Frémondière Pierre

Servat Estelle

Comparison between birth canal size and external pelvimetry: implications for clinical practice

Keywords: Pelvis, Cephalopelvic disproportion, Birth canal, Labor

INTRODUCTION:

Investigating the correlations between the external pelvimetry variables and the birth canal has important implications for clinical practice as well as for the prediction of dystocia. This study aimed to analyze the correlations between several variables of the external and internal pelvimetry.

METHODS : In this single-centered retrospective study, 142 women with a computed tomography pelvimetry were recruited. Four variables of the birth canal (obstetric conjugate, transverse diameters of inlet, midplane and outlet) and five variables of the false pelvis (intertrochanteric, inter antero-superior iliac spines, external conjugate, transverse diagonal, prepubic) were considered in simple and multiple regression models, after adjustment with BMI.

RESULTS :

As expected, the external conjugate moderately correlates with obstetric conjugate (B = 0,65; p=0.01) and weakly correlates with transverse outlet (B=0,21; p=0.03).

In simple model, transverse diagonal weakly correlates with transverse inlet (B=0,29; p=0.01). In multiple models, intertrochanteric correlates with transverse inlet (B=0,36; p=0.01), midplane (B=0,31; p=0.01) and outlet (B=0,37; p=0.01) while transverse diagonal weakly correlates with transverse inlet (B=0,27; p=0.01). Prepubic is not a predictor of birth canal dimensions in simple or multiple models.

CONCLUSION:

Our study confirm that external conjugate is a predictor of obstetric conjugate. The correlation between transverse diagonal and transverse inlet explains that transverse diagonal helps predict cephalopelvic disproportion, while intertrochanteric could be used as a proxy of the transverse diameters. Our data further confirm that birth canal size cannot be inferred from the prepubic diameter.

INTRODUCTION

Obstetrical mechanism and difficulties during childbirth largely results from the pelvic anatomy and the interaction between birth canal and the presenting part of the fetus. The fetalpelvic interaction derives from complex evolutionary processes and results to a unique fetalpelvic tight fit [START_REF] Frémondière | Dynamic finite-element simulations reveal early origin of complex human birth pattern[END_REF]. Accordingly, childbirth is remarkably difficult compared with other primates [START_REF] Rosenberg | Birth, obstetrics and human evolution[END_REF], and obstructed labor is thought to be responsible for 2,8 % of all maternal deaths worldwide [START_REF] Say | Global causes of maternal death: a WHO systematic analysis[END_REF] as well as obstetric fistula formation.

The management of dystocia is based on the recognition of its different etiological factors.

Dystocia can result from a mismatch between the size of the fetal head and the maternal pelvis (i.e. cephalo-pelvic disproportion), malposition of the fetal head or ineffective uterine propulsive forces. These are summarized as the three "Ps" of labor [START_REF] Maharaj | Assessing cephalopelvic disproportion: back to the basics[END_REF]: the Passageway represents the maternal pelvic anatomy, the Passenger corresponds to the fetal anatomy, and Powers are defined by the primary and secondary forces of labor. Among these three "Ps", effective uterine forces are identifiable with a tocograph and ultrasound can determine fetal head size, station, or position [START_REF] Ghi | ISUOG practice guidelines: intrapartum ultrasound[END_REF]. However, the pelvic dimensions are rarely available except in the few cases where an MRI or CT scan has been performed [START_REF] Jaufuraully | Magnetic resonance imaging in late pregnancy to improve labour and delivery outcomesa systematic literature review[END_REF].

The cephalo-pelvic disproportion is an important issue in developing regions [START_REF] Say | Global causes of maternal death: a WHO systematic analysis[END_REF]. The prolonged labor resulting from a cephalo-pelvic disproportion could be a cause of uterine rupture, haemorrhage, maternal sepsis, neonatal asphyxia and septicaemia [START_REF] Say | Global causes of maternal death: a WHO systematic analysis[END_REF][START_REF] Liselele | Maternal height and external pelvimetry to predict cephalopelvic disproportion in nulliparous African women: a cohort study[END_REF]. To prevent the prolonged labor, some authors reported that the transverse diagonal, an external pelvimetry measurement, could increase the sensitivity in predicting dystocia, in addition to the maternal height [START_REF] Liselele | Maternal height and external pelvimetry to predict cephalopelvic disproportion in nulliparous African women: a cohort study[END_REF]. In this context, the identification of correlations between external and internal pelvic dimensions could represent an important contribution to the prediction of dystocia.

The diagonal transverse is the distance between the two depressions on the skin over the sacrum. It is a proxy of the sacral breadth and the transverse diameter of the inlet, and belongs to the "Mickaelis' quadrilateral" first described by [START_REF] Mickaelis | Das enge Becken : nach eigenen Beobachtungen und Untersuchungen[END_REF]. During the same period, [START_REF] Cazeaux | Traité théorique et pratique de l'art des accouchements[END_REF], described the external conjugate (measured with a calliper such as a pelvimeter, Figure 1), as a proxy of the obstetric conjugate. The inter anterosuperior iliac spines and intertrochanteric would be proxies of the transverse diameter of the inlet [START_REF] Cazeaux | Traité théorique et pratique de l'art des accouchements[END_REF]. [START_REF] Trillat | The prepubic triangle; new external measurement of the pelvis[END_REF] described the most recent external pelvimetric measurement (the prepubic diameter of Trillat). It corresponds to the distance between the inguinal folds, above the supra-pubic line, and is supposed to be a proxy of the transverse diameter of the inlet.

Exactly how these external pelvimetry diameters are sufficiently reliable to assess bony birth canal dimensions remains a key question. Modern imaging techniques offer the possibility to assess whether external and internal pelvimetry are really correlated. In this study, two thresholds of the computed tomography segmentation are used to model both the soft tissue and skin surface, but also the bone birth canal. The aim of this work is to find out if external pelvimetries are reliable predictors of internal pelvimetries

MATERIALS AND METHODS

A total of one hundred and forty-one women at Lapeyronie Hospital (Montpellier, France) recruited from January 01, 2015 to October 01, 2016 were included in this single-centered study. Inclusion criteria was singleton birth at term with a computed tomography pelvimetry prior to labor. Body Mass Index (BMI) and birth order were collected from the medical records. Computed tomography pelvimetry was performed with a scanner located in the Medical Imaging Department of the hospital with low-level irradiation in the range of 15-35 mGy/cm. The intersection gap was 0.6-1.0 mm. The study protocol was approved by our institutional review board. As a strict retrospective study, consent to participate in an anonymous retrospective study was not required.

Pelvises were reconstructed and measured using Amira 5.0.0 software (FEI Visualization Sciences Group/Zuse Institute Berlin). Two thresholds of segmentation were used to reconstruct the pelvis (threshold of between 210 and 250 Hounsfield units) and the skin (-610 and -660 Hounsfield units). Figure 2 shows the procedure followed to obtain the external and internal pelvimetry data.

Figure 3 shows the internal and external pelvimetry variables. The four variables of the birth canal (i.e. internal pelvimetry) were the antero-posterior inlet (obstetric conjugate or APi), transverse inlet (TRi), transverse midplane (interspinous or TRm) and transverse outlet (TRo).

The external pelvimetry variables were the intertrochanteric, the inter antero-superior iliac spines (interASIS), the external conjugate, the transverse diagonal and the prepubic.

Measurements were performed with manual placement of callipers following the variable definition (Figure 3). The same operator (ES) measured all the external and internal pelvic variables.

Correlation analyses were performed with SPSS Statistics 17.0. A correlation was significant with a p-value<0.05. The correlations were weak, moderate or strongly positive for a B value (coefficient of correlation) between 0-0.2, 0.2-0.5 and 0.5-0.8 respectively (Zhou et al., 2003).

For statistics, we used the following method: (1) A simple correlation analysis using the Pearson correlation coefficient permitted investigation of the association between birth canal and external pelvimetry variables. (2) Potential confounding effect of birth order and BMI in the association of birth canal and external pelvimetry variables were tested by ANOVA. (3) Multiple regression models were used to find associations between birth canal and external pelvimetry variables. (4) Mediation of these associations was investigated following this process: first, we added one by one the external pelvimetry variables in the models, and then we included all the five external pelvic dimensions. All models were adjusted where relevant for BMI and birth order.

RESULTS

Table 1 shows the descriptive results of our sample. Among these diameters, TRm exhibits the lower range of variation (SD=7,9 mm) while interASIS has the higher range of variation (SD=16,8 mm). External conjugate and prepubic exhibit an important variation, ranged between 166,7mm-256,6mm and 100,4mm-171,6mm respectively.

Table 2 reports correlation coefficients across external and internal pelvimetry variables.

Among the variables of the birth canal, TRi significantly correlates with TRm (B=0.565) and TRo (B=0.468). Among external pelvimetry variables, intertrochanteric significantly correlates with all birth canal variables (B=0.225-0.672) as well as external conjugate (B=0.187-0.655). InterASIS correlates with all birth canal variables (B=0.327-0.461) except for TRm and TRo. Transverse diagonal correlates with intertrochanteric and all birth canal variables except APi (B=0.285-0.343). Prepubic is not correlated with birth canal variables, but correlates with interASIS (B=0.192).

There were significant associations revealed by ANOVA regarding BMI but not birth order with transverse diagonal. Women with moderate obesity (30<BMI<35) had a transverse diagonal 1.7 cm higher than women with severe obesity (35<BMI<40) (95% CI 0.16-34.91).

Thus, in the next part of the statistical analysis, we considered BMI in the multiple regression models.

