Adaptive Reduced-Basis Trust-Region Methods for Defect Identification in Elastic Materials

TL;DR

This paper introduces an adaptive reduced-basis trust-region method to efficiently identify defects in elastic materials by reducing computational complexity in high-dimensional hyperbolic systems.

Contribution

It extends reduced-basis and trust-region techniques to hyperbolic elastic wave problems for defect detection, improving computational efficiency and reliability.

Findings

The method achieves online-efficient surrogate models for forward and adjoint evaluations.

Numerical experiments demonstrate reliable defect detection in elastic materials.

The approach extends previous elliptic and parabolic problem techniques to hyperbolic systems.

Abstract

Monitoring the integrity of elastic structures using ultrasonic waves requires the efficient identification of material parameters from measured surface displacements. The displacement field is governed by Cauchy's equation of motion, i.e., an elastic wave equation. Consequently, defect localization leads to a high-dimensional spatial parameter identification problem for a hyperbolic system with given initial and boundary conditions. Stable parameter reconstructions typically rely on regularization techniques such as the iteratively regularized Gauss--Newton method (IRGNM). However, its practical application is computationally demanding due to the high-dimensional nature of the problem. To address this bottleneck, we propose a reduced-order modeling approach that simultaneously reduces the state and parameter spaces using adaptively constructed reduced-basis spaces. This yields online-efficient surrogate models for both the forward and adjoint evaluations required in derivative-based optimization. To ensure reliability, the IRGNM iteration is embedded into an adaptive, trust-region framework that provides accuracy of the reduced-order approximations. The approach extends our recent contributions, which focus on elliptic and parabolic problems, to the hyperbolic setting. We demonstrate the reliability and effectiveness of the method for defect detection through numerical experiments.