Material properties, such as hardness, yield strength, and ductility, depend on the microstructure of the material. If the microstructural organization can be quantified nondestructively, for example, with ultrasonic scattering techniques, then it may be possible to predict the mechanical performance of a component. Three-dimensional digital microstructures have been increasingly used to investigate the scattering of mechanical waves within a numerical framework. These synthetic microstructures can be generated using different tessellation algorithms that result in different grain shapes. In this study, the variation of ultrasonic scattering is calculated for microstructures of different morphologies for a nickel polycrystal. The ultrasonic properties are calculated for the Voronoi, Laguerre tessellations, and voxel-based synthetic microstructures created by DREAM.3D. The results show that the differences in the two-point statistics and ultrasonic attenuation for different morphologies become more significant at wider size distributions and higher frequencies.

Microstructural features, such as texture, morphology, and grain size statistics, control numerous mechanical properties of materials like the yield strength,1 ductility,2,3 fatigue crack resistance,4,5 and hardness.6 The materials may also be subjected to special heat treatments to achieve a desired microstructure with certain mechanical responses.7,8 These tailored materials are, in fact, optimized for certain applications. A key step in this process is the accurate and efficient characterization of microstructures.

The most common methods of microstructural characterization are the use of linear intercept measurements from optical or scanning electron microscopy (SEM) images9 or analysis of electron backscattering diffraction (EBSD) images. Although these techniques provide a vast amount of microstructural details, they are destructive, limited to surface information from small specimens, and, in general, expensive and time consuming. However, nondestructive techniques are not limited by the size of the sample and preserve the integrity of the part. They are also flexible in terms of the information that can be ascertained at a certain depth within the sample. In a recent study by Keyvani et al.,10 the in situ measurement of grain size evolution using laser ultrasound was investigated in cobalt L605 alloys. A reference signal was required for the measurement and was evaluated from a sample with known grain size and morphology. Some of the well-known nondestructive microstructural characterization methods include the use of attenuation spectra from laser ultrasound,11 spatial resolved acoustic spectroscopy (SRAS),12,13 and the spatial variance of diffuse ultrasonic scattering.14 

Ultrasonic scattering15 can provide information about grain statistics for a polycrystalline material. Scattering events occur due to the impedance mismatch at the grain boundaries.16 The amplitudes and profiles of these events depend on the elastic modulus, density, and grain size statistics of the polycrystals.17,18 Different analytical17,18 and numerical models19,20 have been used to calculate the ultrasonic scattering characteristics from equiaxed21–24 and elongated25–27 polycrystalline materials of different crystal symmetries. For the numerical models, a digital representation of the microstructure is often used.

Researchers have been using different tessellation methods to generate the three-dimensional (3D) microstructures of the polycrystalline materials. Voronoi,28,29 Laguerre,30 and DREAM.3D are the most widely used tessellations for synthetic volume generation. The Voronoi tessellation is popular in the computational acoustics19,32 community. Laguerre tessellations are usually used to model porous structures such as foam.33 A Voronoi tessellation fills the volume with random seed points and the grain boundaries are equidistant from neighboring seed points. This type of tessellation has been widely used to solve the elastodynamic wave equation using the finite element (FE) method.19,32,34 The Laguerre tessellation is based on the Voronoi tessellation and has the ability to create wider grain size statistics. DREAM.3D is an open source software that fills the defined volume with grains of a desired morphology from input grain statistics. The grains are allowed to grow in all directions until they come in contact with other grains such that no voids are present in the volume. Details about these tessellations are described in Sec. III. Due to the randomness inherent to the tessellation algorithms, morphological variations are present across the synthetic microstructures.35 Similarly, in the study by Ryzy et al.,36 based on FE wave simulations, differences in ultrasonic properties were observed for three variants of Voronoi tessellations. The authors noted differences in the two-point spatial correlation (TPSC) function and attenuation among the three variants with a fixed number of grains having different grain statistics. To the best of the authors' knowledge, no study has been performed to compare differences in ultrasonic predictions among different tessellation methods with similar grain statistics.

The possible differences between realizations become more important as newer tessellation methods are being used by the material science community to generate synthetics. The current study performs the comparison by quantifying the changes in the ultrasonic scattering responses resulting from the morphological variations between the three different tessellations and affine invariant that is related to grain shape.37 The affine invariant is a measure that can be used to quantify morphology of grains. In irradiated materials38 and materials with abnormal grain growth,39,40 the microscopy images often show the evolution of complex shaped grains. Here, the study aims to establish a correlation among different tessellations and assess the feasibility of using ultrasonic scattering techniques to quantify abnormal shaped grains in microstructure. This article is divided into four sections. First, the numerical model used to calculate the ultrasonic scattering of synthetic volumes is described in Sec. II. Sections III and IV contain the details of the three tessellations and the grain size statistics of different synthetic volumes, respectively. Finally, the differences between ultrasonic scattering responses of the synthetic volumes are discussed.

Using ensembles of synthetic volumes, the influence of morphology on ultrasonic attenuation is investigated. Norouzian and Turner21 established the foundation to obtain the attenuation values using synthetic microstructures. In their work, the theoretical framework developed by Weaver18 was modified to handle the discrete nature of these synthetic volumes. For completeness, an overview of the technique is included here.

For an incident wave propagating in the p̂ direction with polarization direction û and scattering into ŝ direction with polarization direction v̂, the differential scattering coefficient can be defined as

Γûv̂(p̂,ŝ)=ω432π2ρ2[CIû(p̂)]3[CSv̂(ŝ)]5Λ(r)·ûp̂ŝv̂·ûp̂ŝv̂ei[(ω/CIû)p̂(ω/CSv̂)ŝ]rdr,
(1)

where ω is the frequency, ρ is the material density, and r=xx is the separation vector between two positions, x and x. CIû and CSv̂ define the phase velocity of the incident wave polarized in the û direction and phase velocity of the scattered wave polarized in the v̂ direction, respectively. Λ is the covariance function of the elastic modulus, C, in which information regarding the spatial distribution of the material heterogeneity is defined. The inner product between the covariance function and propagation and polarization vectors included in Eq. (1) is given in direct notation by

Λ(r)·ûp̂ŝv̂·ûp̂ŝv̂=δCijkl(x)δCαβγδ(x+r)ûiûαp̂jp̂βŝkŝγv̂lv̂δ,
(2)

where δC is the modulus fluctuation about the mean. Because the material information is known at discrete locations, a discrete form of Eq. (1) may be defined. This integration is achieved using a Monte Carlo scheme. Hence, Eq. (1) can be rewritten as

Γûjv̂k(p̂,ŝ)=R3ω424πρ2NNp[CIûj(p̂)]3[CSv̂k(ŝ)]5n=1N[m=1NpδC(xm)δC(xm+rn)]·ûjp̂ŝv̂k·ûjp̂ŝv̂keiqrn,
(3)

where x is a randomly selected location in the volume, Np is the total number of locations, N is the number of discrete points used for the integration, and R is the largest distance used to calculate the two-point statistics. The attenuation is calculated by integrating the differential scattering coefficient over all of the scattering directions. For example, the expression of longitudinal attenuation can be found by integrating the three differential scattering coefficients for the three outgoing wave types given by

