Peridynamic Differential Operator Research Articles

Nonlocal modelling of piezoelectric solid with cracks based on peridynamic differential operator

The paper establishes a nonlocal model for piezoelectric solids using the Peridynamic differential operator. The method transforms the displacement-electric potential equation, constitutive equation, and boundary conditions of piezoelectric solids from a local differential form to a nonlocal integral form. The solution for the displacement and electric potential distributions in piezoelectric solids is obtained using variational principles and the Lagrange multiplier method. The correctness and effectiveness of the model are validated through a comparison of the deformation analysis of piezoelectric solid square plates with analytical solutions. Analysis of piezoelectric solid square plates with edge cracks proves the applicability of this model for solving discontinuous problems. Furthermore, the paper explores the suitability of the model in analysing problems involving multiple cracks in piezoelectric solid plates, highlighting its capability to address defective piezoelectric solid problems. This model provides a new perspective for solving problems involving defects under electro-mechanical coupling conditions.

Meshless Generalized Finite Difference Method Based on Nonlocal Differential Operators for Numerical Simulation of Elastostatics

This study proposes an innovative meshless approach that merges the peridynamic differential operator (PDDO) with the generalized finite difference method (GFDM). Based on the PDDO theory, this method introduces a new nonlocal differential operator that aims to reduce the pre-assumption required for the PDDO method and simplify the calculation process. By discretizing through the particle approximation method, this technique proficiently preserves the PDDO’s nonlocal features, enhancing the numerical simulation’s flexibility and usability. Through the numerical simulation of classical elastic static problems, this article focuses on the evaluation of the calculation accuracy, calculation efficiency, robustness, and convergence of the method. This method is significantly stronger than the finite element method in many performance indicators. In fact, this study demonstrates the practicability and superiority of the proposed method in the field of elastic statics and provides a new approach to more complex problems.

An Eulerian–Lagrangian particle method for weakly compressible viscous flows using peridynamic differential operator

This study proposes a peridynamic differential operator (PDDO)-based Eulerian–Lagrangian hybrid particle method for weakly compressible viscous flows. The PDDO is utilized to transform the governing partial differential equations into their integral form; hence, the issue of local non-differentiability is eliminated. Both the Eulerian and Lagrangian formulations of particle methods for solving the governing integral equations are derived and then combined to propose the hybrid method. In the proposed method, the Eulerian and Lagrangian formulation is utilized to solve the governing equations in the inner computational domain and in areas with free surfaces, respectively. The results of these two domains are then merged to arrive at the solution. In such a way, the hybrid particle method effectively balances the computational demands and applicability. The numerical stability and interpolation consistency of the proposed method are shown by introducing the laminar viscosity model. A detailed numerical procedure is provided involving the boundary conditions and time-stepping strategy. The proposed method is validated by several benchmark problems. Furthermore, the method was applied to several benchmark problems including the hydrostatic test, Taylor–Green vortex, and numerical wave generation. The results of these numerical examples suggest that the proposed method is computationally less demanding while maintaining accuracy.

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

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.

A Coupled Thermomechanical Crack Propagation Behavior of Brittle Materials by Peridynamic Differential Operator

A Coupled Thermomechanical Crack Propagation Behavior of Brittle Materials by Peridynamic Differential Operator

Fracture mechanics investigation for 2D orthotropic materials by using ordinary state-based peridynamics

Fracture behaviors of orthotropic plates are studied by using ordinary state-based peridynamic (OSPD) theory. Based on OSPD, a novel nonlocal formulation of interaction integral is proposed by considering the material orthogonality for the fracture parameter evaluation. By employing peridynamic differential operator, the partial differential terms in the formulation can be transformed into corresponding spatial integral form, which contributes to the calculations of stress intensity factors within the framework of OSPD. It has built up a relationship between classical theory and peridynamic theory. Static and dynamic fracture parameters are carefully evaluated. The crack propagation directions are predicted by prototype microelastic brittle (PMB) criterion in OSPD and maximum circumferential stress (MCS) criterion in classical theory. Several pre-cracked plates with orthotropic material are examined, and results are validated by comparing against the reference solutions. The relationship between fiber orientation and crack inclined angle are also examined. Meanwhile, the crack inclination determined by PMB criterion and MCS criterion are compared and discussed. Accuracy of the peridynamic orthotropic model and proposed nonlocal interaction integral are discussed in detail.