Table 3 provides multiple regression models for the prediction of birth canal variables with external pelvimetry variables. When adjusted for BMI, external conjugate was associated with APi (B=0.45, p=0.01). This association remained when all external pelvimetry variables were added to the model (B=0.44 p=0.01). In simple model, external conjugate is weakly associated with TRo (B=0.15, p=0.03). Transverse diagonal appears to predict TRi (B=0.27, p=0.01) in simple model. InterASIS was associated to TRi(B=0.22, p=0.01) and APi (B=0.19, p=0.01) in simple models, while this variable is a predictor of TRm in the multiple model (B=-0.15, p=0.01). Intertrochanteric was associated with all the transverse diameters of the birth canal : TRi(B=0.36, p=0.01), TRm(B=0.31, p=0.01) and TRo (B=0.37, p=0.01) in simple and multiple models. Prepubic was not correlated with any variable of the birth canal in simple or multiple models.

DISCUSSION

Association between external and internal pelvimetry variables

In this study, External conjugate correlates with APi. These two variables have the highest coefficient of correlation in this sample. This is in accordance with the initial description of [START_REF] Cazeaux | Traité théorique et pratique de l'art des accouchements[END_REF]. This author suggested that APi can easily be determined by the following formula: APi = external conjugate-8 cm. Intertrochanteric is highly correlated with TRo. This is in good agreement with previous anthropological studies, where an important covariation between bitrochanteric and Tri, TRm and TRo has been described [START_REF] Ruff | Biomechanics of the hip and birth in early Homo[END_REF]. Transverse diagonal shows a moderate but significant correlation with TRo. This correlation may explain why transverse diagonal measurement is able to predict one out of two cases of cephalopelvic disproportion in nulliparous women [START_REF] Liselele | Maternal height and external pelvimetry to predict cephalopelvic disproportion in nulliparous African women: a cohort study[END_REF]. However, this analysis does not show any significant correlation between prepubic and internal pelvimetry variables. This is in contradiction with the study of Kakoma et al., (2000) where prepubic is a significant predictor of caesarean section. In this study, four women had a prepubic of less than 11 cm and none have had a vaginal delivery, whereas 148 women had a prepubic of at least 11 cm and only 28 (19%) have had a caesarean section (Kakoma et al., 2000). Since prepubic is not correlated with internal pelvimetry variables (i.e. the "size"), it could be associated instead with morphological features of the inlet (i.e. the "shape"), which may also be a risk factor for dystocia [START_REF] Fischer | Covariation between human pelvis shape, stature, and head size alleviates the obstetric dilemma[END_REF]: but see [START_REF] Starrach | Pelvic inlet area is associated with birth mode[END_REF].

Postural adjustment for a narrowed pelvis

Studies have shown that certain birthing positions can change the dimensions of the bony birth canal [START_REF] Michel | MR obstetric pelvimetry: effect of birthing position on pelvic bony dimensions[END_REF]Rietter et al., 2014). It would therefore be possible to compensate for the reduction of some diameters of a narrowed pelvis by encouraging the women to assume different birthing positions. Squatting position is supposed to increase TRm and TRo [START_REF] Michel | MR obstetric pelvimetry: effect of birthing position on pelvic bony dimensions[END_REF] as well as the antero-posterior mid pelvic and anteroposterior outlet (Rietter et al., 2014). However, assuming a supine position would increase the APi [START_REF] Michel | MR obstetric pelvimetry: effect of birthing position on pelvic bony dimensions[END_REF]Rietter et al., 2014). [START_REF] Michel | MR obstetric pelvimetry: effect of birthing position on pelvic bony dimensions[END_REF] found that APi is significantly larger the supine position than in the upright squatting position, but the difference is only 2 ± 4 mm. These studies show that changes in a woman's position led to changes in pelvic diameters [START_REF] Michel | MR obstetric pelvimetry: effect of birthing position on pelvic bony dimensions[END_REF]Rietter et al., 2014). However, these changes in pelvic diameters are moderate (Rietter et al., 2014), suggesting that the change in birthing position may facilitate delivery, but cannot compensate for major pelvic contraction.

Limitations and perspectives

We are aware that our research may have some limitations. In practice, external pelvimetry variables are measured with anthropometric instruments (such as a pelvimeter, or a tape measure). In our study, measurements are performed by choosing the placement of landmark points along the surface of the pelvis. These two methods of measurement are not similar. However, external pelvimetry variables have an unvarying definition, based on anatomical markers (i.e. landmarks). In our opinion, a very close covariation exists between virtual measurements (our study) and measurements with anthropometric tools (in clinical practice). Some studies investigated the variation of the external pelvimetry variables in relation to the postural change [START_REF] Siccardi | A postural approach to the pelvic diameters of obstetrics: the dynamic external pelvimetry test[END_REF]. However, it is unclear whether this variation is similar to the variation of internal pelvimetry variables. Further studies should investigate if the changes in external and internal pelvimetry variables are correlated when the change in birthing position is controlled.

CONCLUSION

The aim of this study is to identify accurate predictor of birth canal variables with external pelvimetry variables. Among these external variables, prepubic is not a predictor of any birth canal variables. Intertrochanteric is a good predictor of transverse diameters of the birth canal while transverse diagonal only predicts the TRi. External conjugate is a predictor of APi.

These correlations may have important clinical implications for enhancing the prediction of bony dystocia.

FIGURE 1 .FIGURE 2 .

 12 FIGURE 1. Photograph of a calliper that can be used as a Pelvimeter

FIGURE 3 .

 3 FIGURE 3. Internal and external pelvimetry variables

TABLE 1 Description of the pelvimetric variables.

 1

	DIAMETER (MM)	MEAN (SD)	MINIMUM-
			MAXIMUM
	APi	124,2 (9,3)	97,2-156,5
	TRo	120,5 (9,4)	101,7-139,9
	TRi	125,8 (8,6)	108,6-148,2
	TRm	108,5 (7,9)	94,8-128,3
	Intertrochanteric	258,7 (14,8)	213,4-290,1
	InterASIS	221,1 (16,8)	184,8-261,0
	External conjugate	207,5 (14,5)	166,7-256,6
	Transverse diagonal	92,5 (10,2)	73,0-128,0
	Prepubic	118,1 (12,1)	100,4-171,6
	MM: millimetre; SD: Standard Deviation; TR:Transverse; AP: Antero-posterior; ASIS:Antero
	superior iliac spines

TABLE 2 . Crude correlations among internal and external pelvimetry.

 2 Transverse; AP: Antero-posterior; ASIS:Antero-superior-iliac-spines; *p < .05; **p < .01.CI: Confident Interval

		T	95	T	95%	T		INTERT	95%	IN	95%	EXT	95%	TRA	95%	P	95%
		R	%	R	CI	R		ROCHA	CI	T	CI	ERN	CI	NSV	CI	R	CI
		I	CI	M		O		NTERIC		E		AL		ERS		E-
										R		CON		E		P
										A		JUG		DIA		U
										SI		ATE		GON		BI
										S				AL		C
	AP inlet	0,	[-	-	[-	0,	[-	0,225 ** [0.03	0,	[0.92	0,655	[0.35	0,106	[-	-	[-
		1	0,48	0,	0.23	11	0.06		7 ;0.	32	;0.2	**	7 ;0.		0.06	0,	0.19
		2	;0,3	01	5 ;0.	1	2 ;0.		246]	7 **	70]		549]		1 ;0.	07	9 ;0.
		4	15]	5	198]		284]								253]	5	81]
	TR inlet	1		0,	[0.44	0,	[0.28	0,672 ** [0.31	0,	[0.16	0,408	[0.14	0,289	[0.10	0,	[-
				56	6 ;0.	46	8 ;0.		7 ;0.	46	0 ;0.	**	7 ;0.	**	7 ;0.	16	0.01
				5 *	756]	8 *	564]		463]	1 **	313]		349]		389]	0	1 ;0.
				*		*											243]
	TR mid			1		0,	[0.57	0,584 ** [0.23	0,	[-	0,187	[0.00	0,336	[0.13	-	[-
	plane					78	4 ;0.		4 ;0.	06	0.05	*	1 ;0.	**	2 ;0.	0,	0.15
						3 *	763]		388]	1	4 ;0.		202		401]	04	0 ;0.
						*					110]					2	095]
	TR outl					1		0,542 ** [0.25	0,	[-	0,209	[0.01	0,343	[0.15	-	[-
	et								2 ;0.	01	0.08	*	7 ;0.	**	8 ;0.	0,	0.22
									436]	8	6 ;0.		250]		450]	11	2 ;0.
											106]					1	052]
	Intert							1		0,	[0.18	0,435	[0.28	0,285	[0.16	0,	[-
	rocha									36	6 ;0.	**	6 ;0.	**	9 ;0.	01	0.20
	nteric									9 **	463]		636]		641]	4	3 ;0.
																	238]
	Inter									1		0,446	[0.33	0,065	[-	0,	[0.02
	ASIS											**	7 ;0.		0.17	19	5 ;0.
													725]		5 ;0.	2 *	511]
															390]
	Exter											1		0,061	[-	0,	[-
	nal														0.19	03	0.19
	conju														2 ;0.	5	6 ;0.
	gate														379]		283]
	Trans													1		-	[-
	verse															0,	0.20
	diago															06	4 ;0.
	nal															3	098]
	TR:


K Kalkowski
email: m.kalkowski@soton.ac.uk

Michael J S Lowe

Vykintas Samaitis

Fabian Schreyer

Sebastien Robert

Weld map tomography for determining local grain orientations in welds from ultrasound

The complex structure of inhomogeneous (e.g. austenitic stainless steel) welds poses a long-standing challenge in ultrasonic non-destructive testing. Elongated grains with spatially varying dominant orientations curve ultrasonic beams hindering the interpretation of inspection data. One way to tackle this problem is to include material information in imaging and signal analysis; however, such information is most often only gathered using destructive methods. This paper reports the development of a weld inversion strategy determining local grain orientations, based on a ray tomography principle. The considered approach does not rely on a macroscopic weld description but may incorporate it to facilitate inversion. Thanks to that, it is more general that other available approaches, but allows for reducing the complexity by accounting for known information at the same time. The methodology is demonstrated on both numerical and experimental examples. The experimental work focuses on mock-up samples from the nuclear industry and a mock-up specifically manufactured for the project sponsoring this research. The 'ground truth' for the latter comes from an EBSD evaluation, which is the most accurate (yet destructive) examination technique available. The paper concludes by presenting the benefit of the material information obtained from the reported inversion model for ultrasonic imaging.