ΓqL(p̂,ŝ)=ΓqLqL(p̂,ŝ)+ΓqLqT1(p̂,ŝ)+ΓqLqT2(p̂,ŝ).
(4)

Here, qL is the quasi-longitudinal wave and qT1 and qT2 represent the two quasi-shear polarizations. The longitudinal attenuation is

αL(p̂)=ΓqL(p̂,ŝ)dŝ.
(5)

The shear attenuation is also calculated in a similar manner. Interested readers can find detailed information in the studies by Norouzian and Turner21 and Norouzian et al.35 

Synthetic microstructures can be created using different tessellations and contain grains of different shapes. Due to variations in tessellations, a difference is recognized in the morphology of the grains within the 3D realizations. For this study, these morphological variations are studied from two different aspects. One objective is to quantify the variations in grain morphology associated with different tessellation algorithms and another is to study the variations due to the shape distribution of the synthetic volumes. The tessellations considered in this study are Voronoi, Laguerre, and DREAM.3D.31 In addition, the shape distribution parameter considered is Ω3. DREAM.3D is used to study the effect of morphology resulting from different distributions of Ω3. A detailed description of these parameters is presented in Secs. III A–III D.

Voronoi tessellations are used in many fields,29 such as image representation in computer graphics,41 galaxy clusters detection in astrophysics,42 and synthetic microstructure generation in material science.43,44 In this technique, the spatial domain is partitioned into several regions. A brief description about the Voronoi tessellation is given here. Consider (i) a spatial domain DRn, (ii) a set of points Gi(xi) within D, and (iii) a norm function denoted by d(·,·). Then, a Voronoi polyhedron, Ci, may be defined as44 

Ci={P(x)D|d(P,Gi)<d(P,Gj)ji}.
(6)

For the Neper software,44 the norm, d(·,·), is the Euclidean distance. If the points used in the Voronoi tessellation are randomly distributed, then the tessellation is known as Poisson-Voronoi. Interested readers are directed to several articles45–47 for detailed information regarding Voronoi tessellations.

The Laguerre tessellation is more versatile than the Voronoi tessellation in the creation of wider grain size distributions. This tessellation is a weighted version of Voronoi and similarly divides the space using a number of seed points. A brief description of the Laguerre tessellation is now provided.

Let pRd be a fixed point and wR+ be the weight of point p. Consider the pair (p,w) as a weighted pair. In place of the Euclidean distance used in the Voronoi formulation, the Laguerre tessellation uses a “power-distance” relation defined as48 

pow(x,(pi,wi))=xpi2w.
(7)

Here, x is a point in the spatial domain of Rd. The term power-distance is not a mathematical distance; it just represents a “degree of farness.”48 The cells associated with the ith generated point, Ci, are defined by

Ci={xRd:pow(x,(pi,wi))pow(x,(pj,wj)),ij}.
(8)

Laguerre tessellations are popular for modeling foam and porous structures. Details about the Laguerre tessellation are given elsewhere.33,49,50

DREAM.3D is an open source software by which 3D voxel-based microstructures can be created. The software allows for control of several parameters to achieve microstructures with certain grain size distribution, grain aspect ratios, and orientation. The process involves the instantiation of grains at random positions inside the volume. These grains are allowed to grow so that the arrangement becomes space-filling. They are, then, iteratively swapped and moved until an arrangement is achieved that is close to the input grain statistics. Interested readers can review the work of Groeber and Jackson,31 regarding the details about synthetic volume creation in DREAM.3D.

Ω3 is a shape determining factor which is derived from the second-order moment-of-inertia tensor. The object moments, μpqr, are defined in 3D Cartesian space as

μpqr=xpyqzrD(x,y,z)dxdydz.

Here, D(x,y,z) is a step function, which is unity inside a grain and zero elsewhere. If the origin of the coordinate system coincides with the center of mass, then the moment-of-inertia tensor, I, can be written as

I=(μ020+μ002μ110μ101μ110μ200+μ002μ011μ101μ011μ200+μ020).
(9)

The invariants of this tensor are

O1=μ200+μ020+μ002,O2=μ200μ020+μ200μ002+μ020μ002μ1102μ1012μ0112,O3=μ200μ020μ002+2μ110μ101μ011μ200μ0112μ020μ1012μ002μ1102.

Ω3 is calculated from the third moment invariant, O3, and is defined as Ω3=V5/3/O3, where V is the grain volume. For this study, the normalized form of Ω3 is used. This factor is obtained by normalizing the Ω3 value for a shape with respect to the maximum theoretical value of Ω3 of a sphere, Ω3s=2000π2/9. Thus, the value of the normalized form of Ω3,Ω¯3=Ω3/Ω3s varies from 0 to 1. The lower value of Ω¯3 represents complex shapes, and Ω¯3=1 denotes a spherical or ellipsoidal shape. Detailed information regarding the role of Ω¯3 in determining shape can be found in Refs. 37, 51, and 52.

The influence of morphology on the ultrasonic responses is investigated using synthetic volumes. The study considers the variation of morphology due to three different tessellations described in Sec. III and the variation of morphology achieved by the change of the normalized affine invariant, Ω¯3, described in Sec. III D. Neper33,44 and DREAM.3D31 were used to create synthetics to study the variations in morphology due to tessellations. To understand the effect of the normalized affine invariant, Ω¯3, on ultrasonic responses, only DREAM.3D was used. DREAM.3D allows the user to control the Ω¯3 distribution of the generated grains.

Neper and DREAM.3D are open source software used by researchers to create synthetic volumes. Voronoi and Laguerre tessellations were generated using Neper. These volumes were then converted to a raster or voxelized format. All microstructures of different tessellations were assumed to be made of nickel (Ni) with single crystal elastic properties of c11=247,c12=153, and c44=122 GPa53 and a density of ρ = 8900 kg/m3. The volumes had a nominal mean grain diameter of D=30μm with four standard deviations (SDs) covering a range from 9 to 18 μm. A set of 100 microstructures was created (25 realizations in each ensemble) for each of the tessellations. These microstructures had a volume of 5003μm3 and contained grains of cubic symmetry with a lognormal distribution, f(d), defined as

f(d)=1dσ2πe(lndμ)2/2σ2,
(10)

