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Automatic generation of alternative build orientations for laser powder bed fusion based on facet clustering Yuchu Qin , Qunfen Qi , Peizhi Shi , Paul J. Scott & Xiangqian Jiang To cite this article: Yuchu Qin , Qunfen Qi , Peizhi Shi , Paul J. Scott & Xiangqian Jiang (2020) Automatic generation of alternative build orientations for laser powder bed fusion based on facet clustering, Virtual and Physical Prototyping, 15:3, 307-324, DOI: 10.1080/17452759.2020.1756086 To link to this article: https://doi.org/10.1080/17452759.2020.1756086

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VIRTUAL AND PHYSICAL PROTOTYPING 2020, VOL. 15, NO. 3, 307–324 https://doi.org/10.1080/17452759.2020.1756086

Automatic generation of alternative build orientations for laser powder bed fusion based on facet clustering Yuchu Qin, Qunfen Qi

, Peizhi Shi, Paul J. Scott and Xiangqian Jiang

EPSRC Future Advanced Metrology Hub, Centre for Precision Technologies, School of Computing and Engineering, University of Huddersfield, Huddersfield, UK ABSTRACT

ARTICLE HISTORY

In this paper, a novel facet clustering based method is proposed to generate alternative build orientations for laser powder bed fusion. This method consists of two steps. First, a hierarchical clustering algorithm is applied to divide facets of the design model in standard tessellation language format into different clusters, each of which shares a similar normal vector. Second, alternatives of each cluster are computed and the final set of alternative build orientations is generated by combining and refining the alternatives from all clusters. To illustrate and validate the method, a set of examples including both regular and freeform surface models are tested, and qualitative and quantitative comparisons between the method and the existing methods are reported. The results suggest that the proposed method is feasible and effective for both regular and freeform surface parts. It is evident that the proposed method can output stable results, provide satisfying efficiency, and work well with facet clusters of varying probability density.

Received 21 January 2020 Accepted 13 April 2020

1. Introduction Laser powder bed fusion (L-PBF), also known as selective laser melting or direct metal laser melting, is an additive manufacturing (AM) process that builds three-dimensional (3D) parts via stacking successive layers of material powders fused by a high power-density laser beam under computer control (Gibson et al. 2015; Chua and Leong 2017). An L-PBF system mainly comprises a powder feeding system, a powder bed on which powder material is distributed, a laser beam source that can produce and apply a laser beam to powder material, a build platform which can move downward, and a roller for spreading powder (ISO 17296-2, 2015).

To use an L-PBF system to build a 3D part, a layer of powder material is firstly spread on the build platform and a laser beam is applied to the layer. Then the build platform is lowered by layer thickness and a new layer of powder material is spread across the previous layer using a roller and the laser beam is applied to the new layer. These steps repeat until the whole object is created.

L-PBF can be applied to produce fully dense parts with high accuracy, strength, and stiffness, which make it

KEYWORDS

Laser powder bed fusion; process planning for additive manufacturing; build orientation determination; alternative build orientation; facet clustering

suitable in high-value industries such as aerospace and defence, automotive industry and medical prosthetics.

In general, the use of the L-PBF process to manufacture a product includes a set of continuous activities, where process planning is an important one (Kim et al. 2015; Qin et al. 2020). This activity refers to the systematic planning of the build orientation, support structure, slicing, process parameters, and tool-path to build a part via LPBF. It mainly includes four successive steps before building the part. Build orientation determination is the first step, which directly influences its three subsequent steps, i.e. support structure generation, 3D model slicing, and path planning (Kulkarni et al. 2000; Jiang et al. 2018, 2019a, 2019b).

In the context of L-PBF AM, the build orientation of a part is a very critical component of the process plans for building the part, as it has a direct or indirect influence on the time and cost to build the part and the accuracy, surface roughness, mechanical properties, and microstructure of the as-built part (Taufik and Jain 2013; Snyder et al. 2015; Wauthle et al. 2015; Brika et al.

2017; Calignano 2018; Salmi et al. 2018; Kuo et al. 2020; Leicht et al. 2020). Build orientation determination for

CONTACT Qunfen Qi q.qi@hud.ac.uk EPSRC Future Advanced Metrology Hub, Centre for Precision Technologies, School of Computing and Engineering, University of Huddersfield, Huddersfield, HD1 3DH, UK Supplemental data for this article can be accessed https://doi.org/10.1080/17452759.2020.1756086 © 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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L-PBF AM is the determination of a desirable build orientation of a part according to the 3D model of the part and specific production requirements and preferences on the part. It generally includes generation of alternative build orientations (ABOs) from an infinite number of theoretical orientations and selection of the optimal build orientation from the generated ABOs. In real workshops, the build orientations of most L-PBF parts are determined by process planners according to their production knowledge and experience. Different planners may possibly determine different build orientations for an identical part under the same conditions. This would have a negative effect on the qualification and production stability of the part.

To automate the process of build orientation determination for L-PBF AM, a method for automatic generation of ABOs and a method for automatic selection of the optimal build orientation are required. In the previous paper of Qin et al. (2019a), a multi-criteria decision making based method to assist selection of the optimal build orientation from a certain finite number of ABOs has been presented. This method assumes that the ABOs are given. However, the ABOs of a part are unknown in actual process planning for L-PBF AM. They need to be identified first. Motivated by this, a facet clustering based method for automatic generation of ABOs for L-PBF AM is proposed in this paper. This method generates meaningful ABOs from the input of a standard tessellation language (STL) model. It firstly uses the hierarchical clustering algorithm of McInnes and Healy (2017) to divide the facets of the STL model into different groups in such a way that facets in the same group (called a cluster) have similar normal vectors. Then the ABOs of each cluster are computed and the ABOs of the part are obtained via combining and refining the computed ABOs of all clusters.

The rest of the paper is organised as follows. An overview of related work is provided in Section 2. Section 3 explains the details of the proposed method. Section 4 presents a set of examples to verify the effectiveness of the method and made comparisons to demonstrate its advantages. A discussion is carried out in Section 5. Section 6 ends the paper with a conclusion.

2. Related work A desired method for the generation of ABOs should be a standardised method. However, the up to date international standards of AM processes ISO 17296-2 (2015) and ISO 17296-3 (2014) have not yet enclosed an effective and efficient method for ABO generation. To achieve generation of ABOs, there have been a number of methods available. These methods can be classified

into exhaustive computation methods, surface decomposition methods, and facet clustering method on the basis of their fundamental principles.

The exhaustive computation methods firstly rotate the STL model of an AM part in 3D space with a specific step size. Specific algorithms are then leveraged to compute the quality degree of the orientation corresponding to each step, and a certain number of ABOs are obtained according to the computation results. In this process, there is a contradictory problem: How to set a suitable rotation step size? If the rotation step size is set too long, the number of computations for the entire generation process can be reduced, but it is very likely that the true optimal build orientation will be missed; if it is set too short, the chance of missing the true optimal build orientation is greatly reduced, but it will greatly increase the number of computations in the entire generation process. Because of this, most of the exhaustive computation methods are difficult to apply in practical build orientation determination due to their high computational cost (Zhang et al. 2018).

Representative exhaustive computation methods are the methods presented by McClurkin and Rosen (1998), Hur and Lee (1998), Hur et al. (2001), Masood et al.

(2003), Thrimurthulu et al. (2004), Pandey et al. (2004), Kim and Lee (2005), Tyagi et al. (2007), Ahn et al.

(2007), Canellidis et al. (2009), Padhye and Deb (2011), Strano et al. (2011), Zhang and Li (2013), Paul and Anand (2015), Delfs et al. (2016), Ahsan and Khoda (2016), Zhang et al. (2017), Brika et al. (2017), Chowdhury et al. (2018), Huang et al. (2018), Jaiswal et al. (2018), Golmohammadi and Khodaygan (2019), Raju et al. (2019), Jiang et al. (2019c, 2019d), Cheng and To (2019), and Shen et al. (2020). The method of McClurkin and Rosen (1998) calculates the quality degree of the orientation corresponding to each step using response surface analysis and generates the ABOs by surface fitting. This method can be used to generate ideal ABOs for lowdimensional surfaces. But for high-dimensional surfaces, it requires a large number of benchmark data sets that are generally difficult to obtain. To avoid this issue, the methods of Hur and Lee (1998), Hur et al. (2001), Masood et al. (2003), Thrimurthulu et al. (2004), Pandey et al. (2004), Kim and Lee (2005), Tyagi et al. (2007), Ahn et al. (2007), Canellidis et al. (2009), Padhye and Deb (2011), Strano et al. (2011), Paul and Anand (2015), Delfs et al. (2016), Ahsan and Khoda (2016), Zhang et al. (2017), Brika et al. (2017), Chowdhury et al. (2018), Huang et al. (2018), Jaiswal et al. (2018), Golmohammadi and Khodaygan (2019), Raju et al. (2019), Jiang et al.

(2019c, 2019d), Cheng and To (2019), and Shen et al.

(2020) adopt specific multi-objective optimisation algorithms (e.g. genetic algorithm, bacterial foraging

VIRTUAL AND PHYSICAL PROTOTYPING

algorithm, particle swarm algorithm) to generate the ABOs. These methods improve the automation of ABO generation to a certain extent, but they have two common shortcomings. One is that they consume a lot of time in calculating meaningless orientations. This will lead to extremely low efficiency when the number of optimised objectives increases. This is why most of the current software tools for AM process planning can only optimise one or two objectives to generate the ABOs. Different from the methods, the method of Zhang and Li (2013) uses unit spheres to replace the planar triangular facets in STL models. This avoids setting the rotation step size. But the amount of computation required by optimisation algorithm increases sharply when the number of triangular facets of STL models increases. Therefore, the method is not practical for complex STL models.

The surface decomposition methods firstly decompose the STL model into a certain number of surface fragments consisting of adjoining planar triangular facets.

They then determine which predefined shape feature each surface fragment belongs to. According to the feature type of each surface fragment, the ABOs of the surface fragment are generated. The ABOs of the part are obtained via combining the ABOs of all surface fragments of its STL model. From such principle, it is not difficult to observe that the continuous surface decomposition methods focus on a finite set of build orientations. They will not consume time in the computation of the meaningless orientations like the exhaustive computation methods. However, how to decompose an STL model to construct predefined shape features is difficult due to the lack of topological information in the model. In addition, automatic recognition of shape features from a complex 3D model with multiple overlapping shape features is still a very tough issue (Shi et al.

2020). Last but not the least, the methods are not applicable for freeform surface models, because it would be impossible to define the shape features for such models.

Typical surface decomposition methods are the methods presented by Frank and Fadel (1995), Lan et al. (1997), Alexander et al. (1998), Pham et al. (1999), Xu et al. (1999), West et al. (2001), Byun and Lee (2006), and Zhang et al. (2016). The methods of Alexander et al. (1998), Xu et al. (1999), and Byun and Lee (2006) generate the ABOs of an AM part based on the convex hull of the STL model. They are simple, intuitive, and easy to implement. However, all of these methods have precision issues since the convex hull is not equivalent to the STL model. In the method of Lan et al. (1997), ABOs are generated as the orientations of reference planes of the STL model, which is simpler and more intuitive. But this method is difficult to achieve desired results

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for the STL models with surface features, because the reference planes of such STL models are usually difficult to find. To solve this issue, the methods of Frank and Fadel (1995), Pham et al. (1999), and West et al. (2001) introduce the concept of AM feature. They firstly determine the correspondence between each type of AM features and build orientations based on production experience. According to the determined correspondence, the rules for generation of ABOs are then designed. Finally, the AM features in the STL model of each AM part are recognised manually and the ABOs of the part are obtained by rule-based reasoning. These methods provide a viable idea for computer-aided build orientation planning, but they do not provide a clear definition and specific classification of AM features.

Aiming at this issue, the method of Zhang et al. (2016) presents the definition and classification of AM features.

The generation methods based on AM features then become more reliable. But the classification is rather simple (AM features are divided into cylinders, planes, cones, and structural units), and this method does not provide the solutions for automatic recognition of AM features and automatic generation of ABOs. The facet clustering method presented by Zhang et al.

(2018) applies the k-means clustering algorithm (Hartigan 1979) with the Davies-Bouldin index (Davies and Bouldin 1979) to automatically divide the STL model of an AM part into a certain finite number of clusters of meaningful discrete planar triangular facets and directly generates the ABOs of each cluster. The ABOs of the part are achieved via combining and refining the ABOs of all clusters of its STL model. Compared to the surface decomposition methods, this method does not require topological information and avoid shape feature recognition. Most importantly, it can work for both regular surface models and freeform surface models. However, it is found from actual applications that the method suffers from the following issues: (1) The ABO generation results of the method are unstable, because the initial k planar triangular facets in it need to be randomly selected and different k initial facets would obtain different clustering results. (2) The efficiency of the method is still an issue for an STL model with a large number of facets, since it will take a lot of time to determine the value of k and to calculate the angles between each normal vector cluster and the remaining normal vectors. (3) The method could generate unreasonable results with clusters of varying density, as the probability density function in it is assumed to obey k Gaussian distributions.

Motivated by the issues above, in this paper, the hierarchical clustering algorithm of McInnes and Healy (2017) is introduced to develop a novel facet clustering

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method for automatic generation of the ABOs of an LPBF part. This algorithm is an unsupervised learning algorithm to produce clusters of a dataset and is an accelerated version of the original HDBSCAN* (hierarchical density-based spatial clustering of applications with noise*) algorithm (Campello et al. 2015). It can improve the classic k-means clustering algorithms at the aspects of requiring assignment of the number of clusters first, generating unstable locally optimal clustering results, sensitive to noise, and implicit assumption that clusters have Gaussian distributions. The algorithm can also make the computational scalability of the original algorithm comparable in efficiency to mainstream clustering algorithms. Because of such characteristics, the developed method has the following advantages compared to Zhang et al. (2018)’s method: (1) The method can generate the same results at any time with guaranteed stability. (2) The method does not need to calculate the number of clusters first. All clustering results can be produced in a short time via executing the accelerated HDBSCAN* algorithm just once. (3) The method can work normally with clusters of different size and probability density.

The major contribution of the paper is proposing a novel facet clustering method for automatic generation of the ABOs of an L-PBF part. From a theoretical perspective, the contributions of the paper include: (1) Developing an effective rule to refine the generated clusters of facets; (2) Demonstrating the effectiveness of the facet clustering rule, cluster refinement rule, and ABO generation rule in benefiting the tensile strength, elongation, Vickers hardness, surface roughness, and support volume of L-PBF Ti6Al4V parts and the surface roughness and support volume of L-PBF parts theoretically; (3) Presenting a distance metric of the normal vectors of facets that can accelerate the facet clustering process; (4) Providing a feasible way to identify meaningful facet clusters from the output of a hierarchical clustering algorithm.

From a practical perspective, a practical method for automatic generation of the ABOs of an L-PBF part is developed. This method provides a framework for automatic generation of the ABOs in all AM processes. Compared to the existing exhaustive computation methods, the developed method does not need to spend time on the computation of meaningless orientations. Compared to the existing surface decomposition methods, the developed method does not require topological information, can avoid shape feature recognition, and is applicable for freeform surface parts. Compared to the existing facet clustering method, the developed method can output stable results, provide desired efficiency, and work well under unknown distributions of facet clusters.

3. Automatic generation method In this section, the proposed automatic generation method of ABOs for L-PBF AM based on facet clustering is described. Its schematic diagram is shown in Figure 1.

The method contains two key steps: clustering of facets and generation of ABOs. In the first key step, the STL model of an L-PBF part, which is represented by a finite number of planar triangular facets, is used as the input of the accelerated HDBSCAN* algorithm. A certain number of meaningful clusters of planar triangular facets are obtained via the algorithm and the k-cluster lifetime partition criterion. In the second key step, whether the resulting clusters need to be further refined is firstly determined by a process planner. If a refinement is no longer required, then the ABO of each cluster will be generated according to a specific generation rule and the ABOs of the part are obtained via combining the ABOs of all clusters. Otherwise, a refinement of clusters will be carried out according to a specific refinement rule and a smaller number of clusters will be achieved to generate the ABOs of the part.

Figure 1. The schematic diagram of the proposed automatic generation method of ABOs for L-PBF AM.

VIRTUAL AND PHYSICAL PROTOTYPING

3.1. Clustering of facets There are a number of formats available for 3D model representation in AM, such as the STL format, 3D manufacturing format, additive manufacturing file format, and Wavefront object format (Qin et al. 2019b). Among these formats, STL is the most used format and has been the actual standard 3D model format in AM. Almost all CAD (computer-aided design) systems can read and write STL files, and almost all AM machines include the support of the STL format. Without loss of generality, the proposed method takes the 3D models encoded by the STL format as input.

The STL format represents a 3D model using tessellation technique. It firstly uses standard surface triangulation algorithm to triangulate the model. Then the surface of the model is covered by a set of adjoining planar triangular facets, where each facet is described by three vertices and one normal vector. Through encoding the vertices and normal vectors of all facets, an STL file of the 3D model can be obtained. The STL format provides two ways to encode the vertices and normal vectors. One is ASCII (American standard code for information interchange) codes and the other is binary codes. An ASCII STL file is both human readable and machine readable.

It is mostly used for testing. A binary STL file is only machine readable and is mainly used for storage, as it requires less storage space than its ASCII version. To ensure completeness, the automatic generation method supports the input of both versions of STL files.

