Exponential time propagators for elastodynamics

TL;DR

This work introduces an adaptive Magnus-based exponential propagator for elastodynamics, reformulating the second-order spatially discretized equations into a first-order system and solving via a Magnus expansion. The method uses a Krylov subspace to efficiently apply the matrix exponential to a vector, with an a priori error estimate guiding subspace size to achieve a desired accuracy at each time step. Theoretical analysis proves second-order convergence for nonlinear problems and energy conservation plus symplecticity for linear dynamics, while numerical results across linear and nonlinear benchmarks demonstrate substantial speedups (up to 10–400×) and strong parallel scalability relative to conventional time-integration schemes. The approach yields accurate, long-time stable simulations for hyperelastic materials (SVK and Yeoh models) with large permissible time steps, highlighting its practical impact for fast elastodynamic computations and potential extensions to other dynamical systems.

Abstract

We propose a computationally efficient and systematically convergent approach for elastodynamics simulations. We recast the second-order dynamical equation of elastodynamics into an equivalent first-order system of coupled equations, so as to express the solution in the form of a Magnus expansion. With any spatial discretization, it entails computing the exponential of a matrix acting upon a vector. We employ an adaptive Krylov subspace approach to inexpensively and and accurately evaluate the action of the exponential matrix on a vector. In particular, we use an apriori error estimate to predict the optimal Kyrlov subspace size required for each time-step size. We show that the Magnus expansion truncated after its first term provides quadratic and superquadratic convergence in the time-step for nonlinear and linear elastodynamics, respectively. We demonstrate the accuracy and efficiency of the proposed method for one linear (linear cantilever beam) and three nonlinear (nonlinear cantilever beam, soft tissue elastomer, and hyperelastic rubber) benchmark systems. For a desired accuracy in energy, displacement, and velocity, our method allows for $10-100\times$ larger time-steps than conventional time-marching schemes such as Newmark-$β$ method. Computationally, it translates to a $\sim$$1000\times$ and $\sim$$10-100\times$ speed-up over conventional time-marching schemes for linear and nonlinear elastodynamics, respectively.

Figures (14)

• Figure 1: Geometry and finite-element mesh of the cantilever beam used for the analysis.
• Figure 2: Geometry and finite-element mesh of the plate used for the analysis.
• Figure 3: Subspace error for various subspace sizes ($m$) and time-step sizes ($\Delta t$). The hollow markers denote the actual error ($\varepsilon_m$) and the corresponding solid markers denote the respective error estimates ($\epsilon_m$).
• Figure 4: Subspace size, computational time (CPU-hrs) vs time-step size ($\Delta t$).
• Figure 5: Evolution of tip displacement with time computed using the exponential propagator (EP) and Newmark-$\beta$ methods.