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

Larsson, Stig; Saedpanah, Fardin

doi:10.1093/imanum/drp014

Cited by 27 publications

(61 citation statements)

References 10 publications

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“…The computed vertical displacements at the point ( x, y, z ) = (1.5, 1.5, 0.0) for different fractional parameters are shown in the left plot of Fig. 6, from which one could notice that smaller fractional order α generally yields less damping, which is consistent with the claims in [42]. …”

Section: Numerical Resultssupporting

confidence: 86%

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Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms

Perdikaris

Karniadakis

2016

Journal of Computational Physics

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We develop efficient numerical methods for fractional order PDEs, and employ them to investigate viscoelastic constitutive laws for arterial wall mechanics. Recent simulations using one-dimensional models [1] have indicated that fractional order models may offer a more powerful alternative for modeling the arterial wall response, exhibiting reduced sensitivity to parametric uncertainties compared with the integer-calculus-based models. Here, we study three-dimensional (3D) fractional PDEs that naturally model the continuous relaxation properties of soft tissue, and for the first time employ them to simulate flow structure interactions for patient-specific brain aneurysms. To deal with the high memory requirements and in order to accelerate the numerical evaluation of hereditary integrals, we employ a fast convolution method [2] that reduces the memory cost to O(log(N)) and the computational complexity to O(N log(N)). Furthermore, we combine the fast convolution with high-order backward differentiation to achieve third-order time integration accuracy. We confirm that in 3D viscoelastic simulations, the integer order models strongly depends on the relaxation parameters, while the fractional order models are less sensitive. As an application to long-time simulations in complex geometries, we also apply the method to modeling fluid–structure interaction of a 3D patient-specific compliant cerebral artery with an aneurysm. Taken together, our findings demonstrate that fractional calculus can be employed effectively in modeling complex behavior of materials in realistic 3D time-dependent problems if properly designed efficient algorithms are employed to overcome the extra memory requirements and computational complexity associated with the non-local character of fractional derivatives.

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Section: Numerical Resultssupporting

confidence: 86%

“…We now consider a simple but realistic example for a two-dimensional structure which has been employed in [42], to show that the model accurately captures the mechanical response and to investigate the accuracy of the fast convolution method when τ ε ≠ 0. The geometry and mesh are illustrated in Fig.…”

Section: Numerical Resultsmentioning

confidence: 99%

Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms

Perdikaris

Karniadakis

2016

Journal of Computational Physics

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“…On the other hand, many authors have obtained that the Galerkin spectral method is a well‐posedness approximation approach when it used to approximate the solution of different types of FDEs. Some of these equations are fractional integro‐differential equation, fractional diffusion problems, linearized time‐fractional KdV equation, variable‐coefficient conservative fractional elliptic differential equations, and others. ()…”

Section: Initial Value Problemmentioning

confidence: 99%

Modified Galerkin algorithm for solving multitype fractional differential equations

Alsuyuti¹,

Doha

Ezz-Eldien

et al. 2019

Math Methods in App Sciences

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The primary point of this manuscript is to dissect and execute a new modified Galerkin algorithm based on the shifted Jacobi polynomials for solving fractional differential equations (FDEs) and system of FDEs (SFDEs) governed by homogeneous and nonhomogeneous initial and boundary conditions. In addition, we apply the new algorithm for solving fractional partial differential equations (FPDEs) with Robin boundary conditions and time‐fractional telegraph equation. The key thought for obtaining such algorithm depends on choosing trial functions satisfying the underlying initial and boundary conditions of such problems. Some illustrative examples are discussed to ascertain the validity and efficiency of the proposed algorithm. Also, some comparisons with some other existing spectral methods in the literature are made to highlight the superiority of the new algorithm.

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“…The convolution kernel is weakly singular and β ∈ L 1 (0, ∞) with ∞ 0 β(t) dt = γ. Well-posedness of the model problem (1.1) and more general form of such equations in fractional order viscoelasticity have been studied in [13], by means of Galerkin approximation methods. Continuous Galerkin methods of order one, both in time and space variables, have been applied to similar problems in [5], [11] and [12]. Discontinuous Galerkin and continuous Galerkin method, respectively, in time and space variables have been applied to a dynamic model problem in linear viscoelasticity (with exponential kernels) in [10].…”

Section: Introductionmentioning

confidence: 99%

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

Larsson

Racheva

Saedpanah

2015

Computer Methods in Applied Mechanics and Engineering

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An integro-differential equation, modeling dynamic fractional order viscoelasticity, with a Mittag-Leffler type convolution kernel is considered. A discontinuous Galerkin method, based on piecewise constant polynomials is formulated for temporal semidiscretization of the problem. Stability estimates of the discrete problem are proved, that are used to prove optimal order a priori error estimates. The theory is illustrated by a numerical example.

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

Cited by 27 publications

References 10 publications

Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms

Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms

Modified Galerkin algorithm for solving multitype fractional differential equations

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

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