Taking an ASCII or a binary STL file as input, the proposed method leverages the accelerated HDBSCAN* algorithm in (McInnes and Healy 2017), which was developed by accelerating the original HDBSCAN* algorithm in (Campello et al. 2015), to automatically divide all planar triangular facets included in the STL file into a set of meaningful clusters. HDBSCAN* is an unsupervised learning algorithm for generating clusters of a given dataset. It starts with an assumption that there are some unknown density functions which can be used to draw the observed data objects. From a density function f defined on a metric space (X, D) (where X is a set of observed data objects and D is a distance metric), a hierarchical cluster structure in which a cluster is a connected subset of {X | X∈(X, D) and f(X ) ≤ λ} can be constructed.

This structure is called a dendrogram. Each cluster is one of its branches, which extends over the range of λ.

The goal of the HDBSCAN* algorithm is to construct a dendrogram that can suitably approximate f in a hierarchical, nested way.

For the clustering of facets, the observed data objects are all planar triangular facets of an input STL model. The distance metric for facets can be customised according

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to specific AM processes and production requirements.

According to the studies of Wauthle et al. (2015), Brika et al. (2017), Cheng and To (2019), Yu et al. (2019a), Griffiths et al. (2019), Ning et al. (2020a), Kuo et al.

(2020), and Leicht et al. (2020), build orientation may affect the tensile strength, elongation, Vickers hardness, surface roughness, support volume, build time, build cost, residual stress, distortion, and microstructure of a part built by L-PBF AM process. The distance metric can be selected to benefit some of these indicators.

From the study of Zhang et al. (2018), any two planar triangular facets that have similar normal vectors (The smaller the angle between the two vectors, the more similar they are) would have similar production results if they are built in the same orientation. However, the study did not explain which production result indicators can benefit from this situation. According to the following theoretical analysis, it is found that the situation can in theory benefit tensile strength, elongation, Vickers hardness, surface roughness, and support volume for L-PBF Ti6Al4V parts and benefit surface roughness and support volume for L-PBF parts: (1) The situation can improve the average tensile strength, average elongation, and average Vickers hardness of an L-PBF Ti6Al4V part theoretically. The average tensile strength, average elongation, and average Vickers hardness of an L-PBF Ti6Al4V part can be predicted according to the post-processing heat treatments on the part and the build orientation of the part (Wauthle et al. 2015). In the study of Brika et al. (2017), the mathematical relationships between the post-processing heat treatments annealing treatment (AT), hot isostatic pressing (HIP), and stress relief (SR) on Ti6Al4V parts built by L-PBF AM process and the build orientation of the part are established as: ATp =

n

[(27.40 − 0.0356ui − 0.00168(ui − 45)2 )

i=1

(WRp /100) + WRp ]/

n

Ai

(1)

i=1

HIPp =

n

[(15.32 − 0.0378ui − 0.00085(ui − 45)2 )

i=1

(WRp /100) + WRp ]/

n

Ai

(2)

i=1

SRp =

n

[(43.07 − 0.0556ui − 0.00102(ui − 45)2 )

i=1 n

Ai (WRp /100) + WRp ]/ i=1

(3)

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where p = 1, 2, 3 respectively stand for tensile strength, elongation, and Vickers hardness, θi∈[0°, 90°] is the angle between the normal vector of the ith facet and the build orientation, WRp is the wrought reference of the p-th mechanical property, and Ai is the area of the i-th facet. Based on these mathematical relationships, the average tensile strength (TS), average elongation (E), and average Vickers hardness (VH) of a Ti6Al4V part built by LPBF AM process are defined by three functions TS = f1(AT1, HIP1, SR1), E = f2(AT2, HIP2, SR2), and VH = f3(AT3, HIP3, SR3), in which the predicted value of each property has positive correlation with the AT, HIP, and SR values of this property. If facets with similar normal vectors are grouped into the same cluster and the unitised central vector of all normal vectors in a cluster or its opposite vector is selected as the build orientation, then θi with respect to this build orientation will be smaller than that of other build orientations. Since Ai remains the same for the same STL model and WRp remains unchanged for the same mechanical property, the values of ATp, HIPp, SRp with respect to the selected build orientation will be greater than that of other build orientations according to Equations (1)–(3). The values of TS, E, and VH with respect to the build orientation will also be greater than that of other build orientations as each of these values has positive correlation with its AT, HIP, and SR values.

(2) The situation can reduce the average surface roughness and total support volume of an L-PBF part theoretically. In general, the average surface roughness of an AM part can be predicted via geometric analysis of the STL model of the part. The following is a prediction model for the average surface roughness of an L-PBF part (Brika et al. 2017): n

Rai Ai Ra = i=1n

Ai

(4)

i=1

where n is the number of all planar triangular facets of the STL model of a part, Ai is the area of the i-th facet, and Rai is the surface roughness of the i-th facet which can be calculated by Rai = 9.4148 + 0.0389ui

(5)

where θi∈[0°, 90°] is the angle between the normal vector of the i-th facet and the build orientation. If facets with similar normal vectors are grouped into the same cluster and the unitised central vector of all normal vectors in a cluster or its opposite vector is selected as the build orientation, then θi with

respect to this build orientation will be smaller than that of other build orientations. Since Ai remains the same for the same STL model, the value of Ra with respect to the selected build orientation will be smaller than that of other build orientations according to Equations (4) and (5). Like the average surface roughness, the total support volume of an AM part can also be predicted via geometric analysis of the STL model of the part. The following is a prediction model for the total support volume of AM parts (Qie et al. 2018):

m

1 2 V= D + zmax ,i − zmin ,i Ai cos ui 3 3 i=1

(6)

where m is the number of all planar triangular facets of all overhangs of the STL model of a part, D is the distance between the build platform and the part, zmax,i and zmin,i are respectively the maximum and minimum z coordinates of the i-th facet, Ai is the area of the i-th facet, and θi∈[0°, 90°] is the angle between the normal vector of the i-th facet and the build orientation. If facets with similar normal vectors are grouped into the same cluster and the unitised central vector of all normal vectors in a cluster or its opposite vector is selected as the build orientation, then the overhangs and their total area with respect to this build orientation will be smaller than that of other build orientations. It can be concluded from Equation (6) that the value of V with respect to the selected build orientation will be smaller than that of other build orientations.

(3) The relationships between the build time, build cost, residual stress, distortion, and microstructure of a part built by L-PBF AM process and the build orientation of this part are more complex, since each of these indicators are synthetically affected by many interactional factors (Du et al. 2018; Li et al. 2018; Yi et al. 2018, 2019a, 2019b; Griffiths et al. 2019; Wei et al. 2019; Yu et al. 2019b; Han and Jiao 2019; Ning et al. 2019a, 2019b, 2019c, 2020a, 2020b; Qi et al. 2020). It is difficult to conclude whether the situation can directly benefit some of the indicators.

Nevertheless, all production result indicators can be optimised concurrently in the determination of the optimal build orientation from the ABOs. For details regarding the optimisation process, please refer to Qin et al. (2019a).

Based on the analysis above, the observed data objects are the normal vectors of all facets of an input STL model and the distance metric is a metric of angle between the normal vectors of facets in 3D space. This

VIRTUAL AND PHYSICAL PROTOTYPING

metric is theoretically helpful to improve the average tensile strength, average elongation, average Vickers hardness of the as-built L-PBF Ti6Al4V parts and to reduce the average surface roughness and total support volume of the as-built L-PBF parts. Formally, suppose an STL model consists of n planar triangular facets and v0 = (x0, y0, z0), v1 = (x1, y1, z1), … , vn−1 = (xn−1, yn−1, zn−1) are their normal vectors. Then the observed data objects are v0, v1, … , vn−1 and the distance metric is a metric of angle between vi = (xi, yi, zi) and vj = (xj, yj, zj) (i, j = 0, 1, … , n−1 and i ≠ j): xi xj + yi yj + zi zj A(vi , vj ) = arccos xi2 + yi2 + zi2 xj2 + yj2 + zj2

(7)

In actual experiments, it is found that using this metric is computationally expensive and fails to produce clustering results in an acceptable time when the number of facets is more than 100,000, while adopting a metric of Euclidean distance between (xi, yi, zi) and (xj, yj, zj) can greatly improve the efficiency:

D(vi , vj ) = (xi − xj )2 + (yi − yj )2 + (zi − zj )2 (8) The reason is that the amount of computation required for Equation (7) is larger than the amount required for Equation (8). To ensure the efficiency of the automatic generation method, D(vi, vj) is used as the distance metric of the accelerated HDBSCAN* algorithm. This is also effective for the generation of ABOs, because D(vi, vj) and A(vi, vj) have the same monotonicity.

To be more specific, they have the following relationship (Please note that both vi and vj are unit vectors): A(vi , vj ) = arccos

2 − D2 (vi , vj ) 2

(9)

Such relationship is depicted in Figure 2. A(vi, vj) increases as D(vi, vj) increases. Based on this, the similarity

Figure 2. The relationship between the angle of two unit vectors and the Euclidean distance of their ends.

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between two normal vectors can also be measured by the Euclidean distance between their ends.

To suitably approximate the unknown density function of v0, v1, … , vn−1 in a hierarchical, nested way, the HDBSCAN* algorithm needs to perform the following six steps (McInnes and Healy 2017): (1) Compute the core distances of v0, v1, … , vn−1 with respect to m (The minimum number of normal vectors in a neighbourhood for a normal vector to be considered a core normal vector for clustering of facets is one, thus m is assigned 1); (2) Compute the mutual reachability distance of each pair of vi and vj in V = {v0, v1, …, vn−1} with respect to m; (3) Construct a mutual reachability graph of V; (4) Produce a minimum spanning tree of the constructed graph; (5) Extend the minimum spanning tree via adding a self edge for each vertex. The weight of the added edge is set as the core distance of the corresponding normal vector; (6) Construct a dendrogram according to the extended minimum spanning tree.

As can be obtained from (McInnes and Healy 2017), the HDBSCAN* algorithm on n planar triangular facets (i.e. n normal vectors) has O(n 2) run-time. This complexity is a bit high for the clustering of facets, because the number of facets for complex STL models, especially freeform surface STL models, is usually very large (e.g.

500,000, 1,000,000). To improve the efficiency of the HDBSCAN* algorithm, space tree algorithms (Ram et al.

2009) and Borůvka’s algorithm (Nešetřil et al. 2001) were respectively leveraged to accelerate the calculation of distances and the production of a minimum spanning tree in (Campello et al. 2015). Such accelerations derived an accelerated HDBSCAN* algorithm, which reduces the time complexity to O(nlogn). To this end, the automatic generation method also uses these algorithms to improve the efficiency of the clustering of facets.

The output of the accelerated HDBSCAN* algorithm for a specific STL model is a dendrogram that depicts the clustering process of the normal vectors of all its planar triangular facets. On the basis of this dendrogram, the proposed method uses the k-cluster lifetime partition criterion (Fred and Jain 2005) to identify k meaningful clusters of facets. The identification process includes the following steps: (1) Convert the metric of the ordinate of the dendrogram from Euclidean distance to angle using Equation (9) to construct a new dendrogram whose abscissa denotes normal vectors and ordinate denotes angle. Although

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Figure 3. The 3D model of a prism which includes 8 planar triangular facets.

the Euclidean distance between two normal vectors and the angle between them have the same monotonicity, their relationship is not linear, which means that the dendrogram constructed by the accelerated HDBSCAN* algorithm cannot be directly used in later computation. Before the computation, Euclidean distance is converted to angle using Equation (9). As an example, a dendrogram constructed by the algorithm for a prism in Figure 3 is shown in Figure 4(a). It is reconstructed in Figure 4(b) via converting Euclidean distance to angle.

(2) Compute k-cluster lifetimes for k = 1, 2, … , n. Lifetime is originally defined as the distance between that a cluster is created and that it merges with other clusters during clustering. For instance, the kcluster lifetimes (k = 1, 2, … , 8) for the dendrogram in Figure 4(b) are computed as follows: 1-cluster lifetime = 00.00°−00.00° = 00.00° 2-cluster lifetime = 00.00°−00.00° = 00.00°

3-cluster lifetime = 00.00°−00.00° = 00.00° 4-cluster lifetime = 90.00°−76.18° = 13.82° 5-cluster lifetime = 76.18°−00.00° = 76.18° 6-cluster lifetime = 00.00°−00.00° = 00.00° 7-cluster lifetime = 00.00°−00.00° = 00.00° 8-cluster lifetime = 00.00°−00.00° = 00.00° (3) Find the largest lifetime from all k-cluster lifetimes and output its corresponding clusters. For example, the largest lifetime for the dendrogram in Figure 4 (b) is 5-cluster lifetime. The 5 clusters are C1 = {v5, v2}, C2 = {v6}, C3 = {v3}, C4 = {v1, v0}, and C5 = {v7, v4}.

If the planar triangular facets in the same cluster are painted the same colour, then the clustering result depicted in Figure 5 will be obtained.

3.2. Generation of ABOs The generation of ABOs starts from judging whether the identified clusters needs further refinement. This

Figure 4. The constructed dendrogram for the prism in Figure 3 and its converted dendrogram.

Figure 5. The clustering result of the 8 planar triangular facets of the prism in Figure 3.

VIRTUAL AND PHYSICAL PROTOTYPING

is because the number of the produced clusters is usually very large for the STL models with complex shapes or freeform surfaces and a large number of clusters will bring a lot of meaningless computations to the subsequent optimal build orientation determination. The basis for the judgment is the number of the output clusters and the type of the input STL model. In general, the number of the output clusters should be within 20 if the input STL model is a regular (non-freeform) surface model and should be within 200 if it is a freeform surface model. A further refinement is required if such conditions are not satisfied. In some cases, such judgment can also be carried out according to the production experience of AM process planners. For example, the output 5 clusters in Figure 5 no longer need further refinement, because the STL model of the prism is obviously a regular surface model and the number of the output clusters is 5. However, a further refinement is required for the output 95 clusters in Figure 6, as the model is also a regular surface model but the number has reached 95.

The produced facet clusters will be used directly to generate meaningful ABOs if a further refinement is not required for them. To generate meaningful ABOs for each facet cluster, a computational method or a rule-based method can be adopted, which is similar to the shape features based methods in (Frank and Fadel 1995; Pham et al. 1999; West et al. 2001; Zhang et al.

2016). The proposed method uses a central normal vector based generation rule defined in the facet clustering method of Zhang et al. (2018): Rule 1 (Generation rule of cluster ABOs). The unitised central vector of all normal vectors in a cluster and its opposite vector directly serve as the ABOs of this cluster.

Using this rule, the ABOs of each cluster of facets of the STL model of a part can be computed. The ABOs of the part are obtained via combining the computed ABOs of all clusters and removing duplicate ABOs among them.

Formally, let Ci = {vi,0, vi,1, … , vi,ni−1} (where vi,0 = (xi,0, yi,0, zi,0), vi,1 = (xi,1, yi,1, zi,1), … , vi,ni−1 = (xi,ni−1, yi,ni−1, zi,ni−1)) be an arbitrary cluster of normal vectors of facets. The unitised central vector of all normal vectors in this

cluster is as follow:

n i −1 j=0

xi,j ,

n i −1

yi,j ,

j=0

n i −1

zi,j

j=0

vc,i =

2

2

2

n n n i −1 i −1 i −1

xi,j + yi,j + zi,j j=0

j=0

315

(10)

j=0

The opposite vector of this unit vector is −vc,i. The vectors ± vc,i are taken as the ABOs of the cluster Ci based on Rule 1. Suppose the facets of the STL model of a part are partitioned into N clusters C1, C2, … , CN and ± vc,1, ±vc,2, … , ±vc,N are respectively the ABOs of the N clusters, the ABOs of the part are generated as ± vc,1, ±vc,2, … , ±vc,N, in which the same vectors are removed. For instance, the ABOs of the 5 clusters C1 = {v5, v2}, C2 = {v6}, C3 = {v3}, C4 = {v1, v0}, and C5 = {v7, v4} depicted in Figure 5 are respectively as follows: +vc,1 = (0.0000, −1.0000, 0.0000) −vc,1 = (0.0000, 1.0000, 0.0000) +vc,2 = (0.0000, 0.0000, 1.0000) −vc,2 = (0.0000, 0.0000, −1.0000) +vc,3 = (0.0000, 0.0000, −1.0000) −vc,3 = (0.0000, 0.0000, 1.0000) +vc,4 = (−0.5652, 0.8250, 0.0000) −vc,4 = (0.5652, −0.8250, 0.0000) +vc,5 = (0.6661, 0.7459, 0.0000) −vc,5 = (−0.6661, −0.7459, 0.0000) The vectors + vc,3 and −vc,3 are removed as they are respectively the same as −vc,2 and + vc,2. Thus the ABOs of the prism in Figure 3 are + vc,1, −vc,1, +vc,2, −vc,2, +vc,4, −vc,4, +vc,5, and −vc,5. The schematic diagram of these ABOs is shown in Figure 7.

The produced facet clusters will be refined to screen out a smaller number of facet clusters to generate meaningful ABOs if a further refinement is needed. To carry out the refinement, customised rules can be applied. A refinement rule aiming to select a certain number of facet clusters with relatively large areas is naturally thought of, the reason has been explained in the theoretical analysis in Section 3.1. The details of the refinement rule are as follow:

Figure 6. The clustering result of the 276 planar triangular facets of an STL model.

