**1. Introduction**

Quantitative non-destructive inspection is important to ensure the reliability and safety of structures such as aircraft and automobiles [1]. Among the non-destructive inspection techniques, ultrasonic inspection has been widely used because ultrasonic waves are highly sensitive to a damaged part and propagate over long distances. The importance of quantitative ultrasonic inspection has been described for more than 30 years [2].

To undertake quantitative evaluation by ultrasonic inspection, it is essential to develop both the measurement techniques and data analyses. In general, ultrasonic signals contain reflected waves, diffracted waves, and mode-converted waves, and some kinds of ultrasonic waves such as Lamb waves and Love waves have dispersive nature [3–5], which makes ultrasonic waveforms challenging to interpret. Consequently, evaluation of the damage is dependent on the skills of engineers, making misreading signals and false recognition of defects inevitable. Furthermore, the conventional ultrasonic inspection process is not automated; thus, the inspections require a lot of time and labor to scan the whole structure. Numerous studies [4,6–13] have been reported about new techniques and data analyses to overcome these difficulties. For example, ultrasonic arrays [6] have improved the inspection quality and have reduced the inspection costs by performing beam steering with a wide viewing angle through controlled transmission of multiple elements. Acoustic emission detection using a fiber-optic

sensor and mode analysis [7–9] has also been developed to achieve a quantitative evaluation of the damage in composite laminates. The wavelet-transform has frequently been incorporated with the signal processing to analyze dispersive waves [4,10,11]. Furthermore, some inverse analyses [4,12,13] have been presented to estimate the damage quantitatively using the artificial neural network and genetic algorithm.

Aiming at further damage visibility and operability, a visualization method of ultrasonic wave propagation [14–29] has been developed. In this method, ultrasonic waves are generated by illuminating a specimen surface with a pulsed laser and are received by a fixed transducer [14]. Based on the reciprocity of wave propagation [15], the amplitude of each waveform at a particular time plotted in a contour map yields a moving diagram of the wave propagation from the receiver. The feasibility of this method was demonstrated by applying it to a crack and an artificial hollow in metallic materials [15], delamination in carbon fiber reinforced plastic (CFRP) laminates [16–20], a crack in welded steel plates [21], and disbonds in adhesively bonded CFRP/aluminum joints [22]. Frequency and/or wavenumber domain analysis using the Fourier- or wavelet-transform was introduced into ultrasonic propagation imaging to easily interpret the visualized results of wave propagation by isolating a specific frequency mode [23–25]. Moreover, a fully non-contact ultrasonic inspection [26–28] was demonstrated by replacing a fixed transducer with a laser Doppler vibrometer, and this method removed ringing due to the resonance of the piezoelectric transducer and made it easy to interpret scattered waves [28].

Although the visualization method of ultrasonic propagation has high damage visibility and excellent operability, it is difficult to evaluate the damage quantitatively. An efficient automatic ultrasonic image analysis has been presented using deep learning [29], but at the moment, the main target of this method is automated damage detection, not quantitative evaluation. A moving diagram of wave propagation includes wave signals at all illuminating points. Therefore, appropriate analysis for all wave signals will have the potential to acquire quantitative information about the damage.

Topology optimization [30] will be suitable for that purpose, but to our knowledge, there are few studies that apply it to damage identification. In topology optimization for structural design [31–33], a design domain is discretized by finite elements, and the material density distribution is assigned as design variables. Similarly, in damage identification problems, the damage severity is the design variable instead of the material density, and the damage distribution is inversely estimated by reproducing an input phenomenon of focus. Based on this idea, Lee et al. [34] demonstrated numerical examples for estimating damage in thin plates and beam models, taking resonant and anti-resonant frequencies as an objective function. Some numerical examples that focused on natural frequencies were also reported [35–37]. Niemann et al. [38–40] estimated the approximate location of the damage in CFRP laminates after impact tests. However, this damage identification focusing on frequency characteristics was not very accurate. The reason for this is the fact that frequency characteristics are not sufficiently sensitive to damage.

This study proposes a damage identification method using the visualization technique of ultrasonic wave propagation. To this end, we incorporate topology optimization with a moving diagram of wave propagation, having high sensitivity to damage. The feature of wave propagation is reproduced in the analytical model by optimizing the distribution of the damage parameters. As a result, quantitative information about the damage is estimated. This study is the first attempt to integrate the inverse analysis based on topology optimization with the ultrasonic imaging inspection. The method is first proposed, and its feasibility is then verified using a two-dimensional case of known damage location and size.
