Component shape optimisation for enhanced non-destructive testing

Determining an optimal component design is fundamental to many applications including many of those in the aerospace, automotive and civil engineering industries. Previously, the compatibility of the component with future non-destructive testing (NDT) requirements has not been considered at the design stage. As a result, NDT operators are often challenged to reliably inspect components with complex geometries. In this paper, a framework is proposed for the optimisation of a component’s shape to maximise the sensitivity and coverage of an interrogating ultrasonic wave, thus enhancing the ability to non-destructively image defects. The design of beam cross-sections are optimised using both a low-dimensional parameterisation with a genetic algorithm, and the level set method which enables more degrees of freedom in the component shape parameterisation. Images of flaws computed using the total focusing method show an 8 dB improvement in the signal to noise ratio for the optimised component as well as a five-fold improvement in the estimate of flaw size. These results show that the NDT-optimised design of components can provide significant improvements in flaw imaging. This in turn assists in extending the lifespan of in-service components and indeed their remanufacturability, which is both environmentally and economically advantageous.

The development of sustainable methods for the manufacture of components is critical across all engineering disciplines. One environmentally friendly and economically viable approach is remanufacturing; the process of bringing a used product to a state that performs at least at the same level as the original product, and is therefore issued an as-new warranty [19,27].
The remanufacturing procedure is as follows: a product is stripped down to its individual components; the components are cleaned, assessed, remanufactured and restored; then the product is rebuilt and performance tested [19]. Lund [27] indicates that 85 % of the mass of a remanufactured product may be sourced from used components, and requires 50-80 % less energy to produce, providing a 20-80 % production cost saving in comparison to conventional manufacturing. The principle of design-for-remanufacturing (DfREM) recognises the importance of the initial component design in aiding the steps involved in the remanufacturing process [15].
During the remanufacturing assessment stage, the detection of any internal defects is critical; both production costs and environmental impact are reduced when the level and reliability of inspection is enhanced, so that defective components are also not propagated through the manufacturing process. Non-destructive evaluation or testing (henceforth, NDT) is a term that describes a range of methods that evaluate components non-invasively [38]. The dominant modality is ultrasonic NDT, which uses high-frequency mechanical waves to image the interior of a component [41]. These methods allow for inexpensive, near real-time, and high resolution images of a component's interior to be generated and any internal defects detected [6,26], even in heterogeneous material [46,47,48]. However, although reliable NDT is key for maintenance and life extension of a component, the suitability of a component for NDT is not currently considered in the optimisation of most component's design [47].
In the last decade, ultrasonic NDT has seen a rapid increase in the use of ultrasonic phased arrays for standardised inspection of components [20,53]. The advantage of using arrays is increased imaging quality as the array can control the directivity and focus of the ultrasound by varying the time delay between firings of the array elements. However, in general, ultrasonic phased arrays are based on piezoelectric transduction, which has certain limitations. For example, the transducers often require the use of a coupling medium between the tested component and the testing equipment and are constructed from a fixed number of elements, with a predefined (normally planar) shape. These constraints mean that the transducer based phased arrays cannot be used on components with complex geometries or in settings where access to the surface of the J o u r n a l P r e -p r o o f Journal Pre-proof component is restricted.
Laser ultrasound offers an alternative, non-contact method, for the transmission and reception of ultrasonic waves [12,45,51]. The light from a pulsed laser induces a mechanical wave in the component through localised heating and thermal expansion at the component surface [40]. The transmitted waves are then detected using laser interferometry, which uses the phase shift of laser light to detect very small displacements at the component surface, offering a completely non-contact ultrasonic inspection system [31]. There are several advantages to using this method. First, as there is no mechanical contact, laser ultrasound is not affected by any impedance mismatch between the component and transducer. Second, laser ultrasound can be used for components with more complex geometries and in testing areas where surface access is restricted, where piezoelectric transducers, with fixed sizes and shapes, constrain the range of testable component shapes [49,54].
Laser phased arrays can be synthesised in post processing: by controlled movement of a generating and receiving pair, an array of sensors can be synthesised [5,44]. This realises a comparable amount of array data to that generated using conventional, transducer based phased arrays. In particular, Stratoudaki et al. [44] captured the full matrix of source-receiver time domain signals (i.e. all NN  ultrasonic generation and detection signal pairs for source/receiver positions), and applied the total focusing method (TFM) imaging algorithm, demonstrating increased ultrasonic imaging quality compared to other laser ultrasonic techniques. Thus, laser ultrasonic phased arrays offer the advantages of remote ultrasonic testing with the added flexibility of choosing the number of elements and their distribution on any complex shape geometry [8,36] and without compromising image quality. It seems appropriate therefore to use laser induced ultrasound in the design-for-testing framework proposed in this paper.
This paper proposes and demonstrates a design-for-testing (DfT) framework for the optimisation of a component's shape with respect to NDT suitability, which is complementary to and may be used in parallel with DfREM and DfAM frameworks. DfT frameworks are commonly implemented in electrical engineering for the design of circuit boards [23] and software engineering for the design of testable programs [39]. However, this is the first implementation of an automated DfT framework for ultrasonic testing. Two algorithms for shape optimisation are implemented within our framework: a genetic algorithm approach for a design parameterisation with a low number of degrees of freedom [52], and a deterministic optimisation approach, coupled with a more complex design parameterisation defined using a level set method [1,28,35,42]. This method is widely used in a range of application areas including aerospace and structural engineering [13,21], and allows smooth perturbations to the boundary of a shape. In this study, the admissible shapes are constrained using a design envelope and the evolution of optimal shapes when the design envelope width is increased (allowing larger deviations from the initial component shape) is investigated.
The paper first describes the optimisation problem, and the analytical approach for generating ultrasonic sensitivity maps for laser induced ultrasound. A suitable objective function is defined and an optimisation method chosen, and the method is tested across a range of two-dimensional cross-sections through various beams which are commonly found in civil engineering applications [4,22]. The total focusing method [16] is then employed, to quantify the improvement in the imaging of a point scatterer embedded in the internal structure of an optimised component. This framework lends itself to being coupled to other design or testing constraints or demands and this paper also discusses the optimal positioning of the laser source/receiver locations at the component's surface.