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Figure 7. The schematic diagram of the 8 generated ABOs of the prism in Figure 3.

Rule 2 (ReďŹ nement rule of facet clusters). The top 20 (200) facet clusters in area are used to generate meaningful ABOs if the input STL model is a regular (freeform) surface model.

Using this rule, 20 or 200 facet clusters are screened out, which can then be used to generate ABOs according to Rule 1. Please note that the numbers 20 and 200 in the rule are two experimental numbers. They are respectively obtained via conducting build orientation determination experiments on 30 dierent regular surface models and 30 dierent freeform surface models using the proposed method and the optimal build orientation determination method in Qin et al. (2019a). The smaller these two numbers, the less computation amount required in determination of the optimal build orientation. In practical use of the proposed method, users can directly use the default values of the two numbers or can adjust their values according to the actual situation.

Formally, let Ci = {vi,0, vi,1, ‌ , vi,ni−1} be an arbitrary cluster of normal vectors of facets and Δi,0 = (Vi,0,1, Vi,0,2, Vi,0,3), Δi,1 = (Vi,1,1, Vi,1,2, Vi,1,3), ‌ , Δi,ni−1 = (Vi,ni−1,1, Vi,ni −1,2, Vi,ni−1,3) be respectively the planar triangles corresponding to vi,0, vi,1, ‌ , vi,ni−1. Then the area of Ci can be computed using the following equation:

n i −1 pi,j ( pi,j − D(Vi,j,1 , Vi,j,2 ))( pi,j − D(Vi,j,1 , Vi,j,3 )) A(Ci ) = ( pi,j − D(Vi,j,2 , Vi,j,3 )) j=0

(11)

where D(Vi,j,1, Vi,j,2), D(Vi,j,1, Vi,j,3), D(Vi,j,2, Vi,j,3) are respectively the lengths of the three edges of the planar triangle Δi,j which can be computed using Equation (8), and 1 pi,j = (D(Vi,j,1 , Vi,j,2 ) + D(Vi,j,1 , Vi,j,3 ) + D(Vi,j,2 , Vi,j,3 )) (12) 2 Assume the facets of the STL model of a part are divided into N clusters C1, C2, ‌ , CN. Then the area of Ci (i = 1, 2, ‌ , N ) is calculated using Equation (11) and the top 20 (200) clusters in area are screened out. The ABOs of the screened out clusters are calculated according to Rule 1. The ABOs of the part are obtained via gathering the calculated ABOs of all screened out clusters and deleting duplicate ABOs among them. For example, the output 95 clusters in Figure 6 are reďŹ ned via Rule 2 and the top 20 clusters in area, as depicted in Figure 8, are screened out.

According to Rule 1, 36 ABOs are generated as the ABOs of the corresponding part. Figure 9 provides the schematic diagram of 8 of the 36 ABOs that are generated by the top 6 clusters in area.

4. Examples and comparisons In this section, a set of examples are ďŹ rstly used to verify the eectiveness of the proposed automatic generation method. Then qualitative and quantitative comparisons to the existing ABO generation methods are

Figure 8. The screened out top 20 clusters in area of the STL model in Figure 6. Note: The remaining facets are displayed in white.

VIRTUAL AND PHYSICAL PROTOTYPING

317

Figure 9. The schematic diagram of 8 ABOs generated by the top 6 clusters in area.

carried out to demonstrate the advantages of the method.

4.1. Examples Nine parts to be built using L-PBF, whose STL models are shown in Figure S1, are taken as examples to validate the proposed method. These parts can be divided into three groups on the basis of the complexity of their shape. The first group consists of Parts 1−3, which respectively correspond to the three basic AM shape features plane, cone, and cylinder defined by Zhang et al. (2016).

These three parts have also been used to demonstrate the facet clustering method presented by Zhang et al.

(2018). Parts 4−6, which respectively contain a certain number of overlapping shape features, form the second group. Part 4 is a classic example for validating a build orientation determination method. It was developed by Cheng et al. (1995) and has been leveraged to

illustrate a number of build orientation determination methods, which include the methods of Pham et al.

(1999), Pandey et al. (2004), Byun and Lee (2006), Canellidis et al. (2009), Zhang et al. (2016), and Qin et al.

(2019a). Part 5 was developed by Zhang et al. (2016) and has been leveraged to demonstrate the optimal build orientation selection method of Qin et al. (2019a).

Part 6 is the well-known NIST AM test artifact developed

by Moylan et al. (2012). The third group consists of three freeform surface parts, i.e. Parts 7−9, where Part 9 has also been used to validate the facet clustering method of Zhang et al. (2018).

Taking the STL model of each part as input, the proposed method can automatically generate meaningful ABOs for each. Some details in the generation process of the ABOs of the 9 parts are listed in Table 1. The generated ABOs of the 9 parts are listed in Table S1. As can be seen from Table 1, the number of the produced facet clusters for each of Parts 7−9 is relatively large, as the

Table 1. Some details in the generation process of the ABOs of the 9 parts in Figure S1.

Part Part 1 Part 2 Part 3 Part 4 Part 5 Part 6 Part 7 Part 8 Part 9

STL model type Regular surface Regular surface Regular surface Regular surface Regular surface Regular surface Freeform surface Freeform surface Freeform surface

STL file version

Number of facets

Clustering result

Number of clusters

Refinement required

Refinement result

Number of ABOs

Diagram of ABOs

ASCII

8

Figure 5

5

No

–

8

Figure 7

ASCII

110

Figure S2

2

No

–

2

Figure S2

ASCII

232

Figure S3

3

No

–

4

Figure S3

ASCII

276

Figure 6

95

Yes

Figure 8

36

Figure 9

Binary

2,426

Figure S4

86

Yes

Figure S4

34

Figure S5

Binary

7,392

Figure S6

86

Yes

Figure S6

34

Figure S7

Binary

13,240

Figure S8

6670

Yes

Figure S8

400

Figure S9

Binary

91,216

Figure S10

55,207

Yes

Figure S10

400

Figure S11

Binary

270,021

Figure S12

199,124

Yes

Figure S12

400

Figure S13

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three parts are freeform surface parts. This number is reduced to 200 after a refinement. According to the refinement rule of facet clusters, the obtained 200 facet clusters for each part (see Figures S8, S10 and S12) are the top 200 facet clusters in area. On the basis of these facet clusters and the generation rule of cluster ABOs, 400 meaningful ABOs of each part are generated. A few examples of the generated ABOs for the three parts are respectively provided in Figures S9, S11 and S13.

In Section 3, the effectiveness of the ABOs generated by the proposed method has been demonstrated theoretically. Such effectiveness can also be verified by practical examples. For instance, through comparing the generated ABOs of Parts 1−3 in Figures 7, S2 and S3 and the designed or generated ABOs of these three parts in (Zhang et al. 2016, 2018), it is found that they are exactly the same. The generated ABOs of Parts 4 and 5 in Figures 9 and S5 include all of the generated ABOs of these two parts in (Zhang et al. 2016). The generated ABOs of Part 6 in Figure S7 involve the optimal build orientation (It is not difficult to determine that O1 in Figure S7 is the optimal build orientation as the part has a base plane) of this part. All of these facts indicate that the proposed method is feasible and effective. It can also be seen that the method can avoid tough shape feature recognition, which is an essential step in (Zhang et al. 2016). As the reference ABOs of Parts 7−9 have not been found from the existing literature, it is impossible to make similar comparisons. Even so, the generated ABOs of these three parts can be considered as effective ABOs according to the theoretical analysis in Section 3, the clustering rule in the proposed method, and the reliability of the accelerated HDBSCAN* algorithm.

4.2. Comparisons 4.2.1. Qualitative comparison A qualitative comparison of different ABO generation methods can be made via comparing their characteristics. Section 2 classified the existing ABO generation methods into exhaustive computation methods (McClurkin and Rosen 1998; Hur and Lee 1998; Hur et al. 2001; Masood et al. 2003; Thrimurthulu et al. 2004; Pandey et al. 2004; Kim and Lee 2005; Tyagi et al. 2007; Ahn et al. 2007; Canellidis et al. 2009; Padhye and Deb 2011; Strano et al. 2011; Zhang and Li 2013; Paul and Anand 2015; Delfs et al. 2016; Ahsan and Khoda 2016; Zhang et al. 2017; Brika et al. 2017; Chowdhury et al. 2018; Huang et al. 2018; Jaiswal et al. 2018; Golmohammadi and Khodaygan 2019; Raju et al. 2019; Jiang et al., 2019c, 2019d; Cheng and To 2019; Shen et al. 2020), surface decomposition methods (Frank and Fadel 1995;

Lan et al. 1997; Alexander et al. 1998; Pham et al. 1999; Xu et al. 1999; West et al. 2001; Byun and Lee 2006; Zhang et al. 2016), and facet clustering method (Zhang et al. 2018) and described their main characteristics.

Briefly, the exhaustive computation methods can be applied to both regular surface parts and freeform surface parts, but they usually require a large amount of computation to obtain desired results. The surface decomposition methods can greatly reduce computation amount, but they may encounter tough overlapping shape feature recognition issue and are not applicable for freeform surface parts. The facet clustering method can overcome the limitations of the surface decomposition methods and maintain the advantage of the exhaustive computation methods, but it suffers from the issues of unstable results, not very high efficiency, and working well under specific distribution.

Compared to the exhaustive computation methods, the proposed method does not need to spend time on the computation of meaningless orientations. Compared to the surface decomposition methods, the proposed method is applicable for freeform surface parts and does not require shape feature recognition. Compared to the facet clustering method presented by Zhang et al. (2018), the proposed method mainly has three advantages: (1) output stable results; (2) provide higher efficiency; (3) work well under unknown distributions.

The first and third advantages are respectively illustrated via the following two experiments. The second advantage will be demonstrated by quantitative comparison in next subsection.

Experiment 1. This experiment aims to illustrate the first advantage. In the experiment, Zhang et al. (2018)’s method was reimplemented (The reimplemented method will also be used in all of the following comparison experiments) and the STL models of Parts 1−3 were respectively taken as the input of this method and the proposed method. Each method was executed 10 times for the same STL model. The clustering results of Zhang et al.’s method are shown in Figure S14. The clustering results of the proposed method have been shown in Figures 5, S3 and S4 (Please note that the clustering results obtained in each execution are the same for the same STL model). As can be seen from Figure S14, the clustering results of Zhang et al.’s method vary in different executions and some of them are unreasonable results. This will result in different ABOs generated in different executions. That is, the results of Zhang et al.’s method are not stable. Unlike Zhang et al.’s method, the proposed method will generate the same clusters and ABOs no matter how many times it is executed.

Thus the stability of the proposed method can be proved.

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319

Figure 10. The clustering results of the two comparison methods in an execution. Note: k = 5, 2 for Part 1, 2, respectively.

Experiment 2. The purpose of this experiment is to illustrate the third advantage. In the experiment, the STL models of Parts 1 and 2 were respectively taken as the input of Zhang et al. (2018)’s method and the proposed method. The result of the experiment is shown in Figure 10. It can be seen from the figure that Zhang et al.’s method could generate unreasonable results due to its random initialisation and distribution assumption (Please note that Gaussian distribution assumption will make the ends of all normal vectors in each cluster roughly in a sphere), while the proposed method can always generated reasonable results because there is no random initialisation and distribution assumption.

4.2.2. Quantitative comparison Theoretically, the time complexity of the k-means clustering algorithm is O(n 2) (where n is the number of planar triangular facets of the input STL model), and the time complexity of the accelerated HDBSCAN* algorithm is O(nlogn) (McInnes and Healy 2017). It can be inferred that the efficiency of the proposed method should be higher than that of Zhang et al. (2018)’s method. To verify this inference, a quantitative comparison of the

execution time of the two methods was made. In this comparison, two experiments were carried out on a machine with Intel(R) Core(TM) i9-7900X CPU and 8 GB RAM. As the difference between the two methods is in facet clustering and both methods adopt the same rule to generated ABOs from clusters, only the time required in producing facet clusters is considered in the experiments for the sake of simplicity. The details of the experiments are as follows: Experiment 3. This experiment was conducted to compare the facet clustering time needed in Zhang et al. (2018)’s method and the proposed method with respect to the number of facets. In the experiment, suppose the minimum Davies Bouldin value is always obtained from k = 2 to k = 40 (This value is deduced from all application examples in (Zhang et al. 2018)) for Zhang et al.’s method and a refinement is always required for the proposed method. The time spent on facet clustering of the two methods from n = 1 to n = 1,000 is shown in Figure 11(a). It is obvious that the efficiency of the proposed method is higher than that of Zhang et al.’s method. The main reasons are as follows: (1) The time complexity of the proposed

Figure 11. The time spent on facet clustering of Zhang et al. (2018)’s method and the proposed method with respect to facet number.

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facets of the two methods, which is listed in Table 2.

As can be seen from the table, the proposed method spends significantly less time than Zhang et al.’s method spends for the STL model of every part. This indicates that the proposed method is more efficient than Zhang et al.’s method.

5. Discussion

Figure 12. The time spent on facet clustering of the proposed method with respect to facet number.

method (O(nlogn)) is lower than that of Zhang et al.’s method (O(n 2)); (2) The computation of Davies Bouldin values to find a suitable k in Zhang et al.’s method is more time consuming than the refinement manipulation in the proposed method. (3) The computation of angles of normal vectors in Zhang et al.’s method also takes a lot of time. It is converted to the more efficient computation of Euclidean distances in the proposed method. As the first two points cannot be changed, it is further assumed that the calculation of angles is transformed to the calculation of Euclidean distances in Zhang et al.’s method. The comparison result under such assumption is shown in Figure 11(b). It is found that the proposed method still outperforms Zhang et al.’s method. To further test the efficiency of the proposed method, n was assigned from 1 to 1,000,000 and the facet clustering time is depicted in Figure 12. As can been seen from the figure, the proposed method can provide acceptable efficiency within n = 1,000,000.

Experiment 4. This experiment was carried out to compare the facet clustering time required in the two methods with respect to specific STL models. In the experiment, the STL models of Parts 1−9 in Figure S1 were respectively taken as the input of Zhang et al.

(2018)’s method and the proposed method, which were respectively executed 10 times. The values of k in Zhang et al.’s method are respectively assigned 5, 2, 3, 20, 20, 20, 200, 200, 200 (These values are obtained from Table 1) for Parts 1−9 to make the output conditions of the two methods the same. The result of the experiment is the average time spent on clustering of

In actual process planning for L-PBF AM, build orientations and other process plans (e.g. support structure, slices, and scan pattern) and parameters (e.g. laser power, scan speed, and hatch space) should be synthetically designed via optimising certain part quality indicators under specific L-PBF AM machine and powder materials. This is because the influence of all these process plans and parameters on part quality is comprehensive and sometimes contradictory and the process plans and parameters usually affect each other. As one example, users always care more about the tensile strength, elongation, Vickers hardness, residual stress, distortion, and microstructure for an L-PBF part. According to the studies of Wauthle et al. (2015), Du et al. (2018), Li et al. (2018), Yi et al. (2018, 2019a, 2019b), Cheng and To (2019), Wei et al. (2019), Yu et al. (2019b), Ning et al.

(2019c, 2020a, 2020b), Qi et al. (2020), Kuo et al. (2020), and Leicht et al. (2020), all of these indicators may be affected by build orientation and may also be influenced by other process plans and parameters. Generation of a certain number of ABOs that can benefit some of the indicators and determination of the optimal build orientation from the generated ABOs are not sufficient, as build orientation could also has an effect on other process plans or parameters, such as support volume, layer thickness, scan pattern, and hatch distance. An ideal approach is to comprehensively design all process plans and parameters via weighing certain part quality indicators, which should be a major function that a systematic computer-aided L-PBF AM process planning software tool needs to implement.

Nevertheless, this does not mean that the proposed method is meaningless. Process planning for L-PBF AM is a very complicated activity. Researchers generally divide the activities into a set of single issues, such as build orientation determination, support structure generation, 3D model slicing, and path planning, and

Table 2. The average time spent on facet clustering of Zhang et al. (2018)’s method and the proposed method with respect to different STL models.

Facet clustering method

Part 1

Part 2

Part 3

Part 4

Part 5

Part 6

Part 7

Part 8

Part 9

Number of facets (n) Zhang et al.’s method The proposed method

8 0.0054s 0.0018s

110 0.0213s 0.0080s

232 0.0614s 0.0158s

276 1.1501s 0.0188s

2,426 10.7656s 0.1418s

7,392 26.6823s 0.4761s

13,240 919.1475s 0.8292s

91,216 6,320.0414s 6.8392s

270,021 20,514.5010s 33.9499s

VIRTUAL AND PHYSICAL PROTOTYPING

address each issue separately. From the perspective of scientific research, separate resolution of all small issues would trigger the development of a systematic approach.

Although the proposed method is developed mainly for L-PBF process, it can be directly used in other AM processes if the objective is to benefit surface roughness or support volume (if required). This is because the prediction models of average surface roughness and total support volume in Section 3 are applicable for all AM processes theoretically. The method can also be used to achieve a specific objective in certain AM process via altering the facet clustering rule, cluster refinement rule, and ABO generation rule based on the objective.

321

determination in L-PBF AM. The developed software tool would be further extended to include support structure generation, 3D model slicing, and path planning to achieve a systematic software tool for computer-aided LPBF AM process planning.

