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The residence time distribution (RTD) has an important impact in the performance of chemical reacting systems. The segregated fluid assumption is commonly used to assess the analysis and design of chemical reactors. Models derived from the segregated fluid assumption for continuously stirred tank reactors (CSTR) rely on batch reactor models. A drawback of such assumption is that the CSTR model structure is not recovered in the limit when the RTD becomes very narrow (e.g., a Dirac delta). The aim of the present work is to propose a nonlocal modeling approach that is consistent with the CSTR model structure. The main idea is to represent the reactor as a continuum of tiny differential CSTRs determined by the RTD. The resulting nonlocal model is expressed as integrodifferential equations representing the interaction of reactive molecules of a given residence time with molecules of all residence times. By doing this, the nonlocal model is reduced to the usual CSTR model in the limit when the RTD is very narrow. The methodology was illustrated with three worked cases of dimerization, autocatalytic and series reaction schemes, and the results were compared with that obtained with the segregated fluid assumption. It was found that nonlocal models predict lower conversions than the segregated fluid models. [ABSTRACT FROM AUTHOR]

Several regularized formulations exist in the literature, such as nonlocal constitutive models, for the objective simulation of localized deformations in quasibrittle materials that prevent the well-known pathological dependence on the employed mesh. Although they usually require a significant discretization to represent the localization zone of a given material, characterized by some length scale linked to its microstructure, scaling techniques exist to derive representative parameters for full-scale simulation of geotechnical problems. However, the softening response observed in conventional triaxial or uniaxial compression tests is not solely a constitutive feature but also the result of a complex boundary value problem (BVP), where the specific localization pattern formed affects the global response of the experiment. Therefore, constitutive softening parameters should be backcalculated through the simulation of a given laboratory test as a BVP. However, as demonstrated in this paper, the cylindrical geometry of the samples in conventional tests hinders the onset of localization, resulting in a stiffer response that underestimates the resulting nominal softening rate. Within this context, the paper aimed to demonstrate the complexity and difficulty in the determination of a softening rate for nonlocal constitutive models from data of conventional triaxial tests performed on cylindrical specimens and to provide relevant insights toward a practical procedure for the determination of parameters in regularized simulations of quasibrittle geomaterials for practical applications. [ABSTRACT FROM AUTHOR]

A nonlocal model for dynamic brittle damage is introduced consisting of two phases, one elastic and the other inelastic. Evolution from the elastic to the inelastic phase depends on material strength. Existence and uniqueness of the displacement-failure set pair follow from an initial value problem describing the evolution. The displacement-failure pair satisfies energy balance. The length of nonlocality ϵ is taken to be small relative to the domain in R d , d = 2 , 3 . The strain is formulated as a difference quotient of the displacement in the nonlocal model. The two point force is expressed in terms of a weighted difference quotient and delivers an evolution on a subset of R d × R d . This evolution provides an energy balance between external energy, elastic energy, and damage energy including fracture energy. For any prescribed loading the deformation energy resulting in material failure over a region R is uniformly bounded as ϵ → 0 . For fixed ϵ , the failure energy is discovered to be is nonzero for d − 1 dimensional regions R associated with flat crack surfaces. Calculation shows, this failure energy is the Griffith fracture energy given by the energy release rate multiplied by area for d = 3 (or length for d = 2 ). The nonlocal field theory is shown to recover a solution of Naiver’s equation outside a propagating flat traction free crack in the limit of vanishing spatial nonlocality. The theory and simulations presented here corroborate the recent experimental findings of (Rozen-Levy et al. in Phys. Rev. Lett. 125(17):175501, 2020) that cracks follow the location of maximum energy dissipation inside the intact material. Simulations show fracture evolution through the generation of a traction free internal boundary seen as a wake left behind a moving strain concentration. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to propose a new boundary element method (BEM) modeling for stress sensitivity of nonlocal thermo-elasto-plastic damage problems. The numerical solution of the heat conduction equation subjected to a non-local condition is described using a boundary element model. The total amount of heat energy contained inside the solid under consideration is specified by the non-local condition. The procedure of solving the heat equation will reveal an unknown control function that governs the temperature on a specific region of the solid's boundary. The initial stress BEM for structures with strain-softening damage is employed in a boundary element program with iterations in each load increment to develop a plasticity model with yield limit deterioration. To avoid the difficulties associated with the numerical calculation of singular integrals, the regularization technique is applicable to integral operators. To validate the physical correctness and efficiency of the suggested formulation, a numerical case is solved. [ABSTRACT FROM AUTHOR]

