ABSTRACT Peridynamics (PD), similarly to some other nonlocal continuum theories, exhibits truncated interaction horizons near free surfaces, cracks, and voids in bounded domains. This loss of neighbors causes artificial surface effects and inconsistencies in the derivative and energy operators that persist under discretization refinement, limiting PD accuracy and robustness. First, the analysis and numerical isolation of the effects of horizon truncation on these operators are carried out, showing how it induces surface softening in bond‐based and stiffening/softening in state‐based PD formulations. Then, a purely PD‐based surface correction is proposed to restore full‐horizon behavior while preserving the original horizon radius and bond topology. Each node is assigned a scalar influence weight, and every bond is scaled by the average of the endpoint weights to maintain symmetry. The nodal influence weights are computed in a preprocessing optimization step enforcing agreement between truncated‐ and full‐horizon derivative and energy operators. Unlike existing approaches, the proposed method does not rely on ghost particles, variable horizons, or reference solutions from classical continuum mechanics (CCM). Benchmarks in one, two, and three dimensions, including a dynamic brittle fracture test, show that the optimized nodal influence weights reduce boundary‐induced surface effects, improve energy consistency, and accelerate convergence of PD solutions. Analytical evaluation of the calibration targets over full‐horizons also improves interior volume integration accuracy. The proposed scheme is simple to implement in existing PD codes, incurs only a modest preprocessing cost, performs robustly for flat and curved boundaries on both regular and irregular point sets, produces reliable crack paths, and applies to both bond‐based and ordinary state‐based PD formulations. An open‐source Python implementation, perifit , is provided.

We present time-domain Dirichlet-type absorbing boundary conditions (ABCs) for 2D ordinary state-based peridynamics using the Linear Peridynamic Solid (LPS) model, aimed at efficiently truncating unbounded elastodynamic domains while retaining the flexibility of arbitrary Poisson ratios. The proposed ABCs are built from a semi-analytical far-field representation expressed as a finite superposition of plane-wave modes satisfying the discrete LPS dispersion relation, so that outgoing waves are absorbed consistently with the numerical P- and S-wave branches and their Poisson-ratio-dependent polarizations. A cloud-based collocation and least-squares procedure eliminates the unknown modal amplitudes locally and yields precomputable updating operators that prescribe boundary displacement and velocity directly within explicit time stepping, avoiding spatial derivatives, transform techniques, auxiliary evolution equations, and surface-correction procedures at truncated horizons. Unlike our earlier bond-based ABC formulations, the present work requires a new state-based far-field theory, a new discrete dispersion operator with coupled P- and S-wave branches for general Poisson ratios, and an additional treatment of the nonlocal dilatation field. Numerical benchmarks show that the resulting ABCs strongly suppress nonphysical reflections over a broad range of material parameters, providing a simple and robust alternative to absorbing layers.

This study introduces a computational framework for simulating the self-assembly of diblock copolymers using a novel peridynamic (PD)-enhanced Fourier spectral method (FSM). Diblock copolymers, composed of two distinct polymer blocks, are capable of forming nanostructured domains with applications in nanoelectronics, photonics, and advanced membranes. Current simulation techniques face challenges in capturing the multiscale dynamics of polymer systems and are often limited by computational inefficiencies. Our approach combines a phase-field model with FSM for spatial discretization and leverages a PD-based diffusion operator to overcome the stability restrictions of explicit time-stepping schemes. This integration allows for larger time steps, ensuring both stability and computational efficiency. The method's scalability is enhanced through parallel implementation using C++ and OpenMP, optimized for multi-core CPUs. Validation through phase diagrams of copolymer melts and simulations of evaporation- induced self-assembly (EISA) processes demonstrates the capability of the proposed method to accurately capture large-scale, dynamic morphologies. Our approach provides a versatile framework and was found in certain examples to improve computational efficiency by more than a factor of 6 compared to forward-Euler FSM approach.

Cerebral blood flow regulation is critical for brain function, and its disruption is implicated in various neurological disorders. Many existing models do not fully capture the complex, multiscale interactions among neuronal activity, astrocytic signaling, and vascular dynamics, especially in key brainstem regions. In this work, we present a 3D-1D-0D multiscale computational framework for modeling the neuro-glial-vascular unit (NGVU) in the dorsal vagal complex (DVC). Our approach integrates a quadripartite synapse model, which captures the dynamic interactions among excitatory and inhibitory neurons, astrocytes, and vascular smooth muscle cells, with a hierarchical description of vascular dynamics that couples a three-dimensional microcirculatory network with a one-dimensional macrocirculatory representation and a zero-dimensional synaptic component. By linking neuronal spiking, astrocytic calcium and gliotransmitter signaling, and vascular tone regulation, our model reproduces key features of neurovascular regulation and elucidates the feedback loops that help maintain cerebral blood flow. Simulation results demonstrate that neurotransmitter release triggers astrocytic responses that modulate vessel radius, thereby influencing local oxygen and nutrient delivery. This integrated framework provides a robust and modular platform for future investigations into the pathophysiology of cerebral blood flow regulation and its role in autonomic control, including the regulation of gastric function.

