# Ultrasonic scattering in polycrystalline materials with elongated grains: A comparative 3D and 2D theoretical and numerical analysis

Juan Camilo Victoria Giraldo, Bing Tie, Jérôme Laurent, Alain Lhémery, Denis Solas

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Juan Camilo Victoria Giraldo, Bing Tie, Jérôme Laurent, Alain Lhémery, Denis Solas. Ultrasonic scattering in polycrystalline materials with elongated grains: A comparative 3D and 2D theoretical and numerical analysis. Ultrasonics, 2025, pp.107642. 10.1016/j.ultras.2025.107642. hal-05004948

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Submitted on 25 Mar 2025

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4 Abstract

In this paper, a previously developed theoretical model for the ultrasonic elastic wave scattering, based on the Stanke and Kino model and applicable to both 2D and 3D single-phase untextured polycrystals, is extended to microstructures with elongated grains. The effect of elongated grains on wave attenuation and phase velocity induced by scattering is investigated, highlighting similarities and discrepancies between the 2D and 3D cases. Additionally, 2D and 3D finite element (FE) models are developed to compare and validate the theoretical predictions under fixed assumptions. The morphology of the numerical polycrystalline samples is characterized using a multi-exponential two-point correlation (TPC) function which, when incorporated with the theoretical model, enables a more direct and accurate comparison. The FE models demonstrate excellent quantitative agreement with the theoretical predictions and, moreover, support the wave propagation's directional dependency in the stochastic scattering region and the 2D-3D dimensionality dissimilarities in the Rayleigh region. It is shown that 2D attenuation can predict 3D behavior in the stochastic limit and provide insights into the estimation of 3D grain morphology in the Rayleigh limit.

Keywords: ultrasonic scattering, attenuation, phase velocity, polycrystalline materials, elastic
wave propagation

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1. Introduction

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The scattering of ultrasonic waves in inhomogeneous media is due to their interactions with local heterogeneities. In polycrystalline materials, these inhomogeneities often manifest themselves as crystallographic misorientation between grains and/or the presence of multiple phases. Knowledge about the influence of such inhomogeneities (grain size and shape, degree of anisotropy, etc.) on the ultrasonic scattering, such as ultrasonic attenuation and alteration of phase velocity, is essential for the advancement of non-destructive testing (NDT) techniques. Indeed, the correlation between ultrasonic scattering and microstructural properties can be used to characterize key material properties [1, 2].

Although ultrasonic scattering in polycrystalline materials has already been extensively studied in 3D and is known as a dimension-dependent physical phenomenon [3, 4, 5], it remains valuable to investigate it in 2D. Certain experimental settings using line transducers can be approximated as 2D [6]; therefore, the interest of 2D theoretical approaches is able to interpret and predict such experimental results. Moreover, the insights gained from 2D investigations can contribute to a deeper understanding of the distinctive scattering characteristics of 3D phenomena [3, 7, 4, 5]. Also, with the rise of data-driven models and inversion methodologies for microstructure characterization, 2D numerical models remain noteworthy due to their reduced computational cost in comparison

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<footnote>*Corresponding author
Email address: bing.tie@centralesupelec.fr (Bing Tie)</footnote>
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with 3D models and still have their role to play in defining, choosing, and developing physics-based strategies and algorithms.

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To date, a number of 3D theoretical models have been developed for the grain scattering-induced attenuation. They are based on the decoupling of the two-point covariance of the elastic moduli into an eight-rank covariance tensor and a spatial two-point correlation (TPC) function. This allows to treat independently the variations of crystallographic orientation and grain morphology. Previous studies, which focused on the case of randomly oriented cubic crystals and equiaxed grains, introduced an isotropic spatial TPC function. In their seminal work, Stanke and Kino [8] proposed a unified theory based on the Karal and Keller [9] second-order approximation on the degree of inhomogeneity of the medium, and capable of accounting for some degree of multiple scattering. Their work led to the explicit formulas for complex longitudinal (L) and transverse (T) wavenumbers, which are valid up to the geometric region. Their analysis demonstrated the well-known power dependence laws of the attenuation coefficient $\alpha$ on frequency $f$ and grain size $d$ (grain diameter). For frequency, a fourth power ($f^4$) dependence in the Rayleigh scattering region, a square power ($f^2$) dependence in the stochastic region, and a frequency-independent behavior in the geometric region. For grain size, a third-power dependence ($d^3$), i.e., grain volume, in the Rayleigh scattering region, a linear power dependence ($d$) in the stochastic region, and an inversely proportional dependence ($d^{-1}$) in the geometric region. In contrast, Weaver [10] based his work on Dyson's equation, which accounts for multiple scattering by the First-Order-Smoothing Approximation of the "self-energy" operator. Additionally, Weaver [10] obtained the same explicit power law behaviors as Stanke and Kino [8] but limited up to the stochastic region due to the introduced Born approximation.

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Both Stanke & Kino and Weaver models also led to the development of extensions considering elongated grains mainly of ellipsoidal shape. For instance, Ahmed et al. [11] built upon the work of Stanke and Kino [8] by introducing a transverse isotropic TPC function for cubic randomly oriented crystals in the Rayleigh scattering region. The implementation of such a function introduces an anisotropy source for the resulting attenuation. By considering an ellipsoid of rotation they already were able to find two distinct grain shape dependencies. In the Rayleigh scattering region, they noticed a direct grain volume dependency, thus identical to the aforementioned equiaxed grain case, and in the stochastic region, a proportional dependence of the attenuation with the grain's aspect ratio. That is, higher attenuation is found with L-waves propagating along the grain's longest axis than along its shortest axis. Yang et al. [12] implemented a general anisotropic TPC function accounting for ellipsoidal grains in Weaver [10] equations and thus obtained a closed-form expression using the Born approximation. Then, they presented attenuation expressed in the Rayleigh and stochastic limits, which show clearly in the Rayleigh limit the $f^4$ and effective grain volume dependence, and in the stochastic limit the $f^2$ and ellipsoidal radius dependence, with the grain's radius in the wave propagation direction. Huang et al. [13] adopted Weavers's approach but opted to solve the non-linear equation for the complex wave number. This strategy enabled them to circumvent the frequency range limitation. Also, Calvet and Margerin [14] proposed a spectral methodology inspired by the work of Weaver [10] and Yang et al. [12], obtaining the same frequency and grain shape dependencies as previous works with results up to the geometric scattering region. These authors' approach is noteworthy in that it considers a transverse isotropic Green function that allows the T-wave to be split into in-plane and out-of-plane components, and reveals a discrepancy in attenuation and phase velocity between these components as the wave propagates perpendicular to the elongated axis. Regarding the phase velocity, Huang et al. [15] were able to show a grain's elongation dependency in the quasi-static limit, encompassing the stochastic and geometric scattering regions. They were able to show in the quasi-static limit a lower phase velocity along the grain's short axis than along the elongated axis. Instead, at the geometric scattering region, this behavior is inverted. Other methodologies for estimating the elastic wave scattering have been recently used. Boström and Ruda [16] suggested utilizing the transition matrix relating the scattered and incident waves in the presence of an anisotropic scatterer surrounded by a reference isotropic medium. Initially in 2D, Boström and Ruda [16] considered the scatterer to be circular and was subsequently extended to 3D by assuming a spherical scatterer [17].

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Numerical modeling of ultrasonic waves in polycrystalline materials is an alternative to theoretical
modeling and experimental testing. Theoretical approaches are limited by the assumptions and approximations required to derive explicit models from generally unsolvable equations. In contrast, numerical simulations employing grain-scale representation of the polycrystalline microstructure through the finite element (FE) method are able to capture the full wave multiple scattering without any simplifying assumptions. Recently, an increasing number of studies have been dedicated to employing the FE method to simulate polycrystalline models with equiaxed grains [18, 19, 20] or elongated grains [15, 13] to estimate the wave phase velocity dispersion and the wave amplitude attenuation. These works have found excellent agreement between their 3D analytical results and their 3D FE results up to the stochastic scattering region. Nevertheless, there is a lack of comprehensive studies assessing the numerical and theoretical comparison in 2D, especially for the case of elongated grains. The works of Van Pamel et al. [4, 5], particularly the numerical simulations in 2D microstructures with equiaxed grains, revealed, regarding the attenuation, a power law of $f^3 d^2$ in the Rayleigh scattering region that differs from the 3D case. Additionally, these simulations demonstrated a lower dispersion in L-wave phase velocity in comparison with the 3D velocity. Bai et al. [21, 22] performed 2D FE simulations with a time-discontinuous space-time Galerkin method to evaluate attenuation and backscattering coefficients. Their findings, especially regarding the attenuation, were consistent with those previously reported by Van Pamel et al. [5]. Furthermore, they generalized Stanke & Kino's model by inserting the 2D Green's function to develop a unified explicit formula of the attenuation in the 2D and 3D cases and to have an analytical comparison between the 3D and 2D elastic wave scattering mechanisms [3]. Excellent comparisons were obtained between 2D theoretical and numerical results. Subsequently, the same research team applied the aforementioned developed theoretical model to propose a multiparametric optimization method to characterize, from attenuation data, the grain size distribution of microstructures with a bimodal size distribution using a space discontinuous Galerkin finite element method [7].

As outlined above, very few studies have addressed the theoretical aspects and numerical modeling necessary to evaluate and compare 2D and 3D cases within a unified framework, making it a topic of significant interest for the authors, as evidenced by our study on bimodal microstructures. In this work, Bai et al. [3] model for bulk waves is recalled and then extended to the general case of elongated grains of ellipsoidal shape for the 2D and 3D cases. The extended theoretical model is then applied to 2D and 3D studies of ultrasonic attenuation and phase velocity. Our main objective is to improve the understanding and the modeling of the effects of the shape and orientation of elongated grains, thereby gaining insights into the feasibility of their inverse characterization and into the usefulness of attenuation and phase velocity in such an inversion.