This study aims to review and highlight the important modelling methodologies used for studying carbon nanotubes (CNTs) and their composites. Understanding appropriate modelling methods for specific applications is crucial as CNTs become integral in achieving lighter, multifunctional composite structures. This paper explores a range of techniques, including finite element modelling (FEM), Molecular Dynamics (MD), Molecular Structural Mechanics (MSM), as well as nonlocal models, and the Cauchy-Born (CB) rule. Emphasis is placed on factors such as interphase effects between CNTs and the matrix, bonding interactions, non-bonded van der Waals (vdW) forces, and dynamic behaviour. Multiscale modelling is extensively discussed as a pivotal approach for efficiently addressing various length scales in nanocomposites. Modelling of failure, damage and its propagation, delamination, and instabilities such as buckling and fracture have been highlighted, and research gaps have been pointed out. [ABSTRACT FROM AUTHOR]

Nonlocal perfectly matched layers (nonlocal PMLs) for nonlocal wave equations and the corresponding numerical discretization to solve the reduced PML problems on bounded domains are studied. The nonlocal PMLs are derived by combining the Laplace transform and the analytical continuation into the complex plane. The discrete scheme defined on the complex space may lead to a spurious imaginary part of the numerical solution and has a large computational cost. Here we design some new nonlocal PMLs and numerical schemes and prove that they are on the real space, which is much efficient comparing with the complex space. The accuracy and effectiveness of our approaches are illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

This paper proposes a method to describe the bi-modular behavior of quasi-brittle materials. The approach is based on using multiple loading functions that are associated with the stress state of the material point. The model also includes degradation laws that control the evolution of the internal variables under tension and compression, which allows the representation of the material response with fidelity. The following approach outlines an objective methodology for improving classical models and provides a theoretical framework for developing new constitutive models. This work also presents an updated version of the Mazars damage model, with improved material parametrization through multiple damage laws. Similarly, the modified von Mises model has been extended to include different functions for damage evolution, combining exponential and continuum polynomial laws for tension and compression. A third model was developed to showcase the full potential of the proposed formulation. This model formulates the Lemaitre and Chaboche criterion, typically used for ductile materials, to describe quasi-brittle materials. The model provides a comprehensive characterization of the tension and compression behavior by utilizing various loading functions and multiple damage laws. Numerical simulations of experimental benchmarks are presented to show the performance of the proposal, including a nonlocal approach to avoid strain localization and mesh dependency problems. • A formulation to represent simply the bi-modularity of quasi-brittle materials by isotropic damage models is presented. • The proposal provides versatility to prescribe any damage law, especially with fill constitutive parameters. • The strategy is applied to classical models present in literature providing an enhancement of the model behavior. • The approach can be extended and applied to other models presented in the literature. [ABSTRACT FROM AUTHOR]

Nonlocal models allow for the description of phenomena which cannot be captured by classical partial differential equations. The availability of efficient solvers is one of the main concerns for the use of nonlocal models in real world engineering applications. We present a domain decomposition solver that is inspired by substructuring methods for classical local equations. In numerical experiments involving finite element discretizations of scalar and vectorial nonlocal equations of integrable and fractional type, we observe improvements in solution time of up to 14.6x compared to commonly used solver strategies. [ABSTRACT FROM AUTHOR]

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to propose a new boundary element method (BEM) modeling for stress sensitivity of nonlocal thermo-elasto-plastic damage problems. The numerical solution of the heat conduction equation subjected to a non-local condition is described using a boundary element model. The total amount of heat energy contained inside the solid under consideration is specified by the non-local condition. The procedure of solving the heat equation will reveal an unknown control function that governs the temperature on a specific region of the solid’s boundary. The initial stress BEM for structures with strain-softening damage is employed in a boundary element program with iterations in each load increment to develop a plasticity model with yield limit deterioration. To avoid the difficulties associated with the numerical calculation of singular integrals, the regularization technique is applicable to integral operators. To validate the physical correctness and efficiency of the suggested formulation, a numerical case is solved.