We introduce a nonlocal extension of the Kim-Kim-Suzuki (KKS) phase-field corrosion model aimed at bridging local and nonlocal corrosion modeling approaches, such as phase-field and peridynamic frameworks. In this formulation, classical gradient operators are replaced with integral operators defined over a finite interaction horizon, naturally embedding an intrinsic length scale that aligns with nonlocal theories like peridynamics. Under precise assumptions on function spaces and kernel functions, we define a nonlocal free energy that integrates a standard bulk free energy density with a nonlocal interaction term. Through differentiation in an appropriate Hilbert space, we derive evolution equations, yielding a nonlocal Allen–Cahn equation for the phase-field and a nonlocal Cahn–Hilliard-type equation for the concentration. The latter is expressed as a gradient flow in a metric induced by the inverse of the nonlocal operator, mirroring the classical H^-1 metric for conserved dynamics. We establish the well-posedness of these equations using Galerkin approximations, uniform energy estimates, and compactness arguments. Furthermore, we prove the convergence of the nonlocal model to its local KKS counterpart as the interaction horizon approaches zero, effectively unifying local and nonlocal perspectives. Numerical experiments, implemented via finite difference spatial discretization and explicit time-stepping, demonstrate the effects of nonlocality and confirm the theoretical convergence, reinforcing the connection between the two modeling paradigms.

This paper provides a comprehensive derivation and application of the nonlocal Nernst-Planck-Poisson (NNPP) system for accurate modeling of electrochemical corrosion with a focus on the biodegradation of magnesium-based implant materials under physiological conditions. The NNPP system extends and generalizes the peridynamic bi-material corrosion model by considering the transport of multiple ionic species due to electromigration. As in the peridynamic corrosion model, the NNPP system naturally accounts for moving boundaries due to the electrochemical dissolution of solid metallic materials in a liquid electrolyte as part of the dissolution process. In addition, we use the concept of a diffusive corrosion layer, which serves as an interface for constitutive corrosion modeling and provides an accurate representation of the kinetics with respect to the corrosion system under consideration. Through the NNPP model, we propose a corrosion modeling approach that incorporates diffusion, electromigration and reaction conditions in a single nonlocal framework. The validity of the NNPP-based corrosion model is illustrated by numerical simulations, including a one-dimensional example of pencil electrode corrosion and a three-dimensional simulation of a Mg-10Gd alloy bone implant screw decomposing in simulated body fluid. The numerical simulations correctly reproduce the corrosion patterns in agreement with macroscopic experimental corrosion data. Using numerical models of corrosion based on the NNPP system, a nonlocal approach to corrosion analysis is proposed, which reduces the gap between experimental observations and computational predictions, particularly in the development of biodegradable implant materials.

Construction of absorbing boundary conditions (ABCs) for nonlocal models is generally challenging, primarily due to the fact that nonlocal operators are commonly associated with volume constrained boundary conditions. Moreover, application of Fourier and Laplace transforms, which are essential for the majority of available methods for ABCs, to nonlocal models is complicated. In this paper, we propose a simple method to construct accurate ABCs for peridynamic scalar wave‐type problems in viscous media. The proposed ABCs are constructed in the time and space domains and are of Dirichlet type. Consequently, their implementation is relatively simple, since no derivatives of the wave field are required. The proposed ABCs are derived at the continuum level, from a semi‐analytical solution of the exterior domain using harmonic exponential basis functions in space and time (plane‐wave modes). The numerical implementation is done using a meshfree collocation approach employed within a boundary layer adjacent to the interior domain boundary. The modes satisfy the peridynamic numerical dispersion relation, resulting in a compatible solution of the interior region (near‐field) with that of the exterior region (far‐field). The accuracy and stability of the proposed ABCs are demonstrated with several numerical examples in two‐dimensional unbounded domains.

