Resonant ultrasonic testing can quantitatively assess the microscopic porosity of complex-shaped additively manufactured AlSi10Mg components.

The utility of resonant ultrasonic testing for quality control of complex-shaped additively manufactured (AM) components in terms of porosity variations is investigated. A fully non-contact test setup is used to investigate differences in the volumetric porosity between AM AlSi10Mg samples. A set of 96 samples with programmatically induced pores varying in nominal total porosity between 0% and 2% is tested: one half of the samples are prismatic, and the other half have a complex internal Triply Periodic Minimal Surface (TPMS) structure. In addition, a subset of the samples is scanned using X-ray micro-computed tomography (  -CT). It is found that the resonance frequency corresponding to the 1 st compressional mode can predict the total nominal porosity even in TPMS samples. From statistical analysis, the smallest detectable porosity difference is found to be 0.25% for the prismatic samples and 0.5% for the TPMS samples. The experimental results agree well with the predictions of finite element (FE) simulations and analytical models. However, X-ray  -CT appears to underestimate the porosity, possibly due to its inability to resolve the small pores. Our findings suggest that resonant ultrasonic testing can quantitatively assess the total porosity of AM parts having complex geometries.


Introduction
A common problem in AM metallic components is the formation of subsurface porosity [1].Porosity in AM metals is attributed to keyhole-mode melting, gas porosity contained within the powder particles prior to deposition, or lack of fusion pores due to poor process parameters [2][3].Presence of pores inside the material may alter its mechanical behavior, an underestimation of the volumetric porosity and pore size may lead to an overestimation of material's strength and hardness [4].When undergoing cyclic loading, pores can act as crack initiation sites thus shortening the fatigue life [5].
Researchers have applied pulse-echo ultrasonic testing using contact [6] and immersion [7], [8] transducers as well as laser ultrasonics [9] to nondestructively inspect the volumetric porosity in metallic composites and AM samples.In these studies, the porosity values obtained by means of scanning acoustic microscopy, optical microscopy or X-ray CT are correlated with the measured wavespeed and attenuation.It is shown that wavespeed has a negative correlation with increasing porosity whereas attenuation presents a positive correlation.Furthermore, phased array ultrasonic testing has been employed for the imaging of porosity in metallic AM components and shows comparable performance to X-ray imaging [10][11].However, ultrasonic wave propagation-based testing is limited when inspecting complex-shaped AM parts.There have been few recent studies, where complex-shaped samples have been encased in a block of ice to allow pulse-echo ultrasonic testing for the detection of internal defects [12][13][14].However, the increased impedance mismatch between the ice and the metallic components still poses a challenge, as the resulting ultrasonic images may contain numerous artifacts.Moreover, such encapsulation may impose additional stresses on the test sample caused by ice solidification.These challenges motivate resonant ultrasonic testing as an alternative method to evaluate the AM parts of arbitrarily complex geometries.Although resonance-based testing cannot provide the spatial distribution of pores, it can yield the total porosity for online quality control.
Resonant ultrasonic testing of complex-shaped objects has been used for two primary objectives.The first is for the material characterization of the test samples, through resonant ultrasound spectroscopy (RUS).This has been successfully implemented in the case of rhombic dodecahedron (RD) lattices [15].Furthermore, RUS has proven effective for estimating the elastic properties of AM 6061-T6 aluminum hollow spheres so long as a representative FE model is available [16].A geometrically accurate digital twin of the spheres is created to generate the FE model to increase the fidelity of the inversion process.The second objective is to detect defects in the test samples.In Ti-553 octet truss lattice cubes, the number of missing lattices is negatively correlated with the resonance frequencies and the Young's modulus and Poisson ratio back-calculated from RUS [17].Furthermore, EOS CoCr pristine lattices have been successfully distinguished from the defective ones where a number of struts were missing, by comparing the resonance frequencies of the first 225 modes [18].Implementing machine learning models such as linear discriminant analysis can extend RUS applications beyond the simple detection of lattices with missing struts to the prediction of the number of missing struts [19].RUS has also been used to investigate the influence of porosity on the elastic properties of metallic foam components [20].The porosity of MAX (MAX represents the chemical formula Mn+1AXn where M is an early transitional metal, A is an A-group element and X is in this case carbon) phase foam parts made of Ti2AlC and Ti3SiC2 has been negatively correlated with the Young's modulus, the shear modulus, and Poisson ratio back-calculated from RUS.Although RUS is finding increasing applications in the inspection of AM metallic parts, to the authors' best knowledge, there have been no attempts to connect the microscopic porosity content with the resonance frequency in samples with complex structures.
In this study, resonant ultrasonic testing is conducted on a set of 96 AlSi10Mg samples with programmatically induced porosity.Half of the samples have a simple prismatic geometry and the rest have an internal TPMS geometry to simulate parts with arbitrarily complex geometries.X-ray -CT is also used to independently quantify the volumetric porosity in a subset of samples.The resonance frequency corresponding to the 1 st compressional mode is correlated with the nominal programmatically induced porosity for both sets of samples; FEM and analytical models are used to interpret the experimental results.Finally, statistical analysis is performed to test the significance of correlation between the programmatically induced porosity and the resonance frequency and to determine the smallest detectable difference in porosity percentage for each sample set.