The outline of the paper is as follows. In Section 2, we present the proposed theoretical framework extended to consider elongated grains of ellipsoidal shape. Then, Section 3 presents the FE models and the methodology followed to estimate the wave attenuation and phase velocity. Finally, in Section 4, the theoretical and numerical results with special interest in the dimensionality analysis between the 2D and 3D elastic scattering behavior are presented and discussed.

## 2. Theoretical ultrasonic elastic wave scattering in 2D and 3D polycrystals with elongated grains

The model of Bai et al. [3] was developed from the theoretical basis of the unified theory of Stanke and Kino [8]. The objective here is to recall its theoretical framework and extend it to take into account microstructures with elongated grains.

Considering an ensemble of possible inhomogeneous media composed of the same single-phase polycrystalline material and occupying $\Omega \subset \mathbf{R}^{\text{dim}}$, $\text{dim} = 2, 3$. The polycrystalline material is defined by a position-dependent elasticity tensor $C(\mathbf{x})$ and a constant density $\rho$. The governing elastodynamic wave equation is known to be:

$$L(\mathbf{u}(\mathbf{x}, \omega)) = \rho\omega^2\mathbf{u}(\mathbf{x}, \omega) + \nabla_x(C(\mathbf{x}) : \varepsilon(\mathbf{u}(\mathbf{x}, \omega))) = \mathbf{0}. \quad (1)$$

We can then seek a solution by splitting the wave equation operator into two parts, one which is also a wave operator but defined in a homogeneous reference medium, and the other which is the remaining part, related to the deviation of the original heterogeneous medium from the reference medium.
With the assumptions of a single-phase material, the deviation here is $\delta C(\mathbf{x}) = C(\mathbf{x}) - C^0$, with the elasticity tensor $C^0$ of the equivalent homogeneous medium chosen as the Voigt average [9], which ensures $\langle \delta C(\mathbf{x}) \rangle = 0$. Furthermore, assuming a randomly oriented crystallographic texture, the reference medium governed by $C^0$ is isotropic.

The idea is then to find the ensemble-average solution $\langle \mathbf{u}(\mathbf{x}) \rangle$ by assuming it remains a plane wave, i.e., $\langle \mathbf{u}(\mathbf{x}) \rangle = U \exp(ik\hat{\mathbf{k}} \cdot \mathbf{x})$, with $k$ the wavenumber, $\hat{\mathbf{k}}$ the unit wavevector, and $U$ the polarization vector. After considering the aforementioned simplifications in the second-order Karal and Keller's approximation [9], we obtain the following non-linear eigenvalue problem:

$$ \Gamma(k\hat{\mathbf{k}}) \cdot U = \rho\omega k^{-2}U, \qquad (2) $$

defined in terms of a perturbed acoustic tensor $\Gamma(k\hat{\mathbf{k}})$, which can also be split into a homogeneous part $\Gamma^0(\hat{\mathbf{k}})$ and a deviation part $\delta\Gamma(k\hat{\mathbf{k}})$:

$$ \Gamma(k\hat{\mathbf{k}}) \cdot U = \Gamma^0(\hat{\mathbf{k}}) \cdot U + \delta\Gamma(k\hat{\mathbf{k}}) \cdot U, \qquad (3a) $$

$$ \Gamma^0(\hat{\mathbf{k}}) \cdot U = (C^0 : (\hat{\mathbf{k}} \otimes_s U)) \cdot \hat{\mathbf{k}}, \qquad (3b) $$

$$ \delta\Gamma(k\hat{\mathbf{k}}) \cdot U = \left( \langle \delta C^g \otimes \delta C^g \rangle : P(k\hat{\mathbf{k}}) : (\hat{\mathbf{k}} \otimes_s U) \right) \cdot \hat{\mathbf{k}}. \qquad (3c) $$

Assuming that the elasticity tensor varies independently from grain to grain, the elastic autocorrelation function can be expressed as follows:

$$ \langle \delta C(\mathbf{x}) \otimes \delta C(\mathbf{x}') \rangle = \langle \delta C^g \otimes \delta C^g \rangle W(\mathbf{r}), \qquad (4) $$

where $\delta C^g$ is the elasticity tensor variation in a grain and is assumed to be constant. Therefore, the ensemble average of the left term is transformed into an average $\langle \delta C^g \otimes \delta C^g \rangle$ within a single grain over all possible crystallographic orientations, and the TPC spatial function $W(\mathbf{r})$ plays a key role in describing the geometry of the grains. Indeed, it estimates the probability that two random points, $\mathbf{x}$ and $\mathbf{x}'$ separated by a distance $\mathbf{r} = \mathbf{x} - \mathbf{x}'$, lie within the same grain.

One approach to solve the non-linear eigenvalue problem (2) for wavenumber $k$ is to assume that the propagating modes, in the case of weakly scattering media, can be approximated by those pure modes of the reference homogeneous media: $U_\beta^0 (\beta = L, T)$, with $U_L^0 \parallel \hat{\mathbf{k}}$ for a longitudinal mode and $U_T^0 \perp \hat{\mathbf{k}}$ for a transverse mode. Then, after applying the scalar product with $U_\beta^0$ to both sides of (2), the following expression (5) for the propagation constant $k_\beta$ can be obtained from the perturbed acoustic tensor:

$$ k_{\beta}^{2} - k_{0\beta}^{2} = k_{\beta}^{2} \frac{\langle \delta C_{j\beta jkl} \delta C_{mnj\beta j} \rangle}{C_{j\beta jj\beta j}^{0}} P_{klmn}(k_{\beta} \hat{\mathbf{k}}), \quad P_{klmn}(k_{\beta} \hat{\mathbf{k}}) = \int_{\mathbb{R}^{dim}} G_{km}(\mathbf{r}) D_{ln}(\mathbf{r}) d\mathbf{r}, \quad (5) $$

where $G_{km}$ denotes the dyadic Green function in the 2D or 3D reference isotropic homogeneous medium (see Equations (16, 17) in Bai et al. [3]), and $D_{ln}(\mathbf{r}) = \frac{\partial^2}{\partial r_n \partial r_l} (W(\mathbf{r}) \exp(ik_\beta \hat{\mathbf{k}} \cdot \mathbf{r}))$ is a second-order derivative tensor of the TPC spatial function. Using the Born approximation, with $k_\beta^2 - k_{0\beta}^2 \approx 2k_{0\beta}(k_\beta - k_{0\beta})$ and $k_\beta \approx k_{0\beta}$. The following expression is obtained for the wavenumber, valid in the 2D and 3D dimensions:

$$ k_{\beta} = k_{0\beta} + \frac{k_{0\beta}}{2C_{j\beta jj\beta j}} \langle \delta C_{j\beta jkl} \delta C_{mnj\beta j} \rangle P_{klmn}(k_{0\beta} \hat{\mathbf{k}}). \qquad (6) $$

Expression (6) is applicable only in the Rayleigh and stochastic scattering regimes as it relies on the Born approximation. In this final expression, the complex wavenumber within the heterogeneous medium is obtained as the sum of the wavenumber in the reference homogeneous medium and a disturbance resulting from grain crystallographic and/or morphological heterogeneities. Finally, the ultrasonic attenuation can be obtained with the imaginary part of the complex wavenumber: $\alpha^\beta = \mathrm{Im}(k_\beta)$, and the phase velocity with its real part: $V^\beta = \omega/\mathrm{Re}(k_\beta)$.
In the present work, denoting $(e_i)_{i=1,2,3}$ the orthonormal basis of a laboratory coordinate system, the direction of wave propagation is fixed parallel to $e_3$, i.e., $\hat{k} = e_3$. This allows us to calculate the autocorrelation function of the elasticity tensors as $\langle \delta C_{33kl} \delta C_{mn33} \rangle$ for L waves and $\langle \delta C_{13kl} \delta C_{mn13} \rangle$ for T waves. Finally, the complex wavenumber in 2D and 3D can be both obtained from the following expressions:

$$k^{\beta\gamma,2D} = k_{0\beta} + \frac{k_{0\beta} \langle \delta C_{j\beta jkl} \delta C_{mnj\beta j} \rangle}{2C_{j\beta jj\beta j}^0} \int_{r=0}^{\infty} \int_{\theta=0}^{2\pi} G_{km}^{\gamma}(r, \theta) D_{ln}^{\beta}(r, \theta) r d\theta dr \quad \beta, \gamma = L, T, \quad (7a)$$

$$k^{\beta\gamma,3D} = k_{0\beta} + \frac{k_{0\beta} \langle \delta C_{j\beta jkl} \delta C_{mnj\beta j} \rangle}{2C_{j\beta jj\beta j}^0} \int_{r=0}^{\infty} \int_{\theta=0}^{\pi} \int_{\varphi=0}^{2\pi} G_{km}^{\gamma}(r, \theta, \varphi) D_{ln}^{\beta}(r, \theta, \varphi) r^2 \sin\theta d\theta d\varphi dr. \quad (7b)$$

The expressions (7a) and (7b) are valid for the general case of arbitrary crystal symmetry and grain shape. The integrals can be solved numerically using any available numerical integration library.

The work of Bai et al. [3] addressed the case of microstructures with equiaxed grains and cubic crystal symmetry by considering an isotropic TPC spatial function of the form $W(r) = \exp(-2r/d)$. In the present work, we retain the cubic crystal symmetry assumption, but now consider the case of microstructures with elongated grains.

In the 3D case, it can be assumed that the elongated grains have an ellipsoidal geometric shape. The assumed grain shape can be represented by the anisotropic TPC function (8), which is the extended form of the equiaxed TPC function. This function has previously been used by Ahmed et al. [11], Yang et al. [12] and Calvet and Margerin [14].

$$W(\mathbf{r}) = \exp(-\sqrt{\mathbf{r}^T \mathbf{A} \mathbf{r}}), \quad \text{with} \quad \mathbf{A} = \sum_{i=1}^{3} \frac{1}{a_i^2} \mathbf{g}_i \otimes \mathbf{g}_i. \qquad (8)$$

In equation (8), $\mathbf{A}$ is a second-order symmetric tensor composed by the inverse of $a_1^2, a_2^2, a_3^2$, which are the ellipsoidal radii in the local orthonormal coordinate system of the ellipsoid ($\mathbf{g}_1, \mathbf{g}_2, \mathbf{g}_3$). In the present work, we will always consider that $a_1 \le a_2 \le a_3$. Moreover, the aspect ratios of the ellipsoid along the axes $\mathbf{g}_3$ and $\mathbf{g}_2$ are defined as $R_3 = a_3/a_1$ and $R_2 = a_2/a_1$, respectively. By tuning the ratios $R_3$ and $R_2$, the grains can assume any shape, e.g. cigar or pancake, on each axis of elongation $\mathbf{g}_3$ and $\mathbf{g}_2$.