where d is the grain diameter and μ and σ are the average and SD of the grain diameter in log-space, respectively. The focus of this study was to understand the influence of morphology on ultrasonic responses. Therefore, it was crucial to have microstructures with nearly identical grain statistics. Creation of these microstructures proved to be challenging because it was computationally expensive and the output grain size statistics of DREAM.3D deviated from the input grain statistics, especially for wider grain statistics. These challenges were overcome by using a trial and error–based method for creation of the DREAM.3D synthetics and the computing resources of the Holland Computing Center (HCC). The Voronoi and Laguerre tessellations closely followed the input statistics. However, Neper required a longer time to generate the synthetic microstructures in comparison with DREAM.3D, especially for those with wider distributions. For instance, a single synthetic volume generation with DREAM.3D using one core in the HCC was more than ten times faster than that of Neper using ten cores on HCC for wider grain statistics. For all three of the tessellations, the grain-size statistics of the boundary grains were statistically larger than the interior grains.21 For this reason, the synthetic volumes were cropped according to their grain size distribution to ensure a homogeneous spatial distribution. The cropping was performed from the center of the 5003μm3 volume such that each face was closer to the center by an amount that was three times the SD, Σ, of the grain diameter. Hence, the final cube dimension became l=(5006Σ)μm. Figure 1 shows example realizations of the three tessellations after cropping. The volumes are from the widest grain-size distribution. Figure 1 exhibits the morphological differences among three tessellations. The generated grains of DREAM.3D are more equiaxed in shape while the Voronoi and Laguerre tessellations consist of grains with sharper edges. The grain statistics of the resulting volumes given in Table I show that the four ensembles of volumes have a very similar average grain diameter. The maximum diameter represents the average of the largest grain diameters across the ensemble. From Table I, it can be observed that the SD to mean grain diameter ratio, (Σ/D), is similar for each of the ensembles within the three different tessellations. With respect to the number of grains, the Voronoi tessellations have more grains in comparison with DREAM.3D for all of the grain statistics. In the case of the Laguerre tessellations, it has more grains in comparison with DREAM.3D except for the narrowest grain statistics. Such a difference is the result of the higher number of small grains in Voronoi and Laguerre tessellations. Figure 2 shows an example histogram from one synthetic volume for all three of the tessellations and the four grain size distribution considered in this study. All of the grain size distributions are lognormal and similar to one another. The wider distributions, depicted in Figs. 2(c) and 2(d), show that DREAM.3D has more grains closer to the mean grain diameter in comparison with Voronoi and Laguerre tessellations. For the widest distribution, the difference in grain statistics becomes more prominent in the histogram. Voronoi and Laguerre have much smaller grains in comparison with DREAM.3D.

FIG. 1.

(Color online) The cropped synthetic volumes created using different tessellations. (a), (b), and (c) represent synthetic microstructures created using DREAM.3D, Voronoi, and Laguerre tessellations, respectively. The colors represent different grains. Voronoi and Laguerre tessellations consist of grains with sharper edges.

FIG. 1.

(Color online) The cropped synthetic volumes created using different tessellations. (a), (b), and (c) represent synthetic microstructures created using DREAM.3D, Voronoi, and Laguerre tessellations, respectively. The colors represent different grains. Voronoi and Laguerre tessellations consist of grains with sharper edges.

Close modal
TABLE I.

The grain statistics of the cropped volumes. For each distribution, D and Σ are the mean and SD, respectively, of grain diameters obtained using the volumes of the new ensembles. From this point forward, all of the calculations are performed using the cropped microstructures.

TessellationsD(μm)Σ(μm)Σ/DMaximum diameter (μm)Number of grains
DREAM.3D 27.40 ± 0.1 10.73 ± 0.1 0.391 ± 0.004 68.6 ± 0.7 5563 ± 46 
27.70 ± 0.1 13.14 ± 0.14 0.475 ± 0.01 86.16 ± 1.5 3984 ± 53 
27.69 ± 0.2 16.07 ± 0.26 0.580 ± 0.01 126.95 ± 6 2684 ± 73 
26.88 ± 0.50 18.32 ± 0.41 0.682 ± 0.02 153.65 ± 10.7 1944 ± 96 
Voronoi 26.88 ± 0.07 10.35 ± 0.1 0.384 ± 0.005 84.14 ± 4.3 5966 ± 34 
26.79 ± 0.1 12.5 ± 0.1 0.467 ± 0.005 104.89 ± 7.4 4459 ± 32 
26.70 ± 0.15 14.87 ± 0.12 0.557 ± 0.007 124.40 ± 5.32 3191 ± 33 
26.80 ± 0.3 17.32 ± 0.2 0.647 ± 0.01 138.82 ± 7.1 2206 ± 43 
Laguerre 27.63 ± 0.06 10.49 ± 0.05 0.380 ± 0.002 83.78 ± 4.68 5537 ± 22 
27.43 ± 0.12 12.74 ± 0.08 0.465 ± 0.005 111.6 ± 7.89 4196 ± 26 
27.39 ± 0.14 14.98 ± 0.11 0.547 ± 0.006 135.61 ± 10.6 3037 ± 28 
27.37 ± 0.24 17.34 ± 0.14 0.634 ± 0.009 166.96 ± 18.97 2115 ± 32 
TessellationsD(μm)Σ(μm)Σ/DMaximum diameter (μm)Number of grains
DREAM.3D 27.40 ± 0.1 10.73 ± 0.1 0.391 ± 0.004 68.6 ± 0.7 5563 ± 46 
27.70 ± 0.1 13.14 ± 0.14 0.475 ± 0.01 86.16 ± 1.5 3984 ± 53 
27.69 ± 0.2 16.07 ± 0.26 0.580 ± 0.01 126.95 ± 6 2684 ± 73 
26.88 ± 0.50 18.32 ± 0.41 0.682 ± 0.02 153.65 ± 10.7 1944 ± 96 
Voronoi 26.88 ± 0.07 10.35 ± 0.1 0.384 ± 0.005 84.14 ± 4.3 5966 ± 34 
26.79 ± 0.1 12.5 ± 0.1 0.467 ± 0.005 104.89 ± 7.4 4459 ± 32 
26.70 ± 0.15 14.87 ± 0.12 0.557 ± 0.007 124.40 ± 5.32 3191 ± 33 
26.80 ± 0.3 17.32 ± 0.2 0.647 ± 0.01 138.82 ± 7.1 2206 ± 43 
Laguerre 27.63 ± 0.06 10.49 ± 0.05 0.380 ± 0.002 83.78 ± 4.68 5537 ± 22 
27.43 ± 0.12 12.74 ± 0.08 0.465 ± 0.005 111.6 ± 7.89 4196 ± 26 
27.39 ± 0.14 14.98 ± 0.11 0.547 ± 0.006 135.61 ± 10.6 3037 ± 28 
27.37 ± 0.24 17.34 ± 0.14 0.634 ± 0.009 166.96 ± 18.97 2115 ± 32 
FIG. 2.

(Color online) Example histograms of DREAM.3D, Voronoi, and Laguerre tessellations for four different distribution widths for a single volume. The distribution width increases gradually from (a) to (d).

FIG. 2.

(Color online) Example histograms of DREAM.3D, Voronoi, and Laguerre tessellations for four different distribution widths for a single volume. The distribution width increases gradually from (a) to (d).