The focus of this paper is on application of peridynamics (PD) to propagation of elastic waves in unbounded domains. We construct absorbing boundary conditions (ABCs) derived from a semi-analytical solution of the PD governing equation at the exterior region. This solution is made up of a finite series of plane waves, as fundamental solutions (modes), which satisfy the PD dispersion relations. The modes are adjusted to transmit the energy from the interior region (near field) to the exterior region (far field). The corresponding unknown coefficients of the series are found in terms of the displacement field at a layer of points adjacent to the absorbing boundary. This is accomplished through a collocation procedure at subregions (clouds) around each absorbing point. The proposed ABCs offer appealing advantages, which facilitate their application to PD. They are of Dirichlet-type, hence their implementation is relatively simple as no derivatives of the field variables are required. They are constructed in the time and space domains and thus application of Fourier and Laplace transforms, cumbersome for nonlocal models, is not required. At the discrete level, the modes satisfy the same numerical dispersion relations of the near field, which makes the far-field solution compatible with that of the near field. We scrutinize the performance of the proposed ABCs through several examples. Our investigation shows that the proposed ABCs perform stably in time with an appropriate level of accuracy even in problems characterized by highly-dispersive propagating waves, including crack propagation in semi-unbounded brittle solids.(c) 2023 Elsevier B.V. All rights reserved.

Peridynamic (PD) models are commonly implemented by exploiting a particle-based method referred to as standard scheme. Compared to numerical methods based on classical theories (e.g., the finite element method), PD models using the meshfree standard scheme are typically computationally more expensive mainly for two reasons. First, the nonlocal nature of PD requires advanced quadrature schemes. Second, non-uniform discretizations of the standard scheme are inaccurate and thus typically avoided. Hence, very fine uniform discretizations are applied in the whole domain even in cases where a fine resolution is per se required only in a small part of it (e.g., close to discontinuities and interfaces). In the present study, a new framework is devised to enhance the computational performance of PD models substantially. It applies the standard scheme only to localized regions where discontinuities and interfaces emerge, and a less demanding quadrature scheme to the rest of the domain. Moreover, it uses a multi-grid approach with a fine grid spacing only in critical regions. Because these regions are identified dynamically over time, our framework is referred to as multi-adaptive . The performance of the proposed approach is examined by means of two real-world problems, the Kalthoff–Winkler experiment and the bio-degradation of a magnesium-based bone implant screw. It is demonstrated that our novel framework can vastly reduce the computational cost (for given accuracy requirements) compared to a simple application of the standard scheme.

Magnesium (Mg) and its alloys are promising materials for temporary bone implants due to their mechanical properties and biocompatibility. The most challenging aspect of Mg-based implants involves adapting the degradation rate to the human body, which requires extensive in vitro and in vivo testing. Given that in vivo tests are significantly more labour-intensive than in vitro and ethics prohibit direct experiments on animals or humans, attempts are commonly undertaken to infer conclusions on in vivo degradation behavior from in vitro experiments. However, there is a wide gap between these tests, and in vitro testing is often a poor predictor of in vivo outcomes. In the development of biodegradable Mg-based implants, considerable efforts are being made to reduce the overall time and cost of in vitro and in vivo testing. Finding a suitable alternative to predict the degradation of Mg alloys, however, remains challenging. We present computational modelling as a possible alternative to bridge the gap between in vitro and in vivo testing, thus reducing overall cost, duration and number of experiments. However, traditional modelling approaches for complex biodegradable systems are still rather time-consuming and require a clear definition of the relations between input parameters and the model result. In this study, Kriging surrogate models based on the peridynamic in vitro degradation model were developed to simulate the degradation behavior for two main alloys, Mg-5Gd and Mg-10Gd, for both in vitro and in vivo cases. Using Kriging surrogate models, the simulation parameters were calibrated to the volume loss data from in vitro and in vivo experiments. In vivo degradation of magnesium has one order of magnitude higher apparent diffusion coefficients than in vitro degradation, thus yielding the higher volume loss observed in vivo than in vitro. On the basis of the diffusivity of the Mg2+ ions modeled under in vitro degradation, Kriging surrogate models were able to simulate the in vivo degradation behavior of Mg-xGd with a ratio between 0.46 and 0.5, indicating that the surrogate-modelling approach is able to bridge the gap between in vitro and in vivo degradation rates for Mg-xGd implants.

This paper proposes a computational framework to describe the biodegradation of magnesium (Mg)-based bone implants. It is based on a sequential combination of two models: an electrochemical corrosion model to compute the mass loss of the implant over several weeks combined with a mechanical model to assess its residual mechanical strength. The first model uses a peridynamic (PD) corrosion model to tackle the complex moving boundary of the corroding material in an efficient manner. The results of this corrosion simulation are mapped to a finite element (FE) model by way of a damage variable. Subsequently, the FE model is used for mechanical analysis. To use PD for such a complex problem, we proposed three innovative improvements compared to state-of-the-art PD models: (1) application of an adaptive multi-grid discretization in space and an implicit time-stepping algorithm enabling an efficient simulation of the complex implant geometry over prolonged periods, (2) novel non-local Dirichlet absorbing boundary conditions to truncate the simulation domain in the close neighborhood of the implant of interest without prohibitive losses of accuracy, and (3) selection of suitable non-local kernel functions and parameter calibration on the basis of experimental data by an evolutionary algorithm. We demonstrate that this framework can capture the loss of implant mass due to corrosion for typical alloys such as Mg-5Gd and Mg-10Gd. Moreover, we point out how this framework can be used in the future to predict the declining mechanical strength of bone screws subject to biocorrosion over several weeks.

