On the solution of hyperbolic equations using the peridynamic differential operator

Bekar, Ali Can; Madenci, Erdogan; Haghighat, Ehsan

doi:10.1016/j.cma.2022.114574

Cited by 18 publications

(6 citation statements)

References 20 publications

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“…The direction of characteristics must be considered when solving hyperbolic equations. Baryt et al [29] constructed an upwind weight function according to the direction of the incoming flow, when G k i ≥ 0 and the weight function is as described in Eq. (11a) (Fig.…”

Section: Ee-pddo Methods For Solving the Population Balance Equationmentioning

confidence: 99%

“…Thus, this method has been successfully applied to the study of stress analysis [26], heat and mass transfer [27], and thermodynamic problems [28]. Recently, Baryt et al [29] modified the weight function of the PDDO based on the characteristics of hyperbolic equations and obtained a generalized PDDO scheme to accurately solve these equations. Herein, we attempt to apply this method to solve the PBE.…”

Section: Introductionmentioning

confidence: 99%

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Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Ruan,

Dong,

Liang

et al. 2024

CMES

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Using Euler's first-order explicit (EE) method and the peridynamic differential operator (PDDO) to discretize the time and internal crystal-size derivatives, respectively, the Euler's first-order explicit method-peridynamic differential operator (EE-PDDO) was obtained for solving the one-dimensional population balance equation in crystallization. Four different conditions during crystallization were studied: size-independent growth, sizedependent growth in a batch process, nucleation and size-independent growth, and nucleation and size-dependent growth in a continuous process. The high accuracy of the EE-PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods. The method is characterized by non-oscillation and high accuracy, especially in the discontinuous and sharp crystal size distribution. The stability of the EE-PDDO method, choice of weight function in the PDDO method, and optimal time step are also discussed.

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Section: Ee-pddo Methods For Solving the Population Balance Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Ruan,

Dong,

Liang

et al. 2024

CMES

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“…In our previous work [20], Euler's first-order explicit (EE) method-Peridynamic Differential Operator (PDDO) was used to solve 1D PBEs. Compared with the second-order upwind [21] and HR-van [22] methods, the EE-PDDO method shows high accuracy in predicting crystal size distributions.…”

Section: Introductionmentioning

confidence: 99%

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization

Dong,

Ruan

2024

Crystals

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The population balance equations (PBEs) serve as the primary governing equations for simulating the crystallization process. Two-dimensional (2D) PBEs pertain to crystals that exhibit anisotropic growth, which is characterized by changes in two internal coordinates. Because PBEs are the hyperbolic equations, it becomes imperative to establish a high-resolution scheme to reduce numerical diffusion and numerical dispersion, thereby ensuring accurate crystal size distribution. This paper uses Euler’s first-order explicit (EE) method–Peridynamic Differential Operator (PDDO) to solve 2D PBE, namely, the EE method for discretizing the time derivative and the PDDO for discretizing the internal crystal-size derivative. Five examples, including size-independent growth with smooth and non-smooth distributions, size-dependent growth, nucleation, and size-independent/dependent growth for batch crystallization are considered. The results show that the EE–PDDO method is more accurate than the HR method and that it is as good as the fifth-order Weighted Essential Non-Oscillatory (WENO) method in solving 2D PBE. This study extends the EE–PDDO method to the simulation of 2D PBE, and the advantages of the EE-PDDO method in dealing with discontinuous and sharp front problems are demonstrated.

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“…[11][12][13][14][15][16] Recently, Reference 17 proposed a peridynamic differential operator to construct nonlocal solutions of differential equations, thus extending the application of peridynamics to more areas. [18][19][20][21] The original bond-based peridynamic formulation by Silling, 4 however, can only describe a fixed Poisson's ratio. Moreover, the force density grows linearly with bond stretch until a threshold value, where the connection between two material points vanishes.…”

Section: Introductionmentioning

confidence: 99%

“…It is thus commonly adopted for modeling fracture processes 11–16 . Recently, Reference 17 proposed a peridynamic differential operator to construct nonlocal solutions of differential equations, thus extending the application of peridynamics to more areas 18–21 . The original bond‐based peridynamic formulation by Silling, 4 however, can only describe a fixed Poisson's ratio.…”

Section: Introductionmentioning

confidence: 99%

The extended peridynamic model for elastoplastic and/or fracture problems

You

et al. 2022

Numerical Meth Engineering

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The strain‐based implementation method for the extended peridynamic model (XPDM) resolves the limitation of standard models where only a fixed Poisson's ratio can be achieved. In this contribution, the XPDM formulation is extended to include bond breakage and/or plasticity mechanisms. The elastoplastic and bond breakage algorithms are elaborated. To capture the fracture process, a shear mechanism is now incorporated to the bond breakage response, in addition to the standard stretching failure mode. It is shown that the shear mechanism is required to accurately reproduce mixed mode fracture behavior observed experimentally. To demonstrate the predictive ability of the strain‐based XPDM, a wide range of quasi‐static and dynamic loading conditions, for both brittle and elasto‐plastic materials, is considered against experimental results or practical engineering scenarios.

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On the solution of hyperbolic equations using the peridynamic differential operator

Cited by 18 publications

References 20 publications

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization

The extended peridynamic model for elastoplastic and/or fracture problems

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