Close modal

As discussed in Sec. III D, Ω¯3 is used to quantify the shape of microstructures. Based on the distribution of Ω¯3, shapes of the grains can be very different. DREAM.3D was used to generate synthetics having different distributions of Ω¯3. The Ω¯3 factor in DREAM.3D follows a beta distribution. The two parameters of the beta distribution, α and β, can be varied to create microstructures with different morphologies. The goal of this study was to quantify the ultrasonic scattering in microstructures that consist of complex grains and establish a comparison with regular equiaxed microstructures. Three different distributions of α and β were considered for this study. The input and output values of α and β distributions in the synthetic volumes were different. The calculated α and β distributions of the three sets of synthetics were (1) α=7.1, β = 7; (2) α=20.5,β=5.6; and (3) α=32.9,β=4.8. These distributions were chosen so that ultrasonic scattering from microstructures with different complex grain shapes can be studied. For each distribution, 30 realizations were created with a volume of 500μm3. These volumes were cropped as described in Sec. IV A and the resulting volumes had final sizes of 410×410×410μm3. The cropped volumes had a distribution width to mean ratio, Σ/D=0.54, with a mean grain diameter of 27.5μm. Figure 3 shows the cross section of three synthetic microstructures with very different Ω¯3 distributions. For Ω¯31, grains are more ellipsoidal or equiaxed. However, for Ω¯3 much smaller than one, the synthetic volumes are mainly comprised of complex shaped grains.

FIG. 3.

(Color online) Slices of three different cropped synthetic microstructures having different distributions of Ω¯3 but similar grain size statistics. (a), (b), and (c) represent the distributions of Ω¯3 of the synthetics used in this study. The corresponding 2D slice of a synthetic microstructure is given in (d), (e), and (f), respectively. The microstructure of (d) is more complex and elongated compared to the microstructures of (e) and (f). The colors in the synthetic microstructure represent different grains and do not illustrate their orientations.

FIG. 3.

(Color online) Slices of three different cropped synthetic microstructures having different distributions of Ω¯3 but similar grain size statistics. (a), (b), and (c) represent the distributions of Ω¯3 of the synthetics used in this study. The corresponding 2D slice of a synthetic microstructure is given in (d), (e), and (f), respectively. The microstructure of (d) is more complex and elongated compared to the microstructures of (e) and (f). The colors in the synthetic microstructure represent different grains and do not illustrate their orientations.

Close modal

Ultrasonic attenuation was calculated for all of the synthetic volumes considering 1000 scattering directions with 5000 discrete points in each scattering direction. The results are divided into two sections. The first is the change in the ultrasonic attenuation due to morphological differences observed in different tessellations and the other is the effect of Ω¯3 distribution on ultrasonic attenuation.

Attenuation was calculated for three different tessellations using the properties of Ni microstructures for a range of frequencies. Neper is capable of generating synthetics in raster or voxelized format. All of the Voronoi and Laguerre synthetics used in the study were in voxelized format. The TPSC function and distribution of Ω¯3 across the tessellations were also studied, and these quantities represent the geometrical and morphological differences of the volumes, respectively.

1. Two-point spatial correlation function and Ω¯3

Across the different tessellation algorithms, there are differences in their TPSCs. Figure 4 shows this function along with the Ω¯3 distribution for DREAM.3D, Voronoi, and Laguerre tessellations. The difference in the TPSC becomes more prominent at the wider distributions. The behavior of the TPSC differs based on the grain size distribution widths. For the narrowest distribution width, all three of the tessellations are nearly identical for shorter separation distances. The differences become noticeable at greater separation distances. Voronoi has the largest correlation length and DREAM.3D has the shortest correlation length because Voronoi has the largest maximum diameter among the three tessellations for the narrowest grain size distribution. In the case of the widest grain size distribution, the difference in TPSC can be noticed from a smaller separation distance. The sharper edges of Voronoi and Laguerre tessellations and observed difference in the distribution of Ω¯3 among the tessellations contribute to this variation. Laguerre has the largest correlation length and Voronoi has the shortest correlation length for the widest grain size distribution, which matches the observed statistics from Table I. The other important factor that is noticed in Fig. 4 is the difference in the Ω¯3 distribution across the three tessellations. Voronoi and Laguerre have a similar distribution of Ω¯3 due to the similarities of their tessellation algorithms. The values of Ω¯3 cover a wider range from 0.4 to 0.95 and most of the grains have an Ω¯3 ≲ 0.8. As described in Sec. III D, the lower the values of Ω¯3, the further away the grain shape is from an equiaxed structure. The Ω¯3 distribution in DREAM.3D is different in comparison from the Voronoi and Laguerre tessellations and comprised of nearly ellipsoidal grains. The differences observed in the two-point statistics and distribution of Ω¯3 contribute to the difference in attenuations among the three tessellations, which is described in Sec. V A 2.

FIG. 4.

(Color online) Spatial correlation function and Ω¯3 distribution for three different tessellations. (a) shows the two-point spatial correlation function of synthetic microstructures for different tessellation algorithms. The results are presented only for the narrowest and widest grain size distributions. (b) represents the distribution of Ω¯3 in DREAM.3D, Voronoi, and Laguerre tessellations for the widest grain size distribution.

FIG. 4.

(Color online) Spatial correlation function and Ω¯3 distribution for three different tessellations. (a) shows the two-point spatial correlation function of synthetic microstructures for different tessellation algorithms. The results are presented only for the narrowest and widest grain size distributions. (b) represents the distribution of Ω¯3 in DREAM.3D, Voronoi, and Laguerre tessellations for the widest grain size distribution.

Close modal

2. Attenuation

The attenuations for these synthetic microstructures were analyzed with respect to frequency. The frequencies considered for this study cover a range from 7.5 to 50 MHz. In comparison with the mean grain size of 27μm, the scattering regime goes from Rayleigh (λD) to stochastic (λD). Here, λ and D represent the wavelength and mean grain diameter, respectively. Figure 5 shows the ultrasonic attenuation for the narrowest and widest grain size distributions considered in this study.

FIG. 5.

(Color online) Ultrasonic attenuation for DREAM.3D, Voronoi, and Laguerre tessellations. (a) and (b) represent the quasi-longitudinal and first quasi-shear attenuation with respect to frequency, respectively. Both of the figures show the attenuation results only for the narrowest and widest grain size distributions considered in the study. The attenuation curve for the narrowest distribution of DREAM.3D appears coincident with the narrowest distribution of the Laguerre tessellation due to the usage of the log scale.

FIG. 5.

(Color online) Ultrasonic attenuation for DREAM.3D, Voronoi, and Laguerre tessellations. (a) and (b) represent the quasi-longitudinal and first quasi-shear attenuation with respect to frequency, respectively. Both of the figures show the attenuation results only for the narrowest and widest grain size distributions considered in the study. The attenuation curve for the narrowest distribution of DREAM.3D appears coincident with the narrowest distribution of the Laguerre tessellation due to the usage of the log scale.

