell

Sandia National Laboratories

Siddharth Nair

Purdue University West Lafayette

Hongfei Zhu

Jiangnan University

Mohit Kumar

Purdue University West Lafayette

Bradley Howell Jared

University of Tennessee at Knoxville

Mihir Sen

University of Notre Dame

Salvatore Buonocore

Italian Institute of Technology

Cooperative Institutions

Purdue University West Lafayette

Sandia National Laboratories

University of Notre Dame

Herrick Technology Laboratories (United States)

Sandia National Laboratories California

Jiangnan University

Changchun University of Science and Technology

Nanjing University

Donghua University

Nanjing Tech University

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Applications of variable-order fractional operators: a review

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences (2020)

Sansit Patnaik John P. Hollkamp Fabio Semperlotti

Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.

10.1098/rspa.2019.0498

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Citations (224)

Variable-order particle dynamics: formulation and application to the simulation of edge dislocations

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences (2020)

Sansit Patnaik Fabio Semperlotti

This study presents the application of variable-order (VO) fractional operators to modelling the dynamics of edge dislocations under the effect of a static state of shear stress. More specifically, a particle dynamic approach is used to simulate the microscopic structure of a material where the constitutive atoms or molecules are modelled via discrete masses and their interaction via inter-particle forces. VO operators are introduced in the formulation in order to capture the complex linear-to-nonlinear dynamic transitions following the translation of dislocations as well as the creation and annihilation of bonds between particles. Remarkably, the motion of the dislocation does not require any

State variable

Coalescence (physics)

Annihilation

Particle system

Particle (ecology)

10.1098/rsta.2019.0290

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Citations (16)

Distillation of non-locality in porous solids

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences (2023)

Sansit Patnaik Mehdi Jokar Wei Ding Fabio Semperlotti

Non-local interactions are intrinsic to multiscale heterogeneous solids. The strength of the interactions exhibits a position-dependent character whose spatial distribution strongly correlates with the underlying microstructure. In this work, a variable-order fractional calculus-based framework to distill the position-dependent non-local effects is developed. By considering the example of porous plates, theoretical and numerical analyses will illustrate the ability of variable-order mechanics to enable parsimonious and causal models via a synthetic variable-order map that naturally measures the heterogeneous non-locality induced by the microstructure; in other terms, the non-classical role of the microstructure in determining the macroscopic response. The characteristic measure for non-locality, different from the local entanglement resulting from the porosity map, allows distillation of the macroscopic non-locality, analogous to quantum non-locality distillation. The macroscopic distillation also enables the additivity of the order map, that is the ability of assembled order maps to capture the response of combined porous plates.

Position (finance)

10.1098/rspa.2022.0770

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Citations (1)

Two-dimensional nonlocal Eshelby’s inclusion theory: eigenstress-driven formulation and applications

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences (2024)

Wei Ding Fabio Semperlotti

The classical Eshelby’s theory, developed based on local linear elasticity, cannot be applied to inclusion problems that involve nonlocal (long range) elastic effects often observed in micromechanical systems. In this study, we introduce the extension of Eshelby’s inclusion theory to nonlocal elasticity. Starting from Eringen’s integral formulation of nonlocal elasticity, an eigenstress-driven nonlocal Eshelby’s inclusion theory is presented. The eigenstress-driven approach is shown to be a valid mathematical extension of the classical eigenstrain-driven approach in the context of nonlocal inclusion problems. Two individual numerical approaches are developed and applied to simulate inclusion problems and numerically extract the corresponding nonlocal Eshelby tensor. The numerical results obtained from both approaches confirm the validity of the derived nonlocal Eshelby tensor and its ability to capture the non-uniform eigenstress distribution within an elliptic inclusion. These results also help reveal the fundamental difference between the mechanical behaviour of the classical local and the nonlocal inclusion problems. The eigenstress-driven nonlocal inclusion theory could provide the necessary theoretical foundation for the development of homogenization methods of nonlocal heterogeneous media.

10.1098/rspa.2023.0842

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Citations (1)

Multiple scattering simulation via physics-informed neural networks

arXiv (Cornell University) (2024)

Siddharth Nair Timothy Walsh Greg Pickrell Fabio Semperlotti

This work presents a physics-driven machine learning framework for the simulation of acoustic scattering problems. The proposed framework relies on a physics-informed neural network (PINN) architecture that leverages prior knowledge based on the physics of the scattering problem as well as a tailored network structure that embodies the concept of the superposition principle of linear wave interaction. The framework can also simulate the scattered field due to rigid scatterers having arbitrary shape as well as high-frequency problems. Unlike conventional data-driven neural networks, the PINN is trained by directly enforcing the governing equations describing the underlying physics, hence without relying on any labeled training dataset. Remarkably, the network model has significantly lower discretization dependence and offers simulation capabilities akin to parallel computation. This feature is particularly beneficial to address computational challenges typically associated with conventional mesh-dependent simulation methods. The performance of the network is investigated via a comprehensive numerical study that explores different application scenarios based on acoustic scattering.