Using a nonlocal second-order traffic flow model we present an approach to control the dynamics toward a steady state. The system is controlled by the leading vehicle driving at a prescribed velocity and also determines the steady state. Thereby, we consider both the microscopic (trajectory based) and macroscopic (density based) scales. We show that the fixed point of the microscopic traffic flow model is (locally) asymptotically stable for any kernel function. To obtain global stabilization, we present Lyapunov functions for both the microscopic and the macroscopic scale and compute the explicit rates at which the vehicles influenced by the nonlocality tend toward the stationary solution. We obtain stabilization results for a constant kernel function and arbitrary initial data or concave kernels and monotone initial data. In particular, the stabilization is exponential in time. Numerical examples demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the study of the mathematical analysis for an optimal control of a nonlocal degenerate aggregation model. This model describes the aggregation of organisms such as pedestrian movements, chemotaxis, animal swarming. We establish the wellposedness (existence and uniqueness) for the weak solution of the direct problem by means of an auxiliary nondegenerate aggregation equation, the Faedo-Galerkin method (for the existence result) and duality method (for the uniqueness). Moreover, for the adjoint problem, we prove the existence result of minimizers and first-order necessary conditions. The main novelty of this work concerns the presence of a control to our nonlocal degenerate aggregation model. Our results are complemented with some numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, an efficient Jacobi spectral collocation method is developed for multi-dimensional weakly singular volume-constrained nonlocal models including both nonlocal diffusion (ND) models and peridynamic (PD) models. The model equation contains a weakly singular integral operator with the singularity located at the center of the integral domain, and the numerical approximation of it becomes an essential difficulty in solving nonlocal models. To approximate such singular nonlocal integrals in an accurate way, a novel nonlocal quadrature rule is constructed to accurately compute these integrals for the numerical solutions produced by spectral methods. Numerical experiments are given to show that spectral accuracy can be obtained by using the proposed Jacobi spectral collocation methods for several different nonlocal models. Besides, we numerically verify that the numerical solution of our Jacobi spectral method can converge to its correct local limit as the nonlocal interactions vanish. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the study of the mathematical analysis for an optimal control of a nonlocal degenerate aggregation model. This model describes the aggregation of organisms such as pedestrian movements, chemotaxis, animal swarming. We establish the wellposedness (existence and uniqueness) for the weak solution of the direct problem by means of an auxiliary nondegenerate aggregation equation, the Faedo–Galerkin method (for the existence result) and duality method (for the uniqueness). Moreover, for the adjoint problem, we prove the existence result of minimizers and first-order necessary conditions. The main novelty of this work concerns the presence of a control to our nonlocal degenerate aggregation model. Our results are complemented with some numerical simulations.

A mathematical framework is provided for a substructuring‐based domain decomposition (DD) approach for nonlocal problems that features interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation (PDE) problems is used in which a computational domain is subdivided into non‐overlapping subdomains. In the nonlocal setting, this approach is substructuring‐based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains. Key results include the equivalence between the global, single‐domain nonlocal problem and its multi‐domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal DD methods. [ABSTRACT FROM AUTHOR]

In this paper, we design efficient quadrature rules for finite element (FE) discretizations of nonlocal diffusion problems with compactly supported kernel functions. Two of the main challenges in nonlocal modeling and simulations are the prohibitive computational cost and the nontrivial implementation of discretization schemes, especially in three‐dimensional settings. In this work, we circumvent both challenges by introducing a parametrized mollifying function that improves the regularity of the integrand, utilizing an adaptive integration technique, and exploiting parallelization. We first show that the "mollified" solution converges to the exact one as the mollifying parameter vanishes, then we illustrate the consistency and accuracy of the proposed method on several two‐ and three‐dimensional test cases. Furthermore, we demonstrate the good scaling properties of the parallel implementation of the adaptive algorithm and we compare the proposed method with recently developed techniques for efficient FE assembly. [ABSTRACT FROM AUTHOR]