Close modal

The difference in ultrasonic attenuation becomes more significant at wider grain size distributions. As shown in Table I, the Voronoi tessellation has smaller maximum grain diameters in comparison with Laguerre and DREAM.3D for the widest distribution. Larger grains increase scattering and attenuation of ultrasonic energy. For this reason, the attenuation values from the Voronoi tessellation are lower in comparison with the DREAM.3D and Laguerre tessellations for the Rayleigh regime for the widest distribution.

Figure 6 shows the difference in the mean attenuation values from the Voronoi and Laguerre tesselations in comparison with DREAM.3D. These percentage differences are calculated with respect to DREAM.3D using

Difference in α%=αD3DαV|LαD3D×100,
(11)

where the subscripts D3D, V, and L represent DREAM.3D, Voronoi and Laguerre, respectively. Additionally, the symbol “|” represents the logical OR. Thus, the negative values in the plot mean that the attenuation of DREAM.3D is lower at that point. For the widest grain size distribution, the percentage difference in longitudinal attenuation is greater between Voronoi and DREAM.3D at the lower frequency and it decreases with the increase in frequency. The sharper edges of Voronoi are sensitive to higher frequency. For the Laguerre tessellation, a similar trend is noticed. With the increase in frequency for the widest distribution, the difference in attenuation decreases with the exception of frequency regime of 10–15 MHz. For the higher frequency regime, the transition from Rayleigh to stochastic is noticed. Although the grains of the Laguerre tessellation are larger, the attenuation of DREAM.3D is higher throughout the frequency regime. This behavior can be explained from the TPSC curve shown in Fig. 4. The area under the curve for DREAM.3D is higher for a correlation length of ∼122 μm. After that value, the TPSC curve of the Laguerre curve crosses the TPSC curve of DREAM.3D. The effect of longer correlation length of Laguerre and its sharper edges contribute to increased attenuation at higher frequencies. For the narrowest distributions, Voronoi and Laguerre tessellations have bigger grains compared to DREAM.3D and their attenuation values are larger in the Rayleigh regime. An inverse trend is noticed in the difference in longitudinal attenuation values for the narrowest distribution. With the increase in frequency, the percentage difference in the longitudinal attenuation increases. Because the grains in the narrowest distribution are not as large as the widest distribution, the smaller sharper edge lengths of the Voronoi and Laguerre tessellations are not sensitive for this frequency range. Moreover, the area under the TPSC curve of DREAM.3D for the narrowest distribution is higher below a correlation length of ∼38 μm. These phenomena contribute to the percent increase in longitudinal attenuation with the increase in frequency. Figure 7 shows the histogram of edge lengths from a single realization of Voronoi and Laguerre tessellations for the narrowest and widest distributions. From the histogram, it is clear that the widest distribution has more edge lengths that are greater than the mean grain diameter in comparison with the narrowest distribution. This effect explains the difference in the ultrasonic predictions for the Voronoi and Laguerre tessellations with respect to the distribution width. Moreover, the short increase and then decrease in the difference of longitudinal attenuation for the Laguerre tessellation for the widest distribution can also be described from Fig. 7. Voronoi tessellations have much larger edges relative to Laguerre tessellations. The scattering due to sharper edges is noticed in the Voronoi tessellation at lower frequencies compared to that in the Laguerre tessellation.

FIG. 6.

(Color online) The differences in mean ultrasonic attenuation for DREAM.3D, Voronoi, and Laguerre tessellations as defined in Eq. (11). (a) and (b) represent the differences in longitudinal and shear attenuation among DREAM.3D and Voronoi and DREAM.3D and Laguerre tessellations, respectively. Both of the figures show the differences in attenuation only for the narrowest and widest grain size distributions considered in the study. The statistics are described in Table I. The difference in attenuation is calculated with respect to the attenuation values of DREAM.3D.

FIG. 6.

(Color online) The differences in mean ultrasonic attenuation for DREAM.3D, Voronoi, and Laguerre tessellations as defined in Eq. (11). (a) and (b) represent the differences in longitudinal and shear attenuation among DREAM.3D and Voronoi and DREAM.3D and Laguerre tessellations, respectively. Both of the figures show the differences in attenuation only for the narrowest and widest grain size distributions considered in the study. The statistics are described in Table I. The difference in attenuation is calculated with respect to the attenuation values of DREAM.3D.

Close modal
FIG. 7.

(Color online) A histogram showing the distribution of edge lengths in Voronoi and Laguerre tessellations for the narrowest and widest distributions of grain sizes. Laguerre tessellations have much smaller edges but fewer larger edges compared to Voronoi tessellations.

FIG. 7.

(Color online) A histogram showing the distribution of edge lengths in Voronoi and Laguerre tessellations for the narrowest and widest distributions of grain sizes. Laguerre tessellations have much smaller edges but fewer larger edges compared to Voronoi tessellations.

Close modal

The shear attenuation is more sensitive to the difference in morphology compared to the longitudinal attenuation. The wave length of a shear wave is also smaller than the longitudinal wave, which means that with increased frequency, the shear wave will be more sensitive to the sharper edges of Voronoi and Laguerre tessellations. The maximum difference in shear attenuation between DREAM.3D and Voronoi is 41% in the Rayleigh regime. In the case of longitudinal attenuation, the maximum difference observed is about 43%. In the case of Laguerre, the maximum differences in longitudinal and shear attenuations are 15% and 14%, respectively. Many metal samples have wide grain size distributions.54 The studies of ultrasonic scattering considering the Voronoi tessellations19,20,23,27,36 mostly consider a narrow distribution and have focused on the longitudinal attenuation only. In the study by Sha et al.,23 comparisons among different analytic and numerical methods for calculating ultrasonic attenuation were made. For triclinic symmetry, the attenuation results between the FE method and analytical model were found to be within 10%. Their results were based on a Voronoi tessellation with a narrow distribution. In another recent study by Liu et al.,55 a wider distribution of grain sizes was considered, but the study was focused only on longitudinal attenuation. The results here indicate that the attenuation difference can vary a significant amount with the types of tessellations and waves. Additional research is needed to quantify the comparison with real materials.

As described in Sec. III D, the affine invariant, Ω¯3, is a shape quantifying factor, and in DREAM.3D, Ω¯3 follows a beta distribution. The first ensemble of synthetics with a beta distribution of α=7.1,β=7 is comprised of grains having complex shapes. Figures 8(c) and 8(d) show an example aspect ratio from each of the three different beta distributions. The complex shaped grains from the first set of synthetics are elongated in both directions. The second ensemble of grains having a beta distribution of α=20.5,β=5.6 is less elongated than the previous one, and the final ensemble of grains with a beta distribution of α=32.9,β=4.8 is close to that of equiaxed grains. Figure 8 also shows the histogram and spatial correlation function for the three sets of synthetics. Although the histograms for all three of the beta distributions are similar, differences in the TPSC function are observed. In the case of the first set of synthetics, the TPSC function drops more rapidly in comparison with the other two sets of synthetics. The TPSC function provides information related to the geometry. As the histograms are similar, the differences observed in the TPSC are the result of the differences in the grain shape and aspect ratio. As shown in Fig. 3, the three different beta distributions result in grains having different morphologies.

