Here is a description of a small FE model that might approximate the case of an infinite number of holes in an infinite plate.

Create a model of a single repeating element with $1/4$ of a hole centered at $x=0, y=0$. Along the boundaries at $x=0$ and $y=0$ apply classical symmetry boundary conditions.

On the two remaining edges, apply constraints on the displacements simply to keep the edges straight and parallel to either the x-axis or y-axis, as appropriate. In other words, all nodes along the boundary $x=a$ would have the same $u$ displacement and all nodes along the boundary $y=b$ would have the same $v$ displacement. This type of boundary condition is often referred to as an MPC (multi-point constraint) (e.g. NASTRAN, ABAQUS).

Either or both of these edges would also have an applied stress loading normal to the edge.

The idea is that the straight-edge constraints would enforce compatibility with the adjoining cells but not restrict the motion unnecessarily.

Here is a description of a small FE model that might approximate the case of an infinite number of holes in an infinite plate.

Create a model of a single repeating element with $1/4$ of a hole centered at $x=0, y=0$. Along the boundaries at $x=0$ and $y=0$ apply classical symmetry boundary conditions.

On the two remaining edges, apply constraints on the displacements simply to keep the edges straight and parallel to either the x-axis or y-axis, as appropriate. In other words, all nodes along the boundary $x=l/2$ would have the same $u$ displacement and all nodes along the boundary $y=l/2$ would have the same $v$ displacement. This type of boundary condition is often referred to as an MPC (multi-point constraint) (e.g. NASTRAN, ABAQUS).

Either or both of these edges would also have an applied stress loading normal to the edge of $\sigma_1$ or $\sigma_2$.

The idea is that the straight-edge constraints would enforce compatibility with the adjoining cells but not restrict the motion unnecessarily.

Here is a description of a small FE model that might approximate the case of an infinite number of holes in an infinite plate.

Create a model of a single repeating element with $1/4$ of a hole centered at $x=0, y=0$. Along the boundaries at $x=0$ and $y=0$ apply classical symmetry boundary conditions.

On the two remaining edges, apply constraints on the displacements simply to keep the edges straight and parallel to either the x-axis or y-axis, as appropriate. This type of boundary condition is often referred to as an MPC (multi-point constraint) (e.g. NASTRAN, ABAQUS).

Either or both of these edges would also have an applied stress loading normal to the edge.

The idea is that the straight-edge constraints would enforce compatibility with the adjoining cells but not restrict the motion unnecessarily.

Here is a description of a small FE model that might approximate the case of an infinite number of holes in an infinite plate.

Create a model of a single repeating element with $1/4$ of a hole centered at $x=0, y=0$. Along the boundaries at $x=0$ and $y=0$ apply classical symmetry boundary conditions.

On the two remaining edges, apply constraints on the displacements simply to keep the edges straight and parallel to either the x-axis or y-axis, as appropriate. In other words, all nodes along the boundary $x=a$ would have the same $u$ displacement and all nodes along the boundary $y=b$ would have the same $v$ displacement. This type of boundary condition is often referred to as an MPC (multi-point constraint) (e.g. NASTRAN, ABAQUS).

Either or both of these edges would also have an applied stress loading normal to the edge.

The idea is that the straight-edge constraints would enforce compatibility with the adjoining cells but not restrict the motion unnecessarily.