Experimental and PDDO-based meshfree numerical study on ductile fracture behavior of steel plate

This paper presents a meshfree approach based on the coupling of peridynamic differential operator (PDDO) and continuum mechanics to predict the ductile fracture of the steel plates. The interpolation function of particles for a shell model is established by applying the PDDO in the 2D case. The weak form of the kinetic equations is derived by discretizing the motion variables. The Gurson-Tvergaard-Needleman (GTN) constitutive model is adopted to characterize the plasticity and ductile fracture behavior of material, in which the material parameters are determined by conducting uniaxial tensile tests and scanning electron microscope analysis. The ductile fracture test of the EH36 steel plate with a prefabricated crack is conducted to validate the proposed numerical approach. The results show that the established meshfree shell model is able to accurately simulate the bearing capacity, crack propagation rate, and ductile fracture characteristics of the structure.

Non-local modelling of multiphase flow wetting and thermo-capillary flow using peridynamic differential operator

AbstractInterfaces in multiphase flows are affected by surface tension, and when temperature gradients occur in the flow domain, tangential surface tensions along the interface also arise. As the behaviour of fluids contacting on a solid surface is also governed by surface tension, the description of the wetting phenomenon is challenging. Peridynamic differential operator (PDDO) can express partial differentials of any order by integral equations. Therefore, the governing equations for multiphase fluid motion, such as the Navier–Stokes equations and energy equations, can be reformulated in terms of integral equations. In this study, a novel non-local method is developed for modelling the multiphase fluid flow motion using the PDDO, and the thermal effect on surface tension force is considered. To describe the surface tension forces in the normal and tangential directions, the non-local form of the continuum surface force (CFS) model is presented. Besides, to overcome the inaccuracy of the unit normal vectors at the three-phase flow intersection region, an additional treatment for this region is presented. Finally, several benchmark multiphase fluid flow cases, such as square droplet deformation, surface wetting, and droplet migration in thermo-capillary flow are presented and validated. The results demonstrate that the developed non-local model can accurately capture the surface tension effect in multiphase fluid flow motion.

Static and dynamic analysis of couple-stress elastic structures by using peridynamic differential operator

In this paper, a peridynamic differential operator (PDDO) method is developed to analyze the static and dynamic responses of size-dependent elastic structures in the frame of a consistent couple stress theory (CCST). The displacement governing equations together with the boundary conditions of the couple-stress static problems are transformed from the local differential form to the nonlocal integral form through a fourth-order PDDO and then solved by a meshfree scheme. For dynamic cases, the time-domain problems are first transformed into frequency domain by Fourier transform, then solved by the PDDO meshless method in the frequency domain. The time-domain results are finally obtained by using an exponential window method (EWM). The size effects on the static and dynamic responses of some typical elastic problems are investigated by the developed PDDO method. A very high level of computational accuracy is achieved by comparing with the analytical solutions and the existing numerical results.

Numerical simulation of 3D fracture propagation problem with reproducing kernel peridynamic method

The non-ordinary state-based peridynamic (NOSB-PD) method has significant advantages in solving crack propagation problems due to its non-local characteristics, but it also faces problems such as numerical oscillations and boundary effects. In order to solve the above defects, the paper first discusses the NOSB-PD method within the Galerkin framework. After that, the peridynamic differential operator (PDDO) approximation was introduced by extending the calculation of the deformation gradient tensor F in NOSB-PD. Since the zero-order approximations in the PDDO approximation and the reproducing kernel (RK) approximation are identical, the distinctions between the two approximations have been thoroughly analysed. Although the two approximations show the same consistency, only the RK approximation satisfies the integrable condition. Therefore, a RK-PD coupling method is proposed in the paper, and an implicit iterative algorithm suited to the RK-PD method is presented as well. At last, several numerical examples demonstrate that the proposed method does not suffer from numerical oscillations and can be used to solve the three-dimensional crack propagation problems.

Peridynamic modeling of void nucleation and growth in metal lines due to electromigration in a finite element framework

This study presents a peridynamic model to predict void formation and growth in metal lines due to electromigration. The nonlocal form of the nonlinear governing equations and boundary conditions for electromigration are derived by using the peridynamic differential operator (PDDO). The equation of motion includes the effect of diffusion strain due to electromigration. Coupled governing equations of atomic concentration and deformation are solved in the Ansys framework with native elements through implicit algorithms. The numerical results show that the current approach can accurately predict the distribution of the deformation, atomic concentration, and hydrostatic stress. Also, the void growth predictions agree well with the experimental predictions.