Material and sample preparation
A set of 96 AlSi10Mg samples with built-in porosity is fabricated using laser powder bed fusion (LPBF) additive manufacturing using an EOS M280 machine with AlSi10Mg feedstock conforming to the DIN EN 1706 (EN AC-43000) standard [21] having a nominal particle size in the range 25-70 μm.The laser power and scanning speed at different printing phases are given in Table 1.The geometry of the samples is designed using the commercial software nTopology [22].As depicted in Fig. 1a, two types of structures are considered, both having an overall length of 50 mm and cross section of 15 mm x 10 mm (see Fig. 1b).Half of the samples have an internal TPMS structure while the rest are solid prismatic samples.To achieve the desired volumetric porosity grading in the samples, cylindrical pores with a diameter of 534 μm and a thickness of 180 μm are introduced using the method described in Cummings, et al [23].For each type of structure (prismatic and TPMS), 48 samples are printed (TPMS samples are numbered 1-48 and the prismatic samples are numbered 49-96), divided in eight 6-sample sets with nominal porosities ranging from 0% (no built-in pores) to 2% (0.0, 0.25, 0.5, 0.75, 1.00, 1.25, 1.50, and 2.00).The samples are removed from the built plate using wire electrical discharge machining (EDM) and tested in their as-deposited state with no post-process heat treatment.

Experimental setup description
The resonance frequency of the AM samples corresponding to the first compressional mode is measured using a fully non-contact resonance ultrasonic testing setup to ensure measurement repeatability.
An arbitrary waveform generator (PXIe-1073) is used to generate a 3 sec chirp sweeping a prespecified 2 kHz frequency range around the expected resonant frequency.For the prismatic samples, this range is set to 49.5-51.5 kHz, while it is 41.5-43.5 kHz for the TPMS samples.
In either case, the sampling frequency is 250 MHz.The waveform generator feeds a 50 kHz non-contact piezoelectric transducer (NCG50-D50, Ultran) outfitted with a 3D-printed horn [24][25].The purpose of the horn is to amplify the emitted wave and guide it as close as possible to one end of the sample without touching it.The particle velocity history at the center of the opposite end of the sample is recorded using a laser

Estimation of resonance frequency and Q-factor
To obtain the resonance frequency, the recorded time-domain signal is converted into the frequency spectrum by applying a Fast Fourier Transform with a frequency resolution of 0.33 Hz (see Fig. 3).To estimate the resonance frequency with higher accuracy, a quadratic polynomial is fitted around the resonance peak to increase resolution.The frequency corresponding to the maximum value of the polynomial is recorded as the resonance frequency.
In addition, the Q-factor is calculated by dividing the estimated resonance frequency by the half-power bandwidth.

Porosity quantification through X-ray imaging
A total of 7 samples (3 prismatic and 4 TPMS) are selected (see details below) and X-ray -CT scanned using a 200 kV General Electric Phoenix v|tome|x L300 scanner.The voxel size is set to 10 μm with x20 magnification.For each sample, a collection of 2014 2D slices is obtained.The commercial software AVIZO 2021 is used for 3D reconstruction and the subsequent image processing.To increase the contrast between the grey areas (solid material) and the darker areas (pores), the default automatic thresholding is applied.Finally, the label analysis tool is implemented to quantify the volume of the void areas inside the material as shown in Figs.4a and 4b.The candidate samples for X-ray -CT are selected as follows.From the TPMS sample set, Sample 6 and Sample 45 are randomly chosen from 0% and 2% nominal porosity samples corresponding to the lowest and highest porosities available.Samples 37 and 40 are also selected because they exhibit the largest variation in resonance frequency among the 1.5% porosity samples (see Fig. 5a) thus representing the variation within samples of the same nominal porosity.Following the same reasoning, the prismatic Samples 83 and 84 corresponding to 1.25% porosity are chosen (see Fig. 5b).To include a representative of 2% porosity from the prismatic samples, Sample 91 is randomly selected.

