Lattice Boltzmann for linear elastodynamics: periodic problems and Dirichlet boundary conditions

Abstract

We propose a new second-order accurate lattice Boltzmann formulation for linear elastodynamics that is stable for arbitrary combinations of material parameters under a CFL-like condition. The construction of the numerical scheme uses an equivalent first-order hyperbolic system of equations as an intermediate step, for which a vectorial lattice Boltzmann formulation is introduced. The only difference to conventional lattice Boltzmann formulations is the usage of vector-valued populations, so that all computational benefits of the algorithm are preserved. Using the asymptotic expansion technique and the notion of pre-stability structures we further establish second-order consistency as well as analytical stability estimates. Lastly, we introduce a second-order consistent initialization of the populations as well as a boundary formulation for Dirichlet boundary conditions on 2D rectangular domains. All theoretical derivations are numerically verified by convergence studies using manufactured solutions and long-term stability tests.

Figures (10)

• Figure 1: D2Q4 velocity set using Miller indices, i. e. $\bar{1}:=-1$.
• Figure 2: Periodicity along the horizontal direction. The outgoing populations on the right side (red dashed arrows) are streamed back into the domain on the left side (red solid arrows), and vice versa from left to right (blue arrows).
• Figure 3: Visualization of node and index sets at the domain boundary. Interior nodes $\mathbb{G}\setminus\mathbb{B}$ -- black dots; boundary nodes $\mathbb{B}$ -- blue dots; normal streaming indices $\mathcal{V}\setminus\mathcal{D}_{\boldsymbol{x}}$ for each $\boldsymbol{x}\in\mathbb{G}$ -- black arrows; missing incoming populations $\mathcal{D}_{\boldsymbol{x}}$ for each $\boldsymbol{x}\in\mathbb{G}$ -- red solid arrows; outgoing populations $-\mathcal{D}_{\boldsymbol{x}}$ for each $\boldsymbol{x}\in\mathbb{G}$ -- red dashed arrows; boundary intersection points $\boldsymbol{x}_b$ -- green crosses.
• Figure 4: Convergence study with periodic boundary conditions. Comparison of displacement error for multiple material parameter combinations.
• Figure 5: Convergence study with periodic boundary conditions. Comparison of Cauchy stress error for multiple material parameter combinations.

Theorems & Definitions (13)

• Proposition 1: Consistency in the domain interior
• proof
• Proposition 2: Second-order consistent Dirichlet boundary formulation
• Proposition 2: Second-order consistent Dirichlet boundary formulation
• proof
• Definition 1: Pre-stability structure
• Definition 2: Weighted L2 grid norm
• Lemma 1
• proof
• Proposition 3: Stability with periodic boundary conditions