Acknowledgements The authors are very grateful to the six anonymous reviewers for their insightful comments for the improvement of the paper. The authors are also very grateful to Dr. Yicha Zhang at the Department of Mechanical Engineering and Design, University of Technology of Belfort-Montbéliard, France for providing the STL files of Parts 4 and 5.

Disclosure statement 6. Conclusion In this paper, a method for automatic generation of ABOs for L-PBF AM based on facet clustering is proposed. This method mainly includes clustering of facets and generation of ABOs. The clustering of facets takes the STL model of an L-PBF part as input and outputs a certain number of meaningful facet clusters. In the generation of ABOs, whether the output facet clusters need to be refined is firstly determined.

If a refinement is not required, the ABOs of the part will be generated directly using a generation rule.

Otherwise, the produced facet clusters will be refined by a refinement rule and the ABOs of the part will be generated according to the refinement result and the generation rule. The paper also reports the validation and comparisons of the proposed method.

The results of the validation and comparisons suggest that the method is effective, stable, and efficient and can work well with facet clusters of varying probability density.

Future work will aim especially at extending the proposed method at the aspect of considering different types of powder materials. The facet clustering rule, cluster refinement rule, and ABO generation rule used in the proposed method are developed via a theoretical analysis of L-PBF Ti6Al4V parts. They may not be used directly on parts made of other types of powder materials. To address this issue, new rules for each type of powder material are needed to be developed. In addition, as the two tasks in build orientation determination for L-PBF AM, the generation of ABOs and the selection of the optimal build orientation, have been respectively addressed in the present paper and the paper (Qin et al. 2019a), future work will focus on combining the proposed methods in the two papers to develop a software tool to assist build orientation

No potential conflict of interest was reported by the author(s).

Data availability statement The related data used in the present paper and implementation code of the two methods in the quantitative comparison have been deposited in the GitHub repository (https://github.com/ YuchuChingQin/GenerationOfABOs).

Notes on contributors Yuchu Qin is currently a PhD candidate and a part-time research fellow at the EPSRC Future Advanced Metrology Hub, University of Huddersfield, UK. Qunfen Qi is currently a senior research fellow at the EPSRC Future Advanced Metrology Hub, University of Huddersfield, UK. Peizhi Shi is currently a research fellow at the EPSRC Future Advanced Metrology Hub, University of Huddersfield, UK. Paul J. Scott is currently a professor at the EPSRC Future Advanced Metrology Hub of the University of Huddersfield.

Xiangqian Jiang is currently the chair professor and the director of the EPSRC Future Advanced Metrology Hub, University of Huddersfield and the Royal Academy of Engineering and Renishaw Chair in Precision Metrology.

ORCID Qunfen Qi

http://orcid.org/0000-0001-5936-1714

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∗ ∗

A framework for knowledge in additive A categorical categorical for formalising formalising 28thframework CIRP Design Conference, May 2018,knowledge Nantes, France in additive manufacturing manufacturing a,* A new methodology to analyze thea,functional physical Qunfen Qia,* , Luca Pagani Paul J Scotta ,and Xiangqian Jiangaarchitecture of Qunfen Qi , Luca Pagania , Paul J Scotta , Xiangqian Jianga EPSRC Future Advanced Metrology Hub, School of Computing and Engineering, University of Huddersfield, Huddersfield, HD1 3DH,UK existing products for an assembly oriented product family identification EPSRC Future Advanced Metrology Hub, School of Computing and Engineering, University of Huddersfield, Huddersfield, HD1 3DH,UK a a

Corresponding author. Tel.: +44-1484-471284; fax: +44-1484-472161. E-mail address: q.qi@hud.ac.uk Corresponding author. Tel.: +44-1484-471284; fax: +44-1484-472161. E-mail address: q.qi@hud.ac.uk

Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat

Abstract École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France Abstract manufacturing (AM) products arepaul.stief@ensam.eu designed, manufactured and measured. It enables the fabrication of components with *Additive Corresponding author. Tel.: +33 3changes 87 37 54the 30;way E-mail address: Additive changes the way products are However designed,traditional manufactured andrules measured.

It enables of components complex manufacturing geometries and(AM) customisable material properties. design or guidelines arethe nofabrication longer applicable for AM.with As geometries and customisable material However traditional rules orguidelines. guidelines It areurges no longer applicable for AM.that As acomplex result design for additive manufacturing lacks properties. of formal and structured design design principles and a comprehensive system acan result for additive manufacturing lacks formal how and structured designdesign principles and guidelines.

It urges a comprehensive helpdesign designers and engineers understand forof example the geometrical and process parameters will affect each other, system and howthat to can help designers and engineers understand for example geometrical and process parameters affect each other, and how to configure process parameters to meet specifications.

In this how paperthe a set of categorydesign ontologies has been developedwill to formalise fundamental/general Abstract configure parameters to meet In this paper aguidelines set of category ontologies has been developed to formalise fundamental/general knowledgeprocess of design and process for specifications.

AM. A collection of design and rules are encapsulated and modelled into categorical structures. knowledge of design and processofforAM AM. A enable collection of design guidelines and rules are encapsulated and modelled into categorical structures.

The formalisation of knowledge will existing fundamental/general knowledge of AM process and state-of-the-art designing cases In today’s business environment, the trend towards more product variety and customization is unbroken. Due to this development, the need of The formalisation of knowledge of AM will enable existing fundamental/general knowledge of AM process and state-of-the-art designing cases computer-readable and toproduction be interrogated andemerged reasoned,toand then canvarious be integrated intoand CAx platforms. agile and reconfigurable systems cope with products product families. To design and optimize production computer-readable andPublished to be interrogated andB.V. reasoned, and then can be integrated into CAx platforms. © 2018 as The Authors.

Elsevier systems well as to choose theby optimal product matches, product analysis methods are needed. Indeed, most of the known methods aim to © Authors.

Published by Elsevier B.V. Committee of the 15th CIRP Conference on Computer Aided Tolerancing - CIRP CAT 2018. © 2018 The under Peer-review responsibility of the Scientific analyze a product orresponsibility one product family on the physical level. of Different product families, however, may differ largely in terms of theCAT number and Peer-review theScientific ScientificCommittee Committee 15th CIRP Conference on Computer Aided Tolerancing - CIRP 2018.

Peer-review under responsibility ofofthe of thethe 15th CIRP Conference on Computer Aided Tolerancing - CIRP CAT 2018. nature of components.

This fact impedes an efficient comparison and choice of appropriate product family combinations for the production Keywords: Additive Manufacturing (AM), geometrical variability, process parameters system.

A new methodology is proposed to analyzevariability, existing products in view of their functional and physical architecture. The aim is to cluster Keywords: Additive Manufacturing (AM), geometrical process parameters these products in new assembly oriented product families for the optimization of existing assembly lines and the creation of future reconfigurable assembly systems. Based on Datum Flow Chain, the physical structure of the products is analyzed. Functional subassemblies are identified, and terrogation.

Withgraph a proper interface, theoutput formalised Introduction a1.functional analysis is performed. Moreover, a hybrid functional and physical architecture (HyFPAG) is the which knowledge depicts the terrogation.

With aplanners proper interface, thedesigners. formalised knowledge 1. Introduction similarity between product families by providing design support to both,can production and product illustrative then be system captured, accessed and interrogate byAn AM designthen be captured, accessed andfamilies interrogate

by AM designexample of a nail-clipper is used to explain the the proposed methodology. industrial case study onwith two product of steering columns of ers/engineers to help decision making regarding product Additive manufacturing (AM) changes way products are Ancan ers/engineers to help with decision making regarding product Additive manufacturing (AM) changes the way products are thyssenkrupp Presta France is then carried out to give a first industrial evaluation of the proposed approach. specification, supporting structures, process parameters, etc. designed, manufactured and measured. It enables the fabrispecification, structures, process parameters, etc. manufactured andcomplex It enables the fabri©designed, 2017 The Authors. Published bymeasured.

Elsevier B.V. The currentsupporting state-of-the-art for formalising AM knowledge cation of components with geometries and customisThe current state-of-the-art for formalising AM knowledge cation of components with complex geometries and customisPeer-review under responsibility of the scientific committee of theand 28th CIRP Designon Conference 2018. is based descriptive logics (DLs) to construct different AM able material properties.

However traditional design rules

able material design rulesmost and guidelines areproperties. no longer However applicabletraditional for AM. In the past the past most research and development efforts in AM have been focused on research and materials development AM have beenMuch focused on new powder and efforts processindevelopment. of the new powder materials and process development.

Much of the existing knowledge body is build upon empirical principles and existing knowledge body build uponmany empirical principles and research [1].isAs a result currently available 1.experimental Introduction experimental research [1].

As a result many currently available AM design guidelines are highly machine-specific or materialAM design guidelines are highly often machine-specific or materialspecific provide compreDue [2]. to Also the the fastguidelines development fail in tothe domain of specific [2].

Also the guidelines often fail to provide comprehensive information help designers theand cacommunication and that an can ongoing trend ofunderstand digitization hensive that can help designers understand thehow capabilitiesinformation and limitations of various type ofare AM processes, digitalization, manufacturing enterprises facing important pabilities and limitations of various type of AM processes, how the geometrical design and process parameters will affect each challenges in today’s market environments: a continuing the geometrical design and process parameters will affect each other, and how to configure process parameters to meet specifitendency towards reduction of product development times and other, and how to configure process parameters to meet specifications. Driven bylifecycles. design functionality, the focus of increasing design for shortened product In addition, there is an cations.

Driven design functionality, the focus of design for AM (DfAM) hasbybeen gradually shifted from process demand of customization, being at the same time in afocused global AM (DfAM) has been gradually shifted from process focused guidelines towith morecompetitors integrated process-geometry competition all over the world.design This guidetrend, guidelines to more integrated process-geometry design guidelines in recent years.

As current DfAM lacks of formal and which is inducing the development from macro to micro lines in recent years.

As current DfAM lacks of formal and structured design principles and guidelines, it urges a compremarkets, results inprinciples diminished lot sizes due to augmenting structured design and guidelines, it urges a comprehensive varieties system that can provide and general [1]. deproduct (high-volume to fundamental low-volume production) hensive system that canguidelines provide fundamental and general design and process control to aid with decision making.

To cope with thiscontrol augmenting variety as with welldecision as to bemaking. able to sign and process guidelines to aid To start with, AM optimization knowledge haspotentials to be formalised to be identify possible in the first existing To start with, AMand knowledge has to be formalised first to inbe machine-readable to enable knowledge reasoning and production system, and it is to important to have a precise knowledge machine-readable enable knowledge reasoning and inKeywords: Familyfor identification guidelinesAssembly; are no Design longermethod; applicable AM. In

is based onsuch descriptive logics (DLs)[3,4] to construct different AM ontologies as design ontology and process ontology ontologies such as design ontology [3,4] and process ontology [5,6]. In each ontology, a set of entities and relation between [5,6]. eachestablished ontology, atosethelp of entities and relation between entitiesInwere AM designers or engineers entities were established to help AM designers or engineers identify relationships and interconnectivity between different identify relationships and interconnectivity between different parameters.

DLs are based on set theory and are best suited to of the product range and characteristics manufactured and/or parameters.

DLs are based on set theory and are best suited to represent relationships between sets.

They are therefore limited assembled in this system. In this context, the main challenge in represent relationships between sets.

They are therefore limited in extent (no of sets) directly merge difmodelling andsets analysis is and nowcannot not only to cope withtwo single in extent (no sets of sets) and cannot directly merge two different ontologies, nor construct complex relationships among products, a limited product range or existing product families, ferent ontologies, nor construct complex relationships among ontologies. thistopaper the and knowledge modelling is based on but also to beInable analyze to compare products to define ontologies.

In this paper the knowledge modelling is based on category theory and the modelling method is updated from aunew product families.

It can be observed that classicalfrom existing category theory and the modelling method is updated authors’ previous [8,9] with redefinedofsyntax semanproduct families work are regrouped in function clientsand or features. thors’ previous work [8,9] with redefined syntax and semantics.

The categorical-based knowledge modelling is substanHowever, assembly oriented product families are hardly to find. tics.

The categorical-based knowledge modelling is substantially distinct from family DL-based ontology.

One of the most in signifOn the product level, products differ mainly two tially distinct from DL-based ontology.

One of the most significant distinguishing features of the categorical-based language main characteristics: (i) the number of components and (ii) the icant distinguishing features ofknowledge the categorical-based language is that itcomponents represents multi-level structures (hierarchitype of (e.g. mechanical, electrical, electronical). is that it represents multi-level knowledge structures (hierarchical) with greatly enhanced searching efficiency.

Classical methodologies considering mainly single products cal) with greatly enhanced searching efficiency. this paper, twoexisting sets of category ontologiesanalyze have been or In solitary, already product families the In this paper, two sets of knowledge. category ontologies have been developed to formalise AM A collection (but product structure on a physical level (components level) which developed to formalise AM knowledge. collection (but non-inclusive) of design guidelines rulesAare encapsulated causes difficulties regarding an and efficient definition and non-inclusive) of design guidelines and rules are encapsulated and modelledofinto categorical structures. proposed catecomparison different product families.The this and modelled into categorical structures.

TheAddressing proposed cate-

2212-8271 2018 The Published

by Elsevier B.V. B.V. 2212-8271©© ©2017 2018The TheAuthors.

Authors.

Published

by Elsevier 2212-8271 by B.V. 2212-8271 ©under 2018responsibility TheAuthors.

Authors.Published Published byElsevier Elsevier B.V. of the 15th CIRP Conference on Computer Aided Tolerancing - CIRP CAT 2018.

Peer-review of the Scientific Committee Peer-review under responsibility of the Scientific Committee of the 15th CIRPConference Conference on Computer Aided Tolerancing - CIRP CAT 2018.

Peer-review under responsibility of the scientific committee ofofthe 2018.

Peer-review under responsibility of the Scientific Committee the28th 15thCIRP CIRPDesign Conference on Computer Aided Tolerancing - CIRP CAT 2018.

10.1016/j.procir.2018.04.076

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Qunfen Qi et al. / Procedia CIRP 75 (2018) 87–91 Q. Qi et al. / Procedia CIRP 00 (2018) 000–000

Table 1: Special properties of morphism f gory ontologies enable formalising AM knowledge at different stages in a higher level, facilitating a systematic approach that Constructor Synatx Semantics can provide rigorous mapping between design and process cate∀A, B, C ∈ C.NO , g ∶ B → C, Epic f ∶A↠B gory ontologies. It is envisaged that the formalisation of knowlh ∶ B → C, {g ○ f = h ○ f } ⇒ g = h edge of AM will enable existing knowledge of AM process and ∀A, B, C ∈ C.NO , g ∶ A → B, Monic f ∶ A▸→ B ad-hoc designing cases machine-readable and to be interrogated h ∶ A → B, { f ○ g = f ○ h} ⇒ g = h as ‘is’, ‘has’, ‘with’, ‘applied to’, ‘assigned to’ and reasoned, and can be integrated into CAx platforms. ∀A, B ∈ C.NO , f ∶ A ↔ B. sections.

The paper is organised as follows. The updated foundation Isomorphic f ∶ A ↔ B ∃ f −1 ∶ B → A, f −1 ○ f = id(A), −1 of category ontology is introduced in Section 2. Using the mod- as ‘is’, ‘has’, ‘with’, ‘applied to’, f ○ f‘assigned = id(B)to’, examples are given in the 2.3.

Morphism structures sections. elling method, a set of general AM design and process category ∀A, B ∈ C.NO , ∃ f −1 ∶ B → A, Retraction f ∶ A●→ B −1 onid(B) categorical concepts, product struc ontologies is then constructed in Section 3, as well as the mapfBased ○f = ping between design and process category ontologies. Section 2.3. Morphism structures angle ∀A,structures B ∈ C.NO , (△), ∃ f −1 rectangle ∶ B → A, structures (◻), p Section f ∶ A◾→ B −1 f ○product f in = id(A) redefined thisstructures paper with enriched details and 4 provides a number of conclusions and future work.

Based on categorical concepts, (×), coproduct structu ∀A, B ∈Note C.N(◻), . f ∶pullback/pushout A ↠ B& f ∶ A▸structure → B structures Figure 2. a morphism allows Othat angle structures (△), rectangle structures ( Epic&Monic f ∶ A▸↠ B &! f ∶ Adetails ↔ B and more deduced structures, a redefined in this paper with enriched ‘is’, ‘has’,‘with’, ‘with’, ‘appliedto’, to’,‘assigned ‘assigned to’, examplesare aregiven givenininthe thefollowing following as ‘is’, Ontology ‘has’, ‘with’, ‘applied ‘assigned to’, examples are given in the following asasto’, ‘is’, ‘has’, ‘applied to’, examples 2. Category O3 Figure 2. Note that a morphism structure O allows nesting of other morphism 3 sections. sections. sections.

u u In this section, a brief introduction of category ontology in O33 O33 2.3.

Morphism structures 2.3.

Morphism structures 2.3.

