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1. Numerical methods for nonlocal and fractional models

   https://doi.org/10.1017/S096249292000001X  

   D’Elia, Marta; Du, Qiang; Glusa, Christian; Gunzburger, Max; Tian, Xiaochuan; Zhou, Zhi (May 2020, Acta Numerica)

   null (Ed.)

   Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling. 

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2. A comparison of 3-D spherical shell thermal convection results at low to moderate Rayleigh number using ASPECT (version 2.2.0) and CitcomS (version 3.3.1)

   https://doi.org/10.5194/gmd-16-3221-2023  

   Euen, Grant T.; Liu, Shangxin; Gassmöller, Rene; Heister, Timo; King, Scott D. (January 2023, Geoscientific Model Development)

   Abstract. Due to the increasing availability of high-performance computing over the past few decades, numerical models have become an important tool for research in geodynamics.Several generations of mantle convection software have been developed, but due to their differing methods and increasing complexity it is important to evaluate the accuracy of each new model generation to ensure published geodynamic research is reliable and reproducible.Here we explore the accuracy of the open-source, finite-element codes ASPECT and CitcomS as a function of mesh spacing using low to moderate-Rayleigh-number models in steady-state thermal convection.ASPECT (Advanced Solver for Problems in Earth's ConvecTion) is a new-generation mantle convection code that enables modeling global mantle convection with realistic parameters and complicated physical processes using adaptive mesh refinement (Kronbichler et al., 2012; Heister et al., 2017).We compare the ASPECT results with calculations from the finite-element code CitcomS (Zhong et al., 2000; Tan et al., 2006; Zhong et al., 2008), which has a long history of use in the geodynamics community.We find that the globally averaged quantities, i.e., root-mean-square (rms) velocity, mean temperature, and Nusselt number at the top and bottom of the shell, agree to within 1 % (and often much better) for calculations with sufficient mesh resolution.We also show that there is excellent agreement of the time evolution of both the rms velocity and the Nusselt numbers between the two codes for otherwise identical parameters.Based on our results, we are optimistic that similar agreement would be achieved for calculations performed at the convective vigor expected for Earth, Venus, and Mars. 

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3. Differentiable Spline Approximations

   Cho, M; Balu, A; Joshi, A; Deva Prasad, A; Khara, B; Sarkar, S; Ganapathysubramanian, B; Krishnamurthy, A; Hegde, C (December 2021, Advances in neural information processing systems)

   The paradigm of differentiable programming has significantly enhanced the scope of machine learning via the judicious use of gradient-based optimization. However, standard differentiable programming methods (such as autodiff) typically require that the machine learning models be differentiable, limiting their applicability. Our goal in this paper is to use a new, principled approach to extend gradient-based optimization to functions well modeled by splines, which encompass a large family of piecewise polynomial models. We derive the form of the (weak) Jacobian of such functions and show that it exhibits a block-sparse structure that can be computed implicitly and efficiently. Overall, we show that leveraging this redesigned Jacobian in the form of a differentiable" layer''in predictive models leads to improved performance in diverse applications such as image segmentation, 3D point cloud reconstruction, and finite element analysis. We also open-source the code at\url {https://github. com/idealab-isu/DSA}. 

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4. An Entropy Dynamics Approach for Deriving and Applying Fractal and Fractional Order Viscoelasticity to Elastomers

   https://doi.org/10.1115/1.4062389  

   Pahari, Basanta R.; Stanisauskis, Eugenia; Mashayekhi, Somayeh; Oates, William (August 2023, Journal of Applied Mechanics)

   Abstract Entropy dynamics is a Bayesian inference methodology that can be used to quantify time-dependent posterior probability densities that guide the development of complex material models using information theory. Here, we expand its application to non-Gaussian processes to evaluate how fractal structure can influence fractional hyperelasticity and viscoelasticity in elastomers. We investigate how kinematic constraints on fractal polymer network deformation influences the form of hyperelastic constitutive behavior and viscoelasticity in soft materials such as dielectric elastomers, which have applications in the development of adaptive structures. The modeling framework is validated on two dielectric elastomers, VHB 4910 and 4949, over a broad range of stretch rates. It is shown that local fractal time derivatives are equally effective at predicting viscoelasticity in these materials in comparison to nonlocal fractional time derivatives under constant stretch rates. We describe the origin of this accuracy that has implications for simulating large-scale problems such as finite element analysis given the differences in computational efficiency of nonlocal fractional derivatives versus local fractal derivatives. 

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5. 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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