Method
Consider a component with spatial domain  , which is to be evaluated using ultrasonic    [40], however in some cases it is desirable to use a line source which can reduce the peak optical intensity, keeping it below the destructive ablation threshold of the material, and facilitate wave directivity [44].

Analytical method for laser induced ultrasound sensitivity
To illustrate the approach, this paper restricts attention to the two-dimensional cross-section of a component. Expressions for the excitation of longitudinal L and transverse T wave displacements in a two-dimensional medium using a thermoelastic source [11], as a function of angle  (measured normal to the cross-section boundary) are given by [3,34]   The sensitivity of detecting longitudinal and transverse waves is given by [29] where 0 F is defined as An example of the transmission and reception sensitivity patterns for longitudinal and transverse waves is shown in Figure 2  commonly used of these is the total focusing method (TFM), see Holmes et al. [16]. In this method, the signals from all source-receiver pairs in the array are summed to synthesise a focus at every point within  in a discretised grid, assuming a constant (homogeneous) and isotropic medium. Take () sr ut to be the signal recorded at receiver r as transmitted at source s . For 2 n source-receiver pairs, the intensity of the TFM image I is given by [16] =1 =1 ( ) =| ( ( ( ))) |, nn sr sr sr where ()  denotes the Hilbert transform. The double summation is over all combinations of source-receiver pairs and the delay term is given by x are the distances from the source and receiver to the point x , respectively.
Using equations (3-6), a sensitivity map () LL E x [44] for longitudinal wave displacements in two dimensions is given by There are a few limitations to this method. For example, only first-order scattering is considered, no reflections from the boundary of the component are accounted for, and only a single homogeneous and isotropic material is considered. This approach also does not account for attenuation of energy during propagation. However, each of these drawbacks can be accounted for in the design process if more accurate prior information about the component's properties is available, and, at the expense of additional computation, the model can be augmented accordingly.
The simplicity of our model allows for rapid computation of sensitivity maps, and this allows for a range of computationally efficient optimisation methods to be used and compared in the absence of detailed prior information about each component.

