Quantifying Manufacturability of Component Geometries Using Shape Descriptors

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Procedia CIRP 00 (2017) 000–000 Procedia CIRP 84 (2019) 462–467 Procedia CIRP 00 (2019) 000–000 Procedia CIRP 00 (2019) 000–000

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29th 29th CIRP CIRP Design Design 2019 2019 (CIRP (CIRP Design Design 2019) 2019)

Quantifying of Geometries Using Shape Quantifying Manufacturability Manufacturability of Component Component Geometries 28th CIRP Design Conference, May 2018, Nantes, France Using Shape Descriptors Descriptors A new methodology to analyze the functional and∗∗ physical architecture of Meng Basu, Meng Liu, Liu, Saurabh Saurabh Basu, Soundar Soundar Kumara Kumara The Harold and Inge Marcus Department of Industrial and Manufacturingoriented Engineering , The Pennsylvania State University, University Park PA 16802, U.S.A. existing products for an assembly family identification The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering , The product Pennsylvania State University, University Park PA 16802, U.S.A. Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat Abstract Abstract École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France A A Do-It-Yourself Do-It-Yourself (DIY) (DIY) framework framework of of part part design design and and manufacturing manufacturing is is proposed. proposed. This This framework framework relies relies on on aa digitally digitally indexed indexed part/process-plan part/process-plan that enable amateur fabricators in manufacturing *library Corresponding Tel.: +33 3 87 37 54 30; E-mail address:and [email protected] library that can canauthor. enable amateur fabricators in designing designing and manufacturing complex complex shapes shapes with with similar similar tolerance tolerance as as professional professional counterparts. counterparts. A A methodology methodology to to do do this this is is proposed, proposed, which which involves involves automated automated identification identification of of parts parts from from the the library library that that has has similar similar functionality functionality as as desired. desired. This step will be followed by automated modification of the corresponding process plans to produce the desired functionality while conforming This step will be followed by automated modification of the corresponding process plans to produce the desired functionality while conforming to to design design and and machine machine tool tool requirements. requirements. Some Some challenges challenges associated associated with with setting setting up up such such frameworks frameworks are are realized. realized. These These include include identifying identifying methodologies for quantifying manufacturability of a final geometry from a starting billet shape using a specified manufacturing Abstract methodologies for quantifying manufacturability of a final geometry from a starting billet shape using a specified manufacturing process. process. The The utility utility of of spherical spherical harmonic harmonic based based shape shape descriptors descriptors is is analyzed analyzed for for the the same. same. Shortcomings Shortcomings of of this this shape shape descriptor descriptor are are discussed. discussed. In today’s business environment, the trend towards more product variety and customization is unbroken. Due to this development, the need of cc 2019  2019 The Authors. Published by Elsevier B.V. agile andThe reconfigurable production systems B.V. emerged to cope with various products and product families. To design and optimize production © Authors. by  2019 The Authors. Published Published by Elsevier Elsevier B.V. Peer-review scientific committee of the CIRP CIRP Design Conference 2019. under responsibility of the scientific of Design Conference 2019. systems as well as to choose the optimal product matches, product analysis methods are needed. Peer-review under responsibility of the scientific committee committee of the the CIRP Design Conference 2019. Indeed, most of the known methods aim to analyze a product or one product family on the physical level. Different product families, however, may differ largely in terms of the number and Keywords: Manufacturing process; Do-It-Yourself; Shape descriptor Manufacturing process; Do-It-Yourself; Shape descriptor Keywords: nature of components. This fact impedes an efficient comparison and choice of appropriate product family combinations for the production system. A new methodology is proposed to analyze existing products in view of their functional and physical architecture. The aim is to cluster these products in new assembly oriented product families for the optimization of existing assembly lines and the creation of future reconfigurable functionality and plan fabricating these design feaassembly systems. Based on Datum Flow Chain, the physical structure of the products is analyzed. Functional subassemblies identified, functionality and aa process process plan for for fabricatingare these design and fea1. Introduction tures. Subsequently, the end-user procures sub-components a1.functional analysis is performed. Moreover, a hybrid functional and physical graph is the output which depicts that the Introduction tures.architecture Subsequently, the(HyFPAG) end-user procures sub-components that assembled to the desired functionality. In more similarity between product families by providing design support to both, are production system planners product designers. An are assembled to provide provide theand desired functionality. In aaillustrative more exexversion of end procures the Distributed manufacturing involves sharing of example of a nail-clipper is used tooften explain the proposed methodology. industrial case study two productthe families of steering columns of treme version of the theonframework, framework, the end user user procures the raw raw Distributed manufacturing often involves sharing of highhigh- An treme thyssenkrupp Presta France is then carriedusing out to give a first industrial evaluation of the proposed approach. material that is then processed in necessary steps to produce the resolution computer-aided designs high-speed digital material that is then processed in necessary steps to produce the resolution computer-aided designs using high-speed digital ©communication 2017 The Authors. Published by Elsevier B.V. part part with with the the desired desired functionality. functionality. communication by by aa source source agent, agent, e.g. e.g. designer. designer. This This informainformaPeer-review responsibility of the e.g. scientific committee of the 28th Conference 2018. The DIY framework of manufacturing is often motivated by tion is usedunder by the receiving agent, producer to fabricate the CIRP Design

