Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity

Larsson, Stig; Racheva, Milena R.; Saedpanah, Fardin

doi:10.1016/j.cma.2014.09.018

Cited by 49 publications

(24 citation statements)

References 17 publications

(20 reference statements)

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“…The main novelty of the presented scheme is the introduction of a fractional‐order aging kernel, that is, the relaxation function , into the viscoelastic formulation. Moreover, a time‐discontinuous Galerkin scheme is hereinafter illustrated by employing linear shape functions for the time discretization, previously used in this context . With regard to the spatial discretization, for the sake of clarity, the classical shape functions related to the Bernoulli‐Navier beam theory are considered.…”

Section: Structural Analysis In Presence Of Fractional‐order Aging Hementioning

confidence: 99%

“…Moreover, a time-discontinuous Galerkin scheme is hereinafter illustrated by employing linear shape functions for the time discretization, previously used in this context. 45 With regard to the spatial discretization, for the sake of clarity, the classical shape functions related to the Bernoulli-Navier beam theory are considered. However, the same FE formulation can be extended to a generic 3D body.…”

Section: Structural Analysis In Presence Of Fractional-order Aging Hementioning

confidence: 99%

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

Beltempo

Bonelli

Bursi

et al. 2019

Numerical Meth Engineering

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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.

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Section: Structural Analysis In Presence Of Fractional‐order Aging Hementioning

confidence: 99%

Section: Structural Analysis In Presence Of Fractional-order Aging Hementioning

confidence: 99%

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

Beltempo

Bonelli

Bursi

et al. 2019

Numerical Meth Engineering

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“…Therefore, following equation can be‘ obtained:

\int_{0}^{t} α (s) ([], \sum_{i = 1}^{normalQ} {boldv}^{T} (s) {boldc}_{i} - \int_{0}^{s} p^{c} (τ) \sum_{i = 1}^{normalQ} {boldw}^{T} (s - τ) {boldb}_{i} dτ) ds = 0

Choosing different weighting functions leads to a different forward model of dynamic load identification, and there are many choosing methods such as collocation method, the subdomain method, the least‐square method and Galerkin method . For the convenience of calculation and simple statement, it might as well make v ( t ) equal to w ( t ).…”

Section: Formulation Of the Forward Problem Based On Tdgmmentioning

confidence: 99%

Time‐domain Galerkin method for dynamic load identification

Liu

Meng

Jiang

et al. 2015

Numerical Meth Engineering

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Summary This paper proposes a new method called time‐domain Galerkin method (TDGM) for investigating the structural dynamic load identification problems. Firstly, the shape functions are adopted to approximate three parameters, such as the dynamic load, kernel function response, and measured structural response Secondly, defining a residual function could be expressed as the difference of the measured response and the computational response. Thirdly, select an appropriate weighting function to multiply the defined residual function and make integral operation with respect to time to be zero. Finally, when the shape functions are chosen as the weighting function, it establishes the forward model called TDGM. Furthermore, the regularization method could have effectiveness in solving the ill‐posed matrix of load reconstruction and obtaining the accurate identified results of the dynamic load. Compared with the traditional Green kernel function method (GKFM), TDGM can effectively overcome the influences of noise and improve the accuracy of the dynamic load identification. Three numerical examples are provided to demonstrate the correctness and advantages of TDGM. Copyright © 2015 John Wiley & Sons, Ltd.

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“…2 There are several numerical methods for the fractional integro-differential equations such as hybrid collocation method, 3 the Jacobi spectral-collocation method, 4 operational Jacobi Tau method, 5 and discontinuous Galerkin method. 6 However, the fractional integro-differential equations with a weakly singular kernel are solved by only a few methods such as Legendre wavelets method 7 and second Chebyshev wavelets methods. 8 The reproducing kernel theory has attracted many scholars' attention and has been applied to many linear and nonlinear equations numerical algorithms.…”

Section: Introductionmentioning

confidence: 99%

Exact solution of a class of fractional integro‐differential equations with the weakly singular kernel based on a new fractional reproducing kernel space

Chen

Yang

2018

Math Methods in App Sciences

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In this paper, we construct a new fractional weighted reproducing kernel space, which is the minimum space containing the exact solution. The closed form of the reproducing kernel is obtained. Using this fractional reproducing kernel space, a class of fractional integro-differential equations with a weakly singular kernel is solved. The error estimation is given. The final numerical experiments demonstrate the correctness of the theory and the effectiveness of the method.KEYWORDS fractional integro-differential equation, fractional reproducing kernel space, weakly singular kernel 1 0 k(x, )u( )d + g(x), m − 1 < ≤ m, u (k) (0) = 0, k = 0, 1, · · · , m − 1,Math Meth Appl Sci. 2018;41:3841-3855.wileyonlinelibrary.com/journal/mma

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Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity

Cited by 49 publications

References 17 publications

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

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

Time‐domain Galerkin method for dynamic load identification

Exact solution of a class of fractional integro‐differential equations with the weakly singular kernel based on a new fractional reproducing kernel space

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