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.

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.

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 Mg${}^{2+}$ 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.