The DIY framework of manufacturing is often motivated by tion is used by the receiving agent, e.g. producer to fabricate the reduced production machine component using aa compatible machine tool at their fareduced production costs costs that that would would otherwise otherwise be be required required for for machine component using compatible machine tool at their faKeywords: Assembly; Design method; Family identification hiring professional designers and fabricators and using cility in a geographically different location [1]. This framework hiring professional designers and fabricators and using industry industry cility in a geographically different location [1]. This framework standard of standard machine machine tools, tools, e.g. e.g. in in conventional conventional distributed distributed manumanuof manufacturing manufacturing relies relies extensively extensively on on the the body body of of knowledge knowledge facturing [4, 5, 6]. However, amateur DIY practitioners of best practices available independently to the designer and facturing [4, 5, 6]. However, amateur DIY practitioners can can face face of best practices available independently to the designer and challenges in complex features simioften 1.fabricator, Introduction the product range and characteristics manufactured significant challenges in producing producing complex features with withand/or simifabricator, often through through their their past past experiences. experiences. Herein, Herein, the the end end ofsignificant larly tolerances and rate as user this system. In the thissame context, main challengeproin larly tight tight in tolerances and at at the same rate the as professionally professionally prouser has has little little say say on on the the formulation formulation of of manufacturing manufacturing process process assembled duced counterparts. These shortcomings of the DIY framework plans. Due to the fast development in the domain of modelling and analysis is shortcomings now not onlyoftothe cope single duced counterparts. These DIYwith framework plans. can from the to Availability high-speed communicacommunication andeconomical an ongoing trend ofdigital digitization and products, a limited productexposure range orof existing product families, can originate originate from limited limited exposure of the end-user end-user to advanced advanced Availability of of economical high-speed digital communicadesign and manufacturing concepts. This shortcoming exaction is however revolutionizing manufacturing by changing the digitalization, manufacturing enterprises are facing important also and to bemanufacturing able to analyzeconcepts. and to compare products tois design This shortcoming isdefine exaction is however revolutionizing manufacturing by changing the but erbated by the non-availability of expensive industrial standard role played by the end user. For instance, widely available challenges environments: a continuing product families. It can be observed thatindustrial classical standard existing erbated by the non-availability of expensive role playedinbytoday’s the endmarket user. For instance, widely available new machine tools the end peer-to-peer digital content, e.g. demonstration videos, have fatendency towards reduction product development times function of clients or features. machinefamilies tools at atare theregrouped end user’s user’sindisposal. disposal. peer-to-peer digital content,of e.g. demonstration videos, haveand fa- product It is anticipated that an digital/indexable cilitated a Do-It-Yourself (DIY) framework of manufacturing shortened lifecycles. In addition, thereofismanufacturing an increasing However, assemblythat oriented product families are hardly tolibrary find. It is anticipated an open-source open-source digital/indexable library cilitated aproduct Do-It-Yourself (DIY) framework of manufacturable part designs and their process plans aid [2, 3]. Here, the end-user not only determines the functionaldemand of customization, at the same time a global the product family level, and products differ mainly in two ofOn manufacturable part designs their process plans can can aid [2, 3]. Here, the end-user being not only determines the infunctionalDIY practitioners. Such a library can be used to search and ity of the desired final product as in the past, but also detercompetition with competitors allasover thepast, world. (i) thea number of components (ii) and the DIY characteristics: practitioners. Such library can be used to and search ity of the desired final product in the butThis also trend, deter- main download existing designs and plans components mines the requisite features will download existing (e.g. designs and process process plans of of components which the development macro the to desired micro type of components mechanical, electrical, electronical). mines is theinducing requisite design design features that that from will provide provide the desired that match specified functional requirements. If similar parts that match specified functional requirements. If similar parts markets, results in diminished lot sizes due to augmenting Classical methodologies considering mainly single products ∗ Corresponding author. Tel.: +1-814-863-2359 ; fax: +1-814-863-4745. ∗ Corresponding author. Tel.: +1-814-863-2359 ; fax: +1-814-863-4745. are indeed found in the library, their process plans can then be are indeed found in the library, their process plans can then be product varieties (high-volume to low-volume production) [1]. or solitary, already existing product families analyze the E-mail address: [email protected] (Soundar Kumara). E-mail address: [email protected] (Soundar Kumara). modified minimally to fabricate the required design features. modified minimally fabricate the(components required design To cope with this augmenting variety as well as to be able to product structure on ato physical level level)features. which If the plans design be If required, required, the process process plans and and perhaps, thedefinition design can canand be c 2019 Theoptimization 2212-8271 possible  Authors. Published by Elsevier B.V. identify potentials in the existing causes difficulties regarding an perhaps, efficientthe c 2019 The Authors. Published 2212-8271  by Elsevier B.V. Peer-review under responsibility of the scientific committee of the CIRP Design Conference 2019. of different product families. Addressing this production system, it is important to have a precise knowledge comparison Peer-review under responsibility of the scientific committee of the CIRP Design Conference 2019. 2212-8271©©2017 2019The The Authors. Published by Elsevier 2212-8271 Authors. Published by Elsevier B.V. B.V. Peer-review under responsibility of scientific the scientific committee theCIRP CIRP Design Conference 2019. Peer-review under responsibility of the committee of the of 28th Design Conference 2018. 10.1016/j.procir.2019.04.308