In the definition of $\mathbf{A}$, the orientation of the grain is implicit, and its local basis depends on the angles of rotation $\tau_1, \varphi_\tau$ and $\tau_2$, which is a set of Euler angles following a classical $Z-X'-Z''$ rotation sequence. The aforementioned angles define the orientation of the axes $\mathbf{g}_1, \mathbf{g}_2, \mathbf{g}_3$ as shown in Figure 1. In contrast with the approach developed by Yang et al. [12], the present method fixes the direction of wave propagation while rotating the grain's principal axes. Both approaches are similar and interchangeable because the key parameter is the angle between $\hat{k}$ and $\mathbf{g}_3$.

The use of the TPC function $W(r)$ defined by (8) leads to the following intrinsic form of the second order tensor $\mathbf{D}^\beta(r)$:

$$\mathbf{D}^{\beta}(\mathbf{r}) = \left[ -\frac{\mathbf{A}}{r} + \left( \frac{1}{r(\hat{\mathbf{r}}^T \mathbf{A} \hat{\mathbf{r}})} + \frac{1}{\sqrt{\hat{\mathbf{r}}^T \mathbf{A} \hat{\mathbf{r}}}} \right) (\mathbf{A} \hat{\mathbf{r}}) \otimes (\mathbf{A} \hat{\mathbf{r}}) - 2ik_{0\beta} \hat{\mathbf{k}} \otimes_s (\mathbf{A} \hat{\mathbf{r}}) \right. \\ \left. - \sqrt{\hat{\mathbf{r}}^T \mathbf{A} \hat{\mathbf{r}}} k_{0\beta}^2 \hat{\mathbf{k}} \otimes \hat{\mathbf{k}} \right] \times \frac{\exp\left( r \left( ik_{0\beta} \hat{\mathbf{k}} \cdot \hat{\mathbf{r}} - \sqrt{\hat{\mathbf{r}}^T \mathbf{A} \hat{\mathbf{r}}} \right) \right)}{\sqrt{\hat{\mathbf{r}}^T \mathbf{A} \hat{\mathbf{r}}}}, \quad (9)$$

with $r = \|\mathbf{r}\|$.

In the 2D case, one can begin by considering an elongated 3D ellipsoidal grain and analyzing a 2D cutting plane section defined by a unit normal vector $\hat{n}$. This means that the crystallographic axes of the 3D grain are rotated in 3D, giving rise to a 3D elasticity tensor, afterwards, a 2D slice is taken, and a projected 2D elasticity tensor is calculated from the full 3D tensor. Geometrically a 2D cutting plane defined solely by its normal vector can give rise to an infinite number of cross-sections in a 3D grain. Here, we simplify by assuming that the 2D plane will always cut the grains
<figure><img src="image_2.png" /><figcaption>Figure 1. Illustration of an ellipsoidal grain, where $a_1, a_2, a_3$ stands for the ellipsoidal radii and $\tau$ and $\phi_\tau$ denote the orientation angles of the ellipsoid's principal axis $g_3$. The angles of the vector $r$ are represented by $\theta$ and $\phi$.</figcaption></figure>

at their center. The intersection of an ellipsoid and a cutting plane will invariably result in an ellipse or even a circle. An orthonormal basis ($t_1, t_2, \hat{n}$) can then be defined with ($t_1, t_2$) belonging to the cutting plane perpendicular to $\hat{n}$. As we decided to fix $\hat{k} = e_3$, it is set herein $t_2 = \hat{k}$ and $t_1 = \hat{k} \times \hat{n}$. It can be demonstrated that $A^{2D}$ can be expressed in terms of $A^{3D}$ and ($t_1, t_2$) as follows:

$$A^{2D} = \sum_{i,j=1}^{2} (t_j A^{3D} t_i) t_i \otimes t_j. \qquad (10)$$

Hence, $A^{2D}$ is also a second-order symmetric tensor and describes an ellipse in the orthonormal basis $t_1, t_2$. Since we decided to fix $t_2 = \hat{k} = e_3$, the options available for $\hat{n}$ are limited to those lying in the plane $e_1, e_2$. By rotating $\hat{n}$, several ellipsoidal cross-sections can be formed, thus facilitating a direct comparison and analysis of the ultrasonic scattering dimensionality in both two and three dimensions.

### 3. Elastic wave propagation in polycrystals with elongated grains: 2D and 3D FE modeling

The numerical simulations are conducted using a discontinuous Galerkin (dG) finite element (FE) solver implemented in an in-house code (OOFE). The solver is based on a space-discontinuous Galerkin (sdG) method in space and an explicit Runge-Kutta scheme in time. The original interest in dG methods arose in the context of fluid mechanics, where it was recognized that in certain problems it was necessary to predict actual physical discontinuities [23]. More recently, dG methods have been applied to address elastodynamic problems involving material discontinuities [24, 25, 26, 27]. The dG solver used in the present study employs a velocity-stress formulation with spatially discontinuous FE basis functions [24].

#### 3.1. Space-discontinuous Galerkin framework for the elastodynamic wave equation

Considering the wave propagation problem in $\Omega$ and in the time interval $[0, T]$, and within the framework of the velocity-stress formulation, the governing equations of the elastic wave problem originally defined by Equation (1) are now expressed in the time domain. This second-order hyperbolic system is reformulated into the following first-order hyperbolic system, with primary unknowns the velocity $v$ and the stress $\sigma$ fields [24]:

$$\rho \partial_t v - \mathrm{Div}_x \sigma = 0, \qquad (11)$$

$$C^{-1} : \partial_t \sigma - \varepsilon(v) = 0,$$

or in a more compact form:

$$M(\partial_t U) + A^{\partial_x}(U) = 0, \qquad (12)$$
with, $\forall \mathbf{W} = (\mathbf{w}, \boldsymbol{\tau})^T$, a generalized field made of $\mathbf{w}$ a velocity field and $\boldsymbol{\tau}$ a stress field,

$$M(\mathbf{W}) = M\left(\frac{\mathbf{w}}{\boldsymbol{\tau}}\right) = \left(\frac{\rho\mathbf{w}}{C^{-1}:\boldsymbol{\tau}}\right), \quad A^{\partial_x}(\mathbf{W}) = A^{\partial_x}\left(\frac{\mathbf{w}}{\boldsymbol{\tau}}\right) = \begin{pmatrix} -\mathrm{Div}_x \boldsymbol{\tau} \\ -\varepsilon(\mathbf{w}) \end{pmatrix}.$$

Then, the idea is to calculate an approximate solution $U_h = (v_h, \sigma_h)^T$ that is discontinuous between elements on a FE mesh discretizing the domain $\Omega$.

Considering a such FE mesh, with $E$ any element and $E'$ any of the neighboring elements of $E$, the space dG variational formulation for $E$ after integration by parts reads as $\forall \mathbf{W}_h = (\mathbf{w}_h, \boldsymbol{\tau}_h)^T$,

$$(\mathbf{W}_h, \mathbf{M}(\partial_t \mathbf{U}_h))_E + (\mathbf{W}_h, A^{\partial_x}(\mathbf{U}_h))_E + \langle \mathbf{W}_h, \hat{\mathbf{F}}_n(\mathbf{U}_h, \mathbf{U}_h') - \mathbf{F}_n(\mathbf{U}_h) \rangle_{\partial E}, \quad (13)$$

where $(\cdot, \cdot)_E$ denotes the integration over the element $E$ of an inner product between two generalized fields defined in terms of the usual inner products between two vector fields or between two tensor fields, and $\langle \cdot, \cdot \rangle_{\partial E}$ denotes integration on the element boundaries. The numerical flux, $\hat{\mathbf{F}}_n(\mathbf{U}_h, \mathbf{U}_h')$, defines the interface condition between neighboring elements $E$ and $E'$. Several numerical fluxes are proposed by Tie and Mouronval [28], and the chosen flux will influence the stability and accuracy of the solution.

The sdG method is conditionally stable, and therefore, the stability is subjected to the Courant-Friedrichs-Lewy condition (CFL). Due to the two-field formulation, the number of degrees of freedom considered is greater than a classical FE method, and consequently, it can be more memory-consuming. However, due to its discontinuity framework, the resulting mass matrices are composed of diagonal element matrices, and together with the explicit time discretization, the parallelization strategies are straightforward.

## 3.2. Numerical setting and estimation of ultrasonic wave amplitude attenuation and phase velocity

The 3D and 2D microstructure samples are generated using the numerical tool Neper developed by Quey et al. [29] and consist of the tessellation of a specified number of grains with an average grain size distribution. To achieve a morphology with elongated grains, a combination of domain scaling and rotation is done, resulting in a final configuration comparable to that observed in Figure 2. The grains are discretized by rasterization of the tessellation as structured FE meshes are used. The discretization process enables the assignment of a set of Euler angles to each element, corresponding to its grain. In 2D, each element is still assigned the three Euler angles of a 3D grain. The 2D elasticity tensor is then derived by projecting the 3D elasticity tensor onto the 2D plane of analysis.

