Abstract

In guided wave structural health monitoring (GW-SHM), a strong need for reliable and fast simulation tools has been expressed throughout the literature to optimize SHM systems or demonstrate performance. Even though guided wave simulations can be conducted with most finite elements software packages, computational and hardware costs are always prohibitive for large simulation campaigns. A novel SHM module has been recently added to the civa software and relies on unassembled high-order finite elements to overcome these limitations. This article focuses on the thorough validation of civa for SHM to identify the limits of the models. After introducing the key elements of the civa SHM solution, a first validation is presented on a stainless steel pipe representative of the oil and gas industry. Second, validation is conducted on a composite panel with and without stiffener representative of some structures in the aerospace industry. Results show a good match between the experimental and simulated datasets, but only if the input parameters are fully determined before the simulations.

1 Introduction

Guided wave structural health monitoring (GW-SHM) relies on the permanent integration of sensors to monitor the health of a structure over time. Despite a large and convincing literature exhibiting various proofs of concepts, GW-SHM has not reached all of its expected objectives: primarily, mainstream preventive maintenance is based on not only real-time health assessment but also structural life extension. Among the limitations to deployment of GW-SHM systems are a systematic and rigorous system optimization and performance demonstration framework.

The uses of simulations in GW-SHM are various. First and foremost, simulations have been conducted to analyze and explain experimental results or simply to develop processing techniques [13] before experimental deployment. Second, simulations in GW-SHM are foreseen to play a significant role in model-assisted probability of detection approach to avoid prohibitively expensive experimental campaigns. More recently, the rise of artificial intelligence has created a need of large databases to train machine learning algorithms to enable efficient and robust diagnostic [4]. However, deep learning requires very large datasets, which, in general, cannot be obtained by experiments and therefore require large simulation campaigns.

Due to the high number of influencing parameters in GW-SHM, conducting such analysis using exclusively experimental data is prohibitively expensive and for this reason, and simulation frameworks have been proposed to simulate guided wave propagation in large structures. Semi-analytical methods have been widely studied in the literature [57] for the propagation of GWs in infinite structures of finite cross sections. In a time-harmonic regime, such techniques have then been coupled to finite elements to create hybrid solvers able to describe large structures with defects [811] but remain somewhat limited in terms of geometry. On the other hand and for most papers in the literature dealing with this topic, full three-dimensional (3D) transient finite element solvers have been used in GW propagation simulations. However, due to small element size and time-stepping requirements intrinsic to GW, finite elements simulations are computationally costly. A benchmark of the most common finite element software platforms was conducted in 2018 on a single composite panel [12] of relatively small dimensions: 120 × 60 × 1 mm with eight plies. The authors compared computational times from 20 h to up to 1 week for a single simulation at 300 kHz of central frequency, with powerful computer hardware and CPU architectures.

Recently, and thanks to the development of affordable and efficient computing hardware, especially GPUs, novel GW simulation techniques have emerged relying on intensive parallelization of mesh groups [1316]. If these methods have shown excellent performance, with improvements of two to three orders of magnitude with respect to traditional finite element methods on CPU, they have three limitations. First, they require a dedicated and expensive hardware, which limits the capacity to run multiple simulations in parallel as GPU computer clusters are not as easily available as CPU clusters. Second, the maximal size of the configurations covered by GPU solvers is limited by the memory available on the GPU. For example, the benchmark [12] required about 40 Go of memory to run on a GPU [16]. Finally, if GPU computations are very fast, communication between the nodes is relatively slow, which leads to a significant slowing of the simulation during data extraction, greatly diminishing the gain associated with GPU for simulations when large data extraction is required.

In a different paradigm, spectral finite element methods (SFEMs) have been proposed to significantly reduce the computational burden through a reduction of the number of degrees-of-freedom (DOFs) and a more efficient computation at each time-step [1719], which is the core of the solver presented in this article. More details on the solver will be presented in Sec. 2, but for comparison purposes, its resolution of the case of the 2020 benchmark [20] took 12 min (for approximately 106 DOFs) on a regular desktop computer and a near negligible memory requirement, enabling parametric studies on realistic time spans with close to no hardware limitations. This solver was recently integrated into the software platform civa for large dissemination toward users by the EXTENDE company.2

Regardless of the simulation procedure, a simulation is useful only if it has been thoroughly validated. Compared to finite elements of order one or two, SFEM have received far less interest (see Ref. [17]) in terms of validation for guided wave propagation. For this reason, the goal of this article is to validate the simulation obtained with civa for the simulation of GW propagation for two applications of interest in GW-SHM: the monitoring of metallic pipes in the pipeline industry and the monitoring of large composite structures in aeronautics. First for the pipe application, data acquired by the authors on a stainless steel pipe were used to validate the simulations. For the composite case, however, the elastic properties of most composite panels being afflicted of large uncertainties, the online database openguidedwaves.de [21], was used. Indeed, the authors of this database have been through a lengthy calibration procedure of the elastic properties of the composite panels under study, which is essential to match experimental and simulated data on composite. The main contribution of this study lies in the combination, integration, and validation of the spectral finite element techniques to obtain an efficient simulation tool, hence enabling parametric analysis and probability of detection studies.

The outline of this article is the following: first, in Sec. 2, the main modeling assumptions of civa for GW-SHM will be presented. Validation is presented for three different use-cases in Secs. 36: (1) a stainless steel pipe, (2) a flat composite panel, and (3) a stiffened composite panel. Conclusions are presented in Sec. 7.