In the framework of elastodynamics, peridynamics is a nonlocal theory able to capture singularities without using partial derivatives. The governing equation is a second order in time partial integro‐differential equation. In this article, we focus on a one‐dimensional nonlinear model of peridynamics and propose a spectral method based on the Chebyshev polynomials to discretize in space. The main capability of the method is that it avoids the assumption of periodic boundary condition in the solution and can benefit of the use of the fast Fourier transform. We show its convergence and find that the method results to be very efficient in terms of accuracy and execution time with respect to spectral methods based on the Fourier trigonometric polynomials associated to a volume penalization technique. [ABSTRACT FROM AUTHOR]

This paper is concerned with the implementations of the meshfree-based reduced-order model (ROM) to time-dependent nonlocal models with inhomogeneous volume constraints. Generally, when using ROM for nonlocal models, the projection of nonlocal volume constraints needs to be computed in every time step to handle the nonlocal boundary conditions. Up to now, only finite element methods (FEM) can work well in constructing ROM for nonlocal models, since the interpolation property of the FEM basis functions makes it easy to obtain such a projection. But if one tries to develop ROM based on existing meshfree methods for nonlocal models, the projection in every time step will lead to a full-order discrete system and is highly time-consuming, since the basis functions of these methods do not meet interpolation property. To overcome the above difficulties, we introduce a mixed reproducing kernel (RK) approximation with nodal interpolation property to develop a meshfree collocation method for nonlocal models and use it to construct ROM. Thanks to the nodal interpolation property, the projection of nonlocal boundary conditions can be obtained explicitly. This ROM is developed using numerical results as snapshots by a full-order model in a small time interval [ 0 , t 1 ]. The surrogate model, which is constructed by POD (proper orthogonal decomposition)-Galerkin approach, leads to a discrete system with far fewer degrees of freedom than the original meshfree method. Numerical experiments for nonlocal problems including nonlocal diffusion and peridynamics are presented to show that our method meets almost the same accuracy with a very small computational cost compared with the full-order meshfree approach. [ABSTRACT FROM AUTHOR]

This study is an attempt towards a better understanding of the length scale effects on the bending response of the granular beams. To this aim, a unidimensional discrete granular chain composed of a finite number of rigid grains is studied. It is assumed that shear and rotational interactions exist at the rigid grain interfaces. This granular model can be classified also as a discrete Cosserat chain with two independent degrees of freedom (DOF) for each grain (the deflection and the rotation). Subsequently, such a discrete model permits to introduce the size effect (grain dimension) in the bending formulation of a microstructured granular beam. It is shown that the bending deformation solutions of this chain asymptotically converge towards the continuum beam model of Bresse–Timoshenko (neglecting the length scale). The exact solutions of this granular model subjected to a uniform distributed loading, are investigated for various boundary conditions which are defined at the grain level. Accordingly, a twin numerical problem is studied to compare the exact analytical results with the numerical ones simulated by discrete element method (DEM). Eventually, through the continualization of the coupled difference equations system governing the discrete beam, a nonlocal elasticity Cosserat continuum model is obtained. The process of continualization consists in approaching the difference equations by differential equations applied either by the polynomial or the rational development in which a length scale appears. It is shown that both the granular model and the nonlocal beam model give very close and eventually coincident results. [ABSTRACT FROM AUTHOR]

A fast algorithm based on reduced-order model (ROM) is proposed for unsteady nonlocal diffusion models. It combines proper orthogonal decomposition (POD) approach and collocation method with local radial basis functions (RBFs), which makes it possible for using ROM to solve nonlocal models. Several numerical experiments showed that this approach significantly reduce the computational cost of nonlocal models while keep the similar convergent behavior compared with the RBF collocation methods. [ABSTRACT FROM AUTHOR]

