# Simulation of ultrasonic inspection of concrete using finite elements and analytical models

Nouhayla Khalid, Manda Ramaniraka, Vincent Dorval, Jean François Chaix, Michel Darmon

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Nouhayla Khalid, Manda Ramaniraka, Vincent Dorval, Jean François Chaix, Michel Darmon. Simulation of ultrasonic inspection of concrete using finite elements and analytical models. e-Journal of Nondestructive Testing, 2025, NDT-CE 2025 - The International Symposium on Nondestructive Testing in Civil Engineering Izmir, Türkiye, September 24-26, 2025, 30 (10), 10.58286/31653. cea-05451107

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# Simulation of ultrasonic inspection of concrete using finite elements and analytical models

Nouhayla KHALID<fnref n="1" />, Manda RAMANIRAKA<fnref n="2" />, Vincent DORVAL<fnref n="1" />, Jean-François CHAIX<fnref n="2" />, Michel DARMON<fnref n="1" />

## Abstract

Ultrasonic non-destructive testing (NDT) simulation plays a key role in the evaluation of concrete structures. Concrete is a dense composite medium whose heterogeneities (aggregates, porosity, micro cracks) cause wave dispersion and attenuation, thus affecting inspection techniques. Classical approaches rely on the study of the coherent wave, characterized by its phase velocity and attenuation. Since 3D numerical models are too computationally expensive, improving existing simplified analytical models is essential. This study proposes a new 2D homogenization model better suited for dense media. The improvement lies in a more accurate description in analytical models of the pair-correlation function of aggregate positions. The new model is compared to a 2D finite elements code integrated in CIVA NDT simulation platform. Results show good agreement between the developed analytical model and numerical simulations. Further work will explore the effects of aggregate granulometry and other material configurations to ultimately extend the model to 3D and improve ultrasonic simulation accuracy for concrete.

**Keywords:** ultrasonic NDT, concrete, multiple scattering, homogenization, CIVA simulation

## 1. Introduction

Concrete is the most widely used construction material, spanning applications from civil engineering to nuclear infrastructures, residential buildings, and skyscrapers. Inspecting concrete structures is essential for detecting and monitoring potential damage. Unlike destructive techniques like coring, which may compromise structural integrity, non-destructive testing (NDT) offers reliable assessment without altering the material.

Among NDT techniques, ultrasonic inspection is widely used due to its ability to characterize concrete's mechanical properties and track aging and its modelling [1] is essential for assess inspections feasibility. However, due to concrete's intrinsic heterogeneity, ultrasonic wave propagation is affected by multiple scattering.

Homogenization theory enables the study of the coherent wave, characterized by its phase velocity and attenuation. This approach replaces the heterogeneous medium with an equivalent homogeneous one, simplifying wave propagation modeling. Dynamic models [2] extract the frequency-dependent effective wavenumber, where the real part represents the phase velocity and the imaginary part represents scattering-induced attenuation. To characterize wave propagation in a strongly heterogeneous medium like concrete, accounting for various scatterers configurations is crucial. Foldy [3] introduced the concept of configurational averaging to statistically consider scatterers positions, later improved by Waterman and Truell [4].

However, these models do not account for scatterers correlation effects. Fikioris and Waterman [5] proposed a correction incorporating the minimum distance between inclusions, known as

<footnote n="1">¹ Université Paris-Saclay, CEA, List, F-91120 Palaiseau, France; e-mail: michel.darmon@cea.fr.</footnote>

<footnote n="2">² LMA, Aix-Marseille Université, CNRS, Centrale Méditerranée, France</footnote>
the Hole Correction pair-correlation function. Later, Linton and Martin [6] developed a model considering cylindrical scatterers, and Conoir and Norris [7] extended it to elastic media.

Although experimentally validated in fluids or solid matrices with air inclusions, their application to solid matrices with solid scatterers remains less studied. This study focuses on evaluating the Conoir and Norris (CN) model for cylindrical aggregates in a cement matrix, analyzing the correlation function's relevance, and exploring improved representations of inclusions distribution in concrete.