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2. Objectives of DAMA

modified to make them compatible with the machine tools at the DIY practitioners disposal. Note that this can also assist the fabrication of components with best tolerances achievable using available tools. Automated modification of process plans requires quantification of manufacturability DM (G0 , G f ) of component features denoted G f from starting billet shape G0 using manufacturing process M. For instance, if part G f cannot be made from available billet shape G0 using manufacturing process M, the process plan and component design can be modified to G∗f that features DM (G0 , G∗f ) < DM (G0 , G f ). A critical component of such analyses is the mathematical representation of arbitrary shapes, e.g. G0 , G f , in order to characterize parameter DM (G0 , G f ). This is done using shape descriptors, which are feature vectors with real-valued components and fixed dimensions that represent the shape of objects. Shape descriptors may be roughly classified as: image-based, spherical-harmonics-based, statistic-based, 3D-geometry-based, and topology-matchingbased [7, 8]. In this article, we delineate some shortcomings of existing shape descriptors while trying to formulate the functional form of DM (G0 , G f ). This exercise is a crucial first step to achieve the overarching goal of implementing the proposed DIY framework. We coin this, the design-anywhere-manufactureanywhere (DAMA) framework. The design-anywhere aspect of this framework will involve automated selection of design features given functionality and dimensional requirements. This will be done using a digital library of design features that will be indexed with respect to their function. The manufacture anywhere aspect of the proposed framework will enable the novice fabricator to produce the aforementioned design features with the best tolerances that are permitted by locally available machine tools. This will be done by modifying and integrating process plans that already exist for the aforementioned design features. These aspects of DAMA naturally require: (i) organization of a digital library of components indexed with respect to their features and functions, and (ii) a platform that enables automated search of similar parts. The objective of this work is to identify whether an existing shape descriptor [9] that can been used in classifying a wide variety of manufactured shapes [10] can be used to characterize ease of manufacturability, i.e. DM (G0 , G f ). The current formulation of such descriptors naturally ignores information such as tolerance and material used. The success of a manufacturing process in fabricating a geometry is intimately governed by this information. However, in this exploratory research, we will also ignore this information. Herein, we implicitly assume that all geometries considered in this work can be manufactured using an ideal material to any tightness of tolerance using the specified manufacturing process. Section 2 attempts to quantify the objective of the DAMA framework. Section 3 summarizes literature review of state of the art shape descriptors that can be used in formulating DM (G0 , G f ). Section 4 demonstrates some shortcomings of a prototypical descriptor in the implementation of DAMA. Finally, section 5 proposed some future work.