Introduction

Ultrasonic inspection of inhomogeneous welds remains a significant challenge in the nuclear industry.

Their distinctive microstructure -long columnar grains with preferential orientations varying across the weld -is responsible for a surge in complexity of ultrasound propagation. First, the ultrasonic beam no longer travels across straight paths, but follow the minimum propagation time principle resulting in curved, preprint complicated rays. Consequently, locating a defect based on the time of flight is no longer viable without detailed material information and a suitable propagation model. The second effect is grain scattering. When passing through the subsequent grain boundaries the ultrasonic energy is scattered leading to the loss of amplitude of the main beam (attenuation) and elevated structural noise. Other issues include, e.g. beam splitting, that is, emergence of a secondary beam at a grain boundary that gives rise to two strong scattered waves.

The above mentioned issues may be at least partially accounted for if material information is available.

Such knowledge allows for correcting ultrasonic images, which could then consider true propagation times.

It would also help interpret conventional inspection acquisitions by revealing true beam paths and potential scatterer locations [START_REF] Connolly | The application of Fermat's principle for imaging anisotropic and inhomogeneous media with application to austenitic steel weld inspection[END_REF]. Material information can be obtained from classical examination (e.g. metallography), microscope imaging (EBSD) or assuming a certain structure of the weld (e.g. the gold-standard MINA model [START_REF] Moysan | Modelling the grain orientation of austenitic stainless steel multipass welds to improve ultrasonic assessment of structural integrity[END_REF] or the Ogilvy map [START_REF] Ogilvy | Computerized ultrasonic ray tracing in austenitic steel[END_REF]). Examination methods are destructive, costly and have no outlook to be applied to components on the plant. On the other hand, simplified weld description models cover limited ranges of cases and their parameters need to be tuned. An alternative is to use inversion based on ultrasound and extract the orientation information from the ultrasonic measurements directly. This concept was one of the premises underpinning the ADVISE project addressing the multi-faceted problem of ultrasonic inspection of complex structured materials, of which this research is a part.

It is widely accepted that to support ultrasonic testing, one does not need to consider detailed grain topology [START_REF] Chassignole | Ultrasonic and structural characterization of anisotropic austenitic stainless steel welds: Towards a higher reliability in ultrasonic non-destructive testing[END_REF][START_REF] Hunter | A study into the effects of an austenitic weld on ultrasonic array imaging performance[END_REF][START_REF] Mark | Investigation of the elastic/crystallographic anisotropy of welds for improved ultrasonic inspections[END_REF]. Instead, a macroscopic description is sufficient, where the grains are locally homogenised with a uniform orientation over a chosen grid (often between 1 and 2 mm). Columnar grains with similar preferential orientations can be approximated as a transversely isotropic material with the axis of material symmetry co-linear with the dominant orientation. Historically, macroscopic weld descriptions were developed first to improve the understanding of inspection results. While no inversion was performed, their parameters were assumed based on the examination data and practical experience. Such geometric models include the famous Ogilvy map [START_REF] Ogilvy | Computerized ultrasonic ray tracing in austenitic steel[END_REF], the layered arrangement [START_REF] Spies | Modeling of transducer fields in inhomogeneous anisotropic materials using Gaussian beam superposition[END_REF] or the more basic approximation using only two angles by Langenberg et al. [START_REF] Langenberg | Application of modeling techniques for ultrasonic austenitic weld inspection[END_REF]. A refined orientation layout, linked to welding parameters came with the MINA model [START_REF] Moysan | Modelling the grain orientation of austenitic stainless steel multipass welds to improve ultrasonic assessment of structural integrity[END_REF] which draws the information from the welder's notebook and captures the solidification processes in a phenomenological manner. All these approaches required assuming some if not all parameters, especially for a weld of which little was known.

The first attempts to use ultrasonic measurements for determining orientations aimed at updating weld description parameters mentioned above. For instance, Gueudre et al. [START_REF] Gueudre | Direct model optimisation for data inversion. Application to ultrasonic characterisation of heterogeneous welds[END_REF] used evolutionary optimisation to guess the 'arbitrary' MINA parameters (remelting rates and tilt angles) based on the echodynamic curve. One problem with this approach is the uncertainty related to transducer coupling when measuring amplitudes, as well as the limited range of configurations it could cover. The same parameters were later determined from the time-of-flight maps by Fan et al. [START_REF] Fan | Nonintrusive estimation of anisotropic stiffness maps of heterogeneous steel welds for the improvement of ultrasonic array inspection[END_REF], also based on a genetic algorithm. Inversion based on Bayesian updating and the Ogilvy map was conducted at the University of Bristol [START_REF] Zhang | Monte carlo inversion of ultrasonic array data to map anisotropic weld properties[END_REF], where this approach estimated parameters of the Ogilvy model. Weld description constraints were removed in another Bayes-based work [START_REF] Tant | A transdimensional Bayesian approach to ultrasonic travel-time tomography for non-destructive testing[END_REF][START_REF] Tant | Effective grain orientation mapping of complex and locally anisotropic media for improved imaging in ultrasonic non-destructive testing[END_REF], where the orientations were identified on a per-cell basis. While delivering promising results, the Bayesian approaches incurred significant computational costs which was deemed as one of their key shortcomings.

Recent advances in the availability of high computing power led to the development of approaches based on soft computing. Singh et al. proposed employing deep neural networks to determine the orientation maps based on a large simulated database of times of flight through different grain layouts [START_REF] Singh | Deep Learning Based Inversion of Locally Anisotropic Weld Properties from Ultrasonic Array Data[END_REF][START_REF] Singh | Real-time super-resolution mapping of locally anisotropic grain orientations for ultrasonic non-destructive evaluation of crystalline material[END_REF]. While this approach gives the promise of real-time application, it still requires a computationally expensive training stage, and does not exploit the physics of wave propagation through the heterogeneous material. An alternative method which takes the advantage of modern optimisation methods was developed by Menard et al. [START_REF] Ménard | Optimization algorithms for ultrasonic array imaging in homogeneous anisotropic steel components with unknown properties[END_REF][START_REF] Menard | Ultrasonic Array Imaging of Nuclear Austenitic V-Shape Welds with Inhomogeneous and Unknown Anisotropic Properties[END_REF]. Their focus is on optimising the amplitude of the TFM image around a postulated defect by adjusting the parameters of a macroscopic weld description. While the main outcome of the method is the enhanced TFM image (with improved defect signature location and the signal-to-noise ratio), a grain orientation map is delivered as a by-product.

Although the developments reported in the previous paragraph provided solutions to some challenges, they were often based on 'black-box' optimisers, requiring significant calculation cost, pre-identified defect area, or limited to a given macro weld description. To the authors' knowledge, physics-based inversion was not fully exploited for the applications of interest. Hence, we developed a ray inversion-based method for characterisation of inhomogeneous welds based on ultrasonic array data acquired at the surface. Compared to the prior work, this is a more general inversion approach with a stronger link to physics of wave propagation. Time-of-flight tomography is a known method applied in a variety of contexts, including the biomedical field, in e.g. breast tomography [START_REF] Hormati | Robust ultrasound travel-time tomography using the bent ray model[END_REF]. However, the materials considered in those problems are locally isotropic which is not the case in complex welds. To account for the material complexity and other practical differences, e.g. the sensor arrangement, we extended the classical ray inversion, and developed and validated a new approach, suited to inhomogeneous welds, referred to as weld map tomography.

The different aspects of the developed method are presented in more detailed in the remainder of this article. In the next section, we outline the Fermat principle-based forward model underpinning the inversion.

Then, we recall the fundamentals of ray tomography for structures comprising isotropic materials and present the extension enabling local anisotropy to be accounted for. This is followed by discussing the details of the implementation of inversion for welds, including our two-stage approach and regularisation. The description is followed by examples -first a numerical example based on grain-scale finite element simulation of a real weld case from EDF; second, a set of experimental examples of industry relevant welds -two mock-ups from EDF and a weld specifically manufactured and characterised for the ADVISE project -based contact transducer array measurements.

Forward model

Weld map tomography matches the simulated ultrasound time-of-flight with the experimental target by adjusting model parameters. The forward model, predicting times-of-flight for a given layout of grain orientations is essential to the inversion. The most accurate, yet computationally expensive route is to use time-domain finite element simulations, extremely versatile with regard to the material structure and geometry. An alternative, offering potentially more efficient solution is to use a ray tracing model. Previous research on ultrasound in inhomogeneous welds [START_REF] Connolly | The application of Fermat's principle for imaging anisotropic and inhomogeneous media with application to austenitic steel weld inspection[END_REF][START_REF] Fan | Nonintrusive estimation of anisotropic stiffness maps of heterogeneous steel welds for the improvement of ultrasonic array inspection[END_REF] used a ray stepping algorithm following scattering at fictitious grain boundaries according to a chosen step size. While accurate, it provided the dominant energy path for a given shoot angle (not a selected transmitter-receiver pair) and hence was difficult to be implemented in an efficient manner. One needed to compute a large range of shooting angles and interpolate to obtain the time-of-arrival at desired location.