Morphism structures which objects, morphisms and morphism structures are introf g O1 ⊔ O2 f Ostructures 1 ×uuO2 Based on categorical concepts, product structures (×), coproduct (⊔),tritrion categorical concepts, product structures (×), coproduct structures (⊔),(×), tri- coproduct Based on concepts, product structures structures (⊔), duced. AsBased the knowledge modelling method iscategorical entirely indeuu angle (△), rectangle structures (◻),(pullback/pushout pullback/pushout structures (2 ∏//∐∐) )are are angle rectangle structures (◻),(△), pullback/pushout structures / ∐) are ∏ angle structures rectangle (◻), structures (p∏ p1 pendent withstructures AM, some(△), readers may find itstructures disconnect with the structures i1 i2 gg O11 ⊔ O ff gg deduced ff details 2 shown O22 O × O 1 2 2 1 2 redefined in this paper with enriched and more structures, as redefined in this paper with enriched details and more deduced structures, as shown in redefined in with enriched details and more deduced structures, as shown inin following section. However the foundation of this the paper knowledge O1 m O1 morphism O2 Figure Note that amorphism morphism structure nestingofofother other morphism Figure that a morphism structure allows of other morphism Figure 2.2.Note anesting structure nesting structures. modelling has2.toNote be represented first otherwise thethat knowledge p1allows pallows ppstructures. m11 1 ii11 ii22structures.

22 structure in the following section can not be understood. mstru (a) Product structure × (b) Coproductm 33 O11 O22 O11 O11 O22 OO33O3 , N M , NS ), OO3 3 A category ontology O3 C is denoted by a triple (N O E where NO is a set of objects, N M is a set of morphisms and NS is (a) Product structure × (b) Coproduct structure ⊔ (c) Triangleq1stru u u (a) Product structure (b) Coproduct structure u u u u u a set of morphism structures. All objects and morphisms satisfy m1 EO1 ππ11 O 2 π1 the set of category g A OO111×⊔×O OO222 ggg f f f OO O⊔2OO2 2 gg Olaws. ff f O OO2qq121 1 × O2 1 1⊔ A q2 uu ◻ π 1 π33 1 Objects Let A pbe anp2object in C, it may also m3 mm△ △ i2p2 of five ◻1 1 pi1p1be pone 1mm2m2 π3 11 2 Om111 i1i1 m11i2i2 m2O22 ππ11 2 π special type of objects with extra properties, written as A.p, π A B 44 m3 qq22 3 mm m 3 C 4 π4 D O23222 OO O O11 1 are: terminalO O1t ∣ i ∣ z ∣ s ∣ e. O O OO where p ∶∶= The types obOOm OO 2 five O 4O C 33 1 133 1 13 222 △22 △11 Om ◻ π 1 π π33 1 π22 qq22 ject (denoted as t), initial object (i), zero object (z), singleton (c) Triangle structure (d) Rectangle structure (a)Coproduct Productstructure structure (b) Coproduct structure (c) Triangle structure △ (a) Product structure × (b) structure (c)Coproduct Triangle structure △ mstructure (d) Rectangle structure ◻ (e) Pullback struc (a) Product ××⊔ (b) ⊔⊔ (c) Triangle structure △ π 44 π 4 4 object (s) and empty object (e).

O33 O44 C D E π1 EE π1π1 A B A B Figure 2: Morphism A B q q1 from obMorphisms A morphism represents a relationship 1 q 1 structure ◻ (d) Rectangle (e) Pullback structure ∏ (f) Pushout stru u ◻ u u π3 1 π2 ◻◻1 1 ject A to B in C, written the π3π3 m1 1 domain of π2π2 m1 as f ∶ A → B. Here A is m O2is the codomain OO11 of f ,πwritten OO22 as Figure 2: 2: Morphism Morphism structures 1 π1π1 Figure structures f , denoted AO1= f (O1 ) and B π qB 1 1B π2.1. q1qstructure A B 44 q2 q2 AAπ4 q2C Definition Product ×(O1 , O2 , p1 , D CC DD morphism set represents all morphisms from obB = f (O2 ).mA m 3 2 mm3 3 π△ m △ △ △2 △1 m2 and O , and two projection morphis objects O ◻ ◻△ 1 2 π ◻ 1 1 11 π π22 3 3 22 2 π 1 π 3 2 u uu jects A to B in C, written as MC (A, B). For any object A ∈ C.NO , q2 q2q2 , p ∶ O ×O → O .

Further, if there is anothe O 1 2 1 2 2 Definition 2.1.

Product structure ×(O , O , p , p ) is constructed by a prod m m 1 1 2 4 4 1 2 1 2 m π4 id(A).

4as π4π4 there is an identity morphismOon O3 OO3denoted OO4D4 3C 4 object A, E f projection C1C DDtwo E11E1and and g, where f ∶ O3 → pO ,g∶O O2 , as morphisms p where p11sho ∶O objects O 1 and O22, and 23, → 2 A morphism f may also has one of six special properties, → O × O , and p ○ u = f , p morphism u ∶ O 3 1 2 1 2m , p ∶ O ×O → O .

Further, if there is another object O has two project O 1 2 1 2 2 3 (e) Pullback structure (f) Pushout structure (d) Rectangle structure ◻ (e) Pullback structure (f) Pushout structure (d) Rectangle structure ◻ (e) Pullback structure (f) Pushout structure 1 2 1 2 2 3 ∏ ∐∏ Rectanglemay structure ◻ (e) Pullback (f) Pushout structure ∐∐ ∏ written as f.p, where p can be null (a(d) morphism not have f and g, where f ∶ O → O , g ∶ O → O , as shown in Figure 2a.

There exis 3 1 3 2 3 1 3 2 any properties), epic (denote as ↠), monic (▸→), isomorphic 2.2. Coproduct structure ⊔(O1 , O2 , Figure2:2:Morphism Morphismstructures structures Figure 2: Morphism structures Fig. 1:Definition Figure ,Morphism and p11 ○structures u = f , p2 ○ u = g . (↔), retraction (●→), section (◾→), both epic and monic but not morphism u ∶ O33 → O11 × O22two objects O1 and2 O2 , and two inclusion morph isomorphic (▸↠), as shown in Table 1. A morphism can only Definition 2.2. Coproduct structure O1 ⊔11,OO22.2,Ifi11,there an object Oby i2 ∶ O2 → ⊔(O 3 with i22) isisconstructed a co have atDefinition most two 2.1. properties. properties a2 ,morphism is two ×(O Definition 2.1.

Product structure ×(O ,aO p1 ,1 ,pp2O)22)2of is constructed by aas product oftwo two ProductThe structure ×(O1of ,2.1.

O pProduct constructed by product two → O , shown in Figure 2b.

There exists g ∶ O Definition , , is constructed by a product of 1O 211,pand 1 , p2 ) isstructure 2 3 , and two inclusion morphisms i and i , where i ∶ O11 objects O 1 2 1 1 2 1 2 1 and p2 , where p1 ∶ O1 × O2 → O1 , p2 ∶ O1 × O2 → O2 . If there of significant to generate results reasonandfrom andptwo morphisms and p2 ,2uobject ,where where p1with ∶OO ×g.

O → details objectsO O1 1and and two projection morphisms andprojection p , where p1 O ∶O Op121and → objects importance O1 and O2 , both = f , u i = More about pro and ○ i OO2 ,2 ,and morphisms p p p O objects 1∶ ○ 1× 2→ 1two 1If×there 1 2 . is an O two inclusion morphisms f iprojection 2 3 1 2 22 ∶2isOanother 22 → O 11 ⊔ 2 3 object O3 has two project morphisms f and g, where ing rules, to understand the nature theO3ghas ×O →OO2object .Further, Further, there is33another another object Ocan has two project morphisms p2 ∶ to O1help ×O2end-users → O2 . Further, another two O project morphisms O1 , and refer to refs [25] p39 and p55. .2of if∶ iffOthere is object O two project morphisms OO1 ,1if,ppthere 1×O 3has , as shown in Figure 2b.

There exists a unique morphism u ∶ O ⊔ 1 2 2∶ ∶OO 1 is 2 2→ 3 2∶2 → 1 O3 → O1 , g ∶ O3 → O2 , there exists a unique morphism relationship. g,where where f∶ ∶OO3 3→→ O1 ,1 ,ggand → O2=,2 ,as as shown inFigure Figure 2a.There There exists aunique unique f and g, where f ∶ O3 → O1 , gf fand ∶ and O3 g, → O inOFigure 2a.

There exists a =unique ∶ ∶OuO O shown in 2a. exists a 3→ 2 , as fshown f , u ○ i g.

More details about product and coproduct in categ ○ i 1 2 3u 1 2 ○ u = f , p2 2.3. ○ u = Triangle g. ∶ O3 → O1 × O2 , and p1 Definition structure △({O1 , O2 , Alsomorphism a morphism with →○Omake andprefer p1 1○○uuto =gg.and . p55. morphism O1 ×assigned O2 ,morphism and p1 ○ auunotion =u∶ ∶O fO ,3p3→ uO1 = gOO .it2 ,2 ,and u ∶ Of 3is→often ==refs f f, ,pp2[25] 1×× 2○○uu= 2to can p39 in between th commutative morphism m 1 and Coproduct structure ⊔(O1 , O2 , i1 , i2 ) is constructed readable, written as f ∶ A(notion) → B. Notions of a morphism byma2 co, O , i , i ) is constructed by a coproduct of Definition 2.2.

Coproduct structure ⊔(O , O , i , i ) is constructed by a coproduct of Definition 2.2.

Coproduct structure ⊔(O = m m1is , asform sho sition morphism of the two m O2 ,2Triangle iobjects bytwo a11,coproduct of Definition 2.2. ⊔(Oof2.3.

2 isOconstructed 1 Coproduct 2‘with’ 1 2 andstructure 2 33○}) could be started with characters such as ‘is’, ‘has’, and morphisms product Definition structure Oinclusion 1 ,1two 1 ,1 i2 ) 1 and O2 ,△({O 22, O33}, {m 11,3 m22, m and O , and two inclusion morphisms i and i , where i ∶ O → O ⊔O , two objects O and O , and two inclusion morphisms i and i , where i ∶ O → O ⊔O , two objects O i1m i21,2m where O111⊔O two objects O1 1and O2 ,2 and 1 2 1 two 2inclusion 1 ‘applied to’. where i1 ∶1 O O ⊔22Oin2 ,between i21 1∶ O12 1→ ⊔ O2 ,22 . If{O there i1 and i2 1,morphisms three objects commutative morphism 11and 112→and 11, O22, O33} an O2 2→→OO1 31⊔⊔ there anobject object with two inclusion morphisms f∶ ∶OO1 1→→ i2∶ ∶O object with inclusion morphisms f of ∶3two O O i2 ∶ O2 → O1 ⊔ O2 . If there isi2an OO2 .2 .two IfIfthere isisan OO3 3with inclusion O∶O ,3 ,1 → 2c.

1 → 3 ,minclusion with two f O is an object O m in Figure sition morphism the two 3 =morphisms 2 ○ mmorphisms 1, asf shown 3O 3 ,In this pap 3 2 1 Morphism Sixinmorphism structures, including 6 → shown Figure 2b.

There exists unique morphism →1OO , 2 → O3 , O2 2→ O3 , as shown Figure There exists a unique morphism u ∶exists O ⊔ Oaa2unique → O3 ,amorphism g ∶ O2 → structures OO3 ,3 ,asas shown ininFigure 2b.

There uu∶ ∶OO1 1⊔⊔OuO2 ∶2→ gg∶ ∶O2b. exists unique morphism O ⊔ g ∶ O 3 ,3O 2 →O 3 ,1there productand structures structures (⊔), triangle struc= f , u ○ i = g.

More details about product and coproduct in category theory and u ○ i f , u ○coproduct i2 = g. More details about product and coproduct in category theory u ○ i1 = (×), in category theory and coand u ○ i1 1= f , u ○ i2 2= g. More details f , u ○ i2and = g.coproduct More details about product and u about ○ i1 = product tures (△), rectangle structures and pullback/pushout struc6 can refertotorefs refs[25] [25]p39 p39 andp55. p55. can refer to refs [25] p39(◻) andcan p55. refer and product in category theory can refer to [7]. tures (∏ / ∐) are redefined based on categorical concepts with △({O enriched details and deduced structures.

Note that Ois {m , by m3two }) is formed Definition 2.3. structure △({O ,structure O2 ,2O , O3by {m , m21,2,m ,O m32}) isformed formed two Definition 2.3.more Triangle structure △({O , O3a},mor{m }) is 1formed two 3 }, 1 , m2by Definition 2.3. structure △({O ,1O }, 3 },{m 3,}) 1 , OTriangle 2Triangle 1 , m2 , m3Triangle 1 ,1 m phism commutative structure allows nesting m of1 other morphism structures. and m in between three by two commutative morphism m and m in between three objects {O , O , O } and a compocommutative morphism m and m in between three objects {O , O , O } and a compomorphism 1 2 three objects {O1 ,1O2 ,2O3 }3 and a compocommutative morphism m1 1and m212in between 2 2 3 Product ×(O , O , O } and a composition morphism of the two objects {O , O2two , p1 ,mpsition ) is constructed by a prod= m ○ m , as shown in Figure 2c.

In this paper, the first sition morphism of the two m = m ○ m , as shown in Figure 2c.

In this paper, the first sitionstructure morphism of1the 1 2 3 2 morphism of the two m3 3= m2 2○ m1 ,1 as shown in Figure 2c. In this paper, the first 3 2 1 m3 = m2 ○ m1 . In this paper, the first object O1 of a triangle uct of two objects O1 and O2 , and two projection morphisms p1

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Qunfen Qi et al. / Procedia CIRP 75 (2018) 87–91 Q. Qi et al. / Procedia CIRP 00 (2018) 000–000

structure is denoted as △(O1 ), the second object as △(O2 ), and the third object as △(O3 ), so does for the morphisms of a triangle structure, written as △(m1 ), △(m2 ) or △(m3 ).

Rectangle structure ◻({O1 , O2 , O3 , O4 }, {m1 , ..., m4 }, △1 , △2 ) is formed by four morphisms and four objects, which also form twoaretriangle △1 ({O1 , O2 , O4 }, {m1 , m2 , m1 ○ m2 }) ’, examples given instructures the following such that and △2 ({O1 , O3 , O4 }, {m3 , m4 , m3 ○ m4 }), △1 (m1 ) = ◻(m1 ), △1 (m2 ) = ◻(m2 ), △2 (m1 ) = ◻(m3 ), e following △2 (m2 ) = ◻(m4 ). And ◻(O1 ) is the staring object of the rectangle structure and ◻(O4 ) is the ending object. ctures (×), coproduct structures (⊔), triPullback structure pullback/pushout structures (∏ /∏ ) are1 , ..., O5 }, {m1 , ..., m9 }, {◻1 , ..., ◻4 }) ∐({O is constructed from a structure ◻1 in which d more structures, as shown rectangle in ures (⊔),deduced tri(△ (m )) or ◻ (△ (m )) is either monic or isomorphic. It ◻ 1 1 2 1 2 2 nesting of other morphism structures. (s∏ / ∐) are consists of a set of five objects and a set of nine morphisms as shown in whose objects and morphisms form four rectangle structures m structures. including ◻1 . The four rectangle structures are listed as follows: ◻1 ({A, B, C, D}, {π1 , π2 , π3 , π4 }, △1 , △2 ), g O2 2 ◻2 ({E, B, C, D}, {q1 , π2 , q2 , π4 }, △3 , △4 ), m2π4 ○ π3 , q2 , π4 }, △5 , △4 ), m1 A, C, D}, {u, ◻3 ({E, {q1 , π2 , u, π4 ○ π3 }, △3 , △5 ), ◻4 ({E, B, A,mD}, 3 O O3 O1 m22 2 where m △(c) B, D}, {π1 , π2△ , π2 ○ π1 }), 1 ({A, ucture ⊔ Triangle structure O33 △2 ({A, C, D}, {π3 , π4 , π4 ○ π3 }), π1 B 1 , π2 , π2 ○ q1 }), △3A({E, B, D}, {q ucture △ ◻1 D}, {q △π43 ({E, C, π22 , π4 , π4 ○ q2 }), ({E, A, D}, {u, π4 q○ π3 , π4 ○ π3 ○ u}) in which two mor△ B 5 π B C (π ○π4 andDπ ○π 1○u) are deduced from the composition phisms 4 3 4 3 π π22 1 π2 rule. u q 2 qq11 D Apart from ◻1 , other rectangle structures (can be also written D E as ◻′ ) all start with object E and end with object D. For any uu cture ∏ ◻′ , morphisms (f) Pushout structure ∐ q2 always form a triangle structure u, π3 and E △ (u, π , q ), so do morphisms u, π and q form △ (u, π , q ).