2 civa Solver for GW-SHM Simulations

This section describes the simulation strategy used in the GW-SHM module of civa. To obtain fast and accurate simulations with low memory requirements, an explicit finite difference scheme in the time domain is coupled with high-order spectral finite elements in space. The main difficulty is then the meshing process to allow the use of such high-order elements in a user friendly software. First, the overall strategy using macro-elements is described before giving the main characteristics of spectral finite elements. Finally, the specifics of the modeling of piezoelectric transducers are given.

2.1 Unassembled Spectral Finite Elements.

The first step of the simulation process is the definition of a macro-scale mesh of the geometry. All geometries under consideration, that is, plates and pipes with added elements (such as stiffeners), can be decomposed into a collection of macro-elements, which are unit cubes up to a determined transform (see Fig. 1). The order of the geometric transform can be high to take into account curvature, for example, in the case of a pipe, a curved plate, or a stiffener. The macro-elements are then subdiscretized at the wavelength scale with respect to the stacking sequence in the case of stratified media, e.g., composites structures. This subdiscretization is parametrically defined, thus having negligible memory footprint and optimal data structures for multithreaded finite element operations.

Fig. 1
Scheme of the macro-element strategy (see Ref. [19] for more details)
Fig. 1
Scheme of the macro-element strategy (see Ref. [19] for more details)

It is then easy to define spectral finite elements on the reference unit cube and use the transform to take the real geometry into account (see Ref. [19] for more details): indeed, in the unit cube, 3D-spectral finite elements are defined as the tensorial product of one-dimensional (1D)-spectral finite elements. These 1D elements are Lagrange polynomials, as in classic low-order finite elements, but using Gauss-Lobatto nodes, defined at order n as the roots of (1 − x2)Pn(x) = 0, where Pn is the Legendre polynomial of order n, instead of equi-spaced nodes. The use of Gauss-Lobatto nodes induces that these elements can be efficiently defined for any given order. The main advantage of spectral finite elements in our case is then the high-order interpolation of the solution, allowing a reduced number of degrees-of-freedom, as well as mass-lumping, drastically reducing the inversion cost in the computation as the mass matrix, which is inverted at every time-step, is diagonal. Its inversion is then straightforward, and there is no need to assemble it.

In the civa for SHM module, the time marching algorithm of the leapfrog time scheme is used. In this context, since the mass matrix is diagonal (at any order), the main computational cost resides in performing the product between the stiffness matrix with the finite element vectors. This operation is performed element by element, in parallel, and on-the-fly (“unassembled” operations). By doing so, no assembling of the global stiffness matrix is required, thus significantly reducing the memory footprint of the solver.

Spectral finite elements require hexahedral meshes, and producing such meshes for a given description of the specimen is known to be a major challenge in computational geometry. Hence, the main drawback of this strategy is the limitation on the available configurations. To circumvent this difficulty, we resort to an analytic parameterization of each configuration. A significant number of geometries are already available in the civa SHM module, and new ones will be added. Note that this is not a limitation of the SFEM code but a restriction of the GUI of civa SHM, which does not require a priori expertise in finite element simulation. In addition, this restriction to parametric configurations facilitates the conduct of sensitivity studies in the civa framework.

The meshing procedure is performed in the following way: first guided modes are computed on the frequency range of the excitation signal and the smallest wavelength of the computation is deduced from it. More precisely, the wavelengths of the propagating modes are computed using a semi-analytical finite element method (SAFE) [5] on the frequency range of the excitation signal. Plate modes are considered in all cases. In the special case of a pipe, circumferential modes, taking into account the bending of the plate to describe the pipe, are also computed. The meshing is then done to achieve a given number of points per wavelength, and this number depends on the order of the used elements. Between 6 to 10 points per wavelength are ensured in the whole mesh. This is sufficient to obtain a good approximation of the solution in the case of SFEM [22]. This value is smaller than the usual “rule of thumb” of 15–20 points per wavelength for low-order methods. Finally, the module shall shortly include damping modeling based on Ref. [23], but all results presented in this article are without damping.

2.2 Piezoelectric Modeling.

To further reduce the computational cost of a configuration, a simplified model is used for the piezoelectric transducers. The source is modeled as a surface load on the geometry, whereas the measure is defined as an integral over the perimeter of the sensor. More precisely, thin piezoelectric disks are considered in this article, which are modeled using the model first introduced in Ref. [24]: the load is axisymmetric and applied on the circumference of the transducer in the radial direction. The measurement is the integral on this circumference of the radial displacement. It is well known that for low frequencies on metallic plates, the results of this model are in good agreements with the experiment. Its in-depth validity in the context of anisotropic structures will be the topic of a follow-up publication. Note that it is needed to define a surface load in the finite element computation, whereas this model gives a load over a line. To circumvent this issue, a small ring is defined at the edge of the sensor, over which the integration takes place.