A Numerical Study of the Peridynamic Differential Operator Discretization of Incompressible Navier–Stokes Problems

We study the incompressible Navier–Stokes equations using the Projection Method. The applications of interest are the classical channel flow problems such as Couette, shear, and Poiseuille. In addition, we consider the Taylor-Green vortex and lid-driven cavity applications. For discretization, we use the Peridynamic Differential Operator (PDDO). The main emphasis of the paper is the performance of the PDDO as a discretization method under these flow problems. We present a careful numerical study with quantifications and report convergence tables with convergence rates. We also study the approximation properties of the PDDO and prove that the N-th order PDDO approximates polynomials of degree at most N exactly. As a result, we prove that the PDDO discretization guarantees the zero row sum property of the arising system matrix.

Quantifying nonlocal bedload transport: A regional-based nonlocal model for bedload transport from local to global scales

Recent studies have emphasized the importance of nonlocal models in characterizing bedload transport in natural rivers, particularly in mixed-size gravel beds or steep hillslopes. Nonlocality denotes that a quantity (flux) at a specific location x is dependent on the conditions in the surrounding area, as opposed to solely at the location itself. This concept applies in bedload transport even in planar flumes, where particles are entrained at an upstream position and travel a finite distance, ultimately contributing nonlocally to the sediment flux. However, existing bedload transport models, such as the advection–diffusion equation (ADE) or the fractional derivative equation (FDE) models, are inadequate in characterizing the nonlocal transport behavior of bedload at a regional scale. Large errors may arise from the lack of an accurate description of the nonlocal bedload transport processes at regional scales. This study proposes a regional-based nonlocal bedload transport model, which is conceptualized from the probabilistic Exner-based equations and the peridynamic (PD) differential operator. The PD model encapsulates the nonlocal motion of bedload sediments on the basis of the PD differential operator, by utilizing a pre-defined weight function and influence domain. Comparisons demonstrate that the PD model serves as a generalized tool connecting the local and the global models with different PD functions and influence domains. Its variability on kernel function and influence domain, enable it conveniently describe sub-, super-, and normal diffusion behaviors of bedload transport.

Peridynamic operator method

We propose a so-called Peridynamic operator method (PDOM) which can convert the local differentiation or its products into the nonlocal integral form. We will show that the peridynamic differential operator (PDDO) or nonlocal operator method (NOM) is a special case of this proposed PDOM. PDOM is derived for elasticity problems in two different ways: a) based on variational principles and b) Lagrange’s equations. For spherical integral domains, PDOM recovers all types of Peridynamics (PD) and Dual-horizon Peridynamics (DH-PD) formulations. We also present a simple and efficient approach without zero-energy modes and develop PDOM for scattered data, ordinary and partial differential equations. One key feature of this approach is that it does not require any special treatment in the presence of discontinuities or singularities. Several numerical examples are presented to verify the capability of the proposed method.

Investigating time-dependent behavior of rocks using kinematic-constraint-inspired non-ordinary state-based peridynamics

In this paper, the kinematic-constraint-inspired non-ordinary state-based peridynamic formulation (KC-NOSBPD) is proposed to investigate the time-dependent behaviors of rocks, which play a critical role in evaluating the long-term stability of the surrounding rocks around underground opening quantitatively. The peridynamic differential operator (PDDO) is introduced to KC-NOSBPD to improve the precision of bond-associated deformation gradient and to suppress the skin effect. On this basis, a unified creep-damage-rupture KC-NOSBPD model is proposed, in which the Burgers-creep viscoplastic constitutive law, a visco-elastoplastic constitutive model, is used to describe the rheological deformation of rocks. A unified damage-rupture framework based on a strain-based isotropic scalar damage variable is established to simulate the creep damage and subsequent creep rupture. The fidelity of this proposed model without damage is verified by comparing it with the benchmark solution for the creep response of a 2D rock specimen subjected to various load histories. Further, the proposed model is employed to simulate uniaxial and triaxial compression creep tests. These present numerical results agree well with the experimental observation, implying that the proposed model successfully captures the accelerating creep behavior of rocks. Moreover, the proposed model performs the time-dependent deformation and failure analysis for the surrounding rocks around the tunnel.