FIG. 8.

(Color online) The grain statistics and two-point spatial correlation functions for three different Ω¯3 distributions. (a) represents the histogram of the diameter of grains, (b) shows the spatial correlation function, and (c) and (d) represent the aspect ratio of grains for three different distributions of Ω¯3. A, B, and C represent the dimensions of grains from longest to shortest.

FIG. 8.

(Color online) The grain statistics and two-point spatial correlation functions for three different Ω¯3 distributions. (a) represents the histogram of the diameter of grains, (b) shows the spatial correlation function, and (c) and (d) represent the aspect ratio of grains for three different distributions of Ω¯3. A, B, and C represent the dimensions of grains from longest to shortest.

Close modal

Attenuation was calculated for all of the synthetics for a frequency of 15 MHz. The calculated quasi-longitudinal and quasi-shear attenuations are presented in Table II. A substantial difference in ultrasonic attenuation is noticed among the three Ω¯3 distributions. Yang et al.56 and Calvet and Margerin57 investigated the effects of grain shapes on ultrasonic attenuation, especially elongated grains. Calvet and Margerin57 focused on the ultrasonic attenuation for shear waves for different shaped grains. Yang et al.56 showed that around the Rayleigh regime, the attenuation in equiaxed grains can be ten times more than the highly elongated grains. From Table II, it is noticed that the attenuation for complex elongated shapes is 56% less than the equiaxed grains. As the grains with a beta distribution of α=7.1,β=7 have elongated grains, which are shown in Fig. 8, the high difference in attenuation is expected and in agreement with previous studies. These results indicate the practicality of detecting or distinguishing complex shaped grains from the regular ellipsoidal and equiaxed microstructures. The studies by Garcin et al.11,58 used an attenuation spectrum model validated with experimental results. The 56% deviation in ultrasonic attenuation with the same grain statistics shows the possibility of detecting the evolution or presence of complex shaped grains using ultrasound. The ratio of maximum diameter to mean grain diameter for the grains of the blocks are 4 and the 56% less attenuation value will definitely fall in the outlier section of the ultrasound attenuation spectrum according to the study by Garcin et al.11 

TABLE II.

The calculated ultrasonic attenuation for the three different Ω¯3 distributions appearing in Fig. 3. The beta distribution parameters and the corresponding attenuation values are represented for an ultrasonic wave of 15 MHz.

Beta distribution parameters (α,β)Mean grain diameter (μm)Quasi-longitudinal attenuation (Np/cm)Quasi-shear attenuation (Np/cm)
α=7.1, β = 7 27.69 ± 0.2 0.27 ± 0.01 0.82 ± 0.05 
α=20.5,β=5.6 27.69 ± 0.2 0.54 ± 0.05 1.65 ± 0.15 
α=32.9,β=4.8 27.69 ± 0.2 0.62 ± 0.06 1.89 ± 0.14 
Beta distribution parameters (α,β)Mean grain diameter (μm)Quasi-longitudinal attenuation (Np/cm)Quasi-shear attenuation (Np/cm)
α=7.1, β = 7 27.69 ± 0.2 0.27 ± 0.01 0.82 ± 0.05 
α=20.5,β=5.6 27.69 ± 0.2 0.54 ± 0.05 1.65 ± 0.15 
α=32.9,β=4.8 27.69 ± 0.2 0.62 ± 0.06 1.89 ± 0.14 

A theoretical study was performed to establish a comparison among the ultrasonic scattering in grains with different morphologies. The findings of this study can be summarized with the following points. The difference in ultrasonic attenuation among DREAM.3D, Laguerre, and Voronoi is maximum in the Rayleigh regime for the widest grain size distribution. For the narrowest distribution, the difference in ultrasonic attenuation is mainly dependent on the size of the maximum grain diameter.

Voronoi tessellations have much larger edges, and for the widest distribution, the Voronoi tessellation has the lowest attenuation among the three tessellations considered in the study. Voronoi tessellations cannot generate bigger grains for the case of wider grain size distribution. The maximum difference in ultrasonic attenuation found in this study is 43% between DREAM.3D and the Voronoi tessellation. The sharper edges of Voronoi and Laguerre tessellations contribute to greater attenuation values at higher frequencies when the transition from Rayleigh to stochastic occurs. Finally, the pattern of longitudinal attenuation and shear attenuation is not the same. Shear attenuation is more sensitive to the morphology of the geometry.

The results of this article show that a significant difference in ultrasonic attenuation will be noticed if there are complex shaped grains within the microstructure. Overall, the attenuation differences due to morphology found in the study should be considered while making comparisons between any ultrasonic attenuation models or methods.

This work was supported by the Air Force Research Laboratory under prime Contract No. FA8650-15-D-5231 and was completed using the HCC of the University of Nebraska, which receives support from the Nebraska Research Initiative.