<figure><img src="image_3.png" /><figcaption>Figure 2. Illustration of (a) 2D and (b) 3D Neper generated microstructures with elongated grains and aspect ratios $R_3 = 5, R_2 = 1$ in 3D and $R_2 = 5$ in 2D.</figcaption></figure>

Simulations are carried out using a Ricker signal applied to the left edge of the sample along with free boundary conditions on the right edge, and symmetric boundary conditions (SBCs) to the remaining edges. The finite-element size $h$ is chosen such that,

$$h \le \lambda(f_1)/20, \quad h^{3D} \le d_{eq}/10, \quad h^{2D} \le d_{eq}/20, \quad (14)$$
with $d_{eq}$ the diameter of a spherical/circular grain with equivalent volume/area, and $\lambda(f_1)$ the wavelength of the L-wave corresponding to $f_1$ the highest frequency of interest. Here, $f_1$ is taken to be equal to the upper frequency corresponding to the half-power bandwidth frequency interval ($f_0, f_1$), which is defined from the frequency content of the Ricker signal. The time discretization is calculated according to the CFL condition due to the employed schema, as follows: $\Delta t = CFL \cdot h/c_{max}$, with $c_{max}$ the fastest L-wave propagation velocity and $CFL = 0.2$ for the specifically used sdG method. In order to satisfy the requirements for each model's parameters, different models are constructed to cover a sufficiently wide frequency bandwidth. For the 2D models, three central frequencies $f_c$ of 2, 3.2 and 10.5 MHz are selected to cover both the Rayleigh and stochastic regions. In contrast, the 3D model operates at a central frequency $f_c$ of 25 MHz, a choice dictated by the necessity of having a smaller grain size, to adequately encompass Rayleigh-to-Stochastic (R-to-S) transition region. A summary of the models used is presented in Table 1.

<table><caption>Table 1. Models in 3D and 2D used in this work, with $\ell_i$ (mm) the sample length in each dimension, R the aspect ratio in the corresponding axis, h (mm) the element size, $f_c$ (MHz) the central frequency of the applied Ricker signal and the covered range of normalized frequency $x_0 = k_{0L}d$, $d = d_2$, $d = d_3$ in 2D and 3D respectively.</caption><thead><tr><th>Model</th><th>ℓ<sub>1</sub> × ℓ<sub>2</sub> × ℓ<sub>3</sub></th><th>R<sub>3</sub> × R<sub>2</sub></th><th>h</th><th>Num. Grains</th><th>f<sub>c</sub></th><th>x<sub>0</sub></th></tr></thead><tbody><tr><td>T3D25</td><td>4 × 0.25 × 1</td><td>5 × 1</td><td>0.01</td><td>3730</td><td>25</td><td>2.1 – 9.6</td></tr><tr><td>T2D10</td><td>33.06 × 16.34</td><td>5</td><td>0.019</td><td>3439</td><td>10.5</td><td>5.2 – 17.3</td></tr><tr><td>T2D3</td><td>33 × 16.5</td><td>5</td><td>0.022</td><td>3466</td><td>3.2</td><td>3.6 – 4.5</td></tr><tr><td>T2D2</td><td>33 × 16.5</td><td>5</td><td>0.022</td><td>3466</td><td>2</td><td>1.0 – 3.3</td></tr><tr><td>E2D10</td><td>33.06 × 16.34</td><td>1</td><td>0.019</td><td>3439</td><td>10.5</td><td>5.2 – 17.3</td></tr><tr><td>E2D3</td><td>33 × 16.5</td><td>1</td><td>0.022</td><td>3466</td><td>3.2</td><td>3.6 – 4.5</td></tr><tr><td>E2D2</td><td>33 × 16.5</td><td>1</td><td>0.022</td><td>3466</td><td>2</td><td>1.0 – 3.3</td></tr></tbody></table>

The frequency-dependent amplitude attenuation and phase velocity are calculated using the Fourier transform of the incident and reflected velocity signals, following the procedure proposed by Kalashnikov and Challis [30] and already widely used by other authors [3, 4, 5, 31, 13, 20]. The incident signal $v^i(x_1 = 0, p_j; t)$ is recorded at the probes $p_j$ at the left edge ($x_1 = 0$), and the reflected signal $v^r(x_1 = \ell_1, p_j; t)^1$ is recorded at the probes $p_j$ at the right edge ($x_1 = \ell_1$). For each individual recorded signal, the discrete Fourier transform is computed after applying an appropriate time window, centered at the maximum amplitude for each incident and reflected signal $\tilde{v}(x_1, p_j; f)$. The coherent velocity signal in the frequency domain is obtained by averaging the Fourier transforms in the plane as follows, $\hat{U}(x_1; f) = \langle \hat{v}(x_1, p; f) \rangle_p$. By comparing the incident and reflected coherent waves in the frequency domain $\hat{U}_r(\ell_1; f)/\hat{U}_i(0; f)$ and denoting the amplitude part of the previous relation as $A(\ell_1; f)$ and the phase part as $\varphi(\ell_1; f)$, the amplitude attenuation and phase velocity can be estimated for a single sample using the following equations,

$$ \alpha(f) = -\frac{\ln A(\ell_1; f)}{2\ell_1}, \quad V(f) = \frac{2\pi f \ell_1}{\varphi(\ell_1; f)}. \qquad (15) $$

The final frequency-dependent attenuation is computed as the mean value obtained from N simulations with the same Neper-generalized grain microstructure but a different random distribution of crystallographic orientations.

Since the numerical evaluation approach is based on the consideration of a random distribution of crystallographic orientations, it is necessary to compute a sufficient number of simulations to achieve sufficient statistical convergence. To ensure this accuracy, this work employs the concept of relative standard error previously used by Huang et al. [31]. For both the amplitude and phase parts, $A(\ell_1; f)$ and $\varphi(\ell_1; f)$ of the coherent wave, their standard deviation and mean values, $\sigma_p$

<footnote n="1">¹Due to the free boundary conditions prescribed at the right edge leading to the superposition of the propagating and reflected waves, the signal $v^r$ should be divided by two.</footnote>
and $\mu_p$ ($p = A, \varphi$), are calculated between all $N$ simulations. Afterward, the relative uncertainties $\varepsilon_p = \sigma_p/\mu_p$ ($p = A, \varphi$) are calculated. As Huang et al. [31] have previously stated, the inherent randomness of the polycrystalline model introduces an element of uncertainty. Therefore, the goal is to reduce it rather than suppress it by increasing the number of simulations for a set of samples with re-randomized crystallographic orientations. At last, the relative standard error is calculated as $\delta_p = \varepsilon_p/\sqrt{N}$ ($p = A, \varphi$). It can be seen that the relative standard error is the uncertainty evaluated with respect to the number of simulations. Furthermore, as was also pointed out by Huang et al. [31], the uncertainty associated with the amplitude is considerably larger than that of the phase, therefore we focus on using the amplitude relative standard error as the parameter to use for increasing the number $N$ until achieving a congruent error with the number of simulations. In this work, performing six simulations $N = 6$ and with the model parameters presented in Table 1, we were able to obtain the frequency dependent errors presented in Figure 3. A maximum amplitude relative standard error of $\delta_A = 1.0\%$ for the 3D model, $\delta_A = 13.6\%$ for T2D10 the 2D model with elongated grains and $\delta_A = 10.5\%$ for E2D10 the 2D model with equiaxed grains. What can be witnessed in the Figure 3 is that the relative error increases significantly as the normalized frequency increases. Very low errors are observed for the models with lower central frequencies, as for the model with $f_c = 10.5$ MHz reaching the maximum error at the maximum frequency value of the considered bandwidth. It is important to note that the increased error observed in 2D simulations results from a higher normalized frequency, leading to a correlated increase in amplitude attenuation at higher frequencies and, consequently, an increase in amplitude uncertainty. Additionally, the discrepancy in the propagation length covered by each model contributes to the overall error. In fact, as the wave propagates through a larger distance, the front wave degenerates proportionally, therefore increasing the amplitude variation observed at the right-edge recording nodes. To elucidate this phenomenon, the error in Figure 3 is normalized using $n_g = \ell_1/d$, representing the number of grains in the propagation direction. As evidenced by the plot, upon normalizing the error of both 3D and 2D models, the errors fall in the same range, thereby enhancing their coherence. Additionally, there is a contribution from the underlying numerical errors due to a reduced number of elements per wavelength in contrast with lower frequency models.

<figure><img src="image_4.png" /><figcaption>Figure 3. Normalized frequency dependent relative standard amplitude error $\delta_{\epsilon_A}$ versus the normalized frequency for the 2D equiaxed and elongated models and the 3D elongated model. The frequency is normalized with $d = d_2$ and $d = d_3$ for the 2D and 3D models respectively. The error is normalized with $n_g = \ell_1/d$ the number of grains in the propagation path.</figcaption></figure>

### 3.3. Improved numerical/theoretical comparison by including microstructural morphological data in the TPC function

In the theoretical development presented here, several simplifications that escape from the numerical simulations were included. The most obvious is the idealized grain shape. The introduced theoretical TPC function describes a grain with a perfect ellipsoidal shape, which is not attainable at all in a numerical sample, as evidenced by Figure 2. Therefore, to accomplish a theoretical
model that more accurately reflects the simulated samples, the actual microstructural grain shape within the simulated samples has to be incorporated into the theoretical TPC function.

To do this, as proposed in [15, 18], the TPC function is now sought as a multi-exponential function of the form $W(r) = \sum_j c_j \exp(-r/a_j)$, with the total number of exponential functions to be chosen iteratively, looking for the best fit. The weighting constants $c_j$ and the parameters of exponential functions $a_j$ are determined by solving the following non-linear least-squares minimization problem: $\min_{\{c_j, a_j\}} \sum_k^M |W(r_k) - F_k|^2$, with the space domain defined by $r$ variable discretized into $M$ values and, for each random point $\boldsymbol{x}$, its pair point $\boldsymbol{x}'$ calculated based on the discretized value $r_k$. Therefore, $F_k$ is the frequency/probability that the pair of points separated by a distance $r_k$ fall within the same grain.