A crucial functionality of a forward model in this context is to determine the time-of-flight between transducers directly, without the need for interpolation. This can be achieved by utilising the Fermat principle stating that the propagation path between two points is always such that the propagation time is minimal. This principle can be effectively used with efficient shortest path finding algorithms popular from the origins of computer science. Nowers et al. [START_REF] Nowers | Novel ray-tracing algorithms in NDE: Application of Dijkstra and A * algorithms to the inspection of an anisotropic weld[END_REF][START_REF] Nowers | Ultrasonic array imaging through an anisotropic austenitic steel weld using an efficient ray-tracing algorithm[END_REF] suggested using various incarnations of the Dijkstra method to model ultrasound in welds. The present model draws from that contribution but with some modifications to improve efficiency. Given its origins, the forward model used here is called the shortest ray path model (SRP).

Nowers et al. [START_REF] Nowers | Novel ray-tracing algorithms in NDE: Application of Dijkstra and A * algorithms to the inspection of an anisotropic weld[END_REF][START_REF] Nowers | Ultrasonic array imaging through an anisotropic austenitic steel weld using an efficient ray-tracing algorithm[END_REF] suggest using a randomly distributed set of nodes and a selected search radius to maximise the accuracy of the prediction. Our approach is based on the fact that the common macroscopic weld descriptions impose a regular grid structure, with local orientations varying from cell to cell.

Propagation across each cell is determined by its assigned orientation and the incident angle. Moreover, the tomography algorithm itself is also typically programmed over a grid. Hence, we decided to use the regular grid arrangement of seeds in the forward model to maintain consistency across different stages of inversion.

For the ray stepping model, edges of each grid cell are seeded according to a chosen resolution, and the paths connecting all points around the cell are candidate propagation path segments. It is worth noting that a single segment only crosses a single cell, that is segments belonging to different cells do not cross, except at their start or end points. Fig. 1 shows an illustrative computational domain with a weld, isotropic parent materials to its left and right, and a set of sources and receivers. The cells generally represent different properties, but for the ray model they are discretised in the same way -using the same number of seeds along each edge. The number of seeds determines the angular accuracy of the model, i.e. the range of allowable angles for a ray passing through the cell. Fig. 2 schematically depicts this setup. The SRP model looks for the shortest propagation time between two chosen points. The propagation times across cells are calculated from the group slowness at given combination angles. The combination angles are a simple function of the local grain orientations and incident angles. Those propagation times are first computed for the whole model and fed into a shortest ray path algorithm (we used the scipy.sparse.csgraph [START_REF] Virtanen | SciPy 1.0 Contributors, SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python[END_REF] implementation).

One immediate advantage of the computational setup chosen is that rays within each cell have the same structure, e.g. they cover the same angles and have equal lengths. Therefore, they can be precomputed for a single cell and then used for the whole model. Further, this arrangement benefits from the common assumption allowing for using a single 'starting' elastic tensor for the weld. The homogenised elastic tensor in material coordinates is the same for the whole weld, but it is rotated on a cell-by-cell basis. The chosen construction of the SRP model only requires calculating a single group slowness as a function of the combination angle and delivers a quick calculation of the entire time-of-flight graph, offering significant computational advantages over the previously reported approaches.

The convergence study is not reported here for brevity, but for most considered cases, having 10 seeds for a 2 mm grid delivered good accuracy when compared with time-domain finite element simulations. This model, despite its high efficiency and versatility also has several limitations, related to its underpinning principlethe shortest propagation time between two points, not accounting for amplitude or energy. Consequently, it is unable to simulate beam splitting and secondary arrival times which sometimes may carry most of the energy. Further, it may output null paths, that is return ray paths associated with negligible energy transfer.

However, these limitations do not affect the inversion algorithm as the inversion relies on measured arrival times. If arrival time for a certain transducer pair cannot be seen in measurements, it is not included in the inversion.

Ray inversion

This section covers the main elements of the ray inversion algorithm. It starts with a description of the better established methodology developed for materials with heterogeneous isotropy. The second subsection introduces the modifications needed to allow for anisotropy and variable grain orientations. The final part gives details about the implementation of this inversion model for an inhomogeneous weld.

Heterogeneous isotropic medium

The inversion model developed in the ADVISE project is motivated by the paper by Hormati et al. [START_REF] Hormati | Robust ultrasound travel-time tomography using the bent ray model[END_REF], who used a bent-ray simulation to construct an ultrasonic breast tomography method. The fundamental concept behind acoustic tomography is to recover local unknown properties from the characteristics of sound propagation through a medium. In a heterogeneous medium (even if locally isotropic) the sound takes generally curved paths between two points, depending on the local properties. It is convenient to use slowness when describing propagation time, and in this case we focus on group slowness, as the method is based on tracking arrival times of finite wave packets generated by the transmitters. A generic time-of-flight is the integral of the group slowness along the propagation path

τ = Γ s dl ( 1
)
where τ is the time, Γ is the propagation path, s is the slowness and dl is the infinitesimal length along the path.

The discretised domain used in the ADVISE SRP model allows for writing it as a discrete sum

τ = N i=0 s i l i (2)
where s i and l i are the local slowness and the local length of a ray passing a cell. As outlined earlier, the set of possible ray lengths is finite and the same for each cell. It is also worth noting, that a single ray typically passes only a small proportion of cells. Hence, if we cast this formulation into the matrix form

τ = Ls (3)
with L being the ray-length matrix and s a vector describing slownesses across the whole model, one notices that L is a sparse matrix.

The inversion minimises a cost function with the aim of determining model parameters that match the simulated times-of-flight with those obtained from a measurement. The cost function used in this work takes the form

C = M i=0 (τ i,model -τ i,exp ) 2 ( 4
)
where M is the total number of rays considered (i.e. the number of transmitter-receiver pairs).

The optimisation relies on the gradient of the cost function with respect to the model parameters (here: slowness distribution). To calculate this gradient, one needs to find the derivative of the time taken by a ray with respect to the slowness of a cell covered by this ray. From Eq. ( 2), we can write

∂τ ∂s j = N i=0 ∂s i ∂s j l i + s i ∂l i ∂s j (5)
A key assumption here is that the ray path remains constant, irrespective of the local properties, i.e.

∂l j /∂s j = 0. This is an approximation, but it simplifies the inversion considerably. The path variations will also be implicitly accounted for in subsequent iterations of the gradient stepping algorithm. The path invariance assumption removes the second component of the above sum, while the first simplifies when one notes that ∂s i /∂s j is only non-zero if s i = s j . Hence, the final expression for the derivative becomes

∂τ ∂s j = l j (6)
meaning that the derivative of the propagation time over the local slowness is simply the local ray length (the length of a section passing the considered cell).

One can now formulate the gradient of the cost function with respect to the local properties, using the chain rule,

∂C ∂s j = M i=0 2 (τ i,model -τ i,exp ) ∂τ i,model ∂s j - ∂τ i,exp ∂s j (7)
We denote the difference between the arrival times from the model and from the measurement as the residual r i and we note that the measured arrival times do not depend on the slowness distribution in the model, which leads to

∂C ∂s j = M i=0 2r i ∂τ i,model ∂s j ( 8
)
where the last term can be replaced by l i for a heterogeneous isotropic medium.

Hormati et al. [START_REF] Hormati | Robust ultrasound travel-time tomography using the bent ray model[END_REF] suggest using the nonlinear conjugate gradient algorithm to step along the gradient toward the desired model properties. This approach calculates the step size at each iteration and is capable of following narrow paths (ill-conditioned), taking the criss-cross pattern. The algorithm starts from an initial guess and calculates the gradient based on the computed ray lengths as outlined above. Each subsequent step follows the local descent direction

∆s n = -∇C(s n ) + γ n ∆s n-1 (9)
where the δ n parameters determines the effect of the preceding step and can be calculated using the Fletcher-Reeves or alternative formulae. It may happen that the stepping will not decrease the cost function. In such cases, a back-tracking line search algorithm ensures that the cost function is decreased enough. The algorithm iteratively decreases the step size by a chosen factor, until the cost function drop is less than a chosen threshold.

From the computational viewpoint, each step of the gradient stepping approach requires solving the forward model once to obtain the gradient. If back-tracking line search is necessary, each iteration takes another forward solution. These offer a very efficient setup, with the number of forward solutions usually not much greater than the number of steps. This is thanks to the convenient expression for the gradient developed above.

Heterogeneous anisotropic medium

The ray inversion method described in the previous section assumes that each cell represents an isotropic medium. On the other hand, anisotropy is one of the key characteristics of inhomogeneous welds, and therefore, the algorithm needs to be extended to be suitable. For heterogeneous anisotropy the local group slowness is a function of the incident angle and the local grain orientation. Hence, the model parameters to be determined using inversion are now local grain orientation, assuming that all cells can be described with one elastic tensor, appropriately rotated cell-by-cell.

To benefit from the characteristics of the grid -identical regular cells -one needs to express the local slowness for a given ray in material coordinates. Since all cells share the same 'basic' tensor, the group slowness curve is simply only rotated from cell to cell. Hence, the combined angle is a difference between the incident angle and the local orientation. Fig. 3 schematically depicts the angles involved and illustrates how the combined angle is formed.

The algorithm requires an expression for the gradient of the cost function with respect to local orientations. Let us first consider the derivative of the time-of-flight along a given ray. The slowness is now a function of both the incident angle and local grain orientation; however, under the path invariance assumption stated in the preceding section, only the dependence on grain orientation is retained. Hence, we write ∂τ ∂θ j = ∂τ ∂s j ∂s j ∂ϕ j ∂ϕ j ∂θ j [START_REF] Fan | Nonintrusive estimation of anisotropic stiffness maps of heterogeneous steel welds for the improvement of ultrasonic array inspection[END_REF] where ∂τ /∂s j = l j , as previously shown, ∂ϕ j /∂θ j = -1, since ϕ j = γ j -θ j , and ∂s j /∂ϕ j can be precomputed at the beginning, and interpolated as a callable function, as it stays the same for all cells. The derivative of the ray time-of-flight over local orientation finally is ∂τ ∂θ j = -l j ∂s j ∂ϕ j at ϕj =γj -θj [START_REF] Zhang | Monte carlo inversion of ultrasonic array data to map anisotropic weld properties[END_REF] This expression determines the form of the cost function gradient expression which is the core of the inversion model:

∂C ∂θ j = M i=0 -2r i l j ∂s j ∂ϕ j at ϕj =γj -θj (12)
It is worth noting that the new term needs to be evaluated for each ray separately at every pixel, which adds complexity and computational cost to the isotropic solution, but these are in part minimised by the computational arrangement that takes the advantage of the regular grid structure.