Shape optimisation algorithm: Parametric optimisation
A common method in design engineering for finding an optimal shape for a component is to take a low degree of freedom parameterisation of a shape (e.g., for the design of airfoils, see Song and Keane [43]), and optimise its properties using, for example, genetic algorithms, differential evolution or CMA-ES methods [25,32]. This approach is taken here for a two-dimensional cross-section through an I-beam (also known as H-beam or universal beam). This is a widely used component in the construction industry, which is therefore widely tested using ultrasonic NDT.
The parameterisation shown in Figure 4  The calculation of the sensitivity map LL E (Equation (9)) contains the distances from a point x to the sources and receivers. As these distances decrease, the sensitivity increases.
Therefore, if a component size was to decrease, the distance from any central point to the boundary decreases and sensitivity increases. Thus, a constraint on the required volume ( req V ) must be included in the optimisation to ensure that the component does not simply shrink to a point mass to maximise sensitivity.   (11)), for varying envelope widths  when finding the optimal length (L), width (W), flange thickness (F), web thickness (T) and corner depth (C). Units are pixels in the images in Figure 5.
The material properties  are constant, and the source/receiver locations  are fixed to the entire boundary  , so that a source/receiver is located at every boundary pixel.  is therefore a function of  ), so we omit the dependence on  and  here. The optimal set of parameters * y , and therefore the optimal shape ** ()  y , are given by * = argmin : ( ( )) K   y yy (11) where the misfit function y is a combination of the volume constraint and the minimum sensitivity, and where  is a weighting factor on the volume constraint, () V  is the solid material volume for the domain  , req V is the required volume and K is the set of admissible shapes. A genetic algorithm [52] is employed to find the set of parameters that solve Equation (11), and information on the genetic algorithm is provided in the appendix. The implementation allows upper and lower bounds for each parameter to be applied, which enables the envelope bounds around an initial shape to be imposed (as in Fig. 1b). The initial I-beam shape (Fig. 5a), is given by the initial parameter values detailed in Table 1. The genetic algorithm optimisation is performed with four increasing envelope widths  (2, 10, 20 and 28 pixels) and using = 0.5  . The optimal shapes using each envelope bound are shown in Figures 5b-e and the parameters are detailed in Table 1; the domains shown are 100 100  pixels. In all cases, the resulting minimum sensitivity * E for the optimal I-beam parameterisation is an improvement from the initial shape and the volume constrained is satisfied (

Shape optimisation algorithm: Level set method
An alternative method for optimising a component's shape is to parameterise the shape using a continuum approach. One such approach is the Level Set Method [1,42,50]. The method expresses a curve (2D) or surface (3D), as a zero level set of a higher-dimensional function (level set function  ) in an implicit manner (the function  is precisely one dimension higher than the domain being optimised). The level set function has the following properties where x is any point in the design domain, and  is the boundary of  .
Any change to the boundary shape during the optimisation is performed via the evolution of the level set function  [42]. Figure 6 illustrates the relationship between the level set function  and a one-dimensional domain  , and shows how perturbations to  affect the length of  . The temporal and spatial evolution of  in the optimisation is governed by where v is a scalar field over the design domain which determines the geometric motion in the normal direction of the boundary of the structure and t denotes the time evolution of  . The level set function  is first initialised as a signed distance function from the boundary of the initial shape 0  (illustrated in Figure 6a). An upwind finite-difference scheme is used to solve Equation (13) numerically (see Challis [7] for algorithm details). The time step t  for the finite difference scheme is constrained by the Courant-Friedrichs-Lewy (CFL) stability condition: , where h is the minimum spatial distance between spatial grid points [42]. The discrete level set topology optimisation algorithm for compliance minimisation of Challis [7] is adapted here to suit an ultrasound sensitivity maximisation problem, by defining  : Illustration describing the relationship between the level set function  (blue) and the component with a one-dimensional shape  (red). The mathematical relationship is described by Equation (20). As the level set function is perturbed so that more points x decrease below zero, the eigen-length of  increases. (14) where () where we have chosen 1 = 0.005  and 2 = 0.01  .

