Exponential time propagators for elastodynamics

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 accurately evaluate the action of the exponential matrix on a vector. In particular, we use an apriori error estimate to predict the optimal Krylov 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 × larger time-steps than conventional time-marching schemes such as Newmark- β method. Computationally, it translates to a ∼ 1000 × and ∼ 10 − 100 × speed-up over conventional time-marching schemes for linear and nonlinear elastodynamics, respectively. • Exponential time propagators can enable fast and accurate long-time elastodynamic simulations. • We construct exponential time propagators for dynamical systems using Magnus expansion. • The exponential propagator is efficiently evaluated using a Krylov subspace approach.