Implementation for an inhomogeneous weld

The weld inversion algorithm described above and the the forward SRP model were developed and coded in Python. The algorithm is generally applicable to generally anisotropic media, however its application to inhomogeneous welds gives a specific flavour to several aspects. They are briefly outlined below.

Material structure and elastic tensor for the weld

Most parent materials joined using the welds of interest to this paper can be modelled as isotropic.

Hence, a typical inversion setup considered in the ADVISE project is solving only for grain orientations within the weld, with the parent material isotropic properties assumed a priori. The 'basic' transversely isotropic elastic tensor governing the columnar grain structure of the weld is considered constant, i.e. only the orientations are allowed to vary. This assumption allows for simplifying the computational domain and discretising only the weld region, considerably reducing the size of the graph along which the shortest path is of interest.

Transducer configuration

Transducer configuration is a challenge for weld map tomography. The classical tomography setup (both optical and acoustic) assumes a circular array of transmitter-receivers, ensuring a full coverage of view angles.

Any setup different from the full view incurs considerable loss in the ability to recover the properties of the medium. Since weld map tomography relies on measured arrival times, one can realistically consider three configurations -(i) a through-transmission setup, with transducers on both the top and bottom surfaces; (ii) a pulse echo setup, with only one array on top of the weld; or (iii) a tandem configuration with two arrays located on the top surface on both sides of the weld. The illustration of the implication of the reduced view for the through-transmission and the pulse-echo setup can be found e.g. in [START_REF] Huthwaite | Quantitative Imaging with Mechanical Waves[END_REF].

The through-transmission setup is straightforward for the considered model, as the sources and receivers are places along the two edges of the computational domain. However, the pulse-echo and the tandem setup cannot be modelled directly. To account for the reflection from the backwall with the SRP model, one needs to considered an extended domain, mirrored around the backwall. Note that not only the nodes but also material properties, including orientation angles need to be flipped. This technique is a direct consequence of the Fermat principle invoked in the preceding section. Fig. 4 schematically depicts the arrangement when through-backwall propagation is considered.

Incorporating the known information and regularisation

Weld map tomography is an ill-posed problem with multiple minima. Compared to the isotropic setup, the added complexity of the anisotropic weld context comes from the fact that the group slowness is not unique with respect to the ray angle. Hence, without incorporating additional information the inversion is likely to fail. Such input is possible both in the form of an educated initial guess/starting point for inversion and regularisation.

The grain layout of inhomogeneous welds is linked to the manufacturing procedure and the physics of solidification. These factors are at the origin of the previously mentioned weld models, such as MINA or the Ogilvy map. Whilst fast and effective, these models are constrained by the assumptions and simplifications.

We suggest to take advantage of these models by using them as initial guesses for inversion. With the expectation that the inversion will target variations of the angles from this starting point, there is a higher chance of convergence compared to the case where the starting angles are random.

Likewise, metallography/EBSD examinations of welds show that the variation of angles is typically smooth with just a few abrupt changes. The orientations, following welding passes, tend to form regions with similar dominant orientations which vary smoothly across the cross section. Hence in many cases adding total variation and smoothness regularisation terms helps achieve convergence and reduce the risk of being trapped in non-physical local minima.

The analysis of destructive examinations suggest that often there exists a dominant orientation of grains.

For the simplest symmetric cases, this will be the vertical direction, i.e. perpendicular to the surface. For other welds it may be a line almost parallel to the chamfer. This dominant orientation affects the propagation path considerably, and one approach to take advantage of this knowledge is to divide the inversion into two stages. Starting from the conventional Ogilvy map, in the first stage the algorithm is only adjusting the dominant orientation, that is, a single angle. When converged, it proceeds to updating local orientations across the weld. This approach was found particularly effective in our work and represents a pragmatic balance between the flexibility of ultrasonic tomography and the knowledge about the structure and the behaviour of welds of interest. This approach required a modification of the Ogilvy model, referred to as the 'generalised Ogilvy map', similar to that used in [START_REF] Menard | Ultrasonic Array Imaging of Nuclear Austenitic V-Shape Welds with Inhomogeneous and Unknown Anisotropic Properties[END_REF], which is described in more detail in Section Appendix A.

Numerical example

Microstructure and forward models

Having outlined the conceptual principles of weld map tomography, the attention now turns to examples.

First, we present a numerical example in which the target times-of-flight came from grain-scale time-domain finite element simulations. The calculations were executed on a graphical processing unit (GPU) using Pogo, a fast GPU finite element solver developed at Imperial College London [START_REF] Huthwaite | Accelerated finite element elastodynamic simulations using the GPU[END_REF]. The example is motivated by a real industrial inspection case at EDF, France. The grain structure of the weld bent and split the incident ultrasonic beam, so that the reflection from the backwall at some locations could be misinterpreted as a defect. At the time, to support the interpretation of the inspection, a macroscopic weld description was created, based on metallography. We used that macroscopic descriptions to create a numeric microstructure, allowing us to capture the rich physics of ultrasound propagation through complex welds in simulation.

The model was generated using free polycrystal generation packageneper [START_REF] Quey | Large-scale 3D random polycrystals for the finite element method: Generation, meshing and remeshing[END_REF]. Weld map tomography, as used in this paper, is based on backwall-reflected first arrivals, that is, longitudinal waves. To simplify extracting them from simulation, we executed an auxiliary calculation in which the backwall was replaced by an absorbing boundary. Substituting the two datasets led to clean first arrivals at the top surface. Such an approach is of course impossible in practice, but helps focus solely on the inversion algorithm, rather than peripheral issues, in a numerical example. Fig. 7 shows an example time trace and both unprocessed and cleaned signals, and the estimated arrival time. The estimation used an implementation of the Akaike Information Criterion [START_REF] St-Onge | Akaike information criterion applied to detecting first arrival times on microseismic data[END_REF] which helps if the first arrival is not the strongest in amplitude. For experimental datasets, obtaining an automated universal procedure is a research challenge on its own, and to keep the focus on inversion, we took the advantage of manual input.

To run the inversion on the simulated ultrasonic data, we first set up the SRP ray tracing model. The domain was cast on a regular 1 mm grid to accommodate weld geometry. The edges of each cell were seeded with ten nodes, and, to take the computational advantage described in Section 2, the algorithm pre-calculated the potential paths across a cell which were then used in all subsequent tracing. The domain was mirrored around the backwall, so that the pulse-echo configuration can be simulated. The SRP model is based on homogenisation, that is, it does not consider individual grains, but assigns local material properties based on the average grain orientation. Regions with columnar grains aligned to the same dominant angle are known to behave like a transversely isotropic material [START_REF] Mark | Investigation of the elastic/crystallographic anisotropy of welds for improved ultrasonic inspections[END_REF]. We exercised a homogenisation calculation based on the numerically generated microstructure, knowing that the dominant orientation should align with the direction of the smallest stiffness. The homogenisation yielded the following transversely isotropic material that the SRP model is set up in the y-z domain with z being the vertical, while the FE model is in the x-y domain.

Inversion

We used the two-stage inversion described above. The algorithm started with a 'standard' Ogilvy map

(T L = T R = n L = n R = 1, α L = α R = 20
) with a vertical dominant orientation (ζ = 0°), converging to -4.7°after six iterations. In the per-cell inversion stage we used smoothing and total variation regularisation, achieving convergence after a further 23 iterations. Fig. 8 shows the obtained weld map and the evolution of the cost function. The vertical line marks the point where the first stage, in which only the dominant angle was varied for a general Ogilvy map, was completed. Individual orientations were then updated to the point where for the three consecutive iterations the cost function dropped by less than 2% (this criterion was adopted for both stages). The weld map can be compared to the original microstructure in Fig. 5.

Detailed error maps for the absolute angular error and the time-of-flight error are presented in Fig. 9.

The mean, median, and maximum absolute angular errors (|θ original -θ inversion |) are 7.28, 5.71, and 33.06°, respectively. From a practical viewpoint, these are good results, compared to commonly reported average errors exceeding 20° [START_REF] Fan | Nonintrusive estimation of anisotropic stiffness maps of heterogeneous steel welds for the improvement of ultrasonic array inspection[END_REF]. We also compared the converged time-of-flight map to that chosen as a target for inversion in Fig. 9(b). The mean, median, and maximum absolute time errors (|τ original -τ inversion |) are 0.03, 0.02, and 0.27 µs, respectively.

The numerical example demonstrates the potential of weld map tomography and quantifies its performance. While errors exist, they are below the commonly reported level (as expected from a simulation-based

Experimental examples

In this section we report exercising weld map tomography on three experimental samples -two industrial mock-ups from EDF and one mock-up specifically manufactured for the ADVISE project. All samples had artificial defects created for testing imaging algorithms. Verifying the effectiveness of the inversion directly is not straightforward, as usually destructive material examinations are taken over a different part of the weld than that used in the measurement. But it is widely accepted in the community that grain orientations do not vary significantly along the weld, except for arc ends. Nevertheless, in a practical scenario, the best evaluation of the obtained weld structure information is its application to imaging. Hence, for all numerical examples, we demonstrate how the result of weld map tomography improves ultrasonic array images and the signatures of target defects.