6 3 2 1 1 7 1 1

ucture ∐

structures

Pushout structure ∐({O1 , ..., O5 }, {m1 , ..., m9 }, {◻1 , ..., ◻4 }) is constructed from a rectangle structure ◻1 in which , p2 ) is constructed product ◻1 (△2of(mtwo ◻1 (△1 (mby 1 ))aor 1 )) is either epic or isomorphic. It sms p1 and p2 , where O1five × Oobjects 1 ∶ of 2 → consists of a pset and a set of nine morphisms, er object Owhose project 3 has two objects andmorphisms morphisms form four rectangle structures duct of two own 2a. There exists a unique including ◻1 . The four rectangle structures is listed as follows: O11 ×inOFigure 22 → ○u=g. ◻1 ({A, B, C, D}, {π1 , π2 , π3 , π4 }, △1 , △2 ), morphisms sts a unique ◻2 ({A, B, C, E}, {π1 , q1 , π3 , q2 }, △3 , △4 ), i1 , i2 ) is constructed by a coproduct of ◻ ({A, B, D, E}, {π , q1 , π2 ○ π1 , u}, △3 , △5 ), hisms i1 and i2 ,3where i1 ∶ O1 → O11 ⊔O , ◻4 ({A, D, C, E}, {π4 ○ π32, u, π3 , q2 }, △5 , △4 , two inclusion morphisms f ∶ O1 → O3 , oproduct of where s→a O unique u ∶ O ⊔O → O , 11⊔O22,morphism △1 ({A, B, D},1 {π1 ,2π2 , π23○ π1 }), oduct and in category theory ∶ O11 → O33coproduct , △ ({A, C, D}, {π3 , π4 , π4 ○ π3 }), 2 ⊔ O22 → O33, △ ({A, B, E}, {π , q , q ○ π }), 3 1 1 1 1 gory theory △ ({A, C, E}, {π , q , q ○ π }), 4 3 }) is formed 3 by 2 two 2 3 , O3 }, {m1 , m2 , m {πa2 compo○ π1 , u, u ○ π2 ○ π1 }). △51({A, hree objects {O , O2 , D, O3E}, } and All rectangle structures in the pushout structure start with own in Figure 2c.

In this paper, the first med by two , the other rectangle structures (◻′ ) all object A apart from ◻ 1 nd a compoend with object E. For any ◻′ , morphisms π2 , u and q1 always per, the first form a triangle structure △6 (π2 , u, q1 ), so do △7 (π4 , u, q2 ).

3. AM design and process category ontologies

In this section, general AM design and process knowledge will be structured into two sets of category ontologies respectively. Mappings between the two sets will then be established,

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that is a set of functors (relationship from one category ontology to another) between the two sets. One of the AM technologies, powder bed fusion (PBF) is selected for the purpose of process modelling.

3.1. Design category ontology To structuralise the design knowledge, different types of designing parameters such as geometrical variability, feature designs including overhanging and extrusion features, and support structures are constructed into a category ontology Design Parameters (DP ) . Objects in the category ontology and morphisms between these objects are then defined as shown in Fig. 2. Here, DP encloses three nested product structures, where object [Geometric] is a product object from [Angular], [Circular], [Spatial] and [Overhang]; [Support Structure] is a product object from [Types] and [Removal]; [Geometrical Variability] is a product object from [Dimensional], [Form], [Orientation], [Location], [Run out] and [Surface Texture].

Design Parameters DP m1: with

m2: decides

Geometric

p1

p2

p3

Angular

Circular

Spatial

Support structure p5

p4

Overhang

Types

p6

Removal

Geometrical Variability p7

Dimensional

p8

p9

p10

p11

p12

Form

Orientation

Location

Run out

Surface texture

Fig. 2: AM design category ontology DP

Note that the objects in DP is non-inclusive, as more objects can be added to form more product structures (×i ). For example, [Surface Texture] can also be a product object, if more surface texture related objects are added into DP . These objects are however not included in this paper as there is yet no evidences of detailed relationships between these objects and objects in the following process category ontologies.

Different from traditional manufacturing processes, support structure is one of the critical designing parameters for AM to extract heat from the part and to provide mechanical anchor to avoid warpage due to thermal stresses during and after the build. The design of support structures is a process to optimise the volume, geometry, location and part-support interface geometry. During the designing process, overhanging features, build orientation, GD&T and the easiness of removal have to be considered. For instance, when a overhanging feature is over 45 degree, or the feature has very large projected areas, a support structure is normally required. Also the need of support structure can be reduced by changing the build orientation, or by designing the support structure on where there is less geometrical accuracy requirements. Therefore there is a morphism m2 :[Overhang](decides) → [Support structure], and the property of m2 is however not yet decidable.

3.2. Process category ontologies For the process control parameters, the following category ontologies are constructed: Environment (PEn ) represents the

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90 4

Design Parameters DP m1: with

m2: decides

Geometric

p1

p2

p3

Angular

Circular

Spatial

Support structure p6

p5

p4

Overhang

Types

Removal

Geometrical Variability p7

Dimensional

p8

p9

p10

p11

p12

Form

Orientation

Location

Run out

Surface texture

Process Control PPC

Environment PEn Inert gases

Oxygen concentration

m3: decides

m4: contributes to

Chamber pressure p13

Gas flow

m5: decides

Particle size distribution

Powder PPo

AM type

p18

p17

Direction

m7: has

Layer thickness

Spot size

p21

Intensity

p22

m8: has

m20: =

Build Orientation m17: contributes to

Scan strategy

m16: impact

Apparent density

m9:has

Flowability

p16

Build

m19: has m18: impact

Internal porosity

p15

Density

m6: contributes to

Rate p14

Machine

Powder

m21: impact

m25: impact

m26: impact

Ignition engergy

m10: decides

Energy beam

p19

p20

Scan velocity

Hatch distance

m14: impact

m15: impact

m22-m24: impact

m13: impact

Power density

Power

m11: has

Mode

m12: decides

Energy PEe

Fig. 3: AM design category ontology DP

chamber environment elements; Powder (PPo ) states parameters of the powder; Energy (PEe ) indicates the parameters of the AM power; and Process Control (PPc ) encloses process parameters as indicated in Fig. 3.

Mappings between design and process category ontologies, can be established by finding relationships between objects in design category ontology and critical objects in process category ontologies. This can also help identify relationships and interconnectivity between different AM models and their parameters.

In the category ontology of Process Control (PPc ), important process parameters including [Build orientation], [Scan strategy], [Layer thickness] and [Scan velocity] have critical impact on the GD&T of the fabricated AM part. The [Build orientation] will impact the form accuracy, for example cylindricity [11]. Surface texture can be affected by the [Layer thickness], [Scan strategy], [Scan velocity], [Hatch distance] and energy beam [Spot size]. For example the setting of medium-high [Scan velocity] together with medium [Hatch distance] is ideal for growth aligning in the build direction and resulting in an

isotropic build thus have a better [Surface texture] [12].

[Layer thickness] is correlated to the [Particle size distribution] (in PPo ). Layer thickness is limited by the mean particle size of the powder and ideally it would be slightly larger than the mean particle size. Normally small [Layer thickness] may result in better [Surface texture]. Along the build direction (Zdirection), the [Layer thickness] is usually affect the [Geometrical variability] (in DP ). Thinner [Layer thickness] together with slower [Scanning velocity] when the total input [Power density] is held constant, will result in narrower track width and improved [Surface texture].

[Scan strategy] is closely related with beam diameter to allow sufficient overlapping of adjacent paths occurs and preventing partial melting. Most metal AM systems employ sophisticated scan strategies to reduce thermal residual stress which can affects the geometry variability. However it is still very difficult to predict accurately the thermal residual stress.

In the category ontology PEn , [Inert gases] such as nitrogen or argon are used to control the build chamber environment and maintain low [Oxygen concentration]. [Oxygen concentration]

Qunfen Qi et al. / Procedia CIRP 75 (2018) 87–91 Q. Qi et al. / Procedia CIRP 00 (2018) 000–000

is closely related to the success of a build process and it is typically maintained below 1-2%. For reactive material such as aluminium and titanium, oxygen content control is of critically importance for safety reasons.

In the category ontology PPo , [Particle size distribution] is closely related to the [Layer thickness] and thus affect [Surface texture] of the fabricated part. The powder [Flowability] will affect powder feeding and raking, and a better [Flowability] can achieve smoother powder layers. Also high [Apparent density] and no [Internal porosity] is preferred for the success of build.

In the category ontology PEe , the [Power density] is closely related to [Scan strategy], [Scan velocity] and [Hatch distance].

The powder [Mode] also decides the geometry of [Energy beam] and [Spot size].

4. Conclusion In this paper a design category ontology and a set of PBF AM process category ontologies were constructed. Mappings between the two sets of category ontologies were also established to represent an abstract framework of AM design and process control. The proposed AM design and process category ontologies can be suited for formalising domain, state-of-the-art and experimental knowledge. With a proper interface, the structured general AM knowledge can then be captured, accessed and interrogate by AM designers/engineers to understand links between different parameters. As this is an abstract framework, the objects in this framework are non-inclusive. More objects and morphisms are expected to identify in the future work. For example, if the designers have to deal with specific processes or specific systems, i.e. laser-based or electron beam-based processes, more specific process-oriented objects can be added into the existing category ontologies. The formalised knowledge can also serve as a training material to help designers and engineers understand the interconnection and complex relationships.

The properties of a morphism is of critical importance for relationship reasoning. As some of the morphisms’ properties cannot be decided, it is desirable to update the properties of the constructed morphisms along with the development of AM technologies and customised case studies.

Acknowledgements The authors gratefully acknowledge the UK’s Engineering and Physical Sciences Research Council (EPSRC) funding of the EPSRC Fellowship in Manufacturing: Controlling Geometrical Variability of Products for Manufacturing (Ref:EP/K037374/1), and funding of Future Manufacturing Research Hubs: Future Advanced Metrology Hub (Ref:EP/P006930/1).

References [1] Yang L, Hsu K, Baughman B, Godfrey D, Medina F, Menon M, Wiener S. Additive Manufacturing of Metals: The Technology, Materials, Design and Production. Springer; 2017.

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[2] https://www.astm.org/cms/drupal-7.51/newsroom/proposedastm-international-guide-create-principles-design-rules-additivemanufacturing-3d [3] Dinar M, Rosen DW. A Design for Additive Manufacturing Ontology. J Comput Inf Sci Eng 2017;17:021013.

[4] Jee H, Witherell P. A method for modularity in design rules for additive manufacturing. Rapid Prototyping J 2017;23:1107-18.

[5] Roh BM, Kumara SR, Simpson TW, et.al. Ontology-Based Laser and Thermal Metamodels for Metal-Based Additive Manufacturing. In: ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Charlotte, North Carolina, USA; 2016.p.V01AT02A043.

[6] Liu X, Rosen DW. Ontology Based Knowledge Modeling and Reuse Approach of Supporting Process Planning in Layer-Based Additive Manufacturing. In: 2010 International Conference on Manufacturing Automation.

Hong Kong; 2010.p.261-266.

[7] Awodey S. Category Theory, volume 52 of Oxford Logic Guides. Oxford: Oxford University Press; 2010.

[8] Qi Q, Jiang X, Scott PJ. Knowledge modeling for specifications and verification in areal surface texture. Precis Eng 2012;36:322-333.

[9] Qi Q, Jiang X, Scott PJ, Lu W. Design and implementation of an integrated surface texture information system for design, manufacture and measurement. Comput Aided Desig 2014;57:41-53.

[10] Lawvere FW, Schanuel SH. Conceptual mathematics: a first introduction to categories. Cambridge University Press; 2009.

[11] Ollison T, Berisso K. Three-Dimensional Printing Build Variables That Impact Cylindricity. J Ind Tech 2010; 26(1).

[12] Arsoy YM, Criales LE, zel T, Lane B, Moylan S, Donmez A. Influence of scan strategy and process parameters on microstructure and its optimization in additively manufactured nickel alloy 625 via laser powder bed fusion. Int J Adv Manuf Tech 2017; 90:1393-1417.

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Advanced Engineering Informatics 39 (2019) 347–358

Contents lists available at ScienceDirect

Advanced Engineering Informatics journal homepage: www.elsevier.com/locate/aei

Full length article

Enabling metrology-oriented specification of geometrical variability – A categorical approach Qunfen Qi, Luca Pagani, Xiangqian Jiang, Paul J. Scott

T

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EPSRC Future Advanced Metrology Hub, School of Computing and Engineering, University of Huddersfield, Huddersfield HD1 3DH, UK

ARTICLE INFO

ABSTRACT

Keywords: Geometrical variability Category theory Knowledge modelling Computed tomography Specification semantics

In this paper a metrology-oriented specification schema is proposed to enrich the specification semantics with sufficient metrological information. It is designed particularly for applications where non-traditional measurement methods are applied; and it can also identify any redundancies, inconsistencies or incompletenesses of a specification. The proposed schema is based on category theoretical semantics which uses category theory as the foundation to model the semantics. A set of verification operations that derived from the measurement process was firstly formalised using the categorical semantics. Then a set of full faithful functors were constructed to map the set of verification operations to a set of specification operations. A set of simplification rules was then developed to deduce all of the necessary specification objects which are independent to each other. Then the residual specification objects provide a compact structure of the specification. Three test cases were conducted to validate the proposed schema. An industrial computed tomography (CT) measurement process for an impeller manufacturing using selective laser sintering (SLS) technique, was modelled and a set of independent specification elements was then deduced. The other two test cases for checking redundancy and incompleteness on general ISO specifications were carried out. The results show that the proposed schema works for proposing semantic enriched specification that are characterised by non-traditional measurement methods and for testing redundancy and incompleteness of specifications based on geometrical product specifications and verification (GPS) standards system.

1. Introduction Advanced manufacturing technologies are evolving towards a digitalised and intelligent era [1–3]. Machines will communicate and exchange large amounts of data to ensure they can work harmoniously and collaboratively with little human intervention. Current machines use digitalised symbolic language to represent and exchange the data, but they cannot directly interpret its meaning. As a result, information loss and incorrect interpretation can often happen during communication. One of many ways to improve manufacturing intelligence, is to enable the manufacturing system to ‘understand’ the data, which we refer to as ‘semantics’ of the data. To start with, the specifications of a product, such as geometric shapes and geometrical variability, can not merely be graphic symbols with text, but its semantics have to be explicitly represented in the machine [4,5], such that its meaning can be unambiguously interpretation to help with decision-making prior to the manufacturing and inspection processes.

Over the past decade, there have been various approaches to represent syntax or semantics of geometrical variability, including the

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currently standardised EXPRESS model [6–8], XML-based model [9], ontology-based approach [10,11], category theory-based model [12,13], relationship model [14,15] and Geospelling [16]. All of these models are either directly or indirectly computer-readable and they all unambiguously represented the syntax of the specification. However this does not promise the representation at a semantic level where the models should have a rigorous structure and be able to be reasoned and interrogated. Among aforementioned models, ontology-based approach is based on the formal semantics of description logics [17], which is well equipped with various reasoning mechanisms. Though categorytheory based approach does not have a formal semantics yet, a natural reasoning mechanism based on categorical laws and concepts can be established. Most importantly, category theory provides a direct solution for the dual mappings between the two totally ordered sets of operations: and they are defined as functors (relationships between categories).

In the meanwhile, with the rising demands of components with complex geometric shapes, emerging manufacturing technologies such as additive manufacturing (AM) [18,19] often require totally different

Corresponding author.

E-mail address: p.j.scott@hud.ac.uk (P.J. Scott).

https://doi.org/10.1016/j.aei.2018.11.001 Received 29 January 2018; Received in revised form 5 July 2018; Accepted 1 November 2018 Available online 25 February 2019 1474-0346/ © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/).

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Nomenclature

f (O1) domain object of morphism f f (O 2 ) codomain object of morphism f id (A) identity morphism of object A ({O1, …, O5}, {m1, …, m9}, { 1, …, 4}) a pushout structure ({O1, O2, O3, O4}, {m1, m2 , m3 , m4}, 1, 2) a rectangle structure ({O1, …O5}, {m1, …m9}, { 1, … 4 }) a pullback structure (O1, O2, p1 , p2 ) a coproduct structure × (O1, O2, p1 , p2 ) a product structure ({O1, O2, O3}, {m1, m2, m3}) a triangle structure F: C1 C2 a functor from the category C1 to the category C2 M a bijective on morphisms functor from category C1 to C2 O a bijective on objects functor from category C1 to C2 a faithful functor from category C1 to C2 a full faithful functor from category C1 to C2 a faithful functor from category C1 to C2 a injective on morphisms functor from category C1 to C2

C a category C . NM the morphism set in category C C . NO the object set in category C C . NS the morphism structure set in category C A, B, C objects A. i A is an initial object A. s A is a singleton object A. t A is a terminal object A. z A is a zero object MC (A, B ) a morphism set from object A to B in category C f: A B a morphism from object A to B f : A B an epimorphism from object A to B f: A B an epimorphism and monomorphism from object A to B f :A B an isomorphism from object A to B f: A B a monomorphism from object A to B B a retraction from object A to B f: A f: A B A section from object A to B composition between morphisms

M O

metrological characteristics, compared to the traditional tactile and optical measurement methods. To measure AM manufactured products with complex geometric shapes (often with internal structures), nontraditional measurement methods such as industrial X-ray computed tomography (CT) have gradually gained in popularity in the past decade [20,21]. However the current specification system which is designed based on traditional measurement characteristics, can not directly be interpreted to make measurement decisions for the emerging measurement process. It urges a new apparatus that can interpret the current specification system in an enriched way regardless of which new measurement method is applied, such that the specification can be decoded and then be further understood to make measurement decisions.

To this end, a metrology-oriented specification schema is proposed.

It is designed to enrich the specification semantics with full sets of measurement information, particularly for applications where nontraditional/merging measurement methods are applied. Whilst a designed specification for a geometrical product can lack essential measurement details, one specification symbol can be characterised by many sequence of methods. This can produce different measurement results leading to large method uncertainty or even legal disputes. It is also desirable that the proposed schema can identify any redundancies, inconsistencies or incompletenesses of the specification.