Defects are inevitable during the manufacturing processes of materials. Presence of these defects and their dynamics significantly influence the responses of materials. A thorough understanding of dislocation dynamics of different types of materials under various conditions is essential for analysing the performance of the materials. Ductility of a material is directly related with the movement and rearrangement of dislocations under applied load. In this work, we look into the dynamics of dislocations in ductile and brittle materials using simplified two dimensional discrete dislocation dynamics (2D-DDD) simulation. We consider Aluminium (Al) and Tungsten (W) as representative examples of ductile and brittle materials respectively. We study the velocity distribution, strain field, dislocation count, and junction formation during interactions of the dislocations within the domain. Furthermore, we study the probability densities of dislocation motion for both materials. In mesoscale, moving dislocations can be considered as particle diffusion, which are often stochastic and super-diffusive. Classical diffusion models fail to account for these phenomena and the long-range interactions of dislocations. Therefore, we propose the nonlocal transport model for the probability density and obtained the parameters of nonlocal operators using a machine learning framework. • We study the simulation-based statistics of dislocation dynamics in ductile/brittle materials. • We investigate the dislocations' PDFs and their behavior subject to different loading conditions. • Using machine learning, we construct nonlocal models for the collective behavior of dislocations. [ABSTRACT FROM AUTHOR]

This study is devoted to the construction of a multifidelity Monte Carlo (MFMC) method for the uncertainty quantification of a nonlocal, non-mass-conserving Cahn-Hilliard model for phase transitions with an obstacle potential. We are interested in estimating the expected value of an output of interest (OoI) that depends on the solution of the nonlocal Cahn-Hilliard model. As opposed to its local counterpart, the nonlocal model captures sharp interfaces without the need for significant mesh refinement. However, the computational cost of the nonlocal Cahn-Hilliard model is higher than that of its local counterpart with similar mesh refinement, inhibiting its use for outer-loop applications such as uncertainty quantification. The MFMC method augments the desired high-fidelity, high-cost OoI with a set of lower-fidelity, lower-cost OoIs to alleviate the computational burden associated with nonlocality. Most of the computational budget is allocated to sampling the cheap surrogate models to achieve speedup, whereas the high-fidelity model is sparsely sampled to maintain accuracy. For the non-mass-conserving nonlocal Cahn-Hilliard model, the use of the MFMC method results in, for a given computational budget, about an order of magnitude reduction in the mean-squared error of the expected value of the OoI relative to that of the Monte Carlo method. • The generalized nonlocal Cahn Hilliard model simulates the phase change process. • Nonlocal models are computationally more expensive than their local counterparts. • The MFMC method lowers the computational cost of UQ on nonlocal models. [ABSTRACT FROM AUTHOR]

Peridynamics is a nonlocal theory for dynamic fracture analysis consisting in a second order in time partial integro-differential equation. In this paper, we consider a nonlinear model of peridynamics in a two-dimensional spatial domain. We implement a spectral method for the space discretization based on the Fourier expansion of the solution while we consider the Newmark- β method for the time marching. This computational approach takes advantages from the convolutional form of the peridynamic operator and from the use of the discrete Fourier transform. We show a convergence result for the fully discrete approximation and study the stability of the method applied to the linear peridynamic model. Finally, we perform several numerical tests and comparisons to validate our results and provide simulations implementing a volume penalization technique to avoid the limitation of periodic boundary conditions due to the spectral approach. [ABSTRACT FROM AUTHOR]

Abstract We develop a fast collocation method for a static bond-based peridynamic model. Based on the analysis of the structure of the stiffness matrix, a fast matrix-vector multiplication technique was found, which can be used in the Krylov subspace iteration method. In this paper, we also present an effective preconditioner to accelerate the convergence of the Krylov subspace iteration method. Using the block-Toeplitz–Toeplitz-block (BTTB)-type structure of the stiffness matrix, we give a block-circulant-circulant-block (BCCB)-type preconditioner. The numerical experiments show the utility of the preconditioned fast collocation method.

The implementation of finite element methods (FEMs) for nonlocal models with a finite range of interaction poses challenges not faced in the partial differential equations (PDEs) setting. For example, one has to deal with weak forms involving double integrals which lead to discrete systems having higher assembly and solving costs due to possibly much lower sparsity compared to that of FEMs for PDEs. In addition, one may encounter nonsmooth integrands. In many nonlocal models, nonlocal interactions are limited to bounded neighborhoods that are ubiquitously chosen to be Euclidean balls, resulting in the challenge of dealing with intersections of such balls with the finite elements. We focus on developing recipes for the efficient assembly of FEM stiffness matrices and on the choice of quadrature rules for the double integrals that contribute to the assembly efficiency and also posses sufficient accuracy. A major feature of our recipes is the use of approximate balls, e.g. several polygonal approximations of Euclidean balls, that, among other advantages, mitigate the challenge of dealing with ball-element intersections. We provide numerical illustrations of the relative accuracy and efficiency of the several approaches we develop. [ABSTRACT FROM AUTHOR]