Weld with uniform orientations

First, we consider a homogeneous Inconel 182 wide chamfer weld from EDF, with approximately uniform orientations, as used in [START_REF] Ménard | Optimization algorithms for ultrasonic array imaging in homogeneous anisotropic steel components with unknown properties[END_REF]. Fig. 10 shows a photograph and a schematic diagram of the experimental pulseecho setup used for inversion. As opposed to [START_REF] Ménard | Optimization algorithms for ultrasonic array imaging in homogeneous anisotropic steel components with unknown properties[END_REF], we used contact measurements, with the array (5 MHz 64-element; pitch 0.6 mm, element: 0.5 mm) placed on the surface of the weld, centrally with respect to the middle defect. This is the simplest test case, where the target of the inversion is a single orientation angle. This means that the objective function depends only on one variable and that the gradient of the objective function is a sum of contributions from all cells. The top and bottom surfaces of the model were reached a realistic value. However, quantitative error analysis is possible for arrival times, as depicted in Fig. 12(b). The mean absolute error was 0.021 µs, the median absolute error -0.015 µs, and the maximum absolute error -0.093 µs. The highest errors are between transducers close to array extremities, suggesting that the geometry of the sample was not captured correctly. Despite that, the result of the tomography is close to the reference provided for the mock-up. We attribute this robustness to the fact that only a single orientation angle is sought for in this example, minimising the impact of locally mismatched definitions. The practically relevant verification of the inversion is to perform imaging with the obtained material information. We used the classical total focusing method (TFM) [START_REF] Holmes | Post-processing of the full matrix of ultrasonic transmit-receive array data for non-destructive evaluation[END_REF] which is a modified, Hilber-envelope-based version of the beamforming algorithm. The imaging function I(x, y) is defined as

I(x, y) = M i=1 M j=1 H u i,j τ i→(x,y) + τ j→(x,y) (13)
where (x, y) is the position in the physical domain, M is the number of elements in the array, u i,j is the response measured by element i with element j exciting, τ i→(x,y) is the traveltime between element i and point (x, y) (called a delay law) and H[] is the Hilbert transform.

Fig. 13 shows TFM images calculated using the same dataset as the one used for inversion. For comparison, we show the 'conventional' image evaluated under the assumption that the material is isotropic, and the 'updated' image which uses the results from inversion presented earlier. Both images are normalised to the maximum amplitude of the middle defect (individually). The circular markers indicate true positions of side drilled holes. To quantify the difference between the two images we calculated the signal-to-noise ratio (SNR) using the following procedure. First, the algorithm searched for a maximum within +/-5 mm of the known defect position. Then, we defined a signal region I sig (solid line square), and the noise probe region I noise (dashed line square), which were squares with sides 4 and 10 mm, respectively, centred around the maximum. The SNR was calculated using the maximum amplitude in the signal region and the rootmean-square amplitude in the noise region (not including the signal part) SNR = 20 log 10 max(I sig )

I 2 noise (14)
The results in Fig. 13 confirm that using the correct material information reduces the overall level of noise in the image, improves the location of defect signatures (by 3.2, 1.1 and 4.6 mm from the left to the right defect, and enhances the SNR (by 5.9, 14.4, and 7.5 dB, respectively), making defect signatures more focused and better pronounced. Moreover, it confirms the effectiveness of weld map tomography and the benefits it provides to imaging.

Weld with a non-planar backwall

In this section we test weld map tomography on a 316L stainless steel industrial mock-up from EDF.

The sample has a typical V-shaped configuration and a non-planar backwall profile. Two target defects (side-drilled holes of 1 mm diameter) located at the centreline (approximately 10 and 20 mm from the top surface) provide a reference for imaging. Compared to the previous sample, the growth of complexity is twofold: first, we update individual local grain orientations; second, the backwall profile has a non-planar profile.

The measurements, taken at CEA, used a contact 5 MHz 64-element array with 0.6 mm pitch and 0.5 mm element width, positioned centrally above the weld. Extracting backwall-reflected arrival times from the recorded dataset required a semi-manual procedure described in Section 5.1. Weld map tomography, as presented in this paper, focuses on determining local grain orientations, that is, material information only. Thus, the geometry was fixed and assumed known, which is a straightforward task for a mock-up. Since this paper focuses on updating material properties, keeping the geometry fixed, we omit the simplified representation of the backwall profile for brevity.

The forward model used in the inversion discretised the domain into 2 × 2 mm cells, seeded with ten nodes per each side. For the parent material, we used ρ parent = 7800 kg m -3 , c 11 = 276 GPa, c 12 = 102 GPa, and c 44 = 87 GPa. The weld material had the same density, and the elasticity matrix with c 11 = 245 GPa, c 12 = 110 GPa, c 13 = 145 GPa, c 22 = 253 GPa, c 33 = 221 GPa, c 44 = 110 GPa, and c 66 = 75 GPa [START_REF] Menard | Ultrasonic Array Imaging of Nuclear Austenitic V-Shape Welds with Inhomogeneous and Unknown Anisotropic Properties[END_REF]. We exercised the two-stage inversion, identical to that in Section 4, where the first stage looks for the optimal dominant orientation angle for the generalised Ogilvy model, and the second stage updates local orientations to further decrease the cost function until convergence. Fig. 14 shows the results of the inversion which started from an Ogilvy map with 0°dominant orientation.

After six iteration the minimisation converged to -20.5°and moved to the second stage where individual angles were further updated, completing after a total of 19 iterations. The evolution of the cost function clearly suggests that the dominant grain orientation had the most significant effect on the inversion. The arrival time errors are higher than in the previous example, with the mean absolute error of 0.034 µs, the median absolute error of 0.029 µs, and the maximum absolute error of 0.185 µs. It is worth noting that the 'white spots' in Fig. 14(b) indicate that for these transducer pairs it was not possible to extract arrival times from the measured data. While this reduction in the available data is undesired for the inversion, this case confirms that weld map tomography does not require the full dataset to work. We also observe that the maximum errors occured for transducer pairs close to the undefined arrival time regions, bound by high uncertainty. The mean error is still low and represents a practically promising performance. In Fig. 15 we offer a visual comparison of the updated orientations with the metallography of the mock-up, confirming the effectiveness of the inversion. The final verification involves calculating the TFM image using the same recorded dataset with and without material information provided by weld map tomography, as shown in Fig. 16. Updated material information reduces the noise in the image and improves the prominence and focus of defect signatures.

While the noise is not eliminated entirely, the SNRs of the target defects improve by 12.3 and 4.4 dB, respectively. We acknowledge artefacts close to the root of the weld; however, their origin, and ways for reducing them, remain the subject of future work. Nevertheless, our proposed inversion technique output local orientations visually agreeing well with the metallography image and providing good enhancement to the ultrasonic image of target defects, despite the complex backwall geometry.

ADVISE mock-up

The final example considered is an austenitic stainless steel mock-up manufactured for the project sponsoring this work. Project partners designed and manufactured a weld of industrial relevance, but with careful recording of welding parameters, materials, and other details necessary for weld formation and phenomenological MINA [START_REF] Moysan | Modelling the grain orientation of austenitic stainless steel multipass welds to improve ultrasonic assessment of structural integrity[END_REF] modelling.

The mock-up was manufactured by joining two stainless steel pipes (1.4550) with wall thickness of 39 mm (outside diameter: 517 mm) -Fig. using the shielded metal arc method with ESAB OK 61.30 filler (1.4316). The cylindrical specimen was then sectioned into several parts along the circumference (Fig. 17), which served other research tasks.

One section of the mock-up was used for material examinations, both traditional metallography and EBSD. For the EBSD measurement, the weld was further sectioned into four parts and each was covered by a grid of smaller measurement areas in the microscope. The obtained dataset, comprising over 100 'tiles' was stitched and analysed using Dream.3D [START_REF] Groeber | DREAM.3D: A Digital Representation Environment for the Analysis of Microstructure in 3D[END_REF], an open-source package for microstructure synthesis, to deliver a numerical microstructure, that can be visualised, analysed, and subsequently used in further modelling or the verification of experimental inversion.

To facilitate measurements, the top and bottom surfaces of the mock-up were machined to achieve flat surface (Fig. 19). This modification helped ensure good contact between the transducer and the sample and allowed focusing on the inversion, rather than the transducer issues. Material evaluations at the University Array measurements using a 64-element, 0.75 mm 2.25 MHz transducer (in contact with the sample, directly on top of the weld, coupled through silicone gel) were taken at KTU using the Gekko system from Eddyfi. The acquisitions were processed using the semi-manual time-of-arrival extraction procedure described in Section 5.1, yielding the target for the weld map tomography algorithm.

The forward model used in inversion assumed known material properties of the sample and known geometry. We used the EBSD dataset to execute homogenisation and determine the macroscopic transversely isotropic elastic tensor based on the grain structure and the crystal elastic tensor. These parameters are listed in Fig. 19. The TFM imaging used the dataset as that produced the time-of-flight target for inversion. Fig. 23 compares images calculated: a) using the conventional isotropic assumption and b) delay laws based on the output of weld map tomography. The effect of the proposed inversion technique is clear and convincing.

First, the sample had two known target defects (side-drilled holes) along the right chamfer -Fig. 23 shows that only one is visible in the conventional TFM image. Including material information coming from inversion raised the bottom chamfer defect from the background noise. The SNRs of the chamfer defects improved by around 4 and 8 dB. We note that if a defect signature is not present in the conventional image, the SNR value has no meaning (as there is no 'signal'); nevertheless, it is calculated in the image for completeness.

Additionally to the intentionally introduced side-drilled holes, two natural defect signatures emerged in Fig. 23b) -one close to the centreline and another in the vicinity of the chamfer. Post-manufacture X-ray scans of the weld indicated the presence of voids in some parts of the weld and it is confirmed with ultrasonic imaging. More importantly, the absence of these natural defects' signatures in the conventional image proves the effectiveness of weld map tomography in supporting imaging and is a compelling proof of the importance of accounting for material information.