It should be noted that this model has a drawback: to limit as much as possible the number of degrees-of-freedom used in the mesh, and because the sensors are usually rather small with respect to the wavelengths, the transducers may be poorly approximated as only a few nodes are included in its support. To avoid this, a special care is given to the sensors mesh and a specific mesh pattern is defined for the corresponding macro-elements, which induces smaller time-steps as the corresponding elements are small. Indeed, for a given material and a fixed time-window, the overall number of time-steps depends on the ratio of the minimum element size over the maximum element size, which is driven by the wavelength. Indeed, the time-step is given by the Courant–Friedrichs–Lewy condition: Δt2/ρ(M1K), where Δt is the maximal time-step, K and M are the stiffness and mass matrices, respectively, and ρ is the spectral radius. While the spectral radius does depend on the ratio of the minimum over maximum element size, the minimum element size is fixed by the geometric singularities (most often the meshing pattern of the sensors, but also sometimes the defect). Conversely, the maximum element size depends on the wavelength.

3 Validation of Wave Propagation in a Stainless Steel Pipe

In this section, comparisons are shown between experimental measurements in a welded stainless steel pipe (see Fig. 2) and the corresponding simulated signals.

Fig. 2
Experimental setup. Picture of the pipe with the bounded piezoelectric transducers.
Fig. 2
Experimental setup. Picture of the pipe with the bounded piezoelectric transducers.

The pipe shown in Fig. 2 is of the following geometry: outer diameter of 254.29 mm, thickness of 2.145 mm, and a length of 2.81 m. Two transducer rings distanced of 40 cm and composed of 15 sensors each are bounded to the structure. The sensors are piezoelectric transducers with a diameter of 8 mm. In all acquisitions, one sensor acts as an actuator, whereas all others act as receivers. The source signal is in all cases a Hanning tone burst emission of 50 kHz generated by a waveform generator (Keysight Technologies) with a peak-to-peak amplitude of ±10 V. The signals measured by the receivers are first filtered analogically and amplified by a low-noise preamplifier (Stanford Research Systems) with a second-order high-pass filter with a cutoff frequency of 10 kHz. Then, the signals are digitized using an oscilloscope (Teledyne LeCroy) and averaged 30 times to increase the signal-to-noise ratio. Multiplexers allow to switch between the piezoelectric transducers. The overall acquisition process is fully automated thanks to a labview code. The experiment was conducted in CEA laboratories.

The exact material properties of the pipe are unknown, but a high-frequency longitudinal bulk wave pulse echo acquisition was performed on various points of the pipe to assess its homogeneity. The measured times of flight exhibit a variation of the order of 5% of the longitudinal waves velocity along the circumference. This variation is in good agreement with standard wall thickness tolerances for welded pipes. To see how this variability affects the propagation, a first comparison was performed on the signals corresponding to the emitter on the first transducer ring and the receiver on the second ring on the same axis. In a perfect pipe with no variability, these signals should be identical up to uncertainties regarding the sensors. The 15 experimental signals are shown in Fig. 3. Note that on those signals, the experimental electromagnetic coupling can be seen around 50 μs, while the wave-packets around 250 μs to 300 μs correspond to the pipe mode L(0, 1) equivalent to A0 in a plate. A high variability of the phase of the signals can be seen for this mode between each measurement. For our interrogating frequency of 50 kHz, such a large phase variation cannot be due to the experimental uncertainties related to the placement of the sensors as this would imply a positioning error greater than 5 mm. However, this phase variation is due to the variability of the geometrical and material properties along to the circumference of the pipe. Conversely, the significant amplitude variation may be due to several factors. First, two signals of very low amplitude and one of very large amplitude compared to the others are visible. After investigation, it was observed that the signal with the smallest amplitude corresponds to the sensors bounded directly on the weld of the pipe. Indeed, as the weld has a different micro structure compared to the rest of the pipe, the wave-packets are scattered for this specific sensor pair path. The two other signals (low amplitude and largest one) are the neighboring couples of sensors at a distance of about 5 cm from the weld. Hence, these amplitude variations seem strongly linked to the weld. Note that the arrangement is not perfectly symmetrical with respect to the weld and that a thickness irregularity, asymmetric with respect to the weld, was observed in the area and might explain the amplitude variation. Second, the amplitude variation may be due to the coupling of the sensors, which is not perfectly reproducible from one sensor to another or the variation of the sensors themselves.

Fig. 3
Experimental signals for axial couples of sensors (emitter in the first ring and receiver in the second one in front of the emitter)
Fig. 3
Experimental signals for axial couples of sensors (emitter in the first ring and receiver in the second one in front of the emitter)

Several simulations have been run with civa SHM for the same geometry, position of sensors, and inspection frequency. A parametric study on the material properties has been performed to identify the best match among the parameters within the interval of values. More precisely, standard values for the Young's modulus and the Poisson's ratio of stainless steel, that is 197 GPa and 0.299 (with ρ = 7890 kg/m3), were taken as an initial guess and a variation of the Young's modulus was performed, corresponding to a ±2.5% of variation of the longitudinal waves velocity. The results are presented in Fig. 4. The set of experimental results is represented by a shaded beam from which the three aberrant signals corresponding to the couples close to the weld are removed from the experimental set. From this result, it can be observed that the best elastic property fit is at E = 188 GPa as represented by the green curve as it intersects quite well the experimental beam whether it is for L(0, 1) (≈ A0 in plates) or for L(0, 2) around 100 μs (≈ S0 in plates).

