Predicting band structure of 3D mechanical metamaterials with complex geometry via XFEM

Abstract

Band structure characterizes the most important property of mechanical metamaterials. However, predicting the band structure of 3D metamaterials with complex microstructures through direct numerical simulation (DNS) is computationally inefficient due to the complexity of meshing. To overcome this issue, an extended finite element method (XFEM)-based method is developed to predict 3D metamaterial band structures. Since the microstructure and material interface are implicitly resolved by the level-set function embedded in the XFEM formulation, a non-conforming (such as uniform) mesh is used in the proposed method to avoid the difficulties in meshing complex geometries. The accuracy and mesh convergence of the proposed method have been validated and verified by studying the band structure of a spherical particle embedded in a cube and comparing the results with DNS. The band structures of 3D metamaterials with different microstructures have been studied using the proposed method with the same finite element mesh, indicating the flexibility of this method. This XFEM-based method opens new opportunities in design and optimization of mechanical metamaterials with target functions, e.g. location and width of the band gap, by eliminating the iterative procedure of re-building and re-meshing microstructures that is required by classical DNS type of methods.

Appendices

Appendix 1: Reciprocal lattice and Brillouin zone

Suppose a periodic structure is characterized by a spatial translation vector $\mathbf {T}$

$$\begin{aligned} \mathbf {T} = n_i \mathbf {a}_i \end{aligned}$$

(25)

where $n_i$ is an arbitrary integer and $\mathbf {a}_i$ are primitive vectors as in Eq. 5. The reciprocal lattice is then defined as a set of vectors $\mathbf {G}$ that satisfy

$$\begin{aligned} \exp (i \mathbf {G} \cdot \mathbf {T}) = 1 \end{aligned}$$

(26)

for all possible lattice position vectors $\mathbf {T}$. Vector $\mathbf {G}$ is also called a reciprocal vector.

For a three-dimensional problem, $\mathbf {G}$ can be written as:

$$\begin{aligned} \mathbf {G} = p_i \mathbf {b}_i \end{aligned}$$

(27)

where $p_i$ is an arbitrary integer and $\mathbf {b}_i$ are primitive vectors in reciprocal space defined as

$$\begin{aligned} \mathbf {b}_1 = 2\pi \frac{\mathbf {a}_2 \times \mathbf {a}_2}{\mathbf {a}_1 \cdot (\mathbf {a}_2 \times \mathbf {a}_3)}\end{aligned}$$

(28)

$$\begin{aligned} \mathbf {b}_2 = 2\pi \frac{\mathbf {a}_3 \times \mathbf {a}_1}{\mathbf {a}_2 \cdot (\mathbf {a}_3 \times \mathbf {a}_1)}\end{aligned}$$

(29)

$$\begin{aligned} \mathbf {b}_3 = 2\pi \frac{\mathbf {a}_1 \times \mathbf {a}_2}{\mathbf {a}_3 \cdot (\mathbf {a}_2 \times \mathbf {a}_2)} \end{aligned}$$

(30)

Any vector that connects two points on the reciprocal lattice is a reciprocal vector. In plane wave propagation problems, Bloch wave vectors $\mathbf {K}$ are in the reciprocal space.

Brillouin zone is the primitive cell in reciprocal space, analogous to the primitive cell in defining the spatial periodicity. For description of a Bloch wave in a periodic medium, the solutions for all the wave vectors in the reciprocal space can be completely characterized by the behavior in the single Brillouin zone. All Bloch wave vectors can be translated to the Brillouin zone by a reciprocal vector.

$$\begin{aligned} \mathbf {K} = \mathbf {K'} + \mathbf {G} \end{aligned}$$

(31)

where $\mathbf {K'}$ is the wave vector in the Brillouin zone.

Due to the symmetries, the Brillouin zone can be further reduced to an irreducible zone. Figure 13 shows the irreducible Brillouin zone for SC, BCC and FCC crystals.

Fig. 13

Original lattice structures and corresponding irreducible Brillouin zones in the reciprocal space for SC, BCC and FCC crystals. The figure is adapted from Ref. [51]

Appendix 2: Solver options for PETSc and SLEPc

The equation to be solved (as in Eq. 18) is a generalized eigenvalue problem with symmetric and definite matrices. A typical example of a generalized eigensolver is available through [21]. The runtime keywords that can be used directly with the example eigensolver are summarized as:

“-eps_type krylovschur -eps_target 10.0 -st_ksp_type minres -st_pc_type bjacobi -st_sub_pc_type icc -st_ksp_rtol 1.e-4 -eps_tol 1.e-4 -eps_nev 40 -st_type sinvert”.