Discussion on the form of construction function in the peridynamic differential operator based on relative function

The peridynamic differential operator (PDDO) adopts nonlocal idea, realizes the solution of any order derivative of a function through the integration of a constructed PD function, and expands the motion equation based on function derivative into the nonlocal peridynamic integral equation. Even at the discontinuity, the peridynamic integral equation still has a definition and the form remains unchanged. When calculating any order derivative of a function f(x), two PDDO formulas can be used, one is the PDDO formula based on proximal function f(x + ξ) and the other is PDDO formula based on relative function f(x + ξ) − f(x). The key to using these two formulas is to construct the PD function. This paper discusses the construction method of the PD function in PDDO formula based on relative function. It is found that it can be constructed by supplementing an equation as well as by reducing a construction term of PD polynomial. After derivation, it is found that it is necessary to reduce a construction term that is even power to all variables, and it is invalid to reduce a construction term with an odd power to a certain variable. Numerical examples show that the two construction methods can lead to high-precision calculation results.

A unified method to simulate electrodeposition and galvanic corrosion using the peridynamic differential operator

This study presents a unified method to model electrodeposition and galvanic corrosion. The governing equations and boundary conditions are recast in their nonlocal form by using the peridynamic differential operator. Electric potential and mass transport equations are solved simultaneously in ANSYS through implicit algorithms for computational efficiency and complex geometry. Deposition and corrosion fronts are established by applying the concept of deposition fraction and examining the concentration values. Element matrices associated with peridynamic interactions are constructed through MATRIX27 elements native to ANSYS. The damping is modeled using combin14 element and the resulting equations are solved in an implicit fashion. Accuracy of this unified method is established by comparison with the published experimental measurements and simulations concerning simple and irregular geometries.

An unsupervised latent/output physics-informed convolutional-LSTM network for solving partial differential equations using peridynamic differential operator

This study presents a novel unsupervised convolutional Neural Network (NN) architecture with nonlocal interactions for solving Partial Differential Equations (PDEs). The nonlocal Peridynamic Differential Operator (PDDO) is employed as a convolutional filter for evaluating derivatives the field variable. The NN captures the time-dynamics in smaller latent space through encoder–decoder layers with a Convolutional Long–short Term Memory (ConvLSTM) layer between them. The ConvLSTM architecture is modified by employing a novel activation function to improve the predictive capability of the learning architecture for physics with periodic behavior. The physics is invoked in the form of governing equations at the output of the NN and in the latent (reduced) space. By considering a few benchmark PDEs, we demonstrate the training performance and extrapolation capability of this novel NN architecture by comparing against Physics Informed Neural Networks (PINN) type solvers. It is more capable of extrapolating the solution for future timesteps than the other existing architectures.

Study on two-dimensional mixed-mode fatigue crack growth employing ordinary state-based peridynamics

Mixed-mode fatigue crack propagation analysis is studied using ordinary state-based peridynamics. By introducing the concept of “remaining life”, the fatigue crack nucleation and propagation can be simulated in the PD framework. The PD fatigue modelling and the crack path prediction is carefully investigated. For the comparison purpose, the maximum circumferential stress criterion is introduced. The interaction integral and the peridynamic differential operator is employed to evaluate the fracture mechanics parameters. Two pre-cracked structures under mixed-mode loading conditions are provided with the validations from reference solutions.

Weak form of bond-associated peridynamic differential operator for thermo-mechanical analysis of orthotropic structures

In this study, a weak form of bond-associated peridynamic differential operator (WBA-PDDO) is presented for thermo-mechanical analysis of orthotropic structures. The proposed method inherits the advantage of the original peridynamic differential operator and eliminates the zero-energy mode by introducing the concept of bond-associated family. The WBA-PDDO for thermo-mechanical equations of orthotropic structures is obtained by using the variational principle. The developed model is applied to heat transfer analysis of orthotropic square plate, thermo-mechanical analysis of orthotropic square plate with a hole and orthotropic connecting rod. By comparing with the exact solutions and finite element results, the convergence and accuracy of the proposed method are demonstrated in uniform and non-uniform discretization. The presented model provides an alternative approach for thermo-mechanical analysis of orthotropic structures.