Periodic Boundary Condition Rve — Design & Simulation

Abstract — Periodic Boundary Condition Rve

To apply periodic BCs, the chosen RVE must allow being stacked together to form a macroscopic body with a periodic structure. One example of such an RVE is a cuboid unit cell as shown in Figure 1 . Figure 1 (right) represents the RVE in its final configuration after the application of periodic boundary conditions.

Analytical & Numerical Modeling Methodology

All the 6 faces can be grouped into 3 pairs, in which each pair contains two opposite faces. The ( Figure 2 ) shows two opposite faces $Z1$ and $Z2$ in final configuration. In each pair, the two faces of the RVE must be related to each other in such a way that they follow the same displacement pattern i.e., the faces should perfectly overlap each other. This is achieved by tying the related DOF together i.e., the difference between the displacements of two opposite faces must be a constant vector. This is expressed for all 3 pairs in the following 3 sets of equations In a finite element mesh, the domain is discretized into elements. As a result, the nodes at corners are shared between 3 faces and the nodes on the edges are shared between two faces. This would lead to some redundant constraints on the shared nodes in the above set of equations $(1)-(3)$. These redundant equations can be eliminated by separating the equations $(1)-(3)$ into 3 distinct sets for corners, inner nodes of edges and inner nodes of faces.

Finite Element Results & Code Compliance

Similarly, equations $(2),(3)$ can also be separated for corners, edges and faces. The complete sets of equations are given below The above set of equations constrain the degrees of freedom of all the boundary nodes except $5, 2, 4, 1$. Hence, the displacement boundary conditions for the periodic RVE are set by prescribing displacement of the corner nodes $5, 2, 4$ whereas node $1$ is constrained to avoid rigid body translations. The equations $(4), (8), (12)$ can be written as follows where $u,v$ and $w$ are the displacements along $X,Y$ and $Z$ directions respectively. The above matrix is equivalent to the displacement gradient tensor multiplied by size of the RVE.

Scope & Practical Applications

• High-rise building and spatial frame analysis under seismic & wind loading
• Finite element continuum modeling in ABAQUS for non-linear material plasticity
• Classical lamination theory (CLT) and failure prediction for composite structures
• Spatial hazard mapping, landslide susceptibility, and catchment terrain analysis in QGIS
• Detailed bar bending schedules (BBS), structural drawings, and calculation booklets