Fig. 4
Comparison between experimental signals and simulated signals for axial couples of sensors and various Young’s modulus
Fig. 4
Comparison between experimental signals and simulated signals for axial couples of sensors and various Young’s modulus

After calibration of the simulation parameters, a comparison is performed for a sensor acting as actuator on the first ring and all 15 sensors of the second ring are acting as receivers. The results are shown in Fig. 5. First, it can be observed that the wave-packets corresponding, respectively, to the direct and first helical modes of L(m, 1) and L(m, 2) have similar time of flights for all signals; hence, the simulation of the group velocity is satisfying. Regarding the phase of the signal, a mismatch is once again observed. More precisely, the phase is close to equal for signals corresponding to sensors aligned or close to aligned along the axis of the pipe, that is the bottom signal and the top one, and the difference grows with the difference in circumferential position between the emitter and the receiver. This can be explained by the variability of thickness, which is not taken into account in the simulation.

Fig. 5
Experimental signals and simulated signals comparison for a full ring of sensors with the calibrated Young’s modulus
Fig. 5
Experimental signals and simulated signals comparison for a full ring of sensors with the calibrated Young’s modulus

4 Validation of Wave Propagation in Composite Through Wavefield Analysis

This section presents results of the validation study of a wavefield propagating in a flat composite panel.

4.1 Experimental Data.

As previously observed, a major difficulty in validating simulations is to obtain a complete and precise knowledge of the influencing parameters. As GWs are especially sensitive to many parameters, this is a significant challenge for validation in general and more specifically for composite materials. Indeed, most often the elastic properties values are not precisely known as manufacturers often provide conservative values for the use of composite as load-carrying structures. Moreover, elastic properties of composite structures can vary over a fairly large range of values due to manufacturing tolerances. For example, carbon fiber content in carbon fiber-reinforced composite (CFRP) might vary up to ±4% [25]. By a quick calculation with the rule of mixture and the SAFE [5], this roughly leads to an error in terms of the group velocity of the A0 mode up to 6% around 100 kHz, which is a far greater effect than the one induced by most defect one might want to detect in such structures. For this reason, a proper knowledge of the elastic properties of the composite is essential for validation.

In this study, the validation is conducted on data provided by the open-guided waves (OGW) initiative [21] as the authors went through an extensive calibration procedure of the elastic properties of the composite sample under consideration, which significantly reduces the main source of potential mismatch between simulations and experiments. This section focuses on reproducing the wavefield measured by a laser vibrometer on a plate. The main parameters of this experiment are copied here, but for more details, the reader is invited to refer to Ref. [21]. The studied composite panel is a 16-layer CFRP panel of dimensions 500 × 500 × 2 mm3. It is instrumented by a piezoelectric actuator in its center, and the propagated wavefield is measured by a 3D laser Doppler vibrometer in one-quarter of the plate on a grid with a 1 mm spatial step. To enhance the quality of the experimental data, a low-pass Butterworth filter of order 5 is applied to every experimental measurement with a cutoff frequency corresponding to three times the central excitation frequency.

4.2 Simulated Configurations.

The previously described configuration is replicated with the civa software with the elastic properties provided by OGW. The simulation is full 3D: no symmetry argument was used, and each layer is modeled by a 3D solid element, resulting in 16 plies of dimensions 500 × 500 mm. The local orientation of the anisotropy within each element is carried by the macro-mesh, which is composed of a single 3D solid element over the entire thickness (that carries the distortion of the reference material) plus a 1D description of the reference material in the stacking (elasticity coefficients, fiber orientation, and thickness in a straight medium). These data are taken directly from the input parameters of the civa GUI.

Simulations are conducted for five-cycle tone bursts of central frequency 50, 100, 150, and 200 kHz. Displacements are extracted from the simulations at a grid of points corresponding to the experimental laser scan. Note that in the OGW dataset, defects are added by using an attached mass, which is not a defect available in civa. The analysis of defective wavefield will therefore not be conducted in this section, but some defect imaging will be shown in the following section.

4.3 Validation Results on Wavefield.

The simplest qualitative validation step observed the wavefield generated by civa and compared it to the measurements as represented in Fig. 6, 100 μs after the 50 kHz excitation. At first glance, the main difference between these results is the noise level, inexistent in the simulations, and dominant in the in-plane components of the experiment. Due to the noisiness of the measurement of the in-plane components, the validation of the wavefield will focus on the comparison of the out-of-plane component for the remaining of this section. Note that it means that this section focuses on the A0 mode, which is mainly out-of-plane, while the S0 mode, dominated by its in-plane components, will be studied in the following section. The comparison of dispersion curves between experimental and simulated data is shown in Fig. 7. Note that for each of these plots, the dispersion curve is compared to the theoretical values provided by the SAFE technique [5] and computed with civa. The left column displays the experimental results from 50 to 200 kHz, and the right column displays the simulated results. As expected, the simulated results are perfectly superimposed with the theoretical calculation provided by SAFE; however, a small shift can be observed in the experimental data (note that the same shift can be observed in Fig. 9 of Ref. [21]). As the elastic properties have been measured with relatively small confidence intervals, this mismatch between experiment and theory is most likely explained by the remaining unknowns of the plate (plate homogeneity, temperature at the time of the measurement versus the one of the calibration, etc.).