Numerical modeling
To help interpret the experimental results, FE models simulating the resonant ultrasonic testing of porous samples are developed using the commercial software ABAQUS Standard/Explicit 2022.To mimic the boundary conditions during the laboratory measurements, a sinusoidally varying pressure load sweeping a pre-specified 2 kHz range is applied to one face of the sample and the nodal displacement at the center of the opposite face is recorded (see Fig. 6).Only the prismatic structure is considered for modelling with the frequency range set to 49-51 kHz.Within each model, randomly distributed pores are designed inside the material volume using Python scripting [26].Though cylindrical pores were programmatically introduced into the experimental coupons, to investigate the influence of pore shape, one set of models introduced spherical pores, and the other set had cylindrical pores; half aligned parallel to the long axis of the model and the other half perpendicular to this axis.The diameter-to-thickness ratio for the cylindrical pores is the same as the nominal ratio used when designing the programmatically induced pores inside the test samples.In both sets of simulations, the volume of individual pores is 5 times larger than each pore inside the actual samples while matching the total nominal porosity.If the size of the cylindrical pores in the actual samples was to be used in the FE models, an extremely fine mesh would be required which would result in very computationally expensive simulations.
To assign the proper material properties in the FE models, the density, Young's modulus, and Poisson ratio of the 6 prismatic samples with 0% built-in porosity are measured.The density of individual samples is evaluated and the average density is used in the FE models (see Table 2).The Young's modulus and Poisson ratio are retrieved from the measurements of longitudinal and shear wave speeds by conducting ultrasonic P-wave and S-wave through-transmission tests [cite?].The average values of the longitudinal and the shear wave speed,   and   respectively, for each sample are also listed in Table 2.The average wave speeds along with the measured average density are used to estimate the Young's modulus (69.51 GPa) and the Poisson ratio (0.337).Next, a mesh convergence study is conducted to define the appropriate element size using the linear tetrahedral element C3D4.The resonance frequency for the case of 2% porosity is used as the convergence variable.Based on the results depicted in Figs.7a and 7b, the element size of /10 is chosen for the model with spherical pores, and the element size of /5 is chosen for the model with cylindrical pores, where  represents the diameter of the spherical pores and  represents the width of the cylindrical pores as shown in Fig. 6.Note that due to memory limitations in the computing cluster [27] used to run the simulations, it was not possible to use smaller element sizes.In total, 16 models (8 with spherical pores and 8 with cylindrical pores) are created, each corresponding to one of the nominal porosity levels in the physical samples.

Analytical modelling
Two analytical models for elastic modulus change due to porosity are considered to explain the experimental resonance measurements.The test sample is modelled as a vibrating rod (no bending occurs) of constant cross-section.We use Love's model [28] for the equation of motion assuming a harmonic excitation and traction-free boundary conditions on both ends of a rod of length L: where  is the displacement in the longitudinal direction of the rod (),  is the Young's modulus,  is the Poisson ratio,  is the linear density of the rod,  is the radius of gyration,  is the excitation frequency,  is the cross-section of the rod, and  is the excitation amplitude.In Eq. ( 6)-( 7),   is used to denote the spatial derivative and   to denote the temporal derivative.For these conditions, the natural frequency  , of the rod is given by: where  is the number of the respective eigenmodes with the first eigenmode corresponding to  = 0. To include the effect of porosity in the samples, both  and  are assumed to be porositydependent.For  we simply assume that: where  is the total volumetric porosity and  0 is the linear density corresponding to 0% porosity.Two different models describing the relation between porosity and Young's modulus are considered.The first one is a simple model proposed by Dewey [29][30][31]: where  0 is the Young's modulus corresponding to 0% porosity.
The second model considered is proposed by Boccaccini [32][33][34], where the orientation and the shape of the pores are also taken into account: 3 )  (Eq.7) where   is the mean axial ratio of pores and cos 2  describes the pore orientation.By substituting either Eq. 5 or Eq. 7 along with Eq. 4 in Eq. 3, we can estimate  , for the porous material.

