Atomistic simulations have been widely used to explore the material properties at the nanoscale. Molecular dynamics (MD) simulations indeed play a crucial role in understanding various material behaviors, such as dislocation initiation and movement [1], grain boundary effects [2,3], crack nucleation and growth [4], and void nucleation, among others. However, length and time scale limitations restrict the application of MD simulation in studying the material behaviors at microscale. Also, the continuum method, as another approach for investigating the mechanical properties of materials, is unable to reveal numerous experimental observations of material behavior, which are dominant at the nanoscale. To bridge this gap, researchers often employ a strategy known as length scaling, where the atomistic domain is coupled with a continuum domain. Length scaling allows for the simulation of nanoscale material properties within a macroscale system. Concurrent multiscale methods are a popular approach to achieve this. In these methods, atomistic simulations are utilized in areas where precise modeling is essential to explore nanoscale material behavior. Meanwhile, a continuum formulation is applied to the rest of the system, effectively scaling the simulation domain and mitigating the effects of boundary conditions.

In other words, the concurrent approach performs atomistic and continuum simulations simultaneously, to consider strong coupling among different scales. Typically, the continuum domain in such simulations employs the finite element method, a widely used technique in computational mechanics. By solving simultaneous algebraic equations for both the atomic and continuum parts, it's possible to obtain stress and strain fields, and providing valuable insights into material behavior across scales [5].

Achieving a seamless connection between atomic and continuum domains is indeed a challenging problem. The differences in material constitutive laws between these two domains can pose significant complications. In molecular dynamics, atoms interact with others within a certain cutoff radius (rcut), which means that atomic bonding has a nonlocal nature. On the other hand, in the finite element method (FEM), each node interacts only with the nodes connected to it, resulting in a local constitutive law in the continuum domain. To find equilibrium in such simulations, researchers commonly employ two approaches: "energy-based" and "force-based" minimizations. The choice between these approaches depends on whether the focus is on minimizing the total energy or the total forces within the systems [6]. Energy-based multiscale methods like the Coupling of Length Scales (CLS) method [7], Bridging Domain (BD) method [8], Quasi-Continuum (QC) method [9], Composite Grid Continuum Method (CACM) [10], and Bridging Scale Method (BSM) [11] are notable for their use of the energy-based formulation. In contrast, force-based multiscale methods, such as the Coupled Atomistic and Discrete Dislocations (CADD) method [12,13], Hybrid Simulation Method (HSM) [14,15], Finite Element-Atomistic (FEAt) method [15,16], and Atomistic-to-Continuum (AtC) [17], rely on force-based formulations. However, it's essential to be aware of some of the challenges that energy-based multiscale methods can face due to the coupling of nonlocal and local materials at both sides of the interface. This coupling can lead to non-physical side effects, often referred to as "ghost forces". These ghost forces need to be addressed and eliminated to ensure the accuracy and reliability of simulations. There's an interesting method discussed in Ref. [9], which introduces a ghost force correction approach. This method involves adding the negative of ghost forces as dead-loads on the affected atoms or nodes to mitigate the influence of these unwanted forces.

If it were feasible to minimize or eliminate the discrepancies in material constitutive properties between the atomic and continuum domains, we could eliminate the occurrence of ghost forces and enhance the accuracy of multiscale methods. Nonlocal elasticity formulation and nonlinear elasticity were employed in the finite element part of Finite-Element/Atomistics method [16] and Hybrid Simulation method [14], respectively. While these methods mitigate the abrupt transition, they often result in complex formulations within the continuum domain.

Since microscopic defects, especially dislocations, play a crucial role in determining the strength and response of materials, the ability to model dislocation nucleation and movement is one the important capabilities of multiscale methods. The study of dislocation nucleation and evolution, as well as plasticity, has received much attention in recent years. The capability of Quasi-coarse-grained dynamics (QCGD) simulations to reproduce the atomic scale evolution of dislocation density fractions is investigated by Agarwal et al. [18]. Also, the phase field approach (PFA) to dislocations is broadly used for modeling plastic deformation at the nanoscale [19]. Han et al. [20] introduced a fully embedded implementation of a full-field crystal plasticity model in an implicit finite element (FE) framework. Furthermore, coupling the two dimensional (2D) discrete dislocation dynamics (DDD) and the extended finite element method (XFEM) [21], the anisotropic Coupled Atomistic/Discrete Dislocation (CADD) model [22], incorporated Taylor-type model and crystal plasticity finite element (CPFE) method at the microstructural scale [23], and the using discrete dislocation dynamics (DDD) simulations and a strain gradient plasticity (SGP) model [24] are some of recent methods for investigation of dislocation in material. However, in all these methods, require different complex formulations for atomic and continuum representations.

Expanding the atomic part and avoiding sudden transitions from nonlocal to local material in interfaces involves a unique approach of maintaining the same nonlocal material behavior as in the atomistic domain. The Generalized Particle multiscale method, which was introduced by Fan [25,26], has the ability to use the atomistic formulations in upper scales. The GP method allows a seamless transition between atomistic and particle domains by maintaining the same nonlocal material behavior. This is achieved using particles with a similar atomic configuration and higher atomic spacing in the particle domain. For example, if atoms in the atomistic region have FCC structure with atom spacing of λ, particles in the particle domain also have FCC structure with particle spacing of k λ. K is defined according to the scaling level of the particle domain relative to the atomic part. This unique approach of expanding the atomic domain and avoiding sudden transitions from nonlocal to local material in interfaces leads to more accurate results and eliminates non-physical forces. The ability to keep the basic material structure at all scales is a key feature of the GP method, enabling the utilization of atomistic formulations in larger scales [27].

Despite the outstanding capabilities of the GP method, this method has been used rarely in literature and has been limited to the author of the original paper on this method. The effects of the coating layer thickness on stress, and the debonding behavior near the interface of coating and substrate were investigated in Ref. [28]. In this research, the GP method was employed for studying aluminum (Al) coated on iron (Fe), utilizing an atomistic domain near the layer interface and Generalized Particles further away from the interface. Furthermore, a hybrid multiscale approach that combines the concurrent Generalized Particle (GP) method and the Hierarchical Cohesive Zone Model (CZM) was employed to explore crack propagation at both the atomistic and mesoscopic scales [29]. Fan et al. [30] developed the XGP multiscale method, which extends the GP method to effectively produce sufficiently large models with high accuracy by using finite element (FE) nodes connected with the outermost particles. He also proposed a multiplication form of energy barrier with a fast decaying function of thermal activation energy as the size-dependent failure initiation criterion using the GP method [31].

Nanoindentation and nanoscratch testing are commonly employed experimental techniques for evaluating the mechanical and tribological properties of thin films and coatings [[32], [33], [34]]. These tests provide valuable information on hardness, Young's modulus, wear rate, coefficient of friction (COF), and plastic behavior of materials. To accurately determine these properties, researchers have used the molecular dynamics methods [35,36] and the multiscale methods in conjunction with nanoindentation [[37], [38], [39], [40], [41]] and nanoscratch [42,43] tests.

This study utilizes the GP multiscale method to examine the mechanical and tribological characteristics of a thin layer of aluminum through nanoindentation and nanoscratch experiments. By performing 3D simulations with one and two levels of particle scaling, the GP multiscale method can validate its accuracy and reliability compared to molecular dynamics (MD) simulations. It can also elucidate the advantages of using this method, such as computational efficiency and the ability to capture larger length and time scales.