A numerical integration approach for fractional‐order viscoelastic analysis of hereditary‐aging structures

Abstract

Summary In this paper, a novel numerical integration scheme is proposed for fractional‐order viscoelastic analysis of hereditary‐aging structures. More precisely, the idea of aging is first introduced through a new phenomenological viscoelastic model characterized by variable‐order fractional operators. Then, the presented fractional‐order viscoelastic model is included in a variational formulation, conceived for any viscous kernel and discretized in time by employing a discontinuous Galerkin method. The accuracy of the resulting finite element (FE) scheme is analyzed through a model problem, whose exact solution is known; and the most significant variables affecting the solution quality, such as the number of Gaussian quadrature points and time subintervals, are then investigated in terms of error and computational cost. Moreover, the proposed FE integration scheme is applied to study the short‐ and long‐term behavior of concrete structures, which, due to the severe aging exhibited during their service life, represents one of the most challenging time‐dependent behavior to be investigated. Eventually, also the Euler implicit method, commonly used in commercial software, is compared.

Convergence Analysis of Iterative Learning Control for Initialized Fractional Order Systems

“…Based on the results of Steps 1 and 2, the convergence property of δu k can ultimately be derived. Applying (37) to substitute ∥ δx k+1 ∥ λ in ( 29) yields…”

“…As discussed in Section 3, the initialization path directly affects the dynamic characteristics of fractional-order systems. In applied disciplines such as secondary battery [36] and viscoelasticity [37], this historical hereditary effect can usually not be neglected. Identification and fitting of history function is crucial for improving system performance [21].…”

Two-scale, non-local diffusion in homogenised heterogeneous media

“…On the other hand, a one-dimensional diffusion problem in a bounded homogeneous medium is studied in [76], wherein Darcy's equation is generalised with a fractional integral in space. Furthermore, in the context of hierarchical materials, such as bones and ligaments, a generalised viscoelastic approach has been proposed to describe their rheological properties by using fractional derivatives and integrals [3,30], while numerical methods have been developed for the case of hereditary-ageing materials in [16]. Additionally, we notice that in [81] the analytical and numerical solution of a generalised heat conduction equation was studied by considering a fractional time derivative instead of the first-order partial time derivative of the temperature.…”

Advanced materials modelling via fractional calculus: challenges and perspectives

“…Furthermore, the use of variable-order fractional calculus has been discussed by Beltempo et al [36] to address ageing of pre-stressed concrete as an alternative to model B3 [37]. Variableorder fractional calculus has been applied to model ageing concrete, obtaining mathematically consistent relaxation functions to be coded into finite-element specific algorithms for a computer simulation of real-case structures [38,39].…”