Time‐parallel implicit integrators for the near‐real‐time prediction of linear structural dynamic responses

Abstract

SUMMARYThe time-parallel framework for constructing parallel implicit time-integration algorithms (PITA) is revisited in the specific context of linear structural dynamics and near-real-time computing. The concepts of decomposing the time-domain in time-slices whose boundaries define a coarse time-grid, generating iteratively seed values of the solution on this coarse time-grid, and using them to timeadvance the solution in each time-slice with embarrassingly parallel time-integrations are maintained. However, the Newton-based corrections of the seed values, which so far have been computed in PITA and related approaches on the coarse time-grid, are eliminated to avoid artificial resonance and numerical instability. Instead, the jumps of the solution on the coarse time-grid are addressed by a projector which makes their propagation on the fine time-grid computationally feasible while avoiding artificial resonance and numerical instability. The new PITA framework is demonstrated for a complex structural dynamics problem from the aircraft industry. Its potential for near-real-time computing is also highlighted with the solution of a relatively small-scale problem on a Linux cluster system.

On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method

“…To fix the stability issue, [10] proposed to improve the coarse solver by reusing information computed at all previous iterations. They applied this idea to the linear hyperbolic problems in structural dynamics.…”

“…To demonstrate the performance of the Krylov subspace parareal method, we use it to solve the linear advection equation, (10). In Figure 4 (left) we show the L ∞ -error at T = 10 against the number of iterations.…”

“…We begin by considering the performance of the reduced basis parareal method and illustrate that it is stable for the 1-D linear advection equation (10). The spatial and temporal discretizations are the same as in Section 2 and the parameters as in (11) are used.…”

“…Another attempt, the Krylov subspace parareal method, builds a new coarse solver by reusing all information from the corresponding fine solver at previous iterations. The stability of this approach was demonstrated for linear problems on linear structural dynamics [10] and a linear 2-D acoustic-advection system [21]. However, the Krylov subspace parareal method appears to be limited to linear problems.…”

Compressed Sensing and Time-Parallel Reduced-Order Modeling for Structural Health Monitoring Using a DDDAS

“…The new aspect of this effort is the generalization to nonlinear second-order hyperbolic problems of the PITA (Parallel Implicit Time-integration Algorithms) framework developed in [4,5] for linear time-dependent problems.…”