In this paper, we consider deterministic, continuous, nonlocal models for the mitochondrial permeability transition, i.e. mitochondrial swelling. Based on seminal papers [1], [2], [3], [5] and the book [4], in which ODE-PDE and PDE-PDE local models for the swelling of mitochondria have been considered, we suggest here new nonlocal models for this process. This new nonlocal deterministic continuous model for mitochondrial swelling scenario contains nonlocal diffusion, nonlocal chemotaxis, as well as nonlocal source term. We would like to especially emphasize that some of the new nonlocal models that we consider in this paper do not have local counterparts in the literature. [ABSTRACT FROM AUTHOR]

Neural operators, which can act as implicit solution operators of hidden governing equations, have recently become popular tools for learning the responses of complex real-world physical systems. Nevertheless, most neural operator applications have thus far been data-driven and neglect the intrinsic preservation of fundamental physical laws in data. In this work, we introduce a novel integral neural operator architecture called the Peridynamic Neural Operator (PNO) that learns a nonlocal constitutive law from data. This neural operator provides a forward model in the form of state-based peridynamics, with objectivity and momentum balance laws automatically guaranteed. As applications, we demonstrate the expressivity and efficacy of our model in learning complex material behaviors from both synthetic and experimental data sets. We also compare the performances with baseline models that use predefined constitutive laws. We show that, owing to its ability to capture complex responses, our learned neural operator achieves improved accuracy and efficiency. Moreover, by preserving the essential physical laws within the neural network architecture, the PNO is robust in treating noisy data. The method shows generalizability to different domain configurations, external loadings, and discretizations. • We proposed PNO, which learns a nonlocal constitutive law from spatial measurements. • It captures complex material responses without prior expert-constructed knowledge. • Meanwhile, the model guarantees the physically required balance laws and objectivity. • Learnt model is generalizable to various resolutions, loading, and domain settings. [ABSTRACT FROM AUTHOR]

This paper presents the steady-state heat conduction analysis in plates with insulated cracks using peridynamic differential operator (PDDO). The PDDO converts the local differentiation to nonlocal integration. Since the PDDO permits differentiation through integration, the equilibrium equations remain valid in the presence of discontinuities such as cracks. The governing equations of the steady state heat equation and boundary conditions were solved by employing the PDDO. The robustness of the PDDO was assessed by considering a plate without cracks under different boundary conditions. The influence of the insulated cracks on the temperature and heat flux distributions was investigated. It was observed that heat flux concentrations developed in the vicinity of the crack tips.

In this paper, we study an optimal control problem for nonlocal Cahn-Hilliard-Brinkman system, which models phase separation of binary fluids in porous media. The system evolves with regular potential in a two-dimensional bounded domain. We extend the existence of weak solution results for the system to prove the existence of strong solution under extra assumptions on the forcing term and initial datum. Further, using our regularity results, we study the tracking type optimal control problem. We prove the existence of optimal control and establish the first-order optimality condition. Lastly, we characterise optimal control in terms of the solution of the corresponding adjoint system. The existence of the solution for the adjoint system is also established. [ABSTRACT FROM AUTHOR]

The peridynamic equation consists in an integro‐differential equation of the second order in time which has been proposed for modeling fractures and damages in the context of nonlocal continuum mechanics. In this article, we study numerical methods for the one‐dimension nonlinear peridynamic problems. In particular we consider spectral Fourier techniques for the spatial domain while we will use the Störmer–Verlet method for the time discretization. In order to overcome the limitation of working on periodic domains due to the spectral techniques we will employ a volume penalization method. The performance of our approach is validated with the study of the convergence with respect to the spatial discretization and the volume penalization. Several tests have been performed to investigate the properties of the solutions. [ABSTRACT FROM AUTHOR]