As defined by the geometrical product specifications and verification (GPS) standards system, the interactions between the specification and the verification are dual (duality principle) [22]. Then a totally ordered set of specification operations can be mapped from/to that of verification operations, assuming the measurement process is topological stable [23]. To represent specifications with enriched semantics, the forward and inverse mappings between the two totally ordered sets have to be represented into a computer-readable format and with a rigorous structure.

In this paper, category theory has been adopted for the knowledge modelling. The modelling method is updated from authors’ previous work [12,13,24], with redefined syntax and semantics based on categorical concepts. Employing the updated categorical modelling method, the measurement procedure is modelled by a top-down approach. In this step, the required verification operations and their total order are identified first, and then the set of operations is modelled into a set of verification categories {Pi } . The designing process of specification elements starts with a set of isomorphic mappings {FSi} from the

a injective on objects functor from category C1 to C2 a surjective on morphisms functor from category C1 to C2 a surjective on objects functor from category C1 to C2

verification categories {Pi } to the specification categories {SPi} ; then follows by a simplification process in which modelled structures are simplified by a set of simplification rules; the residual then forms a new set of specification categories in which necessary specification details have to be designed to reduce uncertainties.

The paper is constructed as follows. A brief introduction to category theory and the updated categorical modelling is introduced in Section 2. How to construct verification categories using GPS concepts is discussed in Section 3.1 and how to construct specification categories from verification is presented in Section 3.2. In Section 4, a CT measurement process for an impeller is modelled and a set of independent specification elements is deduced; and two test cases for checking redundancy and incompleteness on general specifications (from ISO standards) are carried out. Section 5 provides a number of conclusions and future work.

2. Categorical modelling 2.1. Category theory The proposed knowledge modelling is based on category theoretical semantics which uses category theory as its foundation. Category theory is a high-level (abstract) mathematical theory which was invented by Samuel Eilenberg and Saunders Mac lane in 1940s [25]. It was primarily designed to bridge topology and algebra (leading to algebraic topology). Category theory focus on the relationships between different kinds of mathematical objects, and ignores unnecessary details to give general definitions and results.

A category is construed as a collection of objects and a type of relationship from one object to another. The ‘relationship’ is called morphism as shown in Fig. 1a and satisfies the following conditions: Condition 1: For each arrow f there are given objects: dom (f ), cod (f ) called the domain and codomain of f. We write: f

B to indicate that A = dom (f ) and B = cod (f ) ; f :A B or f :A C , that is, with: B and g:B Condition 2: Given morphisms f :A cod (f ) = dom (g ) , there is given an arrow: g f :A C , called the composite of f and g, and is the composition operation; A Condition 3: For each object A, there is an identity arrow idA:A B , idB f = f and satisfying the identity law: for any arrow f : A f idA = f .

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Fig. 1. Basic concepts of category theory.

The sets of objects and morphisms, which have this specific functionality, form a category.

At a higher level, a relationship from one category (C ) to another D , and F satisfies the fol(D ) is called a functor, denoted as F: C lowing conditions:

is a zero object A. z . If A is a singleton object A. s , there is only one element in A. s . In some cases, a singleton object can be also a terminal object. For example in the category of sets, every one-element set is a terminal object, and in the category of topological space, every onepoint space is a terminal object. There are also cases where singleton objects are neither terminal nor initial. For instance in the category of non-empty sets, every non-empty set has a function from a singleton, but this function is in general not unique.

There are also rules that define the properties of objects. As terminal and initial objects are unique up to isomorphism, if two objects A and B are both terminal objects (or both initial objects), there is a unique isomorphism ( ) from object A and B, see Appendix A.

B in C , there is a morphism Condition 1: For each morphism f :A F (f ):F (A) F (B ) in D ; Condition 2: For each object A in C , the equation F (idA) = idFA holds in D ; Condition 3: For each pair of morphisms A equation F (g f ) = F (g ) F (f ) holds in D .

f

B

g

C in C , the

Different commutative functors while obeying certain categorical laws, form various functor structures, in such a way that a higher level of complex structures can be established.

2.2.2. Morphisms A relationship from object A to B in C , is a morphism, written as f :A B . Here A is the domain of f, denoted A = f (O1) and B is the codomain of f, written as B = f (O2 ) . A set represents all morphisms from objects A to B in C , written as MC (A, B ) . For any object A NO , a specified morphism, denoted as id (A) is the identity morphism on object A. A morphism f may also has one of six special properties, written as f . p , where p can be null (a morphism may not have any properties), epic (denote as , definition refer to [26] p31), monic ( , definition refer to [26] p29), isomorphic ( , definition refer to [26] p12), re, also called split epic, definition refer to [27] p49), sectraction ( tion ( , also called split monic, definition refer to [27] p49), both ) as indicated in Table 3. Note epic and monic but not isomorphic ( that a morphism can only have at most two properties such that a morphism can be two types of morphism simultaneously.

B is often assigned with a notion for its readA morphism f :A B . A notion of a morphism often reability, written as f :A (notion) presents the meaning of the relationship between the domain and codomain objects, and it could be started with characters such as ‘is’, ‘has’, ‘with’, ‘applied to’, ‘assigned to’, examples are given in the following sections.

2.2. Categorical modelling The totally ordered set of operations for the verification/specification can be modelled by employing the aforementioned categorical concepts such as categories, objects and morphisms. Multiple commutative morphisms can form geometric shapes such as lines, triangles (as shown in Fig. 1a, b) or rectangles in which one of the four morphisms’ property can generate categorical structures such as pullbacks or pushouts. In this paper those geometric shapes are redefined as “morphism structures”, to facilitate reasoning and to improve the efficiency of knowledge retrieval and other operations. The following terminology of categorical modelling is rooted in category theory, but represents different syntax and semantics comparing to the category theory.

In the categorical modelling, a category C is denoted by a triple (NO , NM , NS ), where NO is a set of objects, NM is a set of morphisms and NS is a set of morphism structures. All objects and morphisms satisfy the set of category laws. Note that the following syntax and semantics of objects and morphisms are different from the original definitions (as introduced in subSection 2.1) from category theory.

2.2.3. Morphism structures As morphism has a predefined path, different numbers of morphisms binding together (which obey categorical laws) form various structures. For example as shown in Fig. 1b, morphisms f , g and composite of f and g form a triangle shaped morphism structure, in such a way that more complex structures can be constructed based on categorical concepts.

In the categorical modelling, product structures (×), coproduct structures ( ), triangle structures ( ), rectangle structures ( ), pullback/pushout structures ( / ) are redefined with enriched details and more deduced structures, as shown in Fig. 2. A morphism structure may allow nesting of other morphism structures, for example, rectangle structure is a nesting of two triangle structures.

Triangle structure, as shown in Fig. 2a, is defined based on the composition rule in category theory, as described in Section 2.1 (morphism condition 2). Some examples of the composition rule can refer to [26] p4 and p9.

2.2.1. Objects Let A be an object in a category C , it may also be one of five special type of objects with extra properties as shown in Table 2, written as A. p , where p :: = t|i|z|s|e . The five types are: terminal object (denoted as t), initial object (i), zero object (z), singleton object (s) and empty object (e). The properties of an object are decided either by the morphisms to/from the object, or decided by the elements in the object.

If A is a terminal object A. t , for every object B in C , there is a unique morphism from B to A. t . A terminal object is of importance within a category, as it provides closure in a category structure and it represents the identity of the category. If A is an initial object A. i , for any object B C there is a unique morphism from A. i to B. For example in the category of sets, is the unique initial object, same as the empty space in the category of topological space. If A is both terminal and initial, A 349

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Fig. 2. Morphism structures.

({O1, O2, O3}, {m1, m2, m3}) is Definition 2.1. Triangle structure formed by two commutative morphism m1 and m2 among three objects {O1, O2, O3} and a composition morphism of the two m3 = m2 m1. The first object O1 of a triangle structure is denoted as , the second object as (O2) , and the third object as (O3) . The first, second and third morphisms are written as (m1), (m2 ) and (m3) respectively.

isomorphic, and pushout when m1 or m3 is either epic or isomorphic.

Therefore it is of importance to define a rectangle structure in order to deduce and retrieve pullback and pushout structures with less computational steps.

({O1, Definition 2.4.

Rectangle structure O2, O3, O4}, {m1, m2 , m3, m4}, 1, 2) is formed by two triangle structures and 1 ({O1, O2, O4}, {m1, m2 , m1 m2}) 2 ({O1, O3, O4 }, {m3, m4 , m3 m4}) , and the two triangle structures share the same (m3). Therefore in total a rectangle structure has four morphisms and four objects. Here 1 (m1) = (m1) , 1 (m2 ) = (m2) , 2 (m1) = (m3) , (O1) is the staring object of the rectangle (m4 ) . And 2 (m2 ) = structure and (O4 ) is the ending object.

Product structure, as shown in Fig. 2b, is redefined based on Product in category theory. Definition and proofs of Product can easily be found in the category literature.

Definition 2.2. Product structure × (O1, O2, p1 , p2 ) is constructed by a product object O1 × O2 , and two projection morphisms p1 and p2 , where p1 : O1 × O2 O1, p2 : O1 × O2 O2 . If there is another object O3 that O1, g : O3 O2 , has two project morphisms f and g, where f : O3 u: O3 O1 × O2 , and there exists a unique morphism p1 u = f , p2 u = g .

Pullback structure, as shown in Fig. 2e, is mainly defined based on pullback in category theory. Definition and proofs of pullback please refer to Ref. [26] p91.

({O1, …, O5}, Definition 2.5.

Pullback structure {m1, …, m 9}, { 1, …, 4}) is constructed from a rectangle structure 1 in which 1 (m2 ) or 1 (m4) is either monic or isomorphic. In total a pullback structure consists of a set of five objects and a set of nine morphisms whose objects and morphisms form four rectangle structures including 1.

As shown in Fig. 2e, the four rectangle structures are listed as follows: where 1 ({A , B , C , D}, { 1, 2, 3, 4}, 1, 2) , 1 ({A , B , D}, { 1, 2, 2 1}) and 2 ({A, C , D}, { 3, 4, 4 3}) ; where 2 ({E , B , C , D}, {q1, 2, q2 , 4 }, 3 , 4) , 3 ({E , B , D}, {q1, 2, 2 q1}) and 4 ({E , C , D}, {q2 , 4 , 4 q2}) ; Note that by applying the composition rule, more triangle structures can be generated from the standard definition of a pullback in category theory. Using the definition of rectangle structure (see Definition 2.4), the generated triangle structures can be used to establish more rectangle structures, i.e. 3 and 4 : where 3 ({E , A , C , D}, {u , 4 3, q2 , 4 }, 5 , 4) , 5 ({E , A , D}, {u, 4 3, 4 3 u}) in which two morphisms ( 4 3 and 4 3 u ) are deduced from the composition rule; 4 ({E , B , A , D}, {q1, 2, u , 4 3}, 3, 5) .

Apart from 1, other rectangle structures (can be also written as ) all start with object E and end with object D. For any , morphisms u, 3 and q2 always form a triangle structure 6 (u, 3, q2 ) , so do

Coproduct structure, as shown in Fig. 2c, is redefined based on Coproduct in category theory. Definition and proofs of Coproduct please refer to Ref. [26] p55.

Definition 2.3. Coproduct structure (O1, O2, i1, i2) is constructed by a coproduct object O1 O2 , and two inclusion morphisms i1 and i2 , where i1: O1 O1 O2, i2: O2 O1 O2 . If there is an object O3 with two O3, g : O2 O3 , there exists a unique inclusion morphisms f : O1 O3 , and u i1 = f , u i2 = g . morphism u : O1 O2 A product structure include two triangles structures in which one of the morphisms domain has to be a product object (condition 1); and one of the three morphisms with the product object as a domain, has to be a projection morphism (condition 2). Whether two arbitrary triangle structures can form a product is fully depending on the two conditions.

Similar conditions apply to the coproduct structure.

Rectangle structure, as shown in Fig. 2d, is defined to establish pullback and pushout structures. Rectangle is never clearly defined in category theory, but it is an essential form to combine two triangle structures whenever they share the same m3 morphism. Most importantly, from a rectangle, one can decide if a pullback or a pushout exists. A pullback/pushout structure is constructed based on a rectangle structure, if and only if in the rectangle structure, morphisms have certain properties: pullback when m2 or m4 is either monic or 350

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morphisms u,

1

and q1 form

7 (u ,

1,

q1) .

workpiece fulfils the specification [22]. It is normally accomplished by first performing a measurement that provides a measurement result with an associated uncertainty. Then the measurement result is compared to the specification limit(s) taking into account GPS principles.

The measurement procedure and the following conformance check together form the verification process. As a specification is sometimes incomplete, the conformance check may not be able to be performed, then the modelling of verification in this paper only deals with the measurement procedure.

As the basis for the measurement procedure, a verification operator is constructed by an ordered set of verification operations which are defined as “specific tools required to obtain features or values of characteristics, their nominal value and their limit(s)” [22]. Eight feature operations are defined in the GPS system. They are ‘partition’, ‘extraction’, ‘filtration’, ‘association’, ‘collection’, ‘construction’, ‘reconstruction’ and ‘evaluation’, each of which can be modelled by a category Pi , where: Partition PPa is to identify bounded features, for example points, straight lines or planes from non-ideal surface features; Extraction PEx is used to identify a finite number of points from a feature with specific rules; Filtration PFi is used to distinguish the feature in different scales, for example in profile surface texture, filtration is used to distinguish roughness, waviness and form; Association PAs is to fit ideal features to non-ideal features according to specific criteria which give an objective for a characteristic and it can set constraints; Collection PCo is to identify and consider some features which together play a functional role; Construction PCn is to build ideal features from other features; Reconstruction PRe is to reconstruct a continuous feature from a finite number of points and is the inverse of extraction; Evaluation PEv is to identify either the value of a characteristic or its nominal value and its limit(s).

The modelling of a verification, includes at least a set of operation categories {Pi } , and the number of operations |{Pi }| is n. Subject to the types of geometrical variability (form, orientation, location, run-out, surface texture), the order of the set of operations in a verification operator differs. Some of the operations need to be performed more than once. A typical example of a verification operator is illustrated in Fig. 3. For tolerances of location, orientation or run-out, there are at least two or more operators. It is because these tolerances are specified with a datum system in which each datum has a corresponding operator as well.

Pushout structure, as shown in Fig. 2f, is mainly defined based on pushout in category theory. Definition and proofs of pushout please refer to Ref. [26] p91.

({O1, …, O5}, Definition 2.6.

Pushout structure {m1, …, m 9}, { 1, …, 4}) is constructed from a rectangle structure 1 in which 1 (m1) or 1 (m3) is either epic or isomorphic. It consists of a set of five objects and a set of nine morphisms, whose objects and morphisms form four rectangle structures including 1.

As shown in Fig. 2f, the four rectangle structures is listed as follows: where 1 ({A , B , C , D}, { 1, 2, 3, 4}, 1, 2) , 1 ({A , B , D}, { 1, 2, 2 1}) and 2 ({A, C , D}, { 3, 4, 4 3}) ; where 2 ({A , B , C , E }, { 1, q1, 3, q2}, 3, 4 ) , 3 ({A , B , E }, { 1, q1, q1 1}) and 4 ({A, C , E }, { 3, q2 , q2 3}) ; where 3 ({A , B , D , E }, { 1, q1, 2 1, u}, 3, 5) , 5 ({A , D , E }, { 2 1, u, u 2 1}) ; 4 ({A , D , C , E }, { 4 3, u , 3, q2}, 5 , 4) .

Note that 3 and 4 are not from the standard pushout definition and they are also deduced from the composition rule and application of rectangle structure.

All rectangle structures in the pushout structure start with an object A apart from 1, the other rectangle structures ( ) all end with object E. For any , morphisms 2, u and q1 always form a triangle structure 6 ( 2, u , q1) , so do 7 ( 4, u , q2 ) .

2.2.4. Functors At a higher level, a morphism can also be a relationship between two categories, and this morphism is called a functor.

Definition 2.7. Functor: relationship F from category C1 to C2 , denoted C2 . It is constructed by a tuple (F O , F M ) where F O is the as F: C1 mapping between the two objects sets C1. NO and C2. NO , and F M is the mapping between the two morphisms sets C1. NM and C2. NM , in such a way that for every pair of objects A, B C1. NO , the functor F represent the mapping from the set of morphisms from A to B in C1 to the set of F (B ) F (A ) C2 , morphisms from to in written as F : MC1 (A, B ) MC2 (F (A), F (B )) .

A functor F may also has a special property. As shown in Table 4, nine properties are listed (there are more properties than nine in category theory, however only nine of them are defined in the categorical modelling in this paper). They are full (denote as ), faithful ( ), full faithful ( ), injective on objects ( ), injective on morph), surjective on objects (O ), sujective on morphisms isms ( (M ), bijective on objects (O ), bijective on morphisms ( M ).

3.2. Constructing specification semantics from verification

3. Designing specification from verification

3.2.1. A set of functors from verification to specification Specifications of a geometrical product, with the form of zone, are expressions of the permissible variation of a non-ideal feature inside a space limited by an ideal feature(s). The modelling of a specification, also includes a set of specification operation categories {SPi} . While the two sets {Pi } and {SPi} are independent, they are however related by the “Duality Principle” in GPS, that is, for any operations in the specification, there are corresponding physical operations in the verification. For example, if the specification operation category SPFi

The proposed metrology-oriented specification schema allows deduction of the structure of essential specification elements from the measurement details. The measurement process has to be modelled first such that the specification can be reasoned from the verification model.