1.
N.
Hansen
, “
Hall–Petch relation and boundary strengthening
,”
Scr. Mater.
51
(
8
),
801
806
(
2004
).
2.
V.
Imayev
,
R.
Imayev
,
G.
Salishchev
,
M.
Shagiev
,
A.
Kuznetsov
, and
K.
Povarova
, “
Effect of strain rate on twinning and room temperature ductility of TiAl with fine equiaxed microstructure
,”
Scr. Mater.
36
(
8
),
891
897
(
1997
).
3.
R.
Imayev
,
N.
Gabdullin
,
G.
Salishchev
,
O.
Senkov
,
V.
Imayev
, and
F.
Froes
, “
Effect of grain size and partial disordering on ductility of Ti3Al in the temperature range of 20–600 °C
,”
Acta Mater.
47
(
6
),
1809
1821
(
1999
).
4.
M. D.
Sangid
,
G. J.
Pataky
,
H.
Sehitoglu
,
R. G.
Rateick
,
T.
Niendorf
, and
H. J.
Maier
, “
Superior fatigue crack growth resistance, irreversibility, and fatigue crack growth–microstructure relationship of nanocrystalline alloys
,”
Acta Mater.
59
(
19
),
7340
7355
(
2011
).
5.
P.
Kumar
and
U.
Ramamurty
, “
Microstructural optimization through heat treatment for enhancing the fracture toughness and fatigue crack growth resistance of selective laser melted Ti-6Al-4V alloy
,”
Acta Mater.
169
,
45
59
(
2019
).
6.
W. M.
Tucho
,
P.
Cuvillier
,
A.
Sjolyst-Kverneland
, and
V.
Hansen
, “
Microstructure and hardness studies of Inconel 718 manufactured by selective laser melting before and after solution heat treatment
,”
Mater. Sci. Eng., A
689
,
220
232
(
2017
).
7.
S. L.
Sing
,
S.
Huang
, and
W. Y.
Yeong
, “
Effect of solution heat treatment on microstructure and mechanical properties of laser powder bed fusion produced cobalt-28chromium-6molybdenum
,”
Mater. Sci. Eng., A
769
,
138511
(
2020
).
8.
O.
Salman
,
C.
Gammer
,
A.
Chaubey
,
J.
Eckert
, and
S.
Scudino
, “
Effect of heat treatment on microstructure and mechanical properties of 316L steel synthesized by selective laser melting
,”
Mater. Sci. Eng., A
748
,
205
212
(
2019
).
9.
K.
Mingard
,
B.
Roebuck
,
E.
Bennett
,
M.
Gee
,
H.
Nordenstrom
,
G.
Sweetman
, and
P.
Chan
, “
Comparison of EBSD and conventional methods of grain size measurement of hardmetals
,”
Int. J. Refractory Met. Hard Mater.
27
(
2
),
213
223
(
2009
).
10.
M.
Keyvani
,
T.
Garcin
,
M.
Militzer
, and
D.
Fabregue
, “
Laser ultrasonic measurement of recrystallization and grain growth in an L605 cobalt superalloy
,”
Mater. Charact.
167
,
110465
(
2020
).
11.
T.
Garcin
,
J. H.
Schmitt
, and
M.
Militzer
, “
In-situ laser ultrasonic grain size measurement in superalloy INCONEL 718
,”
J. Alloys Compd.
670
,
329
336
(
2016
).
12.
M.
Clark
,
A.
Clare
,
P.
Dryburgh
,
W.
Li
,
R.
Patel
,
D.
Pieris
,
S.
Sharples
, and
R.
Smith
, “
Spatially resolved acoustic spectroscopy (SRAS) microstructural imaging
,”
AIP Conf. Proc.
2102
,
020001
(
2019
).
13.
P.
Dryburgh
,
R. J.
Smith
,
P.
Marrow
,
S. J.
Lainé
,
S. D.
Sharples
,
M.
Clark
, and
W.
Li
, “
Determining the crystallographic orientation of hexagonal crystal structure materials with surface acoustic wave velocity measurements
,”
Ultrasonics
108
,
106171
(
2020
).
14.
G.
Ghoshal
and
J. A.
Turner
, “
Diffuse ultrasonic backscatter at normal incidence through a curved interface
,”
J. Acoust. Soc. Am.
128
(
6
),
3449
3458
(
2010
).
15.
J. H.
Rose
, “
Ultrasonic backscatter from microstructure
,”
Rev. Prog. Quant. Nondestr. Eval.
11B
,
1677
1684
(
1992
).
16.
W. P.
Mason
and
H.
McSkimin
, “
Attenuation and scattering of high frequency sound waves in metals and glasses
,”
J. Acoust. Soc. Am.
19
(
3
),
464
473
(
1947
).
17.
F. E.
Stanke
and
G. S.
Kino
, “
A unified theory for elastic wave propagation in polycrystalline materials
,”
J. Acoust. Soc. Am.
75
(
3
),
665
681
(
1984
).
18.
R. L.
Weaver
, “
Diffusivity of ultrasound in polycrystals
,”
J. Mech. Phys. Solids
38
(
1
),
55
86
(
1990
).
19.
A.
Van Pamel
,
G.
Sha
,
S. I.
Rokhlin
, and
M. J.
Lowe
, “
Finite-element modelling of elastic wave propagation and scattering within heterogeneous media
,”
Proc. R. Soc. A
473
(
2197
),
20160738
(
2017
).
20.
M.
Huang
,
G.
Sha
,
P.
Huthwaite
,
S.
Rokhlin
, and
M.
Lowe
, “
Maximizing the accuracy of finite element simulation of elastic wave propagation in polycrystals
,”
J. Acoust. Soc. Am.
148
(
4
),
1890
1910
(
2020
).
21.
M.
Norouzian
and
J. A.
Turner
, “
Ultrasonic wave propagation predictions for polycrystalline materials using three-dimensional synthetic microstructures: Attenuation
,”
J. Acoust. Soc. Am.
145
(
4
),
2181
2191
(
2019
).
22.
M.
Norouzian
and
J. A.
Turner
, “
Ultrasonic wave propagation predictions for polycrystalline materials using three-dimensional synthetic microstructures: Phase velocity variations
,”
J. Acoust. Soc. Am.
145
(
4
),
2171
2180
(
2019
).
23.
G.
Sha
,
M.
Huang
,
M.
Lowe
, and
S.
Rokhlin
, “
Attenuation and velocity of elastic waves in polycrystals with generally anisotropic grains: Analytic and numerical modeling
,”
J. Acoust. Soc. Am.
147
(
4
),
2442
2465
(
2020
).
24.
G.
Sha
, “
Correlation of elastic wave attenuation and scattering with volumetric grain size distribution for polycrystals of statistically equiaxed grains
,”
Wave Motion
83
,
102
110
(
2018
).
25.
M.
Huang
,
G.
Sha
,
P.
Huthwaite
,
S.
Rokhlin
, and
M.
Lowe
, “
Longitudinal wave attenuation in polycrystals with elongated grains: 3D numerical and analytical modeling
,”
J. Acoust. Soc. Am.
149
(
4
),
2377
2394
(
2021
).
26.
M.
Huang
,
S.
Rokhlin
, and
M.
Lowe
, “
Finite element evaluation of a simple model for elastic waves in strongly scattering elongated polycrystals
,”
JASA Express Lett.
1
(
6
),
064002
(
2021
).
27.
M.
Huang
,
G.
Sha
,
P.
Huthwaite
,
S.
Rokhlin
, and
M.
Lowe
, “
Elastic wave velocity dispersion in polycrystals with elongated grains: Theoretical and numerical analysis
,”
J. Acoust. Soc. Am.
148
(
6
),
3645
3662
(
2020
).
28.
M.
Tanemura
,
T.
Ogawa
, and
N.
Ogita
, “
