Sequential subspace optimization for recovering stored energy functions in hyperelastic materials from time-dependent data

TL;DR

This paper introduces a sequential subspace optimization method to efficiently recover stored energy functions in hyperelastic materials from time-dependent data, significantly speeding up the inverse problem solution in structural health monitoring.

Contribution

The paper presents a novel SESOP-based iterative method that accelerates the computation of stored energy functions in hyperelastic materials compared to traditional approaches.

Findings

SESOP method outperforms Landweber iteration in speed

Numerical tests confirm significant acceleration

Method effectively recovers stored energy functions

Abstract

Monitoring structures of elastic materials for defect detection by means of ultrasound waves (Structural Health Monitoring, SHM) demands for an efficient computation of parameters which characterize their mechanical behavior. Hyperelasticity describes a nonlinear elastic behavior where the second Piola-Kirchhoff stress tensor is given as a derivative of a scalar function representing the stored (strain) energy. Since the stored energy encodes all mechanical properties of the underlying material, the inverse problem of computing this energy from measurements of the displacement field is very important regarding SHM. The mathematical model is represented by a high-dimensional parameter identification problem for a nonlinear, hyperbolic system with given initial and boundary values. Iterative methods for solving this problem, such as the Landweber iteration, are very time-consuming. The reason is the fact that such methods demand for several numerical solutions of the hyperbolic system in each iteration step. In this contribution we present an iterative method based on sequential subspace optimization (SESOP) which in general uses more than only one search direction per iteration and explicitly determines the step size. This leads to a significant acceleration compared to the Landweber method, even with only one search direction and an optimized step size. This is demonstrated by means of several numerical tests.