A categorical framework for formalising knowledge in additive manufacturing

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Procedia CIRP 00 (2017) 000–000 Procedia CIRP 75 (2018) 87–91 Procedia CIRP 00 (2018) 000–000 Procedia CIRP 00 (2018) 000–000

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15th CIRP Conference on Computer Aided Tolerancing – CIRP CAT 2018 15th CIRP Conference on Computer Aided Tolerancing – CIRP CAT 2018

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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: [email protected] Corresponding author. Tel.: +44-1484-471284; fax: +44-1484-472161. E-mail address: [email protected]

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 [email protected] 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 guidelines It areurges no longer applicable for AM.that As acomplex result design for additive manufacturing lacks properties.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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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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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]



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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).

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