Conclusions

This paper proposed a method for determining local grain orientations in complex, thick welds from ultrasonic array data. The reconstruction followed the ray tomography principle, accounting for the heterogeneous anisotropy of the welds of interest. The inversion used a shortest-ray-path model as the forward solver and the paper detailed the principles and construction of the model. We discussed the importance of the a priori information about the weld and the advantages of using simple geometrical descriptions for initial stages of inversion. The algorithm was first showcased using a numerical example based on an industrial inspection scenario, with grain-scale time-domain finite element simulations acting as a virtual experiment. The results of the inversion showed good agreement with the numerically generated microstructure. Encouraged by this result, we demonstrated weld map tomography on three mock-ups -two supplied by the industry, and one manufactured specifically for this research. The outcome of the method compared favourably with destructive material examinations (both metalography and EBSD) and provided the input for updating the delay laws for imaging. In all examples, TFM images using material information from inversion showed significant improvement of both the location of defect signatures and SNR, confirming the efficacy of the proposed algorithm. Grain orientations θ defined as the rotation from the vertical positive anticlockwise are calculated using the following expressions

θ(y, z) =            arctan T L |d L +z tan α L | y η L -π 2 , if y < 0 arctan T R |d R +z tan α R | y η R + π 2 , if y > 0 0, if y = 0 (A.1)
The generalised Ogilvy map, as we call it in this paper, adds one more parameter ζ to capture the dominant orientation representing the direction of privileged grain growth during cooling [START_REF] Menard | Ultrasonic Array Imaging of Nuclear Austenitic V-Shape Welds with Inhomogeneous and Unknown Anisotropic Properties[END_REF]. This parameter is the angle between the original z-axis and the new axis along which the grains would align. In the model, or α R it should be capped at the weld angle for the coordinate rotation (for some configurations, e.g. very narrow welds, the limit may need to be less than the weld angle). After the rotation, each point in the original y-z reference frame has its auxiliary counterpart in the y ζ -z ζ frame. Then, the grain orientations are calculated as:

θ(y, z) =              arctan T L| d L +z ζ tan (α L -ζ)| (y ζ ) η L -π 2 + ζ, if y ζ < 0 arctan T R |dR+z ζ tan (α R +ζ)| (y ζ ) η R + π 2 + ζ, if y ζ > 0 ζ, if y ζ = 0 (A.2)
When applying the dominant orientation parameter ζ to an originally symmetric weld (with T L = -T R ), one should adjust these parameters to retain relevance to realistic cases. A rule-of-thumb adjustment based on comparing such maps to metallography images suggests multiplying initial T L and T R parameters by

(1 + 0.01ζ) and (1 -0.01ζ) respectively.

Figure 1 :

 1 Figure 1: Computational representation of a weld in the SRP model. The orange-coloured cells represent those crossed by the depicted ray.

Figure 2 :

 2 Figure 2: Seeding of cells in the SRP model. Two sets of cell rays are shown, for clarity, rays co-linear with the horizontal and vertical grid lines are not drawn.

Figure 3 :

 3 Figure 3: Group slowness for a transversely isotropic material -how the incident angle and grain orientation form a combined ray angle in material coordinates.

Figure 4 :

 4 Figure 4: Auxiliary mirrored domain for inversion through the backwall: a) the setup; b) illustration of the reduction of the view angle between through-transmission and pulse-echo.

-Figure 5 :

 5 Figure 5: The macroscopic weld description with dominant orientations for each zone marked (a), and the generated synthetic microstructure (b) for the numerical example of weld map tomography. The orientations are quoted in degrees, measured from the vertical direction, with positive angles going anti-clockwise

Figure 6 :

 6 Figure 6: Schematic diagram for the FE model for the numerical example of weld map tomography.

Figure 7 :

 7 Figure 7: Illustrative simulated time trace for the numerical example -array element #41 transmitting and element #36 receiving.

constants: c 11 =

 11 248 GPa, c 12 = 91 GPa, c 13 = 133 GPa, c 33 = 206 GPa, c 44 = 119 GPa, c 66 = 77 GPa, and the same density as for the parent material ρ = 8000 kg m -3 . The vertically oriented grains have the 33 material axis aligned with the global z axis. Orientation angles are measured from the vertical, positive anticlockwise. A short note on the orientation convention is included in Section Appendix B. Note also

Figure 8 :

 8 Figure 8: The result of weld map tomography based on time traces from grain-scale finite element simulations: a) obtained grain orientations; b) the cost function.

Figure 9 :

 9 Figure 9: Error maps of weld map tomography based on time traces from grain-scale finite element simulations: a) orientation errors; b) time of flight errors.

  study) and related to the limitations of time-of-flight tomography itself.

16

 16

Figure 10 :

 10 Figure 10: Schematic diagram for experimental weld map tomography on an EDF Inconel 182 V-weld with uniform orientations: a) photograph; b) experimental setup.

Fig. 12 shows

 12 Fig. 12 shows the results of the inversion, based on experimental time traces recorded at CEA. Weld map tomography started from uniform vertical orientation of grains and converged to -8.11 deg after 9 iterations.Determining the exact angular error is not possible for this example, but the metallography performed on an off-cut of the same weld suggests the dominant orientation of -8 deg which confirms that the inversion

Figure 11 :

 11 Figure 11: Extracting time-of-arrival as the target for inversion from experimental data (the map combines time traces across all sensors from single-transducer excitation; the markers show the identified times-of-arrival).

Figure 12 :

 12 Figure 12: Results of weld map tomography for the uniform Inconel weld: a) cost function and orientation angle at different iterations; b) a map of absolute time-of-flight errors (|τ original -τ inversion |).

Figure 13 :

 13 Figure 13: Total focusing method imaging with material information: a) standard isotropic delay laws; b) with delay laws coming from weld map tomography. The SNRs of the defect signatures indicated for comparison.

Figure 14 :

 14 Figure 14: Results of weld map tomography for the stainless steel weld with a non-planar backwall: a) cost function and orientation angle at different iterations; b) a map of absolute time-of-flight errors (|τ original -τ inversion |).

Figure 15 :

 15 Figure 15: Results of weld map tomography for the stainless steel weld with a non-planar backwall: a) updated local grain orientations; b) the metallography image of the mock-up.

Figure 16 :

 16 Figure 16: Total focusing method imaging with material information for the stainless steel weld with a non-planar backwall: a) standard isotropic delay laws; b) with delay laws coming from weld map tomography. The SNRs of the defect signatures indicated for comparison.

17

 17

  . The circumferential weld, in the flat position, was manufactured

Figure 17 :

 17 Figure 17: The ADVISE WP2 mock-up: a photograph and schematic drawings of a cut out section.

Figure 18 :

 18 Figure 18: The workflow for the reconstruction of the microstructure from EBSD measurements.

Figure 19 :

 19 Figure 19: ADVISE WP2 mock-up and the transducer used in weld map tomography: a) a photograph; b) dimensions with material properties.

Figure 20 :Figure 21 :

 2021 Figure 20: Results of weld map tomography for the stainless steel ADVISE mock-up: a) cost function and orientation angle at different iterations; b) a map of absolute time-of-flight errors (|τ original -τ inversion |).

Figure 22 :

 22 Figure 22: Results of weld map tomography for the stainless steel ADVISE mock-up: a) updated local grain orientations; b) microstructure reconstruction based on EBSD measurements.

Fig. 22

 22 Fig. 22 compares grain orientations determined using weld map tomography and the reconstruction of the EBSD measurement taken at the University of Stuttgart. This visual comparison gives a convincing indication of the effectiveness of the method. We note that the errors in orientation angles are mainly concentrated around the chamfer, where the EBSD data had low confidence index. Nevertheless, as in previously presented examples, the final confirmation of the efficacy of weld map tomography is illustrated by applying its output to imaging.

Figure 23 :

 23 Figure 23: Total focusing method imaging with material information for the ADVISE mock-up: a) standard isotropic delay laws; b) with delay laws coming from weld map tomography. The SNRs of the defect signatures indicated for comparison.

Figure A. 1 :

 1 Figure A.1: Schematic diagram for: a) the classical Ogilvy model; b) auxiliary rotation of the coordinate system to account for dominant orientation ζ.

  we account for that by temporarily rotating the y-z system by ζ (see Fig. A.1b)). If |ζ| is greater than α L

Acknowledgement

This research was a part of the ADVISE project (www.advise-h2020.eu) funded from the Euratom research and training programme 2014-2018 under grant agreement no. 755500.

Appendix A. Generalised Ogilvy map

The geometric description of grain orientations proposed by Ogilvy [START_REF] Ogilvy | Computerized ultrasonic ray tracing in austenitic steel[END_REF] is a simple but powerful method.

Throughout recent decades it became the gold standard for modelling local orientations in austenitic stainless steel welds. This paper uses a slightly extended version of the Ogilvy map, accounting for the dominant orientation angle which is relatively common in published examinations (see e.g. Fig. 10a)). What is presented below is conceptually analogous to an approach used in [START_REF] Menard | Ultrasonic Array Imaging of Nuclear Austenitic V-Shape Welds with Inhomogeneous and Unknown Anisotropic Properties[END_REF].

Let us start from the classical Ogilvy map [START_REF] Ogilvy | Computerized ultrasonic ray tracing in austenitic steel[END_REF] for completeness. We consider a simple V-weld, of thickness

Appendix B. Orientation and material properties conventions

The inversion algorithm and the SRP forward model proposed in this paper rely on a macroscopic representation of the weld, in which local properties are governed by a transversely isotropic material rotated according to the orientation angle. The rotation is such that the 'soft' axis of the material aligns with the orientation direction. The SRP forward model and weld map tomography are set up in the y-z domain, hence the 'soft' axis will be the 33 material direction. If an analogous calculation was performed in the x-y domain, e.g. in a 2D finite element simulation, one would need to appropriately rotate the material stiffness tensor, so that the 'soft' axis will appear as the 22 material direction.


