Figure 16. Schematic of the three-scale GP model for a thin plate with a central hole of 4 nm diameter. The loading direction [-110] is along the Y-axis and the X-axis, transverse to the loading direction, is along [110].
2. GP--FEA interface design
An exact interface design must first be drafted before developing a mesh for a GP model. The general design for the interface can be seen in Fig. 17. As mentioned in Section II.B.3. the function of WF and WG is to transfer data “bottom-up” and “top- down” to determine the FEA inner node positions and the outer GP particle positions, respectively, at the interface. Figure 17 shows the details of the ideal interface design.
Figure 17. GP-FEA interface design. Particles in WF average their positions to determine the position of the FE node which are within r1 in the WF domain. As a result, it produces a displacement to the FE node at position d1. The WG particle positions, within r2 in the WG domain, are determined by interpolation with in the FE element.
The depth of the WG domain, r2, from the edge of the GP model, d2, as well as the WF domain width, r1, should be equal to the inter-particle cutoff radius as it follows from the length scaling rule of the GP method shown in eqn. (1). Since Mendelev's Embedded Atom Potential (EAM) is used whose inter-atomic cutoff radius is 0.53 nm,65 by eqn. (1) the inter-particle cutoff radius for S3 when k=2 is calculated to be a3= 2.12 nm (=2(3-1) x 0.53 nm). The nodes of the inner boundary of the FEA mesh should align with the center of the WF domain which is shown in the line d1 of Fig. 17 and is determined by the particles' position average in the WF domain. The depth of the interface, found by subtracting (d2-d0), is important for the scale bridging, and should be larger than two times the inter-particle cutoff radius to allow for a “gap” between WF and WG. This gap prevents undesirable feedback between WF and WG thus hastening the interface convergence, the recommended gap size is the inter-particle cutoff radius. The finite element size in the interface should be uniform about the size of the inter-particle cutoff radius and gradually increase in size farther from the GP domains. Figure 17 shows the interface for the 1D case. This profile is extended all along the edge of the GP model, in a sense tangentially, as illustrated in Fig. 18.
Figure 18. Continuous WF domain design and its relation with high-scale particles and the WG domain for the GP-FEA model interface.
3. FEA Mesh generation
Needless to say, code development is essential for wide applications of multiscale analysis. The GP-FEA method includes two source codes, Multi-Input57.f90 and FEA19.f90, which respectively, directly controls the GP simulation, including the WG- GP subsystem, and couples the calculation of the WF-FEA subsystem. Its corresponding input file is inp57f19.sh. Here, the numbers denote the code version for the GP and FEA trial. The structure and its related function of these codes are complicated and will be introduced elsewhere. Here, we will introduce the constitution of the input file FEAxx.inp. This file will be a part of the main input file but it is specifically designed for the FEA calculation parameters. In other words, when the FEA routine runs it will read this GP-FEA mesh input file. To develop this mesh input file, any FE program, such as ABAQUS, Ansys, etc. may be used to generate the FEA mesh based on the model size, the WF internal boundary and the interface dimensions. The process for the generation and its required format is described in Appendix C.6.
It is important that the GP model be equilibrated before it's connected to the FEA mesh to prevent residual stresses from manifesting at the interface. In the simulation, the NPT ensemble at 300 K was usually used for the equilibration if the system operates in
room temperature. After this separate equilibration the FEA mesh is connected to the GP model for the loading procedure. Here, the FEA mesh is added to the GP hole model such that the entire model size is about 200 nm wide and 500 nm high, nearly an order larger than the GP model. After the geometry has been sketched with the coordinate origin being the same for the GP model and FE mesh, it is best to partition the part around the interface so that it can be seeded to have a uniform element size and shape for both WF and WG domains. Since the element size at the interface is designed to have an edge length equal to the inter-particle cutoff radius, it will create two FE elements overlapping the GP model along the normal direction of the inner FE boundary. It is important to carefully seed the edges of the FE model for nodes by keeping a constant size element at the interface for both WF and WG-FEs then gradually increase the element size away from the interface.
C. Verification of the GP and GP-FEA multiscale methods with Elasticity