Resonant ultrasonic testing results
For each sample, the resonance frequency corresponding to the 1 st compressional mode is obtained for all 3 rounds of measurements and the resulting values are plotted versus the corresponding nominal programmatically induced porosity as shown in Figs.8a and 8b for prismatic and TPMS samples, respectively.In both plots, the intensity of the box plot color is inversely correlated to the size of the programmatically induced porosity in the samples (i.e., the paler the color, the larger the programmatically induced porosity).The red line marks the median value for each set of resonance frequency measurements in samples corresponding to the same programmatically induced porosity value.The insets exhibit the exceptionally high repeatability of the measurements.In both sets of samples, the measured resonance frequency is negatively correlated with programmatically induced porosity.Neither the sample roughness nor the complex geometry of the TPMS samples appears to be affecting the expected correspondence between the measured resonance frequency and nominal programmatically induced porosity.Despite the very different geometries, the percentage decrease of resonance frequency with increasing porosity remains in the same range for both sets of samples (~0.4% decrease), although the data are slightly more scattered for TPMS samples.This increased variability is not necessarily due to larger resonance frequency measurement errors but could be a result of larger variability in manufacturing the TPMS samples of a predefined porosity.Note that the change in resonance frequency due to small changes in porosity (up to 2%) is small.Therefore, it is critical to use a measurement setup with high repeatability in order to reliably capture such small differences.
The Q-factor as a function of nominal porosity within the samples was also considered.No trend was observed for either set of samples since 'damping' at this frequency range (~ 50 kHz) is not affected by low percentage of microscopic porosity due to the high wavelength-to-defectsize ratio [35].

X-ray imaging results
After identifying the pores in the X-ray -CT scans of the 7 tested samples, the total porosity is estimated and compared to the nominal values of built-in porosity according to the following relation: For the prismatic samples,    is set to 0. Table 3 compares the nominal porosities with those estimated by the analysis of X-ray -CT scans.We observe a clear difference between the nominal and the estimated values of the porosities.However, the ratios between the porosities of the different samples are rather similar.For instance, Samples 83 and 84 have a nominal porosity of 1.25% and Sample 91 has a nominal porosity of 2%.The estimated porosities for samples 83 and 84 from the X-ray CT analysis (0.26% and 0.25%) are almost half of that estimated for Sample 91 (0.52%).
Comparing the nominal and estimated porosities, it appears that X-ray -CT may not be able to detect all the internal pores within the samples thus underestimating the total volumetric porosity.Another possible explanation for this difference is that the nominal porosity values are not representative of the actual porosities.In the following section, it is demonstrated that our resonance ultrasonic measurements together with numerical and analytical modelling results support the former hypothesis.

Discussion
To investigate the discrepancy between X-ray -CT and resonant ultrasonic testing results, comparisons are made between the experimental resonance frequency measurements and the numerical and analytical predictions.Finally, we provide a statistical analysis of the resonant testing results to demonstrate the utility of this test for quantitative quality control of complexshaped AM parts, which is the main motivation of this study.

Comparison of resonant ultrasonic testing results with numerical and analytical results
A comparison of the FE-simulated resonance frequencies for nominal porosities with the average values of those obtained from the experimental measurements reveals good agreement, especially when spherical pores are assumed (see Fig. 9a).Similarly, as shown in Fig. 9b, the analytical models also predict resonance frequency values close to those measured experimentally.Dewey model (Eq.5) shows a better agreement than Boccaccini model (Eq.7), where pores are assumed to be cylindrical.These observations support the assertion that the nominal porosity values are close to the total volumetric porosities of the test samples.If the estimated porosity values from X-ray -CT image analysis were true, much smaller changes in resonance frequency would have been measured.This analysis suggests that X-ray -CT is likely to significantly underestimate the total porosities while resonance testing provides accurate estimates of total porosity even for samples with complex geometry.

Statistical analysis of resonance frequency vs. porosity
To examine the significance of the correlation between the programmatically induced porosity and the measured resonance frequency, the Spearman correlation coefficient is used.Based on the models described in Section 5, the resonance frequency is proportional to the square root of Young's modulus, which in turn can be considered to have a linear dependence (Dewey model) or a nonlinear dependence (Boccaccini model) on porosity.Therefore, the Spearman correlation coefficient is appropriate to characterize the correlation since it reveals the significance of a monotonic correlation between two variables rather than their specific dependence (unlike the Pearson correlation coefficient, which assumes that the two variables are linearly dependent).For the case of prismatic samples, the Spearman analysis provides a correlation coefficient of -0.95 while for the case of TPMS samples, the correlation coefficient is calculated as -0.81 reflecting the larger variability in the measurements.In both cases, the correlation coefficients are negative and their absolute value is reasonably close to unity, which signifies a significant negative correlation between porosity and resonance frequency.The correlation is weaker for TPMS samples.
Assuming a normal distribution for the resonance testing results, a one-way analysis of variance (ANOVA) is conducted to investigate the smallest percent porosity difference that can be detected by measuring resonance frequency.Each time, two sets of measurements for the resonance frequency corresponding to two different porosity values are used in the one-way ANOVA and a p-value is calculated.All the repeated measurements are included in this analysis.The results are listed in Table 4 for both the prismatic samples and the TPMS samples respectively.The p-values larger than 0.05 are marked in each table with red.A p-value larger than 0.05 supports the null hypothesis stating that the average values of the two sets are equal.In the prismatic samples, excluding the set 1.0-1.25%(Table 4) which results in p-value larger than 0.05, the smallest detectable percent porosity difference is 0.25%.For a 0.5% difference and larger, the null hypothesis is rejected for all sets.Similarly, in the TPMS samples, excluding the set 0.25-0.75%(Table 4) where p-value is larger than 0.05, the smallest detectable percent porosity difference is 0.5%.For this set, the null hypothesis is rejected for all sets for a difference of 1% and larger.