Michele Linardi
email: michele.linardi@cyu.fr

Vassilis Christophides
email: vassilis.christophides@ensea.fr

Unsupervised Time Series Anomaly Detection: The Road to Effective Explainability

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1 Anomaly Detection in Multivariate Time Series Introduction. Given a large collection of multivariate data sequences1 , or a continuously incoming time series in real-time systems, the anomaly detection task (AD) aims to find data occurrences that describe rare and anomalous events. Anomalies can have different forms; they can occur as single abnormal values (a.k.a. point anomaly) or as complex temporal events described by a contiguous sequence of individual data points (a.k.a sub-sequence anomaly). State-of-the-art. Currently, we dispose of a plethora of heterogeneous and unsupervised AD techniques [START_REF] Freeman | Experimental comparison and survey of twelve time series anomaly detection algorithms[END_REF]), which exhibit shaky performance according to the specificity of the analyzed sequences. In fact, accuracy performance may vary depending to data seasonality, trend, variables correlation drift over time, and the presence of missing values.

In this context, a user that wants to perform anomaly detection must have a deep understanding of the underlying data characteristics in order to know which algorithm represents the best fit in term of accuracy. This is clearly an unrealistic constraint, which severely limit the adoption of the current AD solutions. Moreover, the users require to have the necessary confidence to trust the decisions taken by the model, which cannot be applied as a black-box. In details, AD models must ideally provide explanations that indicate the specific data characteristics that triggered anomaly detection [START_REF] Rojat | Explainable artificial intelligence (XAI) on timeseries data: A survey[END_REF]). In this sense, the ability to explain why a data point (or a sequence) was considered as abnormal (a.k.a Explainable Anomaly Detection) is a strict requirement in a variety of safety-critical domains. Challenges. We consider two open challenges in AD over temporal data. Firstly, we note that typically multivariate time series, represent a difficult-to-interpret data type due to the presence of many different variables (i.e., monitored metrics) that are non-trivially correlated due to synchronization or sampling mismatches. Modern machine/deep learning (DL) models can efficiently learn temporal data evolution but do not provide a robust solution that can learn complex inter-metrics correlations. Furthermore, we need to account that human TS visual interpretation is a hard task also for an expert user. We should note that obtaining a satisfying level of insight remains a more natural task on images, video, or text. Several DL algorithms that obtain state-of-the-art performance in unsupervised TS-AD are black-box applications for the user. This is the case of variational autoencoders that adopt neural-network architectures like LSTM, GRU, and CNN (Su et al. ( 2019 2021)). Using these methods does not provide the possibility to shed any light on the critical factors that affect the decisions taken by the model. Specifically, to increase the confidence in the anomaly detection outcomes, the user may want to understand which feature sub-spaces actively contribute to anomaly classification and which are the root causes that trigger such abnormal event. Even though several recent research efforts [START_REF] Rojat | Explainable artificial intelligence (XAI) on timeseries data: A survey[END_REF]) have tried to adapt explainability techniques used in DL models that were originally applied to images and videos, they currently fail to address the specific requirements of temporal data. To increase user confidence, a model should provide a holistic picture of system events. Hence, in the case of multi-variate TS, we want to obtain information about the temporal evolution of an anomaly, e.g., we want to detect and understand all the events in the past that trigger an anomaly, and along which metrics they occur. Exclusively detect the data occurrences that deviates from normality is a clear limitation of the currently available techniques.

In the next part, we briefly discuss the main elements of our vision of new generation anomaly detection in multivariate time series.

The Road To Explainable Anomaly Detection

Here, we introduce the architecture of the current AD models based on a DL architecture, discussing the research directions we want to follow.

Unsupervised state-of-the-art AD models adopts the architecture we sketch in Figure 1. The first part (Figure 1(a)) consists into an unsupervised learning step over historical multivariate time series using variational autoencoders and recurrent artificial neural networks. In the learning phase, the variational autoencoder maps real data variables to an (estimated) stochastic distribution in a latent space, while minimizing decoding loss, which corresponds to the error of sampling data from the estimated distribution. In this respect, learning to accurately reconstruct the temporal evolution of normal data, while maintaining the existing correlations between the different variables in a latent space becomes crucial to detect anomalies.

In fact, online anomaly detection (Figure 1(b)) considers reconstruction errors of fresh data that are ideally high when decoding abnormal occurrences, and thus the model correctly reconstructs the structure of normal data. Hierarchical variational autoencoder initially proposed for image learning, are recently used to embed time series into a latent space that encodes both the temporal dependency and the inter-variable correlation of multivariate time series. Learning Lagged Correlations. We note that an unsupervised autoencoder learns in essence correlations in a fixed time window. Hence, the user must know not only the window length parameter, but she must expect that inter-variable correlations occur in a well-known timeframe. We thus need to enlarge the capability of current learning algorithm that must be able to learn metrics relations irrespective of their time shift and scale. We consider this as a crucial requirement of models that implement explainability.

To that extent, Long-Run Variance Decomposition Networks (LVDN) are types of complex networks that are typically used to describe lagged relationships [START_REF] Silva | Time series analysis via network science: Concepts and algorithms[END_REF]). LVDN are graphs whose nodes represent TS modeled by vector autoregression, and edges get weighted by the information that one TS contributes to the other. Specifically, each weight connecting TS A and B measures how much the anomalies in A influence the variance of forecasting error of B. Statistical approaches based on LVDN have found high applicability in market analysis to capture long-run dependencies of shocks (abrupt events) [START_REF] Lanne | Generalized forecast error variance decomposition for linear and nonlinear multivariate models[END_REF]). We envision the adoption of complex network representation in AD models as it can serve the building of a discrete and finite structure that encodes scale-invariant correlations. Effective User Interaction. Our second objective is to enable effective user interaction to the learning phase of anomaly detection. This is possible if the user can interpret the relationships learned by the model, e.g., learning complex networks data structures that encode the temporal evolution and relationship of time series metrics. In this sense, we should stress the growing research efforts towards interpretable [START_REF] Yuan | Explainability in graph neural networks: A taxonomic survey[END_REF]) deep learning models applied to graphs. In contrast to applying AD models as black-boxes, graph modeling should permit us to design explanation strategies that highlight relationships between the detected anomalies and their precursors. Anomaly Causal Analysis. One of the crucial requirements in safety-critical domains is causal analysis, namely the detection and study of the events chain that eventually brings the underlying system in an abnormal state. We observe that existing Causal Models (CMs) work over graph structures that encode temporal relationships [START_REF] Moraffah | Causal inference for time series analysis: Problems, methods and evaluation[END_REF]). In this sense, we should consider the theoretical connection between Graph Neural Network and CMs (Zecevic et al. ( 2021)) to propose new applications that can learn anomaly leveraging the powerful graph encoding.

  FIG. 1 -Architecture of unsupervised anomaly detection in multivariate time series. (a) Learning time series representation in a latent space (Variational autoencoder), (b) Detect anomaly based on reconstruction errors.

  ); Li et al. (2021); Pol et al. (2019); Nguyen et al. (

Multivariate time series (TS) are sets of one-dimension variables (a.k.a metrics), namely sequences of real values recorded over time at a fix interval rate.


domain: spi math phys.cond.cm-ms

text:
Numerical analysis of axial compression impact on the hydrodynamics of open-cell foams

Contenu eX-Poro-HydroDynamique lubrication (or XPHD lubrication) presents a different scientific approach to dealing with tribological problems. It's an innovative inter-and multidisciplinary research topic which offers a promising sliding solution for various applications, such as bearings, thrust bearings, various guide components, etc. Its applications in both biological systems, and industrial systems : soft porous squeeze damping or soft porous lubrication are very broad and important. XPHD lubrication is a lubrication mechanism of biomimetic inspiration which features an additional parameter to the system "the porous media". It consists of self-sustained fluid films generated within compressible porous layers (CPL) imbibed with liquids in replacement for using the fluid film only as in the classic lubrication system. Soft and porous structures imbibed with liquids generate a high load support under compression, the load support is generated through the resistance to flow inside the porous material. During compression, the resistance to flow and load support increase, the greater the compression level, the lower the porosity and corresponding permeability. The main objective of this work is to understand the behavior of the fluid flow inside the porous structures when subjected to axial compression stress. In the scientific literature, the works studying the flow in compressible materials are essentially experimental because of their very complex geometrical shape [START_REF] Dasa | Drag and heat transfer closures for realistic numerically generated random open-cell solid foams using an immersed boundary method[END_REF], [START_REF] Bracconi | A systematic procedure for the virtual reconstruction of open-cell foams[END_REF], [START_REF] Bracconi | A fundamental analysis of the influence of the geometrical properties on the effective thermal conductivity of open-cell[END_REF], the CFD (Computational Fluid Dynamics) simulations offer an economical solution to study the performance of this new concept of lubrication. To create the geometry, the morphological structure of foam samples is reconstructed at different levels of compression rates from 3D X-ray microtomography. This is achieved by using a commercial software (Avizo) that allows to process 3D images and create FE/CFD models suitable for numerical analysis [4]. The numerical simulations of flows will be performed with the solvers icoFoam and simpleFoam of the OpenFOAM [5] framework for steady-state incompressible laminar and turbulent flows, making it possible to study different fluids and porous materials. The performed simulations were made with an open-pore polyurethane foam with 96% porosity using five compression rates for creating the different geometries. For the fluid we used water and a hydraulic oil ISO VG 46 to see the impact of the viscosity on the flow. The analysis of the simulations shows the impact of the solid compression on different parameters, such as the decrease in permeability as function of the compression rate, the anisotropy of the flow within the compressible structure and the actual increase in the tortuosity generated by the compression of the solid and the variation of the porosity.

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