Fig. 6
Comparison of the three components of the wavefield at t = 100 μs at 50 kHz: top: experiments data; bottom: simulations; from left to right: in-plane horizontal, in-plane vertical, and out-of-plane components
Fig. 6
Comparison of the three components of the wavefield at t = 100 μs at 50 kHz: top: experiments data; bottom: simulations; from left to right: in-plane horizontal, in-plane vertical, and out-of-plane components
Fig. 7
Comparison of the dispersion curves of the out-of-plane component of the wavefield (left: experiments, right: simulations) from 50 (top) to 200 kHz (bottom)
Fig. 7
Comparison of the dispersion curves of the out-of-plane component of the wavefield (left: experiments, right: simulations) from 50 (top) to 200 kHz (bottom)

To go further in the validation process, three arbitrary points are chosen to compare the out-of-plane components of the wavefield. The emitter is located at the coordinate p0 = (250, 250) mm in Fig. 6 (i.e., the top right corner). The comparison points are selected at the coordinates p1 = (200, 200), p2 = (225, 100), and p3 = (100, 50) mm corresponding to the cylindrical coordinates from the actuator of (62 mm, 42 deg; 147 mm, 7 deg) and (242 mm, 36 deg), respectively. The Ascan comparison for each point is shown in Figs. 810 for the frequencies 50, 100, and 150 kHz, respectively. The 200 kHz comparison is not shown as the experimental data are too noisy to conclude for this frequency. Note that all the data have been normalized to unity to focus on phase and time-of-flight comparison. At 50 kHz in Fig. 8, an excellent envelope match is observed for the three Ascans with a small mismatch for p3 and a phase shift for all the points. These small errors are most likely due to the mismatch observed in the dispersion curve.

Fig. 8
Comparison of the out-of-plane velocity component v3 at 50 kHz for positions p1 = (200, 200), p2 = (225, 100), and p3 = (100, 50) mm
Fig. 8
Comparison of the out-of-plane velocity component v3 at 50 kHz for positions p1 = (200, 200), p2 = (225, 100), and p3 = (100, 50) mm
Fig. 9
Comparison of the out-of-plane velocity component v3 at 100 kHz for positions p1 = (200, 200), p2 = (225, 100), and p3 = (100, 50) mm
Fig. 9
Comparison of the out-of-plane velocity component v3 at 100 kHz for positions p1 = (200, 200), p2 = (225, 100), and p3 = (100, 50) mm
Fig. 10
Comparison of the out-of-plane velocity component v3 at 150 kHz for positions p1 = (200, 200), p2 = (225, 100), and p3 = (100, 50) mm
Fig. 10
Comparison of the out-of-plane velocity component v3 at 150 kHz for positions p1 = (200, 200), p2 = (225, 100), and p3 = (100, 50) mm

At 100 kHz in Fig. 9, a perfect time-of-flight match is observed for the three points along with the reflections from the edges observed in p2 and p3. The amplitude of the reflection in p2 is not properly captured because attenuation effects are neglected in the simulations. The amplitude of the reflection in p3 is roughly correct as it is near the edge of the plate, and the wave-packets have not been attenuated much at this position, unlike in p2. As mentioned earlier, the phase does not match: in opposition for the first two points and roughly in phase for the last one. Once again, this is most likely due to the dispersion curve mismatch previously discussed. At 150 kHz in Fig. 10, the experimental measurements are of too poor quality to conclude. Further analysis leads to the observation that the A0 mode is highly attenuated at these frequencies and beyond, and even the out-of-plane measurement carries very little content above the noise level. No conclusion can therefore be extracted from these data, and for the same reason, the data at 200 kHz are not represented.

4.4 Computational Performance.

Computational efficiency is essential for GW simulations in SHM as usually several simulations are conducted simultaneously or sequentially. For example, to simulate a round-robin scan (i.e., each transducer of a network excites successively a wave while the others act as receivers), optimize SHM systems, or conduct parametric studies on the variations of external parameters. As discussed in Sec. 2, civa SHM is especially efficient in terms of both CPU and memory usage as summarized in Table 1 for the four wavefield simulations of interest. First, it can be observed from this table that the time-step remains constant for the first three frequencies, which is due to the fact that at low frequencies the smallest element is within the sensor area and is independent of the frequency. Second, the number of degrees-of-freedom increases roughly quadratically with the frequency, which is due to the fact that the mesh is refined only in-plane for an increase in frequency, while the out-of-plane mesh is already very fine (due to the multiple plies) and needs no further refinement. Finally, CPU time and RAM usage scale roughly linearly with the number of degrees-of-freedom. It can be concluded that CPU time increases quadratically with the frequency if the time-step remains constant and cubically if the time-step is reduced with the frequency increase.

Table 1

Time-stepping, approximate number of DOFs and computation performance for the wavefield simulation using civa on an Intel Xeon X5690 at 3.47 GHz, 3.46 GHz, 2 × 6 cores, 24 Go RAM

Time-stepDoFCPU usageRAM usage
50 kHz7.8 ns7 × 1061 h 37 mn0.3 Go
100 kHz7.8 ns19 × 1064 h 22 mn0.7 Go
150 kHz7.3 ns36 × 1068 h 26 mn1.4 Go
200 kHz5.5 ns60 × 10617 h 36 mn2.6 Go
Time-stepDoFCPU usageRAM usage
50 kHz7.8 ns7 × 1061 h 37 mn0.3 Go
100 kHz7.8 ns19 × 1064 h 22 mn0.7 Go
150 kHz7.3 ns36 × 1068 h 26 mn1.4 Go
200 kHz5.5 ns60 × 10617 h 36 mn2.6 Go