In the case of elongated grains, the parameters $\{c_j, a_j\}$ clearly depend on the space direction. Hence, the minimization of a 3D multi-exponential function would be considerably more complex, as the number of constants per exponential function to be determined can be up to seven. Indeed, for a grain of general anisotropy shape and orientation, the symmetric tensor $\mathbf{A}$ is determined by three diagonal terms and three off-diagonal terms, with, in addition, the weighting constant $c_j$. To overcome this, we take advantage of the methodology used to generate the microstructure with elongated grains. As stated at the beginning of this section, the elongated grains are obtained after generating equiaxed grains within a scaled domain. And so, the procedure requires fitting an isotropic, so 1D, multi-exponential function in the previously scaled equiaxed domain and then scaling the radii parameters $a_j$ according to the pre-defined grain aspect-ratios as done similarly by Huang et al. [15]. That is, $a_{1j}, a_{2j}, a_{3j} = a_j(1, R_2, R_3)/\sqrt[3]{R_2R_3}$. In terms of the tensor $\mathbf{A}$, we have $\mathbf{A}_{\text{elng}} = \mathbf{A}_{\text{equi}}\mathbf{R}$, where:

$$ \mathbf{R} = \frac{(R_1 R_2 R_3)^{2/3}}{R_i^2} \mathbf{g}_i \otimes \mathbf{g}_i \text{ (in 3D)}, \quad \mathbf{R} = \frac{R_1 R_2}{R_i^2} \mathbf{g}_i \otimes \mathbf{g}_i \text{ (in 2D)}. \qquad (16) $$

Another advantage of pursuing a minimization of the 1D TPC function is that the discrete data is obtained more easily. For this, $n$ points are placed following a uniform random distribution in the domain and their pair are placed based on a discretized $r$ space. For all the pairs of points, the frequency for the cases where both points fall in the same grain for each value $r_k$ is calculated, and the probability is computed by dividing the frequency by the number of pairs $n$. Table 2 presents the parameters of the 3D and 2D equiaxed TPC multi-exponential functions identified for the simulated sample considered in this work. Figure 4a shows the resulting discrete TPC function and the fitted multi-exponential curve for the scaled sample with equiaxed grains for the model T3D (Table 1), and Figure 4b illustrates the measured discrete data for three distinct rotation angles $\theta$ and their corresponding fitted curves obtained with the scaled parameters. It can be observed that the used method yields an excellent fitting function for microstructures with elongated grains.

<table><caption>Table 2. Parameters $c_j$ and $a_j$ (mm) of the equiaxed TPC multi-exponential functions for the 3D and 2D models.</caption><thead><tr><th rowspan="2">Parameters</th><th colspan="2">3D</th><th colspan="2">2D</th></tr><tr><th>c<sub>j</sub></th><th>a<sub>j</sub></th><th>c<sub>j</sub></th><th>a<sub>j</sub></th></tr></thead><tbody><tr><td>1</td><td>-1583.09</td><td>0.0125595</td><td>163.145</td><td>0.0751150</td></tr><tr><td>2</td><td>1400.58</td><td>0.0124722</td><td>163.907</td><td>0.0751178</td></tr><tr><td>3</td><td>-41.7836</td><td>0.0130973</td><td>163.216</td><td>0.0751207</td></tr><tr><td>4</td><td>688.261</td><td>0.0132175</td><td>-2677.88</td><td>0.0773002</td></tr><tr><td>5</td><td>-676.130</td><td>0.0132043</td><td>163.254</td><td>0.0751187</td></tr><tr><td>6</td><td>213.157</td><td>0.0132412</td><td>2025.36</td><td>0.0780598</td></tr></tbody></table>
<figure><img src="image_5.png" /><figcaption>Figure 4. Measured discrete TPC data of the model T3D and its corresponding fitted curve for (a) the scaled sample with equiaxed grains, and (b) the sample with elongated grains ($R_3 = 5, R_2 = 1$), displaying three distinct rotations $\theta$ while keeping $\phi = 0$.</figcaption></figure>

## 4. Similarities and differences between 2D and 3D theoretical and numerical results: dimensionality analysis

This section presents both theoretical and numerical results for the purpose of conducting a dimensionality analysis to identify similarities and differences between 2D and 3D grain-induced wave scattering. We focus on considering the attenuation and phase velocity of both L and T waves in a titanium ($\beta$ phase) material for which the material properties are presented in Table 3.

<table><caption>Table 3. Properties of the considered materials: elastic constants $C_{ij}$ (GPa), density $\rho$ (kg/m$^3$), and Zener's anisotropy factor $A$ [32].</caption><thead><tr><th></th><th>C<sub>11</sub></th><th>C<sub>12</sub></th><th>C<sub>44</sub></th><th>ρ</th><th>A</th></tr></thead><tbody><tr><td>Titanium (β-phase)</td><td>134.0</td><td>110.0</td><td>36.0</td><td>4428</td><td>3</td></tr><tr><td>Reference medium (Voigt average)</td><td>153.0</td><td>100.0</td><td>26.5</td><td>4428</td><td>1</td></tr></tbody></table>

### 4.1. Influence of the grain elongation on the 2D and 3D elastic wave scattering: theoretical results
First, a pure theoretical analysis is performed using the previously presented 3D and 2D theoretical models.

We consider microstructures with elongated grains of average radii $a_1 = 0.1$ mm, $a_2 = 2a_1$, $a_3 = 5a_1$. Additionally, three different grain rotations are considered, with ($\tau_1, \phi_1, \tau_2$) being the rotation angles of the elongated grain (Figure 1) taken respectively equal to: (a) 0 - 0 - 0°, i.e., $g_1 = e_1, g_2 = e_2, g_3 = e_3$; (b) 0 - 90 - 90°, i.e., $g_1 = e_3, g_2 = e_1, g_3 = e_2$; and (c) 0 - 90 - 0°, i.e., $g_1 = e_1, g_2 = e_3, g_3 = e_2$. These three rotations result in three grain sizes parallel to $\hat{k}$, (a) being the longest axis, (b) the intermediate axis and (c) the shortest axis as presented in Table 4. Moreover, we consider two choices of the unit normal vector $\hat{n}$ ($P_{23}$ and $P_{13}$ as shown in Table 4), representing two different 2D slices for each 3D case. The cross-section area of the ellipse represented by $S$ and the grain radius parallel to $\hat{k}$ denoted by $a_{||}$ are the two key parameters to be examined here. These two values for all cases considered are given in Table 4.

#### 4.1.1. Effect of the grain elongation on the wave attenuation

In Figure 5 and Figure 6 we present three plots corresponding to the rotation of each grain axis as detailed in Table 4. Each plot has been normalized with $d_3 = 2a_3$. In the 3D normalized
<table><caption>Table 4. Ellipse cross-section area *S* (mm²) and grain radius parallel to $\hat{k}$ *a*<sub>||</sub> (mm) for the selected 2D normal planes for (a) the longest axis (*a*<sub>3</sub> || $\hat{k}$) (b) intermediate axis (*a*<sub>2</sub> || $\hat{k}$) and (c) the shortest axis (*a*<sub>1</sub> || $\hat{k}$), for the chosen planes *P*<sub>23</sub> ($\hat{n}$ = *e*<sub>1</sub>) and *P*<sub>13</sub> ($\hat{n}$ = *e*<sub>2</sub>).</caption><thead><tr><th></th><th>axis || <i>k</i> (= <i>e</i><sub>3</sub>)</th><th><i>P</i><sub>23</sub>(<i>n̂</i> = <i>e</i><sub>1</sub>)</th><th><i>P</i><sub>13</sub>(<i>n̂</i> = <i>e</i><sub>2</sub>)</th><th><i>a</i><sub>||</sub></th></tr></thead><tbody><tr><td>(a)</td><td>Longest</td><td>π/10</td><td>π/20</td><td>0.5</td></tr><tr><td>(b)</td><td>Intermediate</td><td>π/10</td><td>π/50</td><td>0.2</td></tr><tr><td>(c)</td><td>Shortest</td><td>π/20</td><td>π/50</td><td>0.1</td></tr></tbody></table>

attenuation curves, the obtained results are consistent with those reported in previous studies that considered elongated grains [11, 12, 14]. In the Rayleigh scattering region, both L- and T-wave attenuation exhibit a power behavior of $\alpha_{\beta}^{3D}d \propto (fd)^4$ ($\beta = L, T$), indicating a volume-dependent attenuation. After the R-to-S transition zone, the power behavior shifts to $\alpha_{\beta}^{3D}d \propto (fd)^2$, which indicates an attenuation dependent on the grain's radius parallel to the wave propagation direction $\hat{k}$. These two power laws are also obtained in the case of microstructures with equiaxed grains [3], the difference here is an additional anisotropic attenuation in the higher frequency region, which makes it dependent on the wave propagation direction $\hat{k}$. It is an expected result introduced by the theoretical TPC function (8).

<figure><img src="image_6.png" /><figcaption>Figure 5. Comparison between the normalized 3D and 2D longitudinal attenuation as the wave propagates along the following axes: (a) the longest, (b) the intermediate, and (c) the shortest. Refer to Table 4 for further details. Moreover, the 2D attenuation was plotted for two different cross-sections ($P_{23}$ and $P_{13}$) of the respective 3D grain positioning.</figcaption></figure>

Turning to the 2D normalized attenuation curves, they exhibit a power behavior of $\alpha_{\beta}^{2D}d \propto (fd)^3$ that contrasts with the 3D case in the Rayleigh scattering region. Given that the grain shape is considered to be a tri-axial ellipsoid ($a_3 = 5a_1, a_2 = 2a_1$) for each grain rotation, the two chosen cross-sections are different (see Table 4). It can thus be concluded that the discrepancy between the 2D attenuation curves in the lower scattering region for each grain rotation (see Figures 5a, 5b, 5c and 6a, 6b, 6c) is a consequence of the variation in the cross-section area. What can be also clearly seen is that the attenuation is proportional to *S* as we have that $S_{P_{23}} > S_{P_{13}}$ in all three cases, this behavior is further analyzed in subsection 4.3.