Unconstrained shape optimisation
To illustrate the level set method optimisation approach, we first consider its application to a square initial shape (or cross-section of a square rod, see Fig. 7a) and perform an unconstrained optimisation where the masking term M in Equation (17)  field v (which determines the rate of change of the shape as given by Equation (13)) also tends to zero (Fig. 7n), and this corresponds to the term () C x being minimised (and therefore () being maximised). The minimum sensitivity * LL E increases by approximately 30% from the initial shape (Fig. 7i) and the volume constraint is satisfied (Fig. 7j). In the intermediate stages

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Masking allows for a constraint on the admissible shapes which reduces large deviations from the initial component shape. A constrained optimisation problem for the same square initial shape as in the previous example is now presented. The mask M is prescribed to be a set distance from the x (20) where the envelope width =2d  , measured in pixels (illustrated in Fig. 1b). A mask M where =4  pixels is shown in Figure 8a and the optimisation is performed with the same discretisation and parameters as above. The final shape is shown in Figure 8b and the evolution of the minimum sensitivity * LL E during the optimisation is shown in Figure 8c, where *

LL E increases by
approximately 20% compared to the initial shape. Understandably, the sensitivity for the unconstrained shape (Fig. 7) is higher than for the constrained case, however there is less resemblance to the initial component shape. In these examples, only the ultrasonic sensitivity is optimised. In practice, the strength and stiffness of the beam would also need to be considered, as any change in shape also alters the beam mechanical properties. Such additional constraints can be factored into future optimisations.

TFM Imaging
While the use of ultrasonic sensitivity maps (for example xx (23) where ( , ) Nt  is random Gaussian noise with standard deviation  . A-scans for every source-receiver combination are generated to produce a synthetic full matrix capture dataset.
TFM images were produced using the example initial square component shown in Figure   7a and the optimised disk shape component shown in Figure 7d, (Fig. 10a and 10b,  . An image is then generated using the total focusing method via Equation (7) Fig. 10c. The image for the optimised shape has a higher peak amplitude at the defect location and relative to the background material compared to the image for the initial shape; there is an 8 dB improvement in the flaw image when using the optimised design. When estimating the size of a defect, the -6 dB threshold is commonly used in NDT [10]. Using this approach the estimated defect width in the initial and optimised components is 7.4 mm and 3.9 mm, respectively, where the true defect width is 3.0 mm, so the percentage error in sizing the defect is reduced from 100 (7.