A new algorithm for three-dimensional Voronoi tessellation
,”
J. Comput. Phys.
51
(
2
),
191
207
(
1983
).
29.
Q.
Du
,
V.
Faber
, and
M.
Gunzburger
, “
Centroidal Voronoi tessellations: Applications and algorithms
,”
SIAM Rev.
41
(
4
),
637
676
(
1999
).
30.
C.
Lautensack
and
S.
Zuyev
, “
Random Laguerre tessellations
,”
Adv. Appl. Probab.
40
(
3
),
630
650
(
2008
).
31.
M. A.
Groeber
and
M. A.
Jackson
, “
DREAM.3D: A digital representation environment for the analysis of microstructure in 3D
,”
Integr. Mater. Manuf. Innovation
3
(
1
),
56
72
(
2014
).
32.
A.
Van Pamel
,
C. R.
Brett
,
P.
Huthwaite
, and
M. J.
Lowe
, “
Finite element modelling of elastic wave scattering within a polycrystalline material in two and three dimensions
,”
J. Acoust. Soc. Am.
138
(
4
),
2326
2336
(
2015
).
33.
R.
Quey
and
L.
Renversade
, “
Optimal polyhedral description of 3D polycrystals: Method and application to statistical and synchrotron X-ray diffraction data
,”
Comput. Methods Appl. Mech. Eng.
330
,
308
333
(
2018
).
34.
T.
Grabec
,
M.
Ryzy
, and
I. A.
Veres
, “
Numerical modeling of surface elastic wave scattering in polycrystalline materials
,” in
2017 IEEE International Ultrasonics Symposium (IUS)
, Washington, DC (
IEEE
,
New York
,
2017
), pp.
1
4
.
35.
M.
Norouzian
,
S.
Islam
, and
J. A.
Turner
, “
Influence of microstructural grain-size distribution on ultrasonic scattering
,”
Ultrasonics
102
,
106032
(
2020
).
36.
M.
Ryzy
,
T.
Grabec
,
P.
Sedlák
, and
I. A.
Veres
, “
Influence of grain morphology on ultrasonic wave attenuation in polycrystalline media with statistically equiaxed grains
,”
J. Acoust. Soc. Am.
143
(
1
),
219
229
(
2018
).
37.
J.
MacSleyne
,
J.
Simmons
, and
M.
De Graef
, “
On the use of moment invariants for the automated analysis of 3D particle shapes
,”
Modell. Simul. Mater. Sci. Eng.
16
(
4
),
045008
(
2008
).
38.
Q.
Huang
,
J.
Li
,
R.
Liu
,
L.
Yan
, and
H.
Huang
, “
Surface morphology and microstructure evolution of IG-110 graphite after xenon ion irradiation and subsequent annealing
,”
J. Nucl. Mater.
491
,
213
220
(
2017
).
39.
T.
Omori
,
T.
Kusama
,
S.
Kawata
,
I.
Ohnuma
,
Y.
Sutou
,
Y.
Araki
,
K.
Ishida
, and
R.
Kainuma
, “
Abnormal grain growth induced by cyclic heat treatment
,”
Science
341
(
6153
),
1500
1502
(
2013
).
40.
T.
Omori
,
H.
Iwaizako
, and
R.
Kainuma
, “
Abnormal grain growth induced by cyclic heat treatment in Fe-Mn-Al-Ni superelastic alloy
,”
Mater. Des.
101
,
263
269
(
2016
).
41.
N.
Ahuja
,
B.
An
, and
B.
Schachter
, “
Image representation using Voronoi tessellation
,”
Comput. Vision, Graph., Image Process.
29
(
3
),
286
295
(
1985
).
42.
M.
Ramella
,
W.
Boschin
,
D.
Fadda
, and
M.
Nonino
, “
Finding galaxy clusters using Voronoi tessellations
,”
Astron. Astrophys.
368
(
3
),
776
786
(
2001
).
43.
I.
Benedetti
and
M.
Aliabadi
, “
A three-dimensional grain boundary formulation for microstructural modeling of polycrystalline materials
,”
Comput. Mater. Sci.
67
,
249
260
(
2013
).
44.
R.
Quey
,
P.
Dawson
, and
F.
Barbe
, “
Large-scale 3D random polycrystals for the finite element method: Generation, meshing and remeshing
,”
Comput. Methods Appl. Mech. Eng.
200
(
17-20
),
1729
1745
(
2011
).
45.
F.
Fritzen
,
T.
Böhlke
, and
E.
Schnack
, “
Periodic three-dimensional mesh generation for crystalline aggregates based on Voronoi tessellations
,”
Comput. Mech.
43
(
5
),
701
713
(
2009
).
46.
D.
Dereudre
and
F.
Lavancier
, “
Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction
,”
Comput. Stat. Data Anal.
55
(
1
),
498
519
(
2011
).
47.
S.
Falco
,
P.
Siegkas
,
E.
Barbieri
, and
N.
Petrinic
, “
A new method for the generation of arbitrarily shaped 3D random polycrystalline domains
,”
Comput. Mech.
54
(
6
),
1447
1460
(
2014
).
48.
K.
Sugihara
, “
Laguerre Voronoi diagram on the sphere
,”
J. Geom. Graph.
6
(
1
),
69
81
(
2002
).
49.
Q.
Duan
,
D. P.
Kroese
,
T.
Brereton
,
A.
Spettl
, and
V.
Schmidt
, “
Inverting Laguerre tessellations
,”
Comput. J.
57
(
9
),
1431
1440
(
2014
).
50.
S.
Falco
,
J.
Jiang
,
F.
De Cola
, and
N.
Petrinic
, “
Generation of 3D polycrystalline microstructures with a conditioned Laguerre-Voronoi tessellation technique
,”
Comput. Mater. Sci.
136
,
20
28
(
2017
).
51.
P.
Callahan
,
J.
Simmons
, and
M.
De Graef
, “
A quantitative description of the morphological aspects of materials structures suitable for quantitative comparisons of 3D microstructures
,”
Modell. Simul. Mater. Sci. Eng.
21
(
1
),
015003
(
2013
).
52.
P.
Callahan
,
M.
Groeber
, and
M.
De Graef
, “
Towards a quantitative comparison between experimental and synthetic grain structures
,”
Acta Mater.
111
,
242
252
(
2016
).
53.
A.
Every
and
A.
McCurdy
,
Second and Higher Order Elastic Constants
(
Springer
,
Berlin
,
1992
), Vol.
29
.
54.
A. P.
Arguelles
and
J. A.
Turner
, “
Ultrasonic attenuation of polycrystalline materials with a distribution of grain sizes
,”
J. Acoust. Soc. Am.
141
(
6
),
4347
4353
(
2017
).
55.
Y.
Liu
,
M. K.
Kalkowski
,
M.
Huang
,
M. J.
Lowe
,
V.
Samaitis
,
V.
Cicėnas
, and
A.
Schumm
, “
Can ultrasound attenuation measurement be used to characterise grain statistics in castings?
,”
Ultrasonics
115
,
106441
(
2021
).
56.
L.
Yang
,
O.
Lobkis
, and
S.
Rokhlin
, “
Shape effect of elongated grains on ultrasonic attenuation in polycrystalline materials
,”
Ultrasonics
51
(
6
),
697
708
(
2011
).
57.
M.
Calvet
and
L.
Margerin
, “
Impact of grain shape on seismic attenuation and phase velocity in cubic polycrystalline materials
,”
Wave Motion
65
,
29
43
(
2016
).
58.
T.
Garcin
,
J.-H.
Schmitt
, and
M.
Militzer
, “
Application of laser ultrasonics to monitor microstructure evolution in Inconel 718 superalloy
,” in
MATEC Web of Conferences
(
EDP Sciences, Les Ulis
,
France
,
2014
), Vol.
14
, p.
07001
.