For the L-wave attenuation, as the 2D curve enters the transition zone, the very typical transition bump occurs at lower attenuation values than the 3D case (Figure 5). However, this bump is almost indistinguishable in the T-wave attenuation curves, in both 2D and 3D. Therefore, during the transition to the stochastic region, the 2D curves are closer to the 3D ones if compared with the L-wave (Figure 6). What can also be seen is that the larger the grain radius parallel to $\hat{k}$ the shorter
<figure><img src="image_7.png" /><figcaption>Figure 6. Comparison between the normalized 3D and 2D transverse attenuation as the wave propagates along following axes: (a) the longest, (b) the intermediate, and (c) the shortest. Refer to Table 4 for further details. Moreover, the 2D attenuation is plotted for two different cross-sections ($P_{23}$ and $P_{13}$) of the respective 3D grain positioning.</figcaption></figure>

the R-to-S transition interval (compare Figures 6a and 6c). Within the stochastic scattering region, the 2D L-wave attenuation curve matches the 3D curve, while for the T-wave, the 3D attenuation curve is slightly above the 2D one. Nevertheless, one can see that the attenuation curves for both L and T waves are governed by the same power law $\alpha_{\beta}^{2D,3D} d \propto (fd)^2$ in the stochastic scattering region.

The attenuation power laws in the lower and higher frequency regions can be explained by the underlying wave scattering mechanism. In the Rayleigh scattering region, the main contribution of the scattering is due to the scattered T-waves, and such $(\beta)-(T)$ ($\beta = L, T$) scattering makes it a dimensional-dependent behavior. On the other hand, the stochastic scattering is mainly governed by the mechanism that scatters the waves in those of the same type, $(\beta)-(\beta)$ ($\beta = L, T$). This results in a dimensionless behavior in the case of L-waves: the sole dependence of attenuation is on the grain radius parallel to the wave propagation direction regardless of the 3D and 2D dimensions. In all other cases where scattered T-waves are involved, the scattering mechanism leads to a dimension-dependent attenuation. Indeed, it is important to point out that in 3D, there are two T propagating waves, one in-plane mode and one out-of-plane mode; in contrast, in 2D, there is only one in-plane T mode. The lack of scattering contribution of the second type of T-wave in 2D can explain the underestimation of the 2D T-wave attenuation in the stochastic region, whereas the 2D and 3D L-wave attenuation curves overlap perfectly in the stochastic region as the attenuation is governed by the L-L scattering.

### 4.1.2. Effect of the grain elongation on the wave phase velocity

Figures 7-11 present the phase velocity variation with respect to the Voigt velocity for the three different grain orientations defined in Table 4. For the sake of in-depth analysis, the same L- and T-wave phase velocity curves are compared in two different ways. The pair of Figures 7 and 8 present comparisons between velocities in different directions parallel to the local axes of the ellipsoidal grain for 3D and 2D cases, respectively. The pair of Figures 10 and 11 present comparisons between velocities in 2D and 3D for different propagation directions, respectively.

As a consequence of the elastic wave scattering, the phase velocity is dispersive, except in the Rayleigh scattering region for the L and T waves in both 3D and 2D. In the lower frequency region, the phase velocity is found to be slower in comparison to the Voigt average velocity, a phenomenon that is observed in both 2D and 3D scenarios. This outcome aligns with the established principle that the Voigt velocity serves as a static upper limit, a concept previously articulated by Stanke and Kino [8]. We further note that the 3D velocity has the largest gap in contrast with the 2D one
<figure><img src="image_8.png" /><figcaption>Figure 7. Comparison of the longitudinal phase velocity variation relative to $V_{0L}$ when propagating along the longest, intermediate and shortest axes in (a) 3D and (b) 2D.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 8. Comparison of the transverse phase velocity variation relative to $V_{0T}$ when propagating along the longest, intermediate and shortest axes in (a) 3D and (b) 2D.</figcaption></figure>

(Figures 7 and 8). Upon examination of the quasi-static velocity in 3D for the L curves in Figure 7a and T curves in Figure 8a, it becomes evident that it is proportional to the wave propagation direction: it varies from the slowest in the shortest axis to the fastest in the longest axis, which contrasts with the attenuation behavior observed in the same frequency region. This point has already been shown by Huang et al. [15]. As a new result, the 2D phase velocity variation in the Rayleigh scattering region shows a directional dependence together with a cross-section dependence as for the attenuation. In Figure 7b, one can observe that for each plane $P_{13}$ and $P_{23}$, the L phase velocity varies from the slowest to the fastest as the wave propagates along the shortest axis up to the longest axis, as in the 3D case. Also, if we compare the velocity for each plane, one can see that the L phase velocity is slower in the plane $P_{23}$ than in the plane $P_{13}$. The same behavior is not evident for the T phase velocity (see Figure 8b) as quasi-static velocity in the planes $Longest-P_{13}$, $Longest-P_{23}$ and $Intermediate-P_{13}$ is almost indistinguishable from one another. Nevertheless, one still has that the T phase velocity is slower in the plane $P_{23}$ as the wave propagates along the shortest and intermediate axes. The 2D quasi-static phase velocity directional difference is not, however, as prominent as in the 3D case.

To better illustrate this behavior, Figures 9a and 9b present the variation of respectively the 3D and the 2D L and T phase velocity against the aspect ratio of the elongated grain, while keeping a constant volume/cross-section. In the 3D case, the plot shows that the quasi-static velocity
<figure><img src="image_10.png" /><figcaption>Figure 9. Quasi-static (a) 3D and (b) 2D L and T phase velocity variation relative to $V_{0L}$ and $V_{0T}$ plotted against the aspect ratio that is taken as $R_3 = a_3/a_1$, with $\dot{k}||axis a_3$ and $R_2 = a_2/a_1 = 1$ constant, in 3D and as $R_2 = a_2/a_1$, with $\dot{k}||axis a_2$, in 2D.</figcaption></figure>

variation decreases as the aspect ratio $R_3$ increases, which again is comparable to the results shown also by Huang et al. [15]. Indeed, it can be appreciated the limits in the 3D case when $R_3$ tends to both lower and upper $R_3$ limits of $R_3 \ll 1$ and $R_3 \gg 1$. The variation in 2D, however, does not evidently show these limits and, in contrast, shows a slight decrease after a visible peak, as $R_2$ tends to higher values. The 2D more complex behavior should be further investigated. Finally, in the Rayleigh scattering region, the 2D L and T curves (Figures 10 and 11) exhibit larger phase velocities than the 3D ones. Furthermore, the deviation increases from the longest axis to the shortest axis.

As the frequency increases, the phase velocity displays a markedly more intricate behavior. However, when the wave propagation direction is parallel to the grain's shortest radius $a_1$ one can notice that the phase velocity seems to enter again into a non-dispersive region both in 3D and 2D for both L and T waves (see Figure 10c and Figure 11c). This apparent non-dispersive behavior in the stochastic scattering region can be attributed to the introduction of the Born approximation as previously demonstrated by Stanke and Kino [8]. As the grain radius parallel to the wave propagation increases, one can notice that in this high-frequency region, the phase velocity has again a dispersive behavior, with a sudden decrease proportional to the frequency. This decrease in the phase velocity is more drastic in the longest radius (see Figure 10a) both in 3D and 2D for both L (Figure 10a) and T (Figure 11a) waves.

In contrast with the L-wave phase velocity variation in the longest and intermediate axis, the T-wave does not show a visible peak velocity, and in exchange, there is a faster transition from the Rayleigh to the stochastic region (see Figures 11a and 11b). This behavior is comparable to what was already observed for the attenuation, where the T master attenuation curve shows a shorter R-to-S interval. This can be explained by the stronger wave scattering at higher frequencies, which is also evident in the observed attenuation. Nevertheless, in contrast with the 2D attenuation, the 2D velocity does not seem to be able to predict the 3D counterpart. This phenomenon may be partially explained in the following way. In the stochastic region, the velocity behavior is more complex and is not governed by a 1D mechanism like attenuation, and it depends on the distribution of grains in other directions. Hence, it is greatly influenced by the 2D assumptions. Furthermore, it is noteworthy that the velocity variation spread related to the R-to-S transition in the 3D curves and the 2D curves is also inversely proportional to the grain radius (see Figures 7, 8, 10 and 11). For instance, comparing the 3D curves of Figures 10a, 10b, and 10c, reveals that, as the wave propagates along a shorter grain radius, the variation of velocity in the R-to-S transition zone is the most significant, ranging from the lowest to the highest value.
<figure><img src="image_11.png" /><figcaption>Figure 10. Comparison of the 3D and 2D longitudinal phase velocity variation relative to $V_{0L}$, as the wave propagates along the (a) longest, (b) intermediate and (c) the shortest axes (see Table 4). Moreover, the 2D phase velocity variation is plotted for two different cross-sections ($P_{23}$ and $P_{13}$) of the respective 3D grain positioning.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 11. Comparison of the 3D and 2D transverse phase velocity variation relative to $V_{0T}$, as the wave propagates along the (a) longest, (b) intermediate, and (c) the shortest axes (see Table 4). Moreover, the 2D phase velocity variation is plotted for two different cross-sections ($P_{23}$ and $P_{13}$) of the respective 3D grain positioning.</figcaption></figure>

## 4.2. Validation: comparison between numerical evaluation and theoretical models

In this subsection, we are interested in the comparison of the attenuation and phase velocity obtained with the theoretical model proposed here and the numerical simulations. For this purpose, we consider microstructures comprising elongated grains, either in 3D with aspect ratios $a_3/5 = a_2 = a_1$ and orientation $g_3 || \hat{k}$, or in 2D with aspect ratio $a_2/5 = a_1$ and orientation $t_2 || \hat{k}$.

### 4.2.1. Frequency dependency of the attenuation

The initial consideration is that of the attenuation coefficient. Figure 12a presents the 3D case with the normalized attenuation $\alpha_L d_3$ versus the normalized wavenumber $k_{0L} d_3$, with $d_3 = 2a_3$. Only the loading case of a single Ricker signal of $f_c = 25$ MHz was considered. Despite the high-frequency signal, the numerical results are obtained at the upper limit of the Rayleigh region where they move to the transition zone. This is explained by the relatively small grain equivalent diameter used for this model. Nevertheless, a general good agreement with the theoretical curve is observed: the highest relative difference is $|\alpha_L^{TH} - \alpha_L^{NUM}|/\alpha_L^{NUM} = 65\%$ at the lower frequency
values of the numerical curve around $k_{0L}d_3 \approx 2$.