3.1. Constructing verification categories Verification is the provision of objective evidence that the

Fig. 3. An example of a verification operator.

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represents the filtration operation, the verification operation category PFi will be the physical filtration operation when the specification is interpreted in the verification process.

The duality principle can be understood as the forward mapping from a specification operator to a verification operator, and it can be represented by a set of full faithful functors {FSi} , for example FS1: SPPa PPa . The inverse mapping can then be represented by a set SPi from a verification operator to a specification of functors FVi : Pi operator.

To be able to construct a set of specification operation categories {SPi} from that of verification {Pi } , the specified feature should be known, as the verification operator(s) cannot be formed without knowing the specified feature. The construing process including the following five steps:

then be simplified.

• For any triangle structures • •

Step 1 : Construct a set of full faithful functors FVi : map the actual verification operation categories {Pi } to the specification operation categories {SPi} ; Step 2 : Identify properties of objects, and properties of morphisms in {SPi} ; Step 3 : Determine morphism structures × , , , , and in {SPi} ; Step 4 : Apply a set of simplification rules iteratively to simplify objects in {SPi} until there are no more objects that can be simplified. A set of simplification rules is defined in the following sub-section; Step 5 : Delete categories with no objects, and then determine the new totally ordered set of specification operations {SPm} .

•

4. Case studies 4.1. Providing specification structure for industrial X-ray computed tomography (CT) geometrical measurement

Note that after the simplification process, the number of operations in specification may be smaller than that of verification (P1 Pn ), that is m n . The derived specification should be the simplest solution with minimal number of categories and objects/morphisms.

To validate the proposed method, it is proposed to generate a possible specification structure for a geometrical product that specifically requires CT measurement which is currently one of the most popular technologies for non-destructive full-field scanning.

In this test case, the design template of a turbocharger impeller was supplied as a CAD model (as shown in Fig. 4a) and was manufactured using selective laser sintering (SLS) technique (as shown in Fig. 4b). The product was then measured using an X-Tek XTH225 kv system (Nikon Metrology), where data was acquired at 120 kV and 150 µ A X-ray power, and in total 1583 projections were collected over 360 degrees with 1000 ms exposure time per frame and 2 frames per projections.

After the acquisition the projections were then reconstructed by a commercial software CT Pro 3D (Nikon Metrology). The reconstructed data voxel volume was approximately 93.685 µm3 . The reconstructed volume file was then used to create a triangular mesh surface using the open-source image processing program IsoSurfaceExtraction [30,31].

The mesh surface was saved in mesh format (.ply in this case), and then was compared with the CAD model using the open-source system MeshLab 2016. The two surfaces were initially aligned by manually selecting at least four matching points (as shown in Fig. 5), after reaching either maximum 100 iteration or the target distance of 0.005 mm, the alignment terminated and the deviations for the two steps were then calculated as shown in Fig. 6.

A CT measurement differs from a general GPS operations' sequence, it can be

3.2.2. The simplification rules The set of specification operation categories {SPi} has to be simplified to obtain a specification structure with only necessary details. To ensure the stability of a design, when there are two or more specification elements in the solution, they must be independent according to the axiomatic design theory which was developed by Suh [28,29].

Those independent specification elements can then be mapped back to form a complete specification operator and then construct a complete verification operator using the forward mappings {FSi} . Therefore it can ensure that the forward mapping from the simplified specification to verification is without loss of information.

Simplifying the objects in each SPi is decided by the special type of properties of its objects and morphisms, and decided by the types of morphism structures. A set of simplification rules is defined as follows where the kept objects during the simplification process are called necessary objects, and the emitted are called simplified objects:

• For any initial objects: •

({O1, O2, O3}, {m1, m2, m3}) : – Rule 6: If all O1, O2 and O3 cannot be deduced by other objects, label O2 as the simplified objects, label O1 and O3 as the necessary objects; – Rule 7: If both O1 and O3 are simplified objects, so is O2 ; ({O1, O2, O3, O4}, For any rectangle structures {m1, m2, m3, m4}, 1, 2) , any pullback structures ({O1, …, O5}, {m1, …, m9}, { 1, …, 4}) and any pushout structures ({O1, …, O5}, {m1, …, m 9}, { 1, …, 4}) , Rule 6 and Rule 7 can also be applied; For any isomorphisms in the remaining morphisms from the above simplifications, the following rules are applied: – Rule 8: If either object A or B is a simplified object, the other object will also be a simplified object; – Rule 9: If neither object A or B is a simplified object, only one of the objects has to be the necessary object, and the selection criteria is to choose the simpler object (require less storage space or computation) as the necessary object. This rule is an application of Ockham’s razor (also known as Occams razor). When two competing interpretations make the same predication, the simpler one is better.

For any epic, retraction or section morphisms f: – Rule 10: Label the domains f (O1) as necessary objects and label the codomains f (O2) as simplified objects.

– Rule 1: If any of other objects in the same category are necessary objects, the initial objects are necessary objects; – Rule 2: If all other objects in the same category are simplified objects, so are the initial objects; For any product structures × (O1, O2, p1 , p2 ) : – Rule 3: If both or only one of the objects O1 and O2 can be deduced by other objects other then the product object O1 × O2 , O1 × O2 can be simplified; – Rule 4: If none of the objects O1 and O2 , nor the product object O1 × O2 are simplified objects, only the two objects O1 and O2 can be labelled as necessary objects.

– Rule 5: If the product object O1 × O2 can be deduced by other objects other than objects O1 and O2 , the two objects O1 and O2 can

1. Acquisition PAc : acquire a set of gray value images of the X-ray projections that are collected over 360 degrees of the object; 2. Reconstruction PRe : reconstruct the volume model from the acquired 2D projection images; 3. Extraction PEx : generate a triangular mesh surface model from the volume model; 4. Alignment PAl : align the mesh surface with the nominal CAD file;

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Fig. 4. CAD model and AM part of a turbocharger impeller.

Fig. 5. Alignment points of two stages.

solve the localisation or alignment problem which was caused by the misalignment between the mesh surface and the CAD surface; 5. Partition PPa : this operation may be required to identify various bounded features such as plane, cylindrical surface, or a section of a complex surface, etc. according to the functional importance of the features.

6. Filtration PFi : filter the residual surface or partitioned surface to derive dimensional, form and surface texture; 7. Evaluation PEv : calculate the value of the specified parameter(s).

The elements in the object [Rotation Matrix] are in category PAl :

Mi =

0.8744 0.4849 0.007435 0.04692 0.4851 0.8733 0 0

0.01627 0.9989 0.04464 0

18.11 8.648 64.86 1

and

Mf =

In this case, only the first four operations (PAc , PRe, PEx and PAl ) were modelled to obtain the residual form error without any filtration and evaluation operations.

1 0.003939 0.003254 0

0.003941 0.003252 1 0.0004268 0.0004139 1 0 0

0.01289 0.163 0.0225 1

In the operation categories as shown in Fig. 7, morphism structures

Fig. 6. Alignment results of two stages.

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Fig. 7. Categorical model of the CT verification.

are then identified and listed in Table 1.

After the simplification process, necessary objects are indicated with green background and they are essential in the specification in order to derive an unambiguous measurement procedure. Simplified objects are labelled with grey background. Objects with blue background have to be manually chosen by the inspectors. Objects with white background are known objects prior to the measurement procedure. As a result, the remaining independent objects are [Projections] in the category PAc , Reconstruction slice [Quality] and [Noise reduction] in the category PRe , Threshold [iso-value] in the category PEx , [Points] and [Strategy] in the category PAl and they are recommended to specified in the designing phase.

In this section, two examples of specifications for profile surface texture (PST) and cylindricity were selected to illustrate the detection of redundancy and incompleteness respectively using the proposed method.

4.2.1. For specification redundancy A PST specification symbol, as shown in Fig. 8a, is illustrated in ISO 1302:2002 [33] to explain the meaning of each control elements of a specification symbol. To detect any redundancy or incompleteness of such symbol, the first step is to model the specification based on the general operation categories, and there are four totally ordered operations which are Partition (SPPa ) Extraction (SPEx ) Filtration (SPFi ) Evaluation (SPEv ). The PST model can then be constructed as shown in Fig. 9.

In the category SPPa , there are three product structures which are ×1(Shape, Size, p2 , p3 ); ×2 (Orientation, ×1, p1 , {p2 , p3 } ); and ×3 ( s , n, p5 , p4 ), where object [Orientation] and [Number of Cutoff (n)] cannot be decided by other objects thus they are the necessary objects. Note that in ×3 , the codomain of p5 is in the other category SPFi .

In the category SPEx , there is only one product structures ×4 (Max Sphere Radius, Max Sampling Distance, p8 , p9), where object [Max Sphere Radius] and [Max Sampling Distance] can both be simplified thus there are no necessary objects in this category.

In the category SPFi , there is only one product structures ×5 ( c , s , p6 , p7 ), where object [ c ] and [ s ] can be simplified, and as there is an isomorphism between object [Filter] and [Filter Symbol], the simpler object [Filter Symbol] will be labelled as necessary object.

In the category SPEv , there is only one product structures ×6 (Limit Parameter, Limit Value, p10 , p11), where object [Limit Parameter] and [Limit Value] cannot be simplified: they are essential to decide objects [Sampling length] such that they are labelled as necessary object.

4.2. Checking specifications for redundancy and incompleteness The metrology-oriented method can also be applied on specifications for checking their redundancies, incompletenesses or inconsistencies.

For a specification’s redundancy, the proposed method can identify specification elements that can be deduced from other elements, such that those unnecessary elements can be removed to derive a compact specification.

For a specification’s incompleteness, the proposed method can identify missing specification elements that needs to be defined to deduce a complete measurement plan with a small method uncertainty [32], according to the specification structure which is defined by GPS. This means that it is necessary for a specification to explicitly state which values are intended for use with these specification operations in order for the specification to be unique.

For a specification’s inconsistency, the proposed method can identify any incorrect selection of values in specification elements in the specification model.

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After the simplification process, the result specification is indicated in Fig. 8b.

Table 1 Objects and morphism structures in the CT model.

Category

Prop.

Semantics

PAc

t ×

CT ×1(Voltage, Current, p1 , p2 ) ×2 (Material, Thickness, p3 , p4 ) ×3 (Size, Shape, p6 , p7 )

4.2.2. For specification incompleteness A typical cylindricity specification symbol is shown in Fig. 10a.

From the measurement requirement of cylindricity, there are five totally ordered operation categories which are Partition (SPPa ) Extraction (SPEx ) Association (SPAs ) Filtration (SPFi ) Evaluation (SPEv ). According to the general operation categories the cylindricity model can then be constructed as shown in Fig. 11.

After the simplification process, it can identify that objects [Limit Parameter], [Limit Value] and [Filter Symbol] are necessary objects. To complete the specification as shown in Fig. 10a, the default settings in ISO GPS system should also be considered. In the category of SPFi , the [Filter] can use default Gaussian filter, but in the [Nesting Index], the filter undulation per revolution (UPR) has no default values; and for a cylindricity specification UPR, both generatrix and redial section have to be defined. After the incompleteness check, an example of a complete cylindricity specification is indicated in Fig. 10b.

×4 ({Material}, ×3, p5 , {p6 , p7 } ) 1({(V ,

PRe

t ×

PEx

t ×

I ) , Voxel, Volume}, {m5, m7 , m7 m5} ) Time}, {m6, m9, m9 m6} ) Software ×5 (Quality, Noise reduction, p8 , p9 ) ×6 (Radius, Resolution, p10 , p11 ) 2 (Size,Volume,

3 ({Software, Rotation, Volume}, {m13, m14 , m14 m13} ) Software ×7 (iso-value, Mesh Surface, p12 , p13 ) ×8 (Faces, Vertices, p14 , p15 ) 4 ({Software,

PAl

t ×

iso-value, Material}, {p12 , m16, m16 p12 } )

5 ({Software,iso-value,

Background}, {p12 , m17, m17 p12 } )) Software ×9 (Points, Strategy, p16 , p17 )

×10 (CAD , Mesh', p18 , p19 ) ×11 (Target, Max Iteration, p20 , p22 ) ×12 (Target, Max Iteration, p21 , p23 ) ×13 (Rotation Matrix, Evaluation, p24 , p26 ) ×14 (Rotation Matrix, Evaluation, p25 , p27 )

5. Conclusion In this paper a metrology-oriented specification schema that enriches the specification semantics with full sets of measurement information is proposed. It is designed particularly for applications where non-traditional measurement methods are applied, as well as for identifying any redundancy, inconsistency or incompleteness of a specification. A high-level abstraction mathematical theory - category theory, has been adopted to model the measurement process and generate a proposed specification structure using a set of simplification rules.

For general geometric products, the model helps to generate specifications that comply with GPS specification rules but without redundancy. For complex geometric products whose specification re-

×15 (×11, ×13, {p20 , p22 }, {p24 , p26 } ) ×16 (×12 , ×14 , {p21 , p23 }, {p25 , p27 } ) 6 ({CAD,

Software, Resampling}, {m19, m21, m21 m19} ) Resampling, ×10 ,} m20, m22, m22 m20} ) 8 ({CAD, Software, ×10 }, m19 , m22 m21, m22 m21 m19} ) 9 ({×10 , Initial Align., Final Align.}, m23, m24 , m24 m23} ) 10 ({Initial Align., Final Align., Residual}, m24 , m25, m25 m24} ) 11 ({ ×10 , Final Align., Residual}, m24 m23, m25 , m25 m24 m23} ) 1 ({CAD, Resampling, Software, ×10 }, {m20 , m22, m19, m22 m21}, 7, 8 ) 7 ({CAD,

Fig. 8. A simplification example of specification of profile surface texture.

Fig. 9. Categorical model of the PST specification.

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Fig. 10. Specification examples of cylindricity.

Fig. 11. Categorical model of the cylindricity specification.

quirements are still unknown, the model helps with forming a specification structure that with minimum/independent specification objects.

Three test cases were conducted to validate the proposed schema. A CT measurement process for a turbocharger impeller manufactured using SLS technique, was modelled and a set of independent specification elements was deduced. The other two test cases for checking general specifications (from ISO standards) for redundancy and incompleteness respectively were carried out. The results show that the proposed schema works for proposing enriched specification that are characterised by new measurement methods, and it works for testing the redundancy and incompleteness of specifications that are designed based on ISO standards. It also expects that the proposed method works well for detecting any inconsistency among elements of a specification.

The most challenge part of the proposed scheme is that the verification or specification has to be modelled first, mainly by hand.

However as typical ISO GPS operation categories were constructed, any

new verification/specification models can be edited, rearranged or added into the existing categories. It can also conclude that the completeness of a specification, is not meant to specify all the specification details. A specification should have a minimum number of independent specification objects that can deduce a complete specification operator.

The proposed schema is developed to achieve this goal.

Acknowledgment The authors gratefully acknowledge the UK’s Engineering and Physical Sciences Research Council (EPSRC) funding of the EPSRC Fellowship in Manufacturing: Controlling Geometrical Variability of Products for Manufacturing (Ref:EP/K037374/1), and funding of Future Manufacturing Research Hubs: Future Advanced Metrology Hub (Ref:EP/P006930/1).

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Appendix A. Objects Table 2 Table 2 Special properties of object A. Constructor

Syntax

Semantics

Terminal Object Initial Object Zero Object Singleton Object Empty Object

A. A. A. A. A.

B C . NO, !f : B A B C . NO, !f : A B A. z A. t & A. i id (A): A A , B C . NO, MC (A , B ) = {id (A)} A=

t i z s e

Rules for object properties: A, B

C . NO, A . t & B. t

Then

!f : A

| If A, B

C . NO, A . i & B. i

B

Appendix B. Morphisms Special properties of morphism f : A

B . Table 3.

Table 3 Special properties of morphism f.

Constructor

Synatx

Epic Monic Isomorphic

f: A f: A

B

f: A

B

Retraction

f: A

Epic& Monic

f: A

Section

f: A

Semantics

A, B , C A, B , C

B B

B

B

C . NO, g: B C . NO, g : A

C , h: B B , h: A 1:

A, B

C . NO, f : A

A, B

C . NO,

A, B A, B

C . NO, f 1: B A, f C . NO. f : A B & f: A

f

1:

B.

f

B

A, f f

A, f

B 1

g=h g=h

C , {g f = h f } B, {f g = f h} 1

1

f = id (A), f f

1

= id (B )

= id (B )

f = id (A) B & !f : A

B

Appendix C. Functors Several special properties of functor F: C1 Table 4 Special properties of functor F: C1 Constructor

C2 : Table 4.

C2 .

Synatx

Semantics

Faithful

A, B C1. NO, F : MC 1 (A , B ) injective A, B C1. NO, F : MC 1 (A , B )

Full

surjective A, B C1. NO, F : MC 1 (A , B ) bijective

Full Faithful injectiveO

surjectiveM

bijectiveO

bijectiveM

MC 2 (F (A), F (B )) is MC 2 (F (A), F (B )) is

F O is injective

injectiveM surjectiveO

MC 2 (F (A), F (B )) is

F M is injective

F O is surjective

O

F M is surjective

M

F O is bijective

O

F M is bijective

M

Appendix D. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.aei.2018.11.001.

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