## 2. Ultrasonic Wave Propagation Modeling in Concrete

### 2.1 Multiple Scattering and Homogenization Models

Homogenization theory defines an equivalent medium from a heterogeneous one, where waves behave as coherent waves. Figure 1 illustrates the transition from a heterogeneous medium to a simplified random inclusion distribution and finally to an equivalent homogeneous (geometrical) medium. This enables easy extraction of effective parameters from disordered or heterogeneous media.

<figure><img src="image_3.png" /><figcaption>Figure 1. Homogenization theory</figcaption></figure>

In this study, the CN model is analyzed and applied to randomly distributed N cylindrical aggregates within a mortar matrix. The scatterers are identical with radius a. The exclusion distance b is the minimum allowed center-to-center distance between adjacent particles (Figure 2).
<figure><img src="image_4.png" /><figcaption>Figure 2. Hole correction pair-correlation function used in the CN model</figcaption></figure>

The CN model, developed for uniformly distributed cylindrical scatterers, uses the Hole Correction pair-correlation function to impose equal probability for distances beyond the exclusion distance b. To ensure non-overlapping of hard particles, the exclusion distance must be at least equal to the scatterer diameter a.

The model aims to calculate the effective complex wavenumber for an incident longitudinal wave propagating in the equivalent medium, up to second order in density. The effective characteristics - phase velocity and attenuation- are derived from the complex wavenumber as a function of frequency. The wavenumber expression includes Bessel and Hankel functions and modal coefficients derived from boundary condition resolution for elastic cylindrical scatterers.

The aim of the CN model [7] is thus to calculate the effective complex wavenumber $k_{eff}$ for a longitudinal incident wave propagating in the equivalent homogeneous medium up to second order on density $n_0 = N/V$, with N the number of scatterers embedded in a volume V. An isotropic medium where L longitudinal and T transverse waves may propagate is considered. The effective complex wavenumber $k_{eff}$ is then expressed as follows:

$$k_{eff}^2 = k_L^2 + \delta_1 n_0 + \delta_2 n_0^2 + \dots, \qquad (1)$$

$k_L$ is the wavenumber in the cement matrix, and $\delta_1$ and $\delta_2$ are defined as follows:

$$\delta_1 = -4i \sum_{n=-\infty}^{+\infty} T_{n}^{LL}, \qquad (2)$$

$$\delta_2 = \frac{-16}{k_L^2} \sum_{n,m=-\infty}^{+\infty} D_{m-n}^0 (k_L) T_{n}^{LL} T_{m}^{LL} - \frac{8i\pi}{k_L^2 - k_T^2} \sum_{p,q=-\infty}^{+\infty} N_{p-q}^T (k_L) T_{p}^{TL} T_{q}^{LT}, \qquad (3)$$
where:

$$D_p^0(k) = -\frac{i\pi}{4} \left[ ((kb)^2 - p^2)J_p(kb)H_p^{(1)}(kb) + (kb)^2J'_p(kb)H_p^{(1)'}(kb) \right] \quad (4)$$

and

$$N_p^\alpha(k) = -kbJ'_p(kb)H_p^{(1)}(k_\alpha b) - k_\alpha bJ_p(kb)H_p^{(1)'}(k_\alpha b), \\ \alpha \in \{L, T\}. \quad (5)$$

$J_n(x)$ and $H_n^{(1)}(x)$ are respectively the Bessel and Hankel functions of the first kind. For cylindrical diffusers, the modal coefficients $T_n$ are the T-matrix coefficients [8]. The latter is the matrix that relates the scattered field to the incident field. The expression of the T-matrix in the case of scattering by elastic cylindrical scatterers in an elastic medium is calculated by solving the boundary conditions problem [9].

## 2.2 Numerical Modeling with CIVA and SPECFEM

To assess the CN model for concrete, phase velocity and attenuation are computed and compared to two numerical finite elements tools: CIVA FEM 2D and SPECFEM 2D. CIVA is a NDT/SHM simulation platform with ultrasonic models [10,11], handling various kinds of materials [12,13] and intensively validated [14–16].