Modelling pollutant transport in water is one of the core tasks of computational hydrology, and various physical models including especially the widely used nonlocal transport models have been developed and applied in the last three decades. No studies, however, have been conducted to systematically assess the applicability, limitations and improvement of these nonlocal transport models. To fill this knowledge gap, this study reviewed, tested and improved the state‐of‐the‐art nonlocal transport models, including their physical background, mathematical formula and especially the capability to quantify conservative tracers moving in one‐dimensional sand columns, which represents perhaps the simplest real‐world application. Applications showed that, surprisingly, neither the popular time‐nonlocal transport models (including the multi‐rate mass transfer model, the continuous time random walk framework and the time fractional advection‐dispersion equation), nor the spatiotemporally nonlocal transport model (ST‐fADE) can accurately fit passive tracers moving through a 15‐m‐long heterogeneous sand column documented in literature, if a constant dispersion coefficient or dispersivity is used. This is because pollutant transport in heterogeneous media can be scale‐dependent (represented by a dispersion coefficient or dispersivity increasing with spatiotemporal scales), non‐Fickian (where plume variance increases nonlinearly in time) and/or pre‐asymptotic (with transition between non‐Fickian and Fickian transport). These different properties cannot be simultaneously and accurately modelled by any of the transport models reviewed by this study. To bypass this limitation, five possible corrections were proposed, and two of them were tested successfully, including a time fractional and space Hausdorff fractal model which minimizes the scale‐dependency of the dispersion coefficient in the non‐Euclidean space, and a two‐region time fractional advection‐dispersion equation which accounts for the spatial mixing of solute particles from different mobile domains. Therefore, more efforts are still needed to accurately model transport in non‐ideal porous media, and the five model corrections proposed by this study may shed light on these indispensable modelling efforts. [ABSTRACT FROM AUTHOR]

Nonlocal models provide exceptional simulation fidelity for a broad spectrum of scientific and engineering applications. However, wider deployment of nonlocal models is hindered by several modeling and numerical challenges. Among those, we focus on the nontrivial prescription of nonlocal boundary conditions, or volume constraints, that must be provided on a layer surrounding the domain where the nonlocal equations are posed. The challenge arises from the fact that, in general, data are provided on surfaces (as opposed to volumes) in the form of force or pressure data. In this paper we introduce an efficient, flexible, and physically consistent technique for an automatic conversion of surface (local) data into volumetric data that does not have any constraints on the geometry of the domain or on the regularity of the nonlocal solution and that is not tied to any discretization. We show that our formulation is well-posed and that the limit of the nonlocal solution, as the nonlocality vanishes, is the local solution corresponding to the available surface data. Quadratic convergence rates are proved for the strong energy and $L^2$ convergence. We illustrate the theory with one-dimensional numerical tests whose results provide the groundwork for realistic simulations. [ABSTRACT FROM AUTHOR]

We develop a fast collocation method for a static bond-based peridynamic model. Based on the analysis of the structure of the stiffness matrix, a fast matrix-vector multiplication technique was found, which can be used in the Krylov subspace iteration method. In this paper, we also present an effective preconditioner to accelerate the convergence of the Krylov subspace iteration method. Using the block-Toeplitz–Toeplitz-block (BTTB)-type structure of the stiffness matrix, we give a block-circulant-circulant-block (BCCB)-type preconditioner. The numerical experiments show the utility of the preconditioned fast collocation method. [ABSTRACT FROM AUTHOR]

In 2006 Armstrong, Painter, and Sherratt formulated a nonlocal differential equation model for cell-cell adhesion. For the one-dimensional case on a bounded domain we derive various types of biological boundary conditions, describing adhesive, repulsive, and neutral boundaries. We prove local and global existence and uniqueness for the resulting integrodifferential equations. In numerical simulations we consider adhesive, repulsive, and neutral boundary conditions, and we show that the solutions mimic known behavior of uid adhesion to boundaries. In addition, we observe interior pattern formation due to cell-cell adhesion. [ABSTRACT FROM AUTHOR]