For comparison purposes, a comsol multiphysics simulation was conducted at 100 kHz on the same 16-ply CFRP panel for 300 μs. Because the memory requirements on comsol were larger than the available RAM, a reduced domain was considered: both width and length are reduced from 500 to 50 mm, reducing the overall area of a factor 100. The sensor is modeled in comsol with the same model as the one used in civa. The mesh size and time-steps of comsol are optimized for tetrahedral finite elements of order 1. For the two iterative solvers of comsol called mumps and pardiso, a similar result was obtained of roughly 10 hof computation and 10Go of RAM on the same computer as the one used for civa. According to comsol’s website [26], such large problem scale linearly with the number of degrees-of-freedom. Hence, by applying the area factor of 100, it can be deduced that the comsol software would require approximately 40 days of computation on the same CPU and 1000Go of RAM. This value is to be compared to the actual performance obtained with civa of 4 h and 22 min with a 0.7 Go of RAM, hence a time improvement of a factor 200 on the same hardware. This order of magnitude is compatible with the comparison of the composite benchmark previously discussed [12] and published in Ref. [20].

5 Validation of Wave Propagation in a Finite Composite Panel

As the wavefield analysis presented in the previous section only allows for the validation of the A0 mode to a frequency up to 100 kHz, this section focuses on another OGW dataset to extend the validation range.

5.1 Experimental Data.

The database under consideration in this section is also provided by the OGW initiative [21] and consists of a composite panel of similar material properties instrumented by 12 circular piezoelectric transducers of diameter 10 mm as depicted in Fig. 11. The database contains the round-robin measurement (i.e., each sensor is activated sequentially, while the others are used as emitters) with five-cycle tone bursts of central frequency from 40 to 260 kHz every 20 kHz. Note that the distance from the sensors to the closest edge is of 3 cm only which means it is impossible in this dataset to observe incident wave-packets alone, as wave-packets will systematically be superimposed with the reflected wave-packets from the edges.

Fig. 11
Schematic of the configuration of the composite panel with 12 piezoelectric sensors
Fig. 11
Schematic of the configuration of the composite panel with 12 piezoelectric sensors

5.2 Simulated Configurations.

First, the pristine configurations at 40, 100, 160, and 240 kHz are reproduced as in the OGW database for direct Ascan comparison. The piezoelectric sensors are modeled as described in Sec. 2.2 with a radial load defined in a ring of diameter 9–11 mm. Simulations are run for 800 μs at 40 kHz and 500 μs at the other frequencies to ensure capturing the arrival of the first incident wave-packet of the slowest mode (A0). The mesh refinement is automatically computed by the process described in Sec. 2.1 for the four inspection frequencies.

5.3 Validation Results.

For the graphical comparison of the signals emitted and received by piezoelectric sensors, the path 1–10 is chosen arbitrarily. A more detailed analysis shows that this path exhibits comparable results with the other paths. Furthermore, due to the simplified sensor model described in Sec. 2.2, the amplitude of the signals generated in the simulations are dimensionless; hence, all the signals are normalized to unity within the time window they are represented in. Finally, to simplify the visualization, the validation focuses first on the S0 mode in Fig. 12 with a truncated time window to remove the A0 mode, then the A0 mode is discussed in Fig. 13 with a longer time window. A computation option in civa allows to separate components of the signals corresponding to symmetric and antisymmetric modes. This option is used in the following, in particular to study the A0 mode without the effect of the reflected S0 mode at the same time.

Fig. 12
Comparison of the signal received by sensor 10 while sensor 1 acts as emitter at 40, 100, 160, and 240 kHz focusing on the S0 mode (blue: OGW, red: civa S0 mode). The first shaded area (gray) corresponds to the electronic coupling, while the second shaded area (yellow) is the theoretical time of arrival window of the incident S0 wave-packet, as computed by SAFE and neglecting dispersion. (Color version online.)
Fig. 12
Comparison of the signal received by sensor 10 while sensor 1 acts as emitter at 40, 100, 160, and 240 kHz focusing on the S0 mode (blue: OGW, red: civa S0 mode). The first shaded area (gray) corresponds to the electronic coupling, while the second shaded area (yellow) is the theoretical time of arrival window of the incident S0 wave-packet, as computed by SAFE and neglecting dispersion. (Color version online.)
Fig. 13
Comparison of the signal received by sensor 10 while sensor 1 acts as emitter at 40, 100, 160, and 240 kHz focusing on the A0 mode (blue: OGW, red: civa S0 mode). The shaded area is the theoretical time of arrival window of the incident S0 wave-packet, as computed by SAFE and neglecting dispersion. (Color version online.)
Fig. 13
Comparison of the signal received by sensor 10 while sensor 1 acts as emitter at 40, 100, 160, and 240 kHz focusing on the A0 mode (blue: OGW, red: civa S0 mode). The shaded area is the theoretical time of arrival window of the incident S0 wave-packet, as computed by SAFE and neglecting dispersion. (Color version online.)