<figure><img src="image_13.png" /><figcaption>Figure 12. Numerical and theoretical normalized attenuation curves comparison for (a) 3D grain with $a_3 = 5a_2 = 5a_1$ and $g_3 \| \hat{k}$, and (b) 2D grain with $a_2 = 5a_1$ and $t_2 \| \hat{k}$.</figcaption></figure>

For the 2D case, we also introduce the attenuation curve obtained for a model with equiaxed grains. The objective of this numerical verification is to corroborate the theoretical results previously presented in the 2D case. Figure 12b presents the normalized attenuation versus the normalized wavenumber with the equivalent diameter $d_{eq} = 2a_{eq}$. A very good agreement with the equiaxed and elongated theoretical curves is observed. It can be seen that the Rayleigh limit is well predicted, as is the smooth transition to the stochastic region. As explained in the subsection 3.2, the model with elongated grains is obtained by elongating a microstructure with equiaxed grains by keeping the same cross-section. Therefore, the decision to normalize with the equivalent diameter is intended to show the cross-section dependency in the Rayleigh scattering region by showing the convergence of both equiaxed and elongated curves as they approach the lower frequencies. Around $k_{0L}d_{eq} \approx 2.3$ we find the highest relative difference of 34% is observed for the elongated model. This difference is observed in the overlapping between T2D3 and T2D10 models defined with loadings whose frequency content is respectively centered at 3 MHz and 10 MHz (cf. Table 1). Certainly, there is a visible discontinuity that is induced by the change of model and therefore, different numerical errors. Moreover, this discontinuity occurs in the R-to-S transition zone where the theoretical curve seems to underestimate the attenuation and the scattering phenomenon is more complex. In contrast, it can be seen that the numerical curve for the equiaxed models is smoother and with no visible discontinuities. For this curve, the maximum relative difference obtained is 29% at around $k_{0L}d_{eq} = 0.5$. As for the models of elongated grains, for equiaxed grains, the theoretical attenuation underestimates the numerical results. Comparing both cases, it is evident that the elongated grains have a stronger effect in the numerical simulation. Since the models for equiaxed and elongated grains have a similar numerical setting, it is left to show the observed discontinuity is caused by the morphological difference. In addition, several sources of error can be listed. One of them is the effect of SBCs which mirror boundary grains and can cause stronger dispersion [31]. This effect could be introduced when estimating the numerical TPC function. In this work, this effect is ignored and is left as future work.

In general, numerical results are a better approach since the multiple wave scattering is fully considered in contrast with the theoretical model proposed here. Indeed, Huang et al. [13] have shown that there is an increasing error proportional to the anisotropy constant of the material when theoretical models assume a weakly scattering medium. It is, therefore, anticipated that the material under consideration, which exhibits a relatively high level of anisotropy ($A = 3$), contributes to the obtained difference. Nevertheless, the concordance between the numerical and theoretical results helps to validate the theoretical model under the initially stated assumptions
and limitations.

### 4.2.2. Frequency dependency of the phase velocity variation

<figure><img src="image_14.png" /><figcaption>Figure 13. Numerical and theoretical comparison of the variation of the longitudinal phase velocity w.r.t. the Voigt velocity in the equivalent homogeneous reference medium for (a) 3D grain with $a_3 = 5a_2 = 5a_1$ and $g_3||\hat{k}$, and (b) 2D grain with $a_2 = 5a_1$ and $t_2||\hat{k}$.</figcaption></figure>

Secondly, using the same simulations already considered in the previous subsection 4.2.1, the longitudinal wave phase velocity is estimated and presented in Figure 13 again for both 3D and 2D cases. The plots show the variation of the phase velocity with respect to the longitudinal Voigt velocity in the equivalent homogeneous reference medium $V_{0L}$ versus the normalized wavenumber. The numerical results are compared with the theoretical model presented here. It is noteworthy that the 2D theoretical velocity results in Figure 13b are shifted down by the quasi-static following constant (17).

$$V_L = -\frac{2(C_{11} - C_{12} - 2C_{44})}{375(C_{11}^0)^2} \left(2 + 3\frac{V_{0L}^2}{V_{0T}^2}\right) V_{0L}, \quad (17)$$

which is obtained from the expression for the complex wavenumber in the Rayleigh limit presented by Stanke and Kino [8]. Rokhlin et al. [33] applies this correction due to the assumed far-field Green function. This approximated Green function ignores the radiative terms $(1/r^2)$ and the reactive terms $(1/r^3)$. Therefore, Rokhlin et al. [33] explains that the real part of the perturbed wavenumber is affected in the quasi-static limit. In our case, after analyzing the 2D Green function used [34], it was observed that the involved Hankel functions give an asymptotic behavior that is closer to what is a far-field solution. This behavior can be seen in Figure 14, where the components 11 and 33 for the longitudinal part for the 3D and 2D Green functions are compared. It is clear from the plot that the 2D components have a slower convergence than the 3D ones.

In the 3D case as for the attenuation, a generally good agreement is obtained for the phase velocity (Figure 13a). The numerical results are obtained at the shift from a non-dispersive behavior in the Rayleigh scattering region to a dispersive one which is then to be the transition to the stochastic region. The highest relative error is found at the highest plotted normalized frequency of $k_{0L}d_3 \approx 10$ with a value of 0.35%. In 2D, an excellent agreement is obtained, and the peak velocity is well predicted by the numerical results (Figure 13b). However, the transition from the non-dispersive character to the dispersive one is less well predicted by the numerical results (as in the 3D case). This can be partially explained by the considered material with strong anisotropy, causing a higher scattering captured by the numerical simulations and underestimated by the theoretical model. The dispersive behavior in the stochastic scattering region is not captured by the elongated and equiaxed models and should be further investigated with other models. For the 2D models of elongated grains, the highest relative error is 0.17% at around the normalized frequency of $k_{0L}d_{eq} \approx 7.8$ for the highest plotted frequency. For the 2D models of equiaxed grains,
<figure><img src="image_15.png" /><figcaption>Figure 14. Normalized longitudinal components 11 and 33 of the 3D and 2D Green functions plotted against $r$ normalized using the wavenumber $k_{0L}$.</figcaption></figure>

we obtain an error of 0.2% at around $k_{0L}d_{eq} \approx 2.3$ where the velocity approaches its maximum. In both cases, the numerical curves are very smooth and the highest difference between the theoretical and numerical results is obtained in the dispersive region where the phase velocity increases up to its peak. For both equiaxed and elongated models, the theoretical phase velocity variation underestimates the numerical results as was also observed for the attenuation. In comparison with the numerical results of the attenuation in subsection 4.2.1, the relative error of the phase velocity is around two orders of magnitude lower. The phase part of the FFTs is less susceptible to the numerical errors [31], which can then be noticed in the estimated velocities. However, the sources of error mentioned in the subsection 4.2.1 are also applicable here.

### 4.3. Dimensionality analysis via directional dependency of attenuation

The 2D attenuation shows a cross-section dependency in the Rayleigh scattering region, one can make an in-depth analysis of this phenomenon to find its correlation with the 3D case. For this, we consider again the 3D grain already considered in subsection 4.1 with $a_3 = 5a_1 = 2.5a_2$.

Figure 15 presents the polar plot of the longitudinal normalized attenuation in the 2D dimension versus the angle of rotation of the normal unit vector $\hat{n}$ measured from the axis $e_1$ when it rotates in the anti-clockwise direction for the 3D grain considered previously with axis orientation (a) as in Table 4. With $\hat{n}$ in the plane $e_1 - e_2$, different cross-sections can be formed by different planes falling parallel to the axis $e_3$ and any other axis perpendicular to both $\hat{n}$ and $e_3$. Figure 15 presents three distinct plots, each one of them belonging respectively to the Rayleigh scattering regime 15a, to the transition zone 15b, and to the stochastic regime 15c. In the Rayleigh scattering regime we can see more clearly the cross-section dependency as $\hat{n}$ moves from 0°, corresponding to the maximum cross-section for the ellipse with axes ($g_2, g_3$), to 90°, corresponding to the minimum cross-section for the ellipse with axes ($g_1, g_3$). It is worthwhile to remind that in this configuration, the grain size in the direction of the wave vector $\hat{k}$ remains equal to $d_3$, whatever the direction $\hat{n}$, since one has for all cases $\hat{k} \parallel g_3 \parallel e_3$. Therefore, as the frequency increases, it is clear the cross-section dependency disappears, as the normalized frequency approaches the limit of the stochastic scattering regime. The same analysis can be applied for the T-waves, as shown in Figure 16 that presents the polar plot of the transverse normalized attenuation in the 2D dimension versus the angle of rotation of the normal unit vector $\hat{n}$.

Moreover, it is interesting to point out that in the Rayleigh scattering regime (Figure 15a), the attenuation gives an implicit idea of the grain cross-section shape. For instance, knowing that in the lower frequency domain, the maximum values of attenuation correspond to the maximum cross-section area, the position of the short and elongated axes in the plane parallel to $\hat{n}$, can be deduced from the polar plot. In the case of Figure 15a (plane $e_1 - e_2$), the elongated axis $g_2$ is parallel to $e_2$ (90° in the polar plot).