Limitations
This is the first presentation of a framework for shape optimisation that includes NDT considerations. There are several limitations and potential extensions to the presented method. The forward modelling approach uses an analytical model for the transmission and reception of laser induced ultrasound [3,29,44]. This model is a first-order scattering approximation (waves are reflected only once by a scatterer), and therefore does not consider any boundary reflections. Ray tracing algorithms that account for reflections (e.g., Bai et al. [2]) or numerical simulation of wave propagation [30,33] could be implemented to include second order scattering better representing the physics of wave propagation in practice. In addition to this, the optimisation only considers the propagation of compressional (or longitudinal) waves (Equations (3) and (5)). A weighted combination of compressional and transverse wave components [40] could be included in a future implementation.
As the analytical model for the transmission and reception of laser induced ultrasound only considers wave propagation in two dimensions, the shape optimisations presented here only consider the perturbation of the two-dimensional cross-section of a three-dimensional component (rods and beams) to enhance NDT capability. Current compliance-based level set shape and topology optimisation algorithms are capable of designing intricate three-dimensional structures [21]. In principle, our method can be applied similarly to more complex three-dimensional geometries providing the analytical model that generates the ultrasonic wave propagation can account for three dimensions.
Another simplification as presented here is the lack of topological variations during the level set optimisation (i.e., there are no holes in  and no new holes are nucleated). Optimising for the topology of a component is a major advantage of the level set method [1,7], and is particularly important in compliance-minimisation (stiffness maximisation) problems, where a stiff component needs to be designed with a material volume constraint, so the nucleation of void regions is desirable. For ultrasonic NDT applications, the nucleation of voids introduces reflectors and barriers to wave propagation, and therefore could cause difficulties for NDT. The extension of this method to allow topological variations would be a natural extension to the current method; for example allowing for joint optimisation for compliance and ultrasonic sensitivity.

Experimental Design and Joint Optimisation
In all the examples presented here, there is full coverage of the domain boundary by sources and receivers, that is data is generated and recorded at every point on the boundary of a shape.
Realistically, such dense arrays of source and receiver locations are impractical, both due to the time required to collect data as well as the memory requirements for such large datasets.
Experimental design describes a field of study that seeks to maximise the information in measured data by optimising parameters, such as the number and distribution of sensors, while adhering to some constraint on the cost of the experiment [9,24]. The sensitivity of ultrasonic NDT ( E ) depends on the distribution  of source s and receivers r , the material properties  , and the spatial domain of the component  , and as shown earlier ( , , , ) . This paper has primarily focused on the optimisation of the spatial domain  , however an optimisation of the source and receiver locations  or material properties  , or a joint optimisation for ,  and  could be implemented.
To briefly illustrate this for an arbitrary component, an optimisation on the distribution of sources and receivers  is performed for a randomly generated convex polygon with ten vertices The results are shown for the same component spatial domain  in Figure 11. The iterative grid search method converges with a higher minimum sensitivity * LL E and is less computationally expensive. Such optimisations could be included in the current framework for shape optimisation, such that the shape is optimised to maximise * E , but where the minimum sensitivity * E is calculated for the optimal distribution of a practical number of sources/receiver locations for each shape considered.

Conclusion
A framework for including shape optimisation for ultrasonic non-destructive testing (NDT) in the design of a component has been presented. Ultrasonic sensitivity maps are used to assess the NDT suitability of a component using analytical expressions for the sensitivity of laser induced ultrasound for detecting internal point scatterers. The optimal design of an I-beam cross-section, obtained using a low number of degrees of freedom shape parameterisation and a genetic algorithm, was implemented, and the ultrasonic sensitivity improved by a factor of two while remaining within a design envelope. A level set method was then developed as a high number of degrees of freedom (continuum) approach for shape optimisation for NDT on a range of commonly used beam structures. Up to 20% improvement in ultrasonic sensitivity was achieved in J o u r n a l P r e -p r o o f Journal Pre-proof the case where design envelope and volume constraints were implemented. Flaw images computed using the total focusing method on data arising from the inspection of the optimised shapes were shown to be more robust to the presence of background noise, allowing for more accurate and precise characterisation of internal defects; quantitatively there was an 8 dB improvement in SNR and a five-fold improvement in the estimate of flaw size. Overall, these results show significant potential for the use of a 'Design for Testing' framework to improve NDT capabilities and facilitate efficient maintenance, life extension and remanufacturing processes, in turn minimizing production costs and material waste.

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Appendix: Genetic algorithm parameters

Data Availability
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. Individual MATLAB functions can be made available by request to the authors.

Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: -A design-for-testing framework for ultrasonic non-destructive testing suitability -A novel implementation of the level set method based on ultrasonic coverage maps -Optimised shapes lead to improved flaw detection -Optimised shapes lead to more accurate estimates of defect size