Efficient and accurate calculation of spatial integrals is of major interest in the numerical implementation of peridynamics (PD). The standard way to perform this calculation is a particle-based approach that discretizes the strong form of the PD governing equation. This approach has rapidly been adopted by the PD community since it offers some advantages. It is computationally cheaper than other available schemes, can conveniently handle material separation, and effectively deals with nonlinear PD models. Nevertheless, PD models are still computationally very expensive compared with those based on the classical continuum mechanics theory, particularly for large-scale problems in three dimensions. This results from the nonlocal nature of the PD theory which leads to interactions of each node of a discretized body with multiple surrounding nodes. Here, we propose a new approach to significantly boost the numerical efficiency of PD models. We propose a discretization scheme that employs a simple collocation procedure and is truly meshfree; i.e., it does not depend on any background integration cells. In contrast to the standard scheme, the proposed scheme requires a much smaller set of neighboring nodes (keeping the same physical length scale) to achieve a specific accuracy and is thus computationally more efficient. Our new scheme is applicable to the case of linear PD models and within neighborhoods where the solution can be approximated by smooth basis functions. Therefore, to fully exploit the advantages of both the standard and the proposed schemes, a hybrid discretization is presented that combines both approaches within an adaptive framework. The high performance of the developed framework is illustrated by several numerical examples, including brittle fracture and corrosion problems in two and three dimensions.

Magnesium is attractive for the application as a temporary bone implant due to its inherent biodegradability, non-toxicity and suitable mechanical properties. The degradation process of magnesium in physiological environments is complex and is thought to be a diffusion-limited transport problem. We use a multi-scale imaging approach using micro computed tomography and transmission X-ray microscopy (TXM) at resolutions below 40 nm. Thus, we are able to evaluate the nanoporosity of the degradation layer and infer its impact on the degradation process of pure magnesium in two physiological solutions. Magnesium samples were degraded in simulated body fluid (SBF) or Dulbecco's modified Eagle's medium (DMEM) with 10% fetal bovine serum (FBS) for one to four weeks. TXM reveals the three-dimensional interconnected pore network within the degradation layer for both solutions. The pore network morphology and degradation layer composition are similar for all samples. By contrast, the degradation layer thickness in samples degraded in SBF was significantly higher and more inhomogeneous than in DMEM+10%FBS. Distinct features could be observed within the degradation layer of samples degraded in SBF, suggesting the formation of microgalvanic cells, which are not present in samples degraded in DMEM+10%FBS. The results suggest that the nanoporosity of the degradation layer and the resulting ion diffusion processes therein have a limited influence on the overall degradation process. This indicates that the influence of organic components on the dampening of the degradation rate by the suppression of microgalvanic degradation is much greater in the present study.

Diffusion-type problems in (nearly) unbounded domains play important roles in various fields of fluid dynamics, biology, and materials science. The aim of this paper is to construct accurate absorbing boundary conditions (ABCs) suitable for classical (local) as well as nonlocal peridynamic (PD) diffusion models. The main focus of the present study is on the PD diffusion formulation. The majority of the PD diffusion models proposed so far are applied to bounded domains only. In this study, we propose an effective way to handle unbounded domains both with PD and classical diffusion models. For the former, we employ a meshfree discretization, whereas for the latter the finite element method (FEM) is employed. The proposed ABCs are time-dependent and Dirichlet-type, making the approach easy to implement in the available models. The performance of the approach, in terms of accuracy and stability, is illustrated by numerical examples in 1D, 2D, and 3D.

Magnesium and its alloys attract increasingly wide attention in various fields, ranging from transport to medical solutions, due to their outstanding structural and degradation properties. These properties can be tailored through alloying and thermo-mechanical processing, which is often complex and multi-step, thus requiring in-depth analysis. In this work, we demonstrate the capability of synchrotron-based nanotomographic X-ray imaging methods, namely holotomography and transmission X-ray microscopy, for the quantitative 3D analysis of the evolution of intermetallic precipitate (particle) morphology and distribution in magnesium alloy Mg–5.78Zn–0.44Zr subjected to a complex multi-step processing. A rich history of variation of the intermetallic particle structure in the processed alloy provided a testbed for challenging the analytical capabilities of the imaging modalities studied. The main features of the evolving precipitate structure revealed earlier by traditional light and electron microscopy methods were confirmed by the 3D techniques of synchrotron-based X-ray imaging. We further demonstrated that synchrotron-based X-ray imaging enabled uncovering finer details of the variation of particle morphology and number density at various stages of processing—above and beyond the information provided by visible light and electron microscopy.