2.1. Quantifying functionality of arbitrary shapes The function of an engineered component often motivates its design. We illustrate this concept using a chair, a common device. The primary function of a chair is to provide weight/lumbar support. These requirements naturally prescribe a chairs basic design features, i.e. 3-4 legs supporting a horizontal platform to which a near-vertically aligned flat is connected. Dimensions of these design features are naturally prescribed by some constraints, e.g. space requirements and maximum load bearing ability. Herein, commonality of the aforementioned design features across almost all chairs can be utilized in classifying chair-like shapes within a family of similar shapes, e.g. for shape recognition [9, 10]. Some chair like shapes that feature significant design differences, yet possesses similar features that can classify them as chairs are shown in Fig. 1.

Fig. 1. Chairs featuring different manufacturability and ergonomics.

However, existing frameworks do not enable automated classification of arbitrary shapes based on their function in a manner that is amenable for unsupervised learning. These limitations arise from multi-physics simulations that are often necessary for predicting the function of shapes [11]. For instance, the functional classification of a chair can be complicated by factors such as ergonomic quality. This implies that arbitrary shapes that can serve the basic function of a chair may not necessarily be ideal candidates for the same. On the other hand, several organic shapes that bear lesser resemblance with a basic chair can be better suited for human weight/lumbar support due to superior ergonomics. Automated qualification of arbitrary shapes with respect to their functionality is one overarching goal of DAMA. Using such frameworks, it will be possible for a novice designer to simply specify functionality and retrieve several arbitrary shapes that will meet these requirements. Nonetheless, it can be easily argued that many candidates that are retrieved using such frameworks are poorly suited with respect to their manufacturability. This argument naturally results in the second objective of DAMA, to quantify manufacturability of arbitrary shapes. 2.2. Quantifying manufacturability of arbitrary shapes

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Design for Manufacturability (DFM) considerations specify rules of thumb that must be satisfied for reducing the time and economic resources tied to fabrication. In addition to directly specifying geometric features of a component, DFM considerations can also be used to prescribe machine tool requirements [12] and some material characteristics [13]. This can be

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illustrated by elucidating fabrication of complex surface profiles, e.g. those used in turbine blades [14]. A standard singleaxis machine-tool cannot be used to fabricate such shapes as this requires tool approach from multiple directions. In this regard, DFM considerations under known machine-tool constraints, e.g. non-availability of 5-axis machine-tool, would naturally eliminate complex shapes, e.g. turbine blades from the space of easily manufacturable shapes. Section 3 provides background that is pertinent to the quantification of manufacturability of arbitrary shapes.

3D information [9, 17, 18], e.g. CAD file, and (ii) graph based, wherein connectivity characteristics of component features are quantified into graphs. Some additional techniques that are not feature or graph based also exist, e.g. those based on geometry representation. The efficacy of these shape descriptors in predicting manufacturability of shape G f from shape G0 using process M depends on their efficiency in identifying manufactured features. Here, efficiency depends inversely on the amount of information that is necessary to identify a manufactured feature. The rest of the article describes the efficacy of a spherical harmonic based shape descriptor in predicting manufacturability. The choice of feature-based shape descriptors for this research was based on their established robustness in handling complex shapes, which is necessary for implementing DAMA. Here, robustness refers to: (i) directional invariance, and (ii) ability to work with generic CAD-STL data that does not have pre-processed feature information. In pursuit of identifying the most efficient shape descriptor, we are also working on other approaches.

