B. Linking GP with FEA
3. The “Bottom-Up” and “Top-Down” iteration bridging scheme
The “bottom-up” transformation from particles to FE nodes, denoted by the up blue arrows, is through the particle-node overlapped WF domain and the “top-down”
transformation from FE nodes to particles, denoted by the down green arrows, is through the overlapped node-particle WG domain. The function of WF and WG are similar to the
role of W(n+1)image and W(n)image, (or W2 and W1 domains if n=1) in the GP model. The WF domain transfers data “bottom-up” by averaging particle’s position 𝑟̅(𝑖 = 1, … 𝑙𝑖 𝑛) to determine the position of the corresponding FEA node I in the WF domain by
𝑅̅̅̅ = ∑𝐼 𝑙𝑟̅𝑖
𝑛 𝑙𝑛
𝑖=1 (7) where 𝑙𝑛 is the number of the particles in the NLC of the FE node I. Likewise, the WG domain transfers the data “top-down” by a certain decomposition of the FEA nodes’
position to determine the particle position within the FE elements in the WG domain. The former and the latter processes are important, respectively, to control the deformation of the FEA domain and the motion of the GP domain during the iterative process. In fact, the deformation of the FEA domain is controlled by both the nodes at the remote boundary and the FE nodes in the WF domain; the motion of the GP domain is controlled by both the inside body of the GPs and the particles in the WG domain. To make the two separate but coupling processes of the system functional, it merits note that before any iteration of a load step all FEA nodes in the WF domain are fixed for the WF-FEA process and all particles in the WG domain are fixed for the WG-GP process. In turn, for the first process these nodes serve to be an inner boundary to control the deformation of the FEA domain under remote external loading, thus the GP displacement controls the upper scale FEA node motion in the WF domain to carry on the “bottom-up” transition.
On the other hand, the above controlled deformation process of the FEA domain will change the position of the FEA nodes which overlap the WG domain. In turn, these displaced FEA node coordinates will be used for determining the new coordinates of particles in the WG domain. The principle for assigning the new position of particles with the new coordinates of the FE nodes is based on the FEA isoparametric formulation.51 The latter is simply to use the FE shape function matrix [N (] to determine the coordinates {x ( )} of any particle inside of the element with the coordinate matrix {X} of the FE element nodes. Here are the natural (or non-dimensional) coordinates of the particle and will not change during the FE node displacements. Since after the iteration the node position {X} is known, shape functions are given then the new particle position can be obtained by the matrix product of [N] and {X}. Then the position of these
particles will be held fixed as the external boundary of the GP domain to start the new WG-GP sub-system process for its relaxations of particles and atoms. Thus, the FE node controls the particle motion in the WG domain to make the “top-down” transition realized. Note, shape functions can be linear or bi-linear, using the bi-linear functions is more accurate but also more complicated, the unique solution for bi-linear functions developed by Hua52 to determine the new coordinates of these particles from the element node coordinates should be used.
Specifically, after the equilibration of the GP model and the FE mesh design (see Section 3.5.3) is completed all initial positions, x(j) and y(j), of GP particle j (j=1...nj)
and initial position XLJ, YLJ of element node J (J=1...NJ, L=1..4) in the WG domain are determined. Where, nj and NJ denotes the total number, respectively, of GP particles and FE elements in that domain, L denotes the node number of a given element which varies from 1 to 4 for the quadrilateral element. Using the inverse transformation method developed by Hua, one can find the natural coordinates ξ (x, y)x, y) for each generic particle, j, based on the node coordinates of the element that particle j belongs to. With the deformation process of the WF-FEA subsystem, the node coordinates XLJ, YLJ
change, the corresponding position, x and y of the particle j also changes. The new position x and y of the particle in each iteration is important to formulate the WG-GP external boundary for particle relaxation. It can be determined by shape functions [N] as (Remark: the superscript J of the element ID is dropped for simplicity)
{𝑥(𝜉, 𝜂)
𝑦(𝜉, 𝜂)} =[𝑁]{𝑋},
{𝑋} = {𝑋1, 𝑌1, 𝑋2, 𝑌2, 𝑋3, 𝑌3, 𝑋4, 𝑌4}𝑇
(8, 9) [𝑁] = [𝑁1 0 𝑁2
0 𝑁1 0
0 𝑁3 0
𝑁2 0 𝑁3 𝑁4 0 0 𝑁4]
(10)
Where
𝑁1 = 1
4(1 − 𝜉)(1 − 𝜂), 𝑁2 = 1
4(1 + 𝜉)(1 − 𝜂),
𝑁3 =14(1 + 𝜉)(1 + 𝜂), 𝑁4 = 14(1 − 𝜉)(1 + 𝜂) (11)
The key here in determining the particle position in the WG domain, based on the position (or) displacement of the FE nodes, is to make the natural coordinates (ξ,
constant so the distribution pattern of particles inside the element is fixed as required naturally for a given material domain.
It may be interesting to look at the difference of the WF domain from the WG domain in treating the relationship of FE node with the particles. For the former, this relation is quantified by eqn. (7). In practice, after the initial GP equilibration the first step is to find what particles is inside the NLC of that FE node I and determine their position vectors, 𝑟̅(𝑖 = 1, … 𝑙𝑖 𝑛). These particles’ identification number (ID) will be constant during the whole deformation process (i. e., the particle constituents of the NLC for the FE node I is fixed). Each of these individual particles move during iteration, the spatio-temporal average of their position determines the new position of the generic FE node is through the equation (7). It is truly a lumping process but not a one-to-one fixed relation between FE node and particle. This is natural since the determination of the FE node positions in the WF domain is controlled by a relaxation process of the WG-GP subsystem. Thus, the particles should move to make the system, and particularly the particles surrounding those FE nodes, equilibrated under the fixed WG BC. This is a dynamic process so one needs to integrate Newton’s equation of motion for the system by, say, the Velocity Verlet method.19 However, for the WG the situation is completely different, the particles change its role from “master” in the WF domain to “slave” in the WG. Their positions are controlled by the element node positions which are determined by the motion of the WF-FEA subsystem. Obviously, the efficient way to make this control is to fix the relation of these FE node positions with the internal particle through the fixed natural coordinates ξ, determined after the equilibration In a sense particle position is solely determined by the FE node displacements through an interpolation function, the WG method for the top-down transition may be considered as a decomposition of the FE node displacements to a generic particle.