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1. Fractional-Order Shell Theory: Formulation and Application to the Analysis of Nonlocal Cylindrical Panels

   https://doi.org/10.1115/1.4054677  

   Sidhardh, Sai; Patnaik, Sansit; Semperlotti, Fabio (August 2022, Journal of Applied Mechanics)

   Abstract We present a theoretical and computational framework based on fractional calculus for the analysis of the nonlocal static response of cylindrical shell panels. The differ-integral nature of fractional derivatives allows an efficient and accurate methodology to account for the effect of long-range (nonlocal) interactions in curved structures. More specifically, the use of frame-invariant fractional-order kinematic relations enables a physically, mathematically, and thermodynamically consistent formulation to model the nonlocal elastic interactions. To evaluate the response of these nonlocal shells under practical scenarios involving generalized loads and boundary conditions, the fractional-finite element method (f-FEM) is extended to incorporate shell elements based on the first-order shear-deformable displacement theory. Finally, numerical studies are performed exploring both the linear and the geometrically nonlinear static response of nonlocal cylindrical shell panels. This study is intended to provide a general foundation to investigate the nonlocal behavior of curved structures by means of fractional-order models. 

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2. On the role of the microstructure in the deformation of porous solids

   https://doi.org/10.1038/s41524-022-00840-5  

   Patnaik, Sansit; Jokar, Mehdi; Ding, Wei; Semperlotti, Fabio (December 2022, npj Computational Materials)

   Abstract This study explores the role that the microstructure plays in determining the macroscopic static response of porous elastic continua and exposes the occurrence of position-dependent nonlocal effects that are strictly correlated to the configuration of the microstructure. Then, a nonlocal continuum theory based on variable-order fractional calculus is developed in order to accurately capture the complex spatially distributed nonlocal response. The remarkable potential of the fractional approach is illustrated by simulating the nonlinear thermoelastic response of porous beams. The performance, evaluated both in terms of accuracy and computational efficiency, is directly contrasted with high-fidelity finite element models that fully resolve the pores’ geometry. Results indicate that the reduced-order representation of the porous microstructure, captured by the synthetic variable-order parameter, offers a robust and accurate representation of the multiscale material architecture that largely outperforms classical approaches based on the concept of average porosity. 

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3. Impact of nuclear vibrations on van der Waals and Casimir interactions at zero and finite temperature

   https://doi.org/10.1126/sciadv.aaw0456  

   Venkataram, Prashanth S.; Hermann, Jan; Vongkovit, Teerit J.; Tkatchenko, Alexandre; Rodriguez, Alejandro W. (November 2019, Science Advances)

   Recent advances in measuring van der Waals (vdW) interactions have probed forces on molecules at nanometric separations from metal surfaces and demonstrated the importance of infrared nonlocal polarization response and temperature effects, yet predictive theories for these systems remain lacking. We present a theoretical framework for computing vdW interactions among molecular structures, accounting for geometry, short-range electronic delocalization, dissipation, and collective nuclear vibrations (phonons) at atomic scales, along with long-range electromagnetic interactions in arbitrary macroscopic environments. We primarily consider experimentally relevant low-dimensional carbon allotropes, including fullerenes, carbyne, and graphene, and find that phonons couple strongly with long-range electromagnetic fields depending on molecular dimensionality and dissipation, especially at nanometric scales, creating delocalized phonon polaritons that substantially modify infrared molecular response. These polaritons, in turn, alter vdW interaction energies between molecular and macroscopic structures, producing nonmonotonic power laws and nontrivial temperature variations at nanometric separations feasible in current experiments. 

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4. A data-driven framework for discovering fractional differential equations in complex systems

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   Xiangnan, Yu; Hao, Xu; Zhiping, Mao; Hongguang, Sun; Yong, Zhang; Zhibo, Chen; Dongxiao, Zhang; Yuntian, Chen (May 2025, Nonlinear Dynamics)

   In complex physical systems, conventional differential equations fall short in capturing non-local and memory effects. Fractional differential equations (FDEs) effectively model long-range interactions with fewer parameters. However, deriving FDEs from physical principles remains a significant challenge. This study introduces a stepwise data-driven framework to discover explicit expressions of FDEs directly from data. The proposed framework combines deep neural networks for data reconstruction and automatic differentiation with Gauss-Jacobi quadrature for fractional derivative approximation, effectively handling singularities while achieving fast, high-precision computations across large temporal/spatial scales. To optimize both linear coefficients and the nonlinear fractional orders, we employ an alternating optimization approach that combines sparse regression with global optimization techniques. We validate the framework on various datasets, including synthetic anomalous diffusion data, experimental data on the creep behavior of frozen soils, and single-particle trajectories modeled by Lévy motion. Results demonstrate the framework’s robustness in identifying FDE structures across diverse noise levels and its ability to capture integer order dynamics, offering a flexible approach for modeling memory effects in complex systems. 

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5. Freeform Mode-Engineered Metasurfaces

   https://doi.org/10.1021/acs.nanolett.5c06075  

   Jiang, Zhongjun; Dai, Tianxiang; Guo, Shuwei; Sohag, Soyaib H; Shao, Yixuan; Mao, Chenkai; Alù, Andrea; Fan, Jonathan A; Zhou, You (March 2026, Nano Letters)

   Nanophotonic technologies inherently rely on tailoring light–matter interactions through the excitation and interference of deeply confined optical resonances. However, existing concepts in optical mode engineering remain heuristic and are challenging to extend toward complex and multifunctional resonant phenomena. We introduce an inverse design framework that optimizes near-field distributions, ideally suited to tailoring Mie-type modes within dielectric nanophotonic structures, and we demonstrate its application to the discovery of new classes of nonlocal metasurfaces. We show that freeform nonlocal metasurfaces supporting accidental bound states in the continuum can be readily optimized for tailored illumination conditions, modal properties, and quality factors. We further generalize the framework to higher-order and multifunctional mode engineering and experimentally demonstrate freeform planar nonlocal multiwavelength and chiral metasurfaces. Our versatile framework for freeform mode engineering has applications in broad high-quality-factor nanophotonic platforms relevant to sensing, nonlinear optics, optomechanics, and quantum information processing. 

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