3. Background The objective of delineating manufacturability of arbitrary shapes is analogous to finding a dissimilarity measure DM (G0 , G f ) that quantifies the manufacturability of final shape G f with respect to starting shape G0 if the manufacturing process M is used. Much like dissimilarity measure of shape descriptors [15], the measure DM should feature: (i) identity, i.e. DM (G0 , Gi ) = DM (G0 , G j ), when Gi = G j , and (ii) positivity, i.e. DM (G0 , Gi ) > 0 ∀ G0  Gi . However, the function of DM naturally cannot be symmetric, i.e. DM (G0 , Gi )  DM (Gi , G0 ). This gives the function D M a sense of directionality that is often exhibited in manufacturing processes. This suggests that fabricating a shape Gi , e.g. a cylinder with a smaller diameter from another shape G0 , e.g. a cylinder with a larger diameter, is much easier than going in the reverse direction, e.g. when the turning process is used. The function DM should also exhibit the triangle inequality in many, but not all cases. The triangle inequality suggests that for three shapes G0 , Gi , G j , DM (G0 , G j ) ≤ DM (G0 , Gi ) + DM (Gi , G j ). This rule can be illustrated using turning of a cylindrical billet of larger diameter G0 to a smaller diameter, i.e. G j in two steps, the cylinder of intermediary diameter being represented as Gi . We argue that turning a cylinder G0 to cylinder Gi is as challenging as turning cylinder Gi to a cylinder G j . In this regard, DM (G0 , G j ) = DM (G0 , Gi ) = DM (Gi , G j ) and hence, the triangle inequality is naturally satisfied. The previous paragraph suggests that delineating manufacturability of an arbitrary shape G f from a starting billet shape G0 using manufacturing process M is closely tied to the quantitative description of its shape. Recent investigations have delineated methodologies for the same using process plans of component geometries [16]. In this research, it will be assumed that the process plans of prospective part designs are not available readily. This is a highly plausible scenario, e.g. when the DIY practitioner is searching for designs with similar functionality and equivalent manufacturability but may not have access to process plan related information. It is also realized that the function DM should depend on factors such as materials used and tolerance and these will be discussed in the forthcoming section on future work. In this article, we will limit the discussion to geometry effects. Towards the same, shape descriptors can be primarily classified as [7, 8]: (i) feature based, wherein features associated with the shape are directly extracted into high-dimensional vectors using available

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4. Delineating manufacturability using spherical harmonic based shape descriptor

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A brief of spherical harmonic based shaped descriptors is provided here and illustrated in Figure 2. More details are available in ref. [9]. The descriptor works by normalizing the largest dimension of a shape to 1 and discretizing the shape into a voxel representation. Voxels that lie within the volume of the shape are labeled 1 and those lying outside are labeled 0. Subsequently, the growth of a voxelized sphere located on the centroid voxel of the shape is simulated in discrete steps of progressively increasing radii. Voxels on the surface of the sphere are assigned the same value, e.g. 1 or 0, based on their positions with respect to the volume of the shape, e.g. inside or outside, respectively. These are denoted using yellow and blue color, respectively, under simulated sphere growth in Fig. 2. These spherical surface maps are then fitted to spherical harmonic functions. Amplitudes ||A(n, m)|| of individual wave components are characterized with respect to their quantum numbers n, m and radius of the sphere. Eventually, a radius vs. wave number vs. amplitude field is generated. This field provides the rotational invariant representation R(G) of any arbitrary shape G. Thereafter, shape search and identification are performed by finding Euclidean distances between representations available in a database with respect to the representation of an input shape. In order to delineate the efficacy of a spherical harmonic based shape descriptor in characterizing metric DM (G0 , G j ), we performed two numerical experiments. These experiments involved M = {m1 , m2 }, with m1 = turning of solid cylinders (e.g. G0 ) to cones (G1 (θ)) of specified included angle θ and m2 = boring of solid cylinders (e.g. G0 ) to hollowed cylinders (e.g. G2 (r, z)) featuring internal cylindrical holes of specified depth z and internal radius (IR) r, respectively. We subsequently find the distance parameters D M (G0 , G f ) = ||R(G f ) − R(G0 )|| for the



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Spherical Harmonic Decomposition ||A(n,m)|| ||A(n,l,m)||

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Fig. 2. Spherical harmonics based shape descriptor.

