Structure-preserving dynamical low-rank approximation for parametric elastic guided waves
2026-06-29 • Computational Engineering, Finance, and Science
Computational Engineering, Finance, and Science
AI summaryⓘ
The authors focus on monitoring structural health using special waves that travel and reflect in materials. They address how to make simulations of these waves faster and more efficient by using a smart method called Dynamical Low Rank Approximation (DLRA), which keeps the energy behavior accurate. They build a reduced model in two phases: first preparing it with training data, then using a formula to predict wave behaviors quickly without heavy computations. Their tests on a 2D elastic wave problem show this method works well, saving time and maintaining accuracy over long runs.
Elastic guided wavesStructural Health MonitoringProjection-based reduced order modelingDynamical Low Rank ApproximationHamiltonian structureSymplectic reduced basisDispersive wavesParametric modelingEnergy conservation
Authors
Dimitri Goutaudier
Abstract
Elastic guided waves are widely used in Structural Health Monitoring (SHM). In many-query settings, the computational cost of high-fidelity simulations motivates the use of projection-based reduced order modeling (ROM). However, the transport-dominated and dispersive nature of guided waves challenges static linear subspaces. In addition, preserving the Hamiltonian structure of the equations for energy conservation necessitates dedicated projection techniques. While the Dynamical Low Rank Approximation (DLRA) has proven effective for other wave equations, its application to elastic guided waves in SHM has remained unexplored. In this work, we introduce a structure-preserving parametric ROM framework that leverages the DLRA in an off-line/on-line strategy. During the off-line stage, a time-dependent symplectic reduced basis is constructed from training simulations. For a simplified class of parameter dependencies, we derive a closed-form solution of the nonlinear basis evolution equation. This analytical result yields a closed-form, energy-preserving reduced propagator during wave propagation, eliminating on-line time integration after the loading phase. We validate our approach on a 2D elasticity problem featuring dispersive guided waves interacting with a damage. The results demonstrate high compression ratios (rank $\sim 10-30$), low full field reconstruction errors ($\sim 10^{-3}-10^{-2}$), speedups of two to three orders of magnitude, and excellent long-time energy conservation.