10.1007/s00366-024-02038-3

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A generalized fractional-order elastodynamic theory for non-local attenuating media

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences (2020)

Sansit Patnaik Fabio Semperlotti

This study presents a generalized elastodynamic theory, based on fractional-order operators, capable of modelling the propagation of elastic waves in non-local attenuating solids and across complex non-local interfaces. Classical elastodynamics cannot capture hybrid field transport processes that are characterized by simultaneous propagation and diffusion. The proposed continuum mechanics formulation, which combines fractional operators in both time and space, offers unparalleled capabilities to predict the most diverse combinations of multiscale, non-local, dissipative and attenuating elastic energy transport mechanisms. Despite the many features of this theory and the broad range of applications, this work focuses on the behaviour and modelling capabilities of the space-fractional term and on its effect on the elastodynamics of solids. We also derive a generalized fractional-order version of Snell’s Law of refraction and of the corresponding Fresnel’s coefficients. This formulation allows predicting the behaviour of fully coupled elastic waves interacting with non-local interfaces. The theoretical results are validated via direct numerical simulations.

10.1098/rspa.2020.0200

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Citations (44)

On the role of geometric phase in the dynamics of elastic waveguides

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences (2024)

Mohit Kumar Fabio Semperlotti

The geometric phase provides important mathematical insights to understand the fundamental nature and evolution of the dynamic response in a wide spectrum of systems ranging from quantum to classical mechanics. While the concept of geometric phase, which is an additional phase factor occurring in dynamical systems, holds the same meaning across different fields of application, its use and interpretation can acquire important nuances specific to the system of interest. In recent years, the development of quantum topological materials and its extension to classical mechanical systems have renewed the interest in the concept of geometric phase. This review revisits the concept of geometric phase and discusses, by means of either established or original results, its critical role in the design and dynamic behaviour of elastic waveguides. Concepts of differential geometry and topology are put forward to provide a theoretical understanding of the geometric phase and its connection to the physical properties of the system. Then, the concept of geometric phase is applied to different types of elastic waveguides to explain how either topologically trivial or non-trivial behaviour can emerge based on the geometric features of the waveguide. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

Theme (computing)

10.1098/rsta.2023.0357

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Citations (1)

Stochastic scattering model of anomalous diffusion in arrays of steady vortices

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences (2020)

Salvatore Buonocore Mihir Sen Fabio Semperlotti

We investigate the occurrence of anomalous transport phenomena associated with tracer particles propagating through arrays of steady vortices. The mechanism responsible for the occurrence of anomalous transport is identified in the particle dynamic, which is characterized by long collision-less trajectories (Lévy flights) interrupted by chaotic interactions with vortices. The process is studied via stochastic molecular models that are able to capture the underlying non-local nature of the transport mechanism. These models, however, are not well suited for problems where computational efficiency is an enabling factor. We show that fractional-order continuum models provide an excellent alternative that is able to capture the non-local nature of anomalous transport processes in turbulent environments. The equivalence between stochastic molecular and fractional continuum models is demonstrated both theoretically and numerically. In particular, the onset and the temporal evolution of heavy-tailed diffused fields are shown to be accurately captured, from a macroscopic perspective, by a fractional diffusion equation. The resulting anomalous transport mechanism, for the selected ranges of density of the vortices, shows a superdiffusive nature.

Anomalous Diffusion

Stochastic modelling

Convection–diffusion equation

Time evolution

10.1098/rspa.2020.0183

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Citations (3)

Acoustic Valley-Hall Edge States in phononic elastic waveguides

arXiv (Cornell University) (2017)

Ting‐Wei Liu Fabio Semperlotti

This study investigates the occurrence of acoustic topological edge states in a 2D phononic elastic waveguide due to a phenomenon that is the acoustic analogue of the quantum valley Hall effect. We show that a topological transition takes place between two lattices having broken space inversion symmetry due to the application of a tunable strain field. This condition leads to the formation of gapless edge states at the domain walls, as further illustrated by the analysis of the bulk-edge correspondence and of the associated topological invariants. Although time reversal symmetry is still intact in these systems, the edge states are topologically protected when inter-valley mixing is either weak or negligible. Interestingly, topological edge states can also be triggered at the boundary of a single domain if boundary conditions are properly selected. We also show that the static modulation of the strain field allows tuning the response of the material between the different supported edge states.

Gapless playback

Point reflection

Source

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Citations (3)

Nonlocal elastic metasurfaces: Enabling broadband wave control via intentional nonlocality

Proceedings of the National Academy of Sciences (2020)

Hongfei Zhu Sansit Patnaik Timothy Walsh Bradley Howell Jared Fabio Semperlotti

While elastic metasurfaces offer a remarkable and very effective approach to the subwalength control of stress waves, their use in practical applications is severely hindered by intrinsically narrow band performance. This work introduces the concept of intentional nonlocality as a fundamental mechanism to design passive elastic metasurfaces capable of an exceptionally broadband operating range. The nonlocal behavior is achieved by exploiting nonlocal forces, conceptually akin to long-range interactions in nonlocal material microstructures, between subsets of resonant unit cells forming the metasurface. These long-range forces are obtained via carefully crafted flexible elements whose specific geometry and local dynamics are designed to create remarkably complex transfer functions between multiple units. The resulting nonlocal coupling forces enable achieving phase gradient profiles that are function of the wavenumber of the incident wave.The identification of relevant design parameters and the assessment of their impact on performance are explored via a combination of semi-analytical and numerical models. The nonlocal metasurface concept is tested, both numerically and experimentally, by embedding a total-internal-reflection design in a thin plate waveguide. Results confirm the feasibility of the intentionally nonlocal design concept and its ability to achieve a fully passive and broadband wave control.

10.1073/pnas.2004753117

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Citations (71)