5.3.1 S0 Analysis.

In Fig. 12, the experimental versus simulation signals are shown for the four frequencies of interest, with a time window focusing on the S0 mode (before the A0 first wave-packet). It must be reminded that the first harmonic packet in the experimental data at t = 0 and shaded in gray corresponds to the electronic coupling, which is not an elastic wave to be reproduced in the simulations. The first observation on these results is that the time of flight of the incident wave-packet at the four simulation frequencies matches very well, both between experiment and simulation, but also with the theoretical time-of-flight value computed by the SAFE technique and represented by the yellow-shaded area. However, it must be noticed that at 100 and 240 kHz, the numerically calculated incident wave-packet is separated into two nearly overlapping wave-packets, visible by the two nearby envelope maximums, while this is not the case in the experimental results. The two maximums in the simulations are the incident and its immediate reflection from the edge near the sensor, which are somewhat merged in the experiment, which may be due to a smoother reflection from the edge. Moreover, a nonnegligible time-of-flight shift is observed at 240 kHz: the experimental time of flight is 111 μs and is outside the theoretical time of arrival, while it measured at 107 μs in simulations. The error between the two maximums is equal to one cycle at the excitation frequency 240 kHz. This effect will be further discussed in Sec. 5.4.

The second major observation concerns the amplitude of the wave-packets. Indeed, at the two highest frequencies, within the observed time window, the maximum of the signal is due to the incident wave-packet, which is not the case at 100 kHz, due to the symmetry of the configuration and the constructive interferences from the various edges. Moreover, it must be reminded that the simulations do not include attenuation, which is why the amplitude is seen to increase in the 40 kHz plot, while it is mostly stable in the experiment due to a balance between attenuation and constructive interferences.

5.3.2 A0 Analysis.

In Fig. 13, the experimental versus simulated signals are shown for the four frequencies of interest, with a time window focusing on the A0 mode. To simplify the visualization, the signals are truncated to focus on the A0 mode. The A0 simulated mode is therefore compared to the full experimental signal (i.e., A0 + S0) as modes cannot be separated in the experiment. First, it must be observed that at frequencies 100, 160, and 240 kHz, no clear A0 wave-packet is visible in the experimental data within the theoretical window of arrival (yellow-shaded area). This is because the A0 mode is highly attenuated at these frequencies and therefore is largely hidden within the multiple S0 reflections. Hence, the only conclusion to be drawn from these frequencies is that the simulated first wave-packet of the A0 mode falls within the theoretical time of arrival window.

At 40 kHz, however, the A0 mode dominates the experimental signals, and it can be observed that its time of flight does not match the simulated one, which does not arrive within the theoretical time of arrival window. The experimental time of flight, measured as the maximum of the envelope, is 476 μs, while it is measured at 501 μs in the simulation, which is a significant shift, equal to roughly one cycle at the excitation frequency 40 kHz. This effect will be further discussed in the incoming section.

5.4 Discussion.

A nonnegligible time shift of approximately one period of the central frequency of the excitation was observed for the A0 mode at 40 kHz and for the S0 mode at 240 kHz. It is thought that this shift is majorly due to a different behaviors of the edges of the plate between the model and the experiment. To demonstrate this, another simulation was conducted on a larger 700 × 700 × 2 mm3 plate, while all the other parameters remain the same. This is equivalent to having an infinite plate for the A0 mode within the time window and the path under consideration at 40 kHz. The result of the A0 mode as received by the sensor 10 in this infinite plate is represented in green in the Fig. 14, as well as the simulated signal in the finite plate (i.e., result presented in Fig. 13) and the experimental signal. Note that in this picture, both simulated signals are normalized with respect to the maximum of the amplitude of the simulated signal in the extended panel. Two major effects can be observed after removing the effect of the edges: the time of flight falls perfectly within the theoretical time window of arrival and the maximum amplitude is decreased by a factor of 4 with respect to the one in the finite simulated panel. This result is in accordance with the expectation: in the finite plate, in addition to the direct path from sensor 1 to 10, three other paths with a single edge reflection have similar time of flights and combine constructively to create an amplitude of 4. This explains both the difference of time of flight and amplitude between the simulations. Moreover, the experimental time of flight lands half-way between the signals from the finite and the extended plates. Our interpretation is that the edges do have an influence on the experimentally measured signals, but the reflections are neither fully constructive nor destructive. This difference in the interaction of the reflections may come from higher uncertainties regarding the edges of plates, which may not be well taken into account in the simulations.

Fig. 14
Comparison of the signal received by sensor 10, while sensor 1 acts as emitter at 40 kHz (blue: OGW, red: civa on the finite plate, green: CIVA in the extended plate) (Color version online.)
Fig. 14
Comparison of the signal received by sensor 10, while sensor 1 acts as emitter at 40 kHz (blue: OGW, red: civa on the finite plate, green: CIVA in the extended plate) (Color version online.)

Interestingly enough, according to the dispersion curves of the composite panel, the effect described in this section is observed when the S0 wavelength at 240 kHz is equal to 24 mm while the one of the A0 mode at 40 kHz is 19 mm, which correspond roughly to twice the back-and-forth distance between the edge and the actuator. This effect might therefore be interpreted by some sort of resonance mechanism between the edge of the plate and the actuator.