CIVA FEM 2D is a transient elastodynamic FEM module within CIVA, capable of simulating crack-type defects with a more precise prediction [17,18] of head waves than semi-analytical models [19–24] developed for canonical shapes. It is also adapted for concrete modelling [25]. A new 2D/3D version exists [26] for concrete, though not used here.

SPECFEM 2D, based on the spectral elements method [27], has also been applied to wave propagation in concrete [28]. Both tools (CIVA 2D FEM and 2D SPECFEM) employ high-order 2D finite elements.

## 2.3 Concrete Properties

Concrete is modeled as a biphasic material with mortar matrix and aggregates. Table 1 shows the density and wave velocities of both phases. Cylindrical aggregates with 6 mm radius are considered, with a volume fraction up to 42%
<table><caption>Table 1. Concrete Properties</caption><thead><tr><td>MATERIAL</td><td>ρ(Kg.m<sup>-3</sup>)</td><td>c<sub>L</sub>(m.s<sup>-1</sup>)</td><td>c<sub>T</sub>(m.s<sup>-1</sup>)</td></tr></thead><tbody><tr><td>MORTAR</td><td>2050</td><td>3950</td><td>2250</td></tr><tr><td>AGGREGATES</td><td>2610</td><td>4300</td><td>2475</td></tr></tbody></table>

### 3. CN Model Evaluation for Concrete Structures

The CN model evaluation (Eq. 1) is performed for aggregate concentrations of 12%, 24%, and 42%, with the same exclusion distance $b=2a+0.5b = 2a + 0.5b=2a+0.5$ mm as in numerical models. Numerical simulations cover a 400 mm × 600 mm domain.

Effective attenuation and phase velocity are compared to CIVA FEM 2D and SPECFEM 2D simulations. Good agreement is observed for 12% and 24% concentrations. However, discrepancies increase at 42%, with unphysical negative attenuation in the low-frequency regime and underestimation in the high-frequency range.

<figure><img src="image_5.png" /></figure>
<figure><img src="image_6.png" /><figcaption>Figure 3: attenuation (a) and phase velocity (b) versus frequency for a longitudinal incident wave. Comparison between CN (solid lines), 2D CIVA FEM (dashed lines), and 2D SPECFEM (stars).</figcaption></figure>

**4. Analysis of Scatterers Correlation**

Preliminary studies showed that neither applying low-frequency approximations to the CN model [29] nor modifying exclusion distance improves its validity at high concentrations [30].

Thus, the pair-correlation function—simplified in the CN model to impose equal probability—is revisited for a more realistic description of inter-scatterers distances. A new analytical homogenization model generalizes CN for arbitrary correlation functions [31,32].

In low-frequency regimes, the CN model predicts negative attenuation at high concentrations. The new model, using a realistic correlation function, corrects these nonphysical values. Very good agreement with CIVA FEM simulations is achieved for both attenuation and phase velocity.

<figure><img src="image_7.png" /></figure>
<figure><img src="image_8.png" /><figcaption>Figure 4: attenuation (a) and phase velocity (b) versus frequency for a longitudinal incident wave. Comparison between new analytical model (solid lines), CN (dashed lines), and 2D CIVA FEM (stars).</figcaption></figure>

**4.7. Conclusions**

The propagation of longitudinal waves in dense heterogeneous media (concrete) has been analyzed. The CN model's predictions were compared with CIVA FEM and SPECFEM. At low frequencies and high concentrations, CN predicts unphysical negative attenuation.

This issue stems from oversimplified pair-correlation assumptions (Hole Correction). Enhancing this description with a more realistic distance distribution corrects the problem. The modified CN model we have proposed [31,32] accurately takes into account scatterers correlation and eliminates the negative attenuation artifact.

Further validation, including the effect of aggregate size distribution, is ongoing [31].

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