This paper systematically evaluates (in the one-dimensional setting) the performance of a new type of integral nonlocal averaging scheme, initially motivated by the idea of internal time that reflects the reduction of the elastic wave speed in a damaged material. The formulation dealing with internal time is replaced by the equivalent concept of a modified spatial metric leading to a damage-dependent interaction distance. This modification has a favorable effect on the evolution of the active part of damage zone and leads to its gradual shrinking, which naturally describes the transition from a thin process zone to a fully localized crack. However, when a pure damage model (with no permanent strain) is considered, the resulting load-displacement diagrams exhibit dramatic snapbacks and excessively brittle behavior in the final stages of failure. The concept of damage-dependent interaction distances is therefore extended to damage-plastic models and damage models with inelastic (permanent) strain. It is shown that, for formulations that consider a part of the strain as irreversible, the overall stress-displacement response becomes realistic for quasi-brittle materials such as concrete, for which the diagram typically exhibits a long tail. [ABSTRACT FROM AUTHOR]

We propose a new numerical technique to deal with nonlinear terms in gradient ows. By introducing a scalar auxiliary variable (SAV), we construct effcient and robust energy stable schemes for a large class of gradient ows. The SAV approach is not restricted to specific forms of the nonlinear part of the free energy and only requires solving decoupled linear equations with constant coeffcients. We use this technique to deal with several challenging applications which cannot be easily handled by existing approaches, and we present convincing numerical results to show that our schemes are not only much more effcient and easy to implement, but can also better capture the physical properties in these models. Based on this SAV approach, we can construct unconditionally second-order energy stable schemes, and we can easily construct even third- or fourth-order BDF schemes which, although not unconditionally stable, are very robust in practice. In particular, when coupled with an adaptive time stepping strategy, the SAV approach can be extremely effcient and accurate. [ABSTRACT FROM AUTHOR]

This paper presents the steady-state heat conduction analysis in plates with insulated cracks using peridynamic differential operator (PDDO). The PDDO converts the local differentiation to nonlocal integration. Since the PDDO permits differentiation through integration, the equilibrium equations remain valid in the presence of discontinuities such as cracks. The governing equations of the steady state heat equation and boundary conditions were solved by employing the PDDO. The robustness of the PDDO was assessed by considering a plate without cracks under different boundary conditions. The influence of the insulated cracks on the temperature and heat flux distributions was investigated. It was observed that heat flux concentrations developed in the vicinity of the crack tips. [ABSTRACT FROM AUTHOR]

We consider several nonlocal models for traffic flow, including both microscopic ODE models and macroscopic PDE models. The ODE models describe the movement of individual cars, where each driver adjusts the speed according to the road condition over an interval in the front of the car. These models are known as the FtLs (Follow-the-Leaders) models. The corresponding PDE models, describing the evolution for the density of cars, are conservation laws with nonlocal flux functions. For both types of models, we study stationary traveling wave profiles and stationary discrete traveling wave profiles. (See definitions 1.1 and 1.2, respectively.) We derive delay differential equations satisfied by the profiles for the FtLs models, and delay integro-differential equations for the traveling waves of the nonlocal PDE models. The existence and uniqueness (up to horizontal shifts) of the stationary traveling wave profiles are established. Furthermore, we show that the traveling wave profiles are time asymptotic limits for the corresponding Cauchy problems, under mild assumptions on the smooth initial condition. [ABSTRACT FROM AUTHOR]

Nonlocal diffusion equations and their numerical approximations have attracted much attention in the literature as nonlocal modeling becomes popular in various applications. This paper continues the study of robust discretization schemes for the numerical solution of nonlocal models. In particular, we present quadrature-based finite difference approximations of some linear nonlocal diffusion equations in multidimensions. These approximations are able to preserve various nice properties of the nonlocal continuum models such as the maximum principle and they are shown to be asymptotically compatible in the sense that as the nonlocality vanishes, the numerical solutions can give consistent local limits. The approximation errors are proved to be of optimal order in both nonlocal and asymptotically local settings. The numerical schemes involve a unique design of quadrature weights that reflect the multidimensional nature and require technical estimates on nonconventional divided differences for their numerical analysis. We also study numerical approximations of nonlocal Green's functions associated with nonlocal models. Unlike their local counterparts, nonlocal Green's functions might become singular measures that are not well defined pointwise. We demonstrate how to combine a splitting technique with the asymptotically compatible schemes to provide effective numerical approximations of these singular measures. [ABSTRACT FROM AUTHOR]