5.5 Defect Imaging.

The same OGW database also contains scans with an added mass to simulate a defect at 28 positions. However, if this type of defect is often used in the laboratory to avoid permanently damaging the specimens, it is not necessary to go to such lengths in simulations as holes and delaminations can be created at will. For this reason and for qualitative comparison of the defect detection, simulations with defects were conducted in CIVA, in which the defect was modeled by a circular through-hole of the same diameter as the added mass. The imaging was performed using the reconstruction algorithm for probabilistic inspection of defects [27] using the same parameters as in Ref. [21], that is, β = 1.1, and using only datasets corresponding to sensors located on opposite sides. Results are shown in Fig. 15 for two central frequencies for both the simulated and the experimental data. There is a good match between imaging results in both cases, showing the interest of the simulation for the study and design of an SHM system. This results shows that representative and useful SHM data can be generated albeit the minor data mismatch, principally because the mismatch is mitigated by the use of a baseline in the imaging algorithm.

Fig. 15
Images obtained using RAPID on experimental (left) and simulated (right) data for defect D14 (black cross). Top row: center frequency of 40 kHz. Bottom row: center frequency of 100 kHz.
Fig. 15
Images obtained using RAPID on experimental (left) and simulated (right) data for defect D14 (black cross). Top row: center frequency of 40 kHz. Bottom row: center frequency of 100 kHz.

6 Validation of Wave Propagation in a Stiffened Composite Panel

Finally, stiffened panels were considered using the same experimental protocol and sensor configurations in OGW [28]. Those configurations were simulated in CIVA and are depicted in Fig. 16.

Fig. 16
View of the stiffened panel in CIVA, SHM plate case (left) and wavefield case (right). Defect (delamination) are shown in red (bottom right), and sensors are shown by circles (yellow). (Color version online.)
Fig. 16
View of the stiffened panel in CIVA, SHM plate case (left) and wavefield case (right). Defect (delamination) are shown in red (bottom right), and sensors are shown by circles (yellow). (Color version online.)

6.1 Wavefield Plate.

First, the wavefield case is considered and a simulation for a center frequency of 40 kHz is performed. The wavefield is extracted on the bottom-left quarter of the plate as in the data available in OGW. Note that a defect is simulated in this part of the plate using an added mass of radius 10 mm in OGW and a circular delamination at a depth of 0.875 mm and of the same size in CIVA (delamination is obtained by disconnecting the adjacent 3D solid elements). Snapshots of the resulting wavefield are plotted in Fig. 17. A good qualitative agreement is seen between simulation and experiment. In particular, a wave-packet travelling back from the stiffener (i.e., the wave propagating from top to bottom in the upper half) is seen. The wave front’s position is the same in both cases, but its relative amplitude is smaller in the simulation. This may come from the earlier mentioned simple model used for the edges of plates and here of the stiffener or from uncertainties on the geometry of the stiffener. The effect of the defect on the wave’s propagation can be seen in the simulated wavefield at t = 100 μs but not in the experimental one. The Ascan comparison is not shown in this case as the previous comparisons apply: for wave-packets that did not cross the stiffener, a good match is observed for the time of flight but not the phase, as already presented in Sec. 4, and for wave-packets that were reflected by the stiffener there is a good match for the time of flight but not the phase nor the relative amplitude because of the lack of attenuation and the aforementioned uncertainties on the stiffener.

Fig. 17
Snapshots of the wavefield scan for experimental (left) and simulated (right) data, out-of-plane displacement component at t = 200 μs
Fig. 17
Snapshots of the wavefield scan for experimental (left) and simulated (right) data, out-of-plane displacement component at t = 200 μs

6.2 Structural Health Monitoring Plate.

To conclude this section, simulations were performed for the stiffened panel with 12 piezoelectric sensors for a center frequency of 40 kHz. As mentioned earlier in the case without stiffener, a mismatch exists between the simulation and the experiment because of the edges of the plate, which hinder the comparison. To circumvent this difficulty, Ascans for three emitter–receiver couples with and without the stiffener are shown in Fig. 18. The same effect of the stiffener is seen in both cases, that is a loss in amplitude of the S0 mode and a small delay. Wave-packets at later times are harder to compare because of the reflections on the edges of the plate.

Fig. 18
Ascans of the SHM plate from experiment (top) and simulation (bottom) for sensor 3 as emitter and, respectively, sensor 7, 10, and 12 as receiver for the left, middle, and right plot. Blue, signals without stiffener; red, signals with stiffener; first shaded area (gray), sensor’s coupling. (Color version online.)
Fig. 18
Ascans of the SHM plate from experiment (top) and simulation (bottom) for sensor 3 as emitter and, respectively, sensor 7, 10, and 12 as receiver for the left, middle, and right plot. Blue, signals without stiffener; red, signals with stiffener; first shaded area (gray), sensor’s coupling. (Color version online.)

7 Conclusions

This article focuses on the validation of the guided wave propagation simulated by CIVA for SHM in three use-cases: a stainless steel pipe, a flat composite panel, and a stiffened composite panel. For the first use-case, the elastic properties were empirically calibrated, while for the other two, the properties were carefully measured by the data providers. Good match between experiments and simulations were observed in the three cases. All the mismatches observed between simulations and experiments are attributed to unknown parameters: thickness variation in the pipe, edge reflections in the composite panel, and stiffener geometry and coupling. The computational performance of CIVA was compared with comsol, and a significant improvement was shown on the same hardware.

Subsequent validation work will focus on the validation of the transduction of the sensor and the response from defects. Future versions of CIVA for SHM will include attenuation and more accurate sensor models and will lift the meshing constraints to handle arbitrary geometries.

Footnote

Conflict of Interest

There are no conflicts of interest.

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