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Updated: Sep 27, 2025

Extracting Metrics for Three-dimensional Root Systems: Volume and Surface Analysis from In-soil X-ray Computed Tomography Data

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Solving ultrasonic ray tracing in parts with multiple material layers through Root-Finding methods.

Carmelo Mineo¹, Donatella Cerniglia², Ehsan Mohseni³

¹High Performance Computing and Networking Institute (ICAR), National Research Council (CNR), 90146 Palermo, Italy.

|April 11, 2022

Summary

Ultrasonic testing faces challenges with wave refraction in complex structures. This study optimizes ray-tracing for multi-layered materials, significantly reducing computation time for accurate inspection.

Keywords:

Iterative root-finding methods Multi-layered structures Refraction Wave propagation

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Area of Science:

Physics
Materials Science
Engineering

Background:

Ultrasonic testing, used since the 1930s, is advancing with modern electronics and software.
Wave refraction complicates ultrasonic inspection of geometrically complex parts and multi-layered materials.
Understanding ultrasound wave propagation is critical for interpreting inspection data.

Purpose of the Study:

To develop and present effective approaches for solving the ray-tracing problem in multi-layered structures.
To address the challenge of wave refraction in ultrasonic testing.
To improve the efficiency and accuracy of ultrasonic inspection data analysis.

Main Methods:

Implementing ray-tracing algorithms for multi-layered structures.
Utilizing the Newton-Raphson root-finding method for computational efficiency.
Comparing Newton-Raphson methods with bisection-based methods.

Main Results:

The Newton-Raphson method reduces computation time threefold compared to bisection methods.
A combined approach of Newton-Raphson and bisection iterations is proposed.
Accurate benchmarking validates the proposed ray-tracing solutions.

Conclusions:

The developed ray-tracing methods enhance ultrasonic inspection of complex geometries.
Optimized computation times enable more efficient material analysis.
Findings are applicable beyond ultrasonic testing to other wave propagation fields.