To better illustrate this, Figure 17a plots the inverse of the 2D longitudinal attenuation normalized by $d_3$ for three different rotation planes to which $\hat{n}$ belongs: (i) $P_{12}$ formed by ($e_1, e_2$), (ii) $P_{23}$
<figure><img src="image_16.png" /><figcaption>Figure 15. Polar plot of the normalized longitudinal attenuation in the 2D dimension versus the angle of the normal unit vector $\hat{n}$ measured from the axis $e_1$ for a 3D grain with $a_3 = 5a_1, a_2 = 2a_1$, for the normalized frequencies $k_{0L}d_3$ of (a) 0.295, (b) 2.45 and (c) 155.09.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 16. Polar plot of the normalized transverse attenuation in the 2D dimension versus the angle of the normal unit vector $\hat{n}$ measured from the axis $e_1$ for a 3D grain with $a_3 = 5a_1, a_2 = 2a_1$, for the normalized frequencies $k_{0T}d_3$ of (a) 0.709, (b) 5.88 and (c) 372.65.</figcaption></figure>

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formed by ($e_2, e_3$), and (iii) $P_{13}$ formed by ($e_1, e_3$). Last, another case (iv) for a 3D grain with the same radii as considered before but with $g_2$ rotated by 45° with respect to the axis $e_1$ with $\hat{n}$ belonging to $P_{23}$ is also considered. The above-mentioned planes of rotation are chosen with the intention of “scanning” a 3D grain from its three main planes as follows.

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First, we can see that by looking at the inverse of the attenuation, one can clearly find the grain's rotation and also indicate the grain's elongated axis 135° and the short axis 45° in the plane of $\hat{n}$ (cf. Case (iv) in Figure 17a). Then, it follows that by “scanning” one single 3D grain from its three principal planes, it is possible to have an idea of the full 3D grain shape by the 2D attenuation in the Rayleigh scattering regime. To do this, we consider the 3D grain with the three principal axes aligned with the global axes by carrying out the following analysis steps.

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1. Study of the case (i). We choose $\hat{n} \in P_{12}$, at 0° with $\hat{n} = e_1$, the cross-section is composed by the two remaining axes $e_2, e_3$, and at 90° with $\hat{n} = e_2$, the cross-section is composed by $e_1, e_3$. As $\hat{k} = e_3$, the only difference between the maximum and minimum attenuation relies on the grain's radius in the axis perpendicular to $\hat{k}$ either $e_1$ or $e_2$. Therefore, from the polar plot (see the curve “case (i)” in Figure 17a) one can infer that the grain's radii $a_{90} > a_0$ in the plane of $\hat{n}$. In the next steps, the same analysis for the other two principal planes of inspection is to be done.

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2. Study of the case (ii). We choose $\hat{n} \in P_{23}$ and rotate it in the anti-clockwise sense from the
<figure><img src="image_18.png" /><figcaption>Figure 17. (a) Polar plot of the inverse of the longitudinal normalized attenuation for (i) $\hat{n} \in P_{12}$, (ii) $\hat{n} \in P_{23}$, (iii) $\hat{n} \in P_{13}$, and (iv) $\hat{n} \in P_{12}$ for a grain rotated 45°; (b) Ratio of the longitudinal attenuation when $\hat{k}$ propagates along the shortest and longest axis versus the grain aspect ratio between them $R_3$ for elongated grains with $R_2 = 1$ and three different values of $a_1 = 0.1$ mm, 0.05 mm, and 0.01 mm, which result in three different values of normalized $k_{0L}d_1 = 0.0590, 0.0295$, and 0.0059.</figcaption></figure>

axis $e_2$. One again can see the same behavior, at 0° the resulting ellipse is composed by the axes $e_1, e_3$, and at 90° by the axes $e_1, e_2$, therefore $a_{90} > a_0$ (see the curve "case (ii)" in Figure 17a).

3. Study of the case (iii). The same analysis applies to case (iii) from which it can be deduced that $a_{90} > a_0$. In the latter, we also see the coincidence at 0° with case (i) and at 90° with case (ii).

4. Finally, considering all three cases (i)-(iii) together, one can arrive to the conclusion that $a_{0ii} > a_{0iii} > a_{0i}$ and that $a_{90iii} = a_{90ii} > a_{90i}$. Then by correlation of the angles of rotation and the planes of $\hat{n}$, one can see that $a_{90iii} = a_{90ii} = a_3$, $a_{0ii} = a_{90i} = a_2$ and $a_{0ii} = a_{0i} = a_1$, thus $a_3 > a_2 > a_1$.

In addition to the obvious influence of the shortest and longest radii in the polar plot of normalized attenuation, it is noteworthy that proportionality can be observed between the ratio of attenuation along these two axes and the inverse of the grain aspect ratio between them $R_3$, for a value of $a_1$ and the fixing of $R_2 = 1$. It should be noted that the condition $R_2 = 1$ is not necessary, but we have chosen this fixed value to ease our analysis without loss of generality. In fact, this is an expected relation due to the already determined cross-section dependency. To show this, the ratio of attenuation at 90° in the two cases with respectively $\hat{n} \in P_{12}$ and $\hat{n} \in P_{23}$ is calculated for a grain with increasing $R_3$ and constant $R_2 = 1$ for three different values of $a_1$: 0.1 mm, 0.05 mm and 0.01 mm, resulting in $k_{0L}d_1$: 0.0590, 0.0295, and 0.0059, respectively (Figure 17b). This setting finally makes it possible to compare the attenuation when the wave propagates in the elongated axis $a_3$ and short axis $a_1 = a_2$. Indeed, Figure 17b shows, first, an explicit proportional relationship with a slope equal to 1 between the ratio of attenuation and the ratio of longest/shortest radii as the average equivalent grain size approaches the limit of the Rayleigh scattering regime. Furthermore, it is also shown that this proportionality tends towards 1:1 behavior, as the normalized frequency with $d_1$, which gives an idea of which scattering regime the equivalent grain size approaches the limit of the Rayleigh scattering regime (with $a_1$ decreasing). Here, $d_1$ is chosen for frequency normalization instead of $d_3$ because the latter varies. Nevertheless, it can be seen in the case of $k_{0L}d_1 = 0.0590$ (larger equivalent grain size) that as $R_3$ increases, and therefore $d_3$ increases the grain size/wavelength ratio increases and moves upwards in the normalized frequency region, and therefore the 1:1 proportionality seen at lower equivalent grain size (lower $R_3$) starts to disappear.
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This is an expected behavior since as the grain size increases, it moves out of the Rayleigh scattering frequency, and therefore, the validity of the cross-section dependency does not apply anymore.

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# 5. Conclusions

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In this work, the theoretical model for the longitudinal and transverse wave scattering valid for 2D and 3D developed in [3] was extended to account for microstructures comprising elongated grains, assuming an ellipsoidal shape. Moreover, the theoretical model was validated through numerical simulations using a titanium (β-phase) material by comparing the ultrasonic wave attenuation and phase velocity for longitudinal wave propagation.

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The results are consistent with previous works; that is, in the 3D case, a volume-dependent attenuation was observed in the Rayleigh scattering region. In contrast, in 2D, the attenuation is cross-section dependent. As the frequency increases, it was shown that in the transition region, the attenuation in 3D exceeds that in 2D. However, as the frequency reaches the stochastic scattering region, both in 3D and 2D, the attenuation converges to the same values, demonstrating a directional behavior proportional to the grain radius parallel to the wave propagation direction $\hat{k}$. Regarding the wave phase velocity, both 3D and 2D cases exhibited a non-dispersive behavior in the Rayleigh limit, and, except for the wave propagating in the longest grain radius, also in the stochastic scattering region. Nonetheless, this non-dispersive plateau is suspected to be caused by the introduced Born approximation.

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Simulations were performed of 2D and 3D numerical models of a titanium (β phase) polycrystalline microstructure with elongated grains. The numerical results were in very good agreement with the theoretical ones despite the high anisotropy of the considered material. The comparison between the numerical and theoretical results helps to validate the theoretical assumptions in the specific limited cases of cubic-crystalline symmetry, untextured, and single-phase medium. But beyond this, it helps to show that the numerical models are a viable and versatile approach to studying the very complex ultrasonic wave scattering without any required assumptions.

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Finally, through dimensionality analysis, it was shown that the cross-section dependency of the 2D attenuation in the Rayleigh region could be exploited in the case of microstructures with elongated grains by doing a three-plane scanning to estimate the full 3D grain shape. Moreover, the findings of this study demonstrate the potential of 2D attenuation to predict the 3D attenuation in the stochastic limit, emphasizing its intrinsic relationship with grain elongation. This outcome offers a novel perspective on the feasibility of extracting grain radii using high-frequency ultrasonic signals in polycrystalline microstructures. However, the 2D phase velocity exhibits no direct or clear relation with the 3D one, requiring additional investigation on the matter. Overall, the results herein presented are a process aimed at deeply comprehending the 2D and 3D elastic wave scattering mechanisms and their essential correlation with polycrystalline morphology. The developed theoretical and numerical approaches can also be useful for further investigations on the characterization of polycrystals with inversion methodologies. While further experimental work is needed to fully prove the here presented results, this theoretical work lays the groundwork for future promising studies of interest to NDT work in additive manufacturing. Indeed, the proposed methodologies have the potential to be further extended to study more complex cases, such as strongly marked microstructures, including columnar grains.

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# CRediT authorship contribution statement

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**Juan Camilo Victoria Giraldo:** Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data Curation, Writing - Original Draft, Writing - Review & Editing, Visualization. **Bing Tie:** Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data Curation, Writing - Original Draft, Writing - Review & Editing, Visualization, Supervision, Funding acquisition. **Jérôme Laurent:** Conceptualization, Validation, Investigation, Writing - Review & Editing, Supervision, Funding acquisition, Project administration.

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**Alain Lhémery:** Conceptualization, Validation, Investigation, Writing - Review & Editing, Supervision. **Denis Solas:** Conceptualization, Validation, Investigation, Writing - Review & Editing, Supervision, Funding acquisition.
Declaration of competing interest 700

The authors declare that they have no known competing financial interests or personal relationships 701
that could have appeared to influence the work reported in this paper. 702

Acknowledgments 703

This work was part of Juan Camilo Victoria Giraldo’s doctoral study. This research work was 704 funded/supported by the Agence Nationale de la Recherche, France (No. ANR-21-CE08-0026, 705 Research Project COLUMBO). Computations were performed using computational resources from 706 the “Mésocentre” computing center of Université Paris-Saclay, CentraleSupélec and École Normale 707 Supérieure Paris-Saclay supported by CNRS and Région Île-de-France (https://mesocentre. 708 universite-paris-saclay.fr/). 709

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