Bloch mode synthesis: Ultrafast methodology for elastic band-structure calculations

We present a methodology for fast band-structure calculations that is generally applicable to problems of elastic wave propagation in periodic media. The methodology, called Bloch mode synthesis, represents an extension of component mode synthesis, a set of substructuring techniques originally developed for structural dynamics analysis. In Bloch mode synthesis, the unit cell is divided into interior and boundary degrees-of-freedom, which are described, respectively, by a set of normal modes and a set of constraint modes. A combination of these mode sets then forms a reduced basis for the band structure eigenvalue problem. The reduction is demonstrated on a phononic-crystal model and a locally resonant elastic-metamaterial model and is shown to accurately predict the frequencies and Bloch mode shapes with a dramatic decrease in computation time in excess of two orders of magnitude.

Figure 1

Flowchart describing the BMS process. The thumbnail images show dotted lines for boundary DOF that are not included in the model, and solid lines for boundary DOF that are included in the model. The symbol $\mathbf{L}$ denotes a lattice vector of the periodic material.

Figure 2

Diagram of a free 2D unit cell indicating the locations of different DOF sets. The density, Young's modulus, and Poisson's ratio are denoted by ${\rho }_j,E_j$, and ${\nu }_j$, respectively, where $j=1$ for fiber and $j=2$ for matrix.

Figure 3

Dispersion diagram for full model (16 200 DOF) compared to BMS-IR model (100 DOF). The frequency is defined as $\Omega =\omega L\sqrt{E_2/{\rho }_2}$. Points A, B, and C, and the fourth branch along the $\mathrm{M}$–$\Gamma$ wave-vector path, are marked for later analysis.

Figure 4

Bloch mode shape for point C (see Fig. 3) is shown for the full model (top) and the BMS-IR model (bottom).

Figure 5

Frequency error (top) and Bloch mode-shape error (bottom) plotted against ratio of the reduced-model computation time to the full-model computation time for points marked in Fig. 3.

Figure 6

Reduced computation time as full-model size is increased. The dotted line shows the computation time for 100 FIM. The dash-dotted line shows the time to solve for 10 frequencies using the reduced model at one k point. Both lines are divided by the time to solve for 10 full-model frequencies at one k point. The reduction is based on BMS-IR.

Figure 7

Estimated reduced computation time and full computation time (top) and ratio of reduced to full computation times (bottom) as k point resolution increases. Results are for a full-model size of 54 450 DOF. The reduction is based on BMS-IR.

Figure 8

Dispersion diagram for full model (20 172 DOF) compared to BMS model (330 DOF). The frequency is defined as $\Omega =\omega L\sqrt{E/\rho }$, where $L,E$, and $\rho$ denote unit-cell side length, Young's modulus, and density, respectively. Points D, E, and F, and the fourth branch along the $\mathrm{M}$–$\Gamma$ wave-vector path, are marked for later analysis.

Figure 9

Bloch mode shapes for point F (see Fig. 8) is shown for the full model (top) and the BMS model (bottom).

Figure 10

Frequency error (top) and Bloch mode-shape error (bottom) plotted against ratio of the reduced-model computation time to the full-model computation time for points marked in Fig. 8.

Figure 11

Maximum error in $\mathrm{M}$–$\Gamma$ portion of fourth dispersion branch for RBME, BMS-IR, and simple BMS models versus ratio of the reduced-model computation time to the full-model computation time.

Figure 12

Maximum error in $\mathrm{M}$–$\Gamma$ portion of fourth dispersion branch for RBME, and BMS models versus ratio of the reduced-model computation time to the full-model computation time.

Figure 13

Schematic of performance map for BMS and RBME [17] fast band-structure calculation methodologies.