An efficient and accurate semi-implicit time integration scheme for dynamics in nearly- and fully-incompressible hyperelastic solids

Abstract

The choice of numerical integrator in approximating solutions to dynamic partial differential equations depends on the smallest time-scale of the problem at hand. Large-scale deformations in elastic solids contain both shear waves and bulk waves, the latter of which can travel infinitely fast in incompressible materials. Explicit schemes, which are favored for their efficiency in resolving low-speed dynamics, are bound by time step size restrictions that inversely scale with the fastest wave speed. Implicit schemes can enable larger time step sizes regardless of the wave speeds present, though they are much more computationally expensive. Semi-implicit methods, which are more stable than explicit methods and more efficient than implicit methods, are emerging in the literature, though their applicability to nonlinear elasticity is not extensively studied. In this research, we develop and investigate the functionality of two time integration schemes for the resolution of large-scale dynamics in nearly- and fully-incompressible hyperelastic solids: a Modified Semi-implicit Backward Differentiation Formula integrator (MSBDF2) and a forward Euler / Semi-implicit Backward Differentiation Formula Runge-Kutta integrator (FEBDF2). We prove and empirically verify second order accuracy for both schemes. The stability properties of both methods are derived and numerically verified. We find FEBDF2 has a maximum time step size that inversely scales with the shear wave speed and is unaffected by the bulk wave speed -- the desired stability property of a semi-implicit scheme. Finally, we empirically determine that semi-implicit schemes struggle to preserve volume globally when using nonlinear incompressibility conditions, even under temporal and spatial refinement.

Figures (11)

• Figure 1: Maximum $\omega$ obtained for varying values of $c$, $\lambda$, and $\Delta t$. Values for $c$ were approximated as scalar multiples of $\lambda$.
• Figure 2: Maximum time step size, $\Delta t_{max}$ for FEBDF2 calculated with various values for $\lambda$ and $c$.
• Figure 3: Force diagrams for the three unit square simulations. Blue edges indicate homogeneous Dirichlet boundary conditions. (a) depicts a unit square equipped with a constant body force with magnitude $a=25\text{g cm}/ \text{s}^3$. (b) depicts a equipped with an initial velocity field defined by Equation (\ref{['initial_velocity']}). (c) depicts a unit square with a downward constant traction applied to the top face, where $a=-2.5\text{dyn}/\text{cm}^2$.
• Figure 4: Convergence in behavior for deformations of a unit square as computed by MSBDF2. Each row depicts $\|\varepsilon\|_{L^p}$ values obtained from body force, initial velocity, and traction respectively.
• Figure 5: Convergence in behavior for deformations of a unit square as computed by FEBDF2. Each row depicts $\|\varepsilon\|_{L^p}$ values obtained from body force, initial velocity, and traction respectively.