Conclusions
Resonant ultrasonic testing was performed using a fully non-contact experimental setup on 96 LPBF AlSiMg10 samples of prismatic and complex (TPMS) geometries with programmatically induced porosity in the range 0-2%.The measured resonance frequency corresponding to the 1 st compressional mode was shown to strongly correlate with the increasing programmatically induced porosity for both sets of samples.No correlation was observed with the Q-factor.The experimental results were supported with the corresponding FE analysis and analytical models.
The main findings of the study are: • The percent change in the resonance frequency can serve as an effective indicator of changes in the porosity of AM samples, regardless of the complexity of its geometry.• The resonant ultrasonic testing may surpass X-ray imaging in the prediction of the total volumetric porosity of the samples.The percent changes in the measured resonant frequency correspond to higher porosity levels (as confirmed by FE and analytical modelling) than those retrieved from the analysis of the X-ray -CT images.Further investigation is warranted to confirm this promising result.• The smallest percent difference of built-in porosity that can be detected using as indicator the resonance frequency corresponding to the 1 st compressional mode is 0.25% for the prismatic samples and 0.5% for the TPMS samples.The use of a noncontact setup allowed for the acquisition of measurements with high repeatability, thus providing a clear distinction between the samples, even at these low volumetric porosities.
The novelty of this study lies in using the resonance frequency of only one mode to clearly differentiate complex-shaped AM samples with different programmatically induced porosity levels, as well as the comparison to X-ray -CT image analysis results.Importantly, our findings suggest that resonant testing may provide a significantly more accurate estimate of porosity than X-ray -CT, though additional characterization may be required to confirm.Nevertheless, the resonant ultrasonic testing method could potentially be applied for a fast and economical quality control of AM parts at industrial scale.Investigation of resonant ultrasound spectroscopy (RUS) with multiple modal frequencies to directly obtain the elastic constants and an estimation of porosity from the corresponding FE or analytical models is the natural next step for this line of investigation.
Fig 1 (a) Samples in their as-built state on the build plate; (b) CAD model of TPMS samples showing the programmatically induced cylindrical pores

Fig. 2 (
Fig. 2 (a) Experimental setup; (b) Schematic representation of the experimental setup for the non-contact resonant ultrasonic testing.The numbers associate the components in (a) with the schematic in (b)

Fig. 3
Fig. 3 Workflow for estimating the resonance frequency; (a) The time-domain signal; (b) The frequency spectrum; (c) Fitting of a quadratic polynomial (red line) around the resonance peak to increase resolution.The frequency corresponding to the maximum value of the polynomial is recorded as the resonance frequency (black dashed line) Example of 2D-slice from the X-ray μ-CT imaging of a TPMS sample (grey represents detected material); (b) Pores within the TPMS geometry (cyan) are detected through label analysis; (c) Detailed view of pores detected with the TPMS material (smaller colored voids)

Fig. 6
Fig. 6 Details of the FE model with (a) spherical and (b) cylindrical pores

Fig. 7
Fig. 7 Mesh convergence study results for: (a) spherical pores and (b) cylindrical pores.The two mode shapes shown exhibit the modal displacements at each node parallel to the long axis of the model (z).The dotted vertical black lines mark the selected element size for subsequent simulations

Fig. 8
Fig. 8 Measured resonance frequency vs programmatically induced porosity for: (a) Prismatic samples and (b) TPMS samples

Fig. 9
Fig. 9 Comparison of experimentally measured resonance frequencies in the prismatic samples with those obtained from (a) numerical models and (b) analytical models

Table 1
Laser power and scanning speed used for the different phases of printing

Table 2
Density and wave speed measurements for the prismatic samples with 0% programmatically induced porosity

Table 3
Comparison between nominal programmatically induced porosity and porosity estimated by X-ray imaging for the prismatic and TPMS samples

Table 4 p
-values from one-way ANOVA for the prismatic and TPMS samples