set of cones and bored cylinders, respectively. For the same, we use two distance formulations featuring p = 1 norm, and p = 2 norm. Shapes G0 , G f were enclosed in voxel cubes of dimension 42x42x42. The native resolution of the shapes was ≈ 10−2 voxel. The angular resolution associated with the simulated sphere was ≈ 2◦ . This voxel resolution (e.g. 42x42x42) was inspired from minimum (32x32x32) and maximum (64x64x64) resolutions used in ref. [9, 10]. Coarser resolutions provide computational efficiency at the cost of geometric information. The radius of the simulated sphere was increased by ≈ 2 voxel units in every step of its simulated growth. Cone and boredcylindrical shapes were generated in MATLAB. In both cases, the final maximum diameter and height of the cylinder that enclosed the fabricated shape was set as the initial diameter and height (cf. cone insets in Fig. 3a).

the less, the quadratic norm, i.e. p = 2, provides a better sense of manufacturability in comparison with the 1−norm, i.e. p = 1 by exhibiting a more gradual gradient in distance DM , with respect to changing included angle. This suggests that the spherical harmonic shape descriptor is a better predictor of manufacturability when quadratic norm p = 2 is used, when identifying manufacturability of cones from a starting cylindrical shape.

5. Results and Discussion 5.1. Cone Figure 3a illustrates the manufacturability of cones from cylindrical billets as delineated using the function D M (G0 , G f ) = ||R(G f (θ))−R(G0 )||. Figure 3b shows the derivative of Euclidean p-norm distances with respect to tan θ. Insets on the top of the figure 3a illustrate cone shapes featuring half angle tangents tan θ = 0 − 0.45 that correspond to half angles θ = 0 − 24.2◦ . For constructing this trend, these cones were decomposed to their spherical harmonics based representation R(G f ) as described in section 4. We argue that it is as challenging (or easy) to manufacture a cone of angle θ + dθ if a cone of angle θ is manufacturable by a DIY practitioner. However, figure 3 suggests that the ease of manufacturability DM continues to increase as the cone angle is increased, starting from a cylinder. The trend in parameter DM shows a decline when quadratic norm, i.e. p = 2. This is also evidenced in its decreasing derivaM tive ∂∂D tan θ (cf. Fig. 3b) for p = 2. We also intermittently see spikes in this curve that appear to be amplified for p = 1, presumably being consequences of numerical differentiation. None

Fig. 3. (a) Distance, and (b) Derivative of manufacturability of cones with half angle θ vs. tan θ

5.2. Bored hole

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Figure 4 shows distances of the shape descriptors of cylinders featuring internal bored holes of specified radii r and depths z. These distances are calculated using p = 1, norm e.g. Fig. 4b, and p = 2 norm, e.g. Fig. 4c, with respect to a starting cylindrical shape G0 featuring radius rc and hight zc as shown in Fig. 4a, respectively. Contrary to the trend produced in the parameter DM with p = 1 when manufacturability of cones is assessed, the shape descriptor exhibits a flattening slope in parameter DM with respect to differential changes in shape parameters, e.g. (r, z). This phenomenon is most evident in the zone marked using two crossing arrows in Fig. 4b around (r = 9, z = 4). We do note a sudden change, e.g. increase in manufacturing difficulty for extreme values of depths and radii,

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Fig. 4. Distance featuring (a)Solid cylinder (left); Cylinder with an internal hole (middle) (b) p = 1, and (right) (c) p = 2 norm of shapes featuring bored holes of specified depth and radius with respect to a starting shape, which was cylindrical.

e.g. (r = 9, z = 10). In comparison, the distance field calculated with p = 2 exhibits an ever increasing slope at an analogous location, this pointed using a ∗ in Fig. 4c. Ideally, we anticipate the slope to flatten out as boring internal holes featuring parameters (r, z) is as challenging as that featuring (r + dr, z + dz). In this regard, p = 1, as opposed to p = 2 provides a better measure of manufacturability here.

cess M directly determines how slender a final cylinder can be. For final cylindrical shapes that are fat or moderately slender, choice of M = m1, where m1 = turning should suffice. However, the performance of the turning process deteriorates when fabricating moderate to highly slender shapes due to mechanical instabilities that arise from low structural rigidity of highly slender objects. In such situations, drawing must be used. Shape descriptors that are used in this research do not encapsulate these insights.

5.3. Advantages and limitations of spherical harmonics based descriptors of manufacturability Sections 5.1 and 5.2 suggest that spherical harmonics based descriptors can be used to perform a preliminary analysis of manufacturability of shapes denoted G f from starting billet shapes G0 . This is evident from Figs. 3a, 4b, and 4c that exhibit changes in function DM (G0 , G f ) with respect to changes in G f . However, straightforward formulations using p − norm, where p = 1, or p = 2 provide inaccurate delineations of manufacturability DM (G0 , G f ). This conclusion is drawn from sections 5.1 and 5.2, wherein the distance between the starting cylindrical shape G0 and the final cone shapes G(θ) and bored cylinder shapes G(r, z) are characterized as D M (G0 , G(θ)) = ||R(G(θ)) − R(G0 )||, and D M (G0 , G(r, z)) = ||R(G(r, z)) − R(G0 )||, respectively. The value of parameters D M (G0 , G(θ)) and D M (G0 , G(r, z)) become larger as θ, and r and z increase. This is due to increasing dissimilarity across the corresponding visual shapes. However, realistically their manufacturability DM should be similar. This implies that at worst, the parameter DM should exhibit a monotonic trend with a decaying gradient. It appears from our numerical simulations that the nature of the norm, i.e. value of p can be tuned to achieve this criterion. However, it is evident from Figs. 3 and 4 that ideal choice of the parameter p in order to achieve the aforementioned criterion depends on characteristics of the initial G0 and final shapes G f . Automated selection of the optimum value of p based on these factors was not pursued in this research and will be pursued in the future. Further, the function DM in the present form is symmetric and does not encapsulate effects of the manufacturing process M. This can have profound negative implications. A simple illustration of such implications is obtained by considering quantification of manufacturability of slender cylindrical shapes G slender with respect to fat starting cylinders G f at . Here, the choice of the manufacturing pro-

6. Future Work In addition to limitations identified in section 5.3 we note the following list of factors that must be integrated with the function D M before the DAMA framework can be fully implemented. • Manufacturability can depend intimately on the material used through linkages of manufacturing process performance on materials physics. Hence, the function D M must additionally also depend on the materials physics of manufacturing. • Manufacturing process routes are determined by tolerance requirements, wherein tighter requirements generally make part fabrication, a more challenging endeavor. • Ease of manufacturing can be affected by the use of extraneous elements such as lubricants and coated tools.

6.1. Practical Implementation of DAMA

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A practical implementation of the shape search aspect of the DAMA framework is envisioned in Figure 5. The framework contains a query engine, a database with a multiple-index structure, a matching system, and a ranking system. In order to efficiently search a suitable design, some necessary measures in addition to the 3D shape index are required, with respect to which, the database would be indexed. This would enable users to tune their search by offering more information. The matching system should be able to measure the similarity of multiple indices, and the ranking system should be able to sort designs



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[8] Fig. 5. Conceptual framework of DAMA.

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and their process plans based on parameters in addition to geometric similarity such as cost and feasibility. [10]

7. Conclusion This article discusses benefits and limitations of spherical harmonic based shape descriptors R(G) in quantifying manufacturability of part geometry G f with respect to starting shape G0 . Quantification of manufacturability requires calculation of difference between the desired shape G f and reference shape G0 . This difference was calculated using the p − norm, as D M (G0 , G f ) = ||R(G(θ)) − R(G0 )||. Benefits and limitations of using parameters p = 1 or p = 2, respectively, were discussed to identify manufacturability of simple geometries. Preliminary analyses showed that that p = 2 provided a better measure of manufacturability of cones from a starting cylindrical shape. On the contrary, p = 1 provided a better measure of manufacturability of internally bored cylinders. It was concluded that shape descriptors by themselves do not provide adequate measures of manufacturability since this is also influenced by required tolerances and mechanics of the material used. Quantification of manufacturability is crucial towards its automated delineation in arbitrary shapes. Some necessary characteristics of manufacturability measures were identified, viz. identity, asymmetry and triangle inequality.

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[14]

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