Honeycomb pattern on thin wall object with grain based 3D printing

(1)

ScienceDirect

Available online at www.sciencedirect.comAvailable online at www.sciencedirect.com

ScienceDirect

Procedia Manufacturing 00 (2017) 000–000

www.elsevier.com/locate/procedia

* Paulo Afonso. Tel.: +351 253 510 761; fax: +351 253 604 741 E-mail address: [email protected]

Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017.

Manufacturing Engineering Society International Conference 2017, MESIC 2017, 28-30 June 2017, Vigo (Pontevedra), Spain

Costing models for capacity optimization in Industry 4.0: Trade-off between used capacity and operational efficiency

A. Santana

^a

, P. Afonso

^a,*

, A. Zanin

^b

, R. Wernke

^b

a University of Minho, 4800-058 Guimarães, Portugal

bUnochapecó, 89809-000 Chapecó, SC, Brazil

Abstract

Under the concept of "Industry 4.0", production processes will be pushed to be increasingly interconnected, information based on a real time basis and, necessarily, much more efficient. In this context, capacity optimization goes beyond the traditional aim of capacity maximization, contributing also for organization’s profitability and value.

Indeed, lean management and continuous improvement approaches suggest capacity optimization instead of maximization. The study of capacity optimization and costing models is an important research topic that deserves contributions from both the practical and theoretical perspectives. This paper presents and discusses a mathematical model for capacity management based on different costing models (ABC and TDABC). A generic model has been developed and it was used to analyze idle capacity and to design strategies towards the maximization of organization’s value. The trade-off capacity maximization vs operational efficiency is highlighted and it is shown that capacity optimization might hide operational inefficiency.

Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017.

Keywords: Cost Models; ABC; TDABC; Capacity Management; Idle Capacity; Operational Efficiency

1. Introduction

The cost of idle capacity is a fundamental information for companies and their management of extreme importance in modern production systems. In general, it is defined as unused capacity or production potential and can be measured in several ways: tons of production, available hours of manufacturing, etc. The management of the idle capacity

Peer-review under responsibility of the scientific committee of the 46th SME North American Manufacturing Research Conference.

10.1016/j.promfg.2018.07.117

10.1016/j.promfg.2018.07.117 2351-9789

Peer-review under responsibility of the scientific committee of the 46th SME North American Manufacturing Research Conference.

ScienceDirect

Peer-review under responsibility of the scientific committee of NAMRI/SME.

46th SME North American Manufacturing Research Conference, NAMRC 46, Texas, USA

Honeycomb pattern on thin wall object with grain based 3d printing

AMM Nazmul Ahsan, and Bashir Khoda, PhD*

Industrial and Manufacturing Engineering Department, North Dakota State University, Fargo 58102, USA

* Corresponding author. Tel.:+0-701-231-8071; fax: +0- 701-231-7195.

E-mail address: [email protected]

Abstract

Thin wall free form shaped objects frequently encounter printing difficulties including significantly higher build time, increased support structure, and trapped support material. Limited envelop volume of the existing 3D Printers narrows down the printability of larger objects. In this paper, we propose a novel grain based segmentation approach suitable for curved free-form shaped thin wall objects. Surface points from mesh are clustered into homogeneous groups based on the attributes including facet normal, neighboring facet, locations, and spatial variation of functional response. The boundary defined by clustered facets is called grain here which is constructed with modified K-mean facet clustering algorithm. The grain geometry is optimized for maximum uniformity in their surface curvature to reduce their fabrication complexity, support volume and fabrication time. Once the grains are generated, they are optimally oriented on the base plane to reduce the deviation caused by the surface curvature. The grain is then tessellated with variational hexagonal honeycomb cells to capture functional variation. The thin wall lattice grains are manufactured with the optimal orientations and assembled together with glue. The proposed methodology is implemented on two freeform shaped custom objects– helmet and hand cast, and tested on commercially available printers for fabrication. Comparative results indicate a significant reduction in total build time and support volume for both objects.

Keywords: Thin wall object, Object segmentation, Honeycomb pattern, Object grain orientation.

1.Introduction

Additive manufacturing, also known as 3D printing, enables direct fabrication of a 3D physical object from its digital model by successively printing

and stacking the layers of material one upon another.

The layer geometries are generated through slicing the digital model of the object. Materials are then deposited or cured inside these layers and stacked as it goes along the build direction. The succeeding layers Available online at www.sciencedirect.com

ScienceDirect

46th SME North American Manufacturing Research Conference, NAMRC 46, Texas, USA

Honeycomb pattern on thin wall object with grain based 3d printing

AMM Nazmul Ahsan, and Bashir Khoda, PhD*

Abstract

1. Introduction

The layer geometries are generated through slicing the digital model of the object. Materials are then deposited or cured inside these layers and stacked as it goes along the build direction. The succeeding layers

2 AMM Nazmul Ahsan and Bashir Khoda/ Procedia Manufacturing 00 (2018) 000–000 are supported by the previous layers during the whole

fabrication process. Depending upon the object geometry and the process plan, the object may require additional sacrificial support material to fabricate the overhangs.

To enhance the functionality and characteristics of 3D printed parts, research on novel structure design methods and techniques has received significant attention over the last few years. For instance, one common approach to reduce the deposition volume and material usage in this bi-modal layers pattern is hollowing the object and creating a scaled-down version inside it [1]. Another hollowing approach to reduce material wastage along with structural stability enhancement is a skin-frame structure design [2]

where frame structure is placed underneath the skin.

With the same objectives, Lu et al. [3] demonstrated another hollowing technique where the object interior was filled with voronoi honeycomb structure.

However, the hollowing techniques will introduce uncured trapped material inside the object for processes like powder bed fusion, vat polymerization, binder jetting etc. Removing the uncured material from the internal space can be tedious and time consuming and may often time require minor design modification to avoid trapped volume.

Hollowing the curved free form shaped thin wall objects can be done by placing repetitive patterns on the object surface and removing the interior of the patterns. These repetitive cutout patterns on thin wall objects are used for decorative purpose, easy uncured material removal, light weighting, or other functionality such as ventilation/air pass. The by- example pattern synthesis algorithm demonstrated by Dumas et al. [4] generates patterns as surface texture based on a given example pattern considering both appearance and structural properties of the object.

Zehnder et al. [5] proposed a decorative surface patterning technique formulated as energy minimization which creates ornamental curve networks on the object surface based on user-defined spline curves as the design primitives. Schumacher et al. [6] also presented an example pattern based cutout pattern/stencil generation technique taking aesthetics, stability, and material efﬁciency into account. The distribution and scaling of the patterns on object shell were determined solving an energy minimization problem. A stochastic search and a boosting algorithm proposed by Chen et al. [7] were used to cover an object surface with thin patterns termed as filigree.

This filigree synthesis was posed as a packing problem

with a focus on appearance. Besides providing some advantages, the thin wall objects with cutout patterns, however, suffers from fabrication complexity. For example, the overhang sections of the hollow cutout patterns frequently require increased amount of support structure from both outside and inside of the object. Again, support removal becomes another challenge because of the thin hollow sections of the object wall.

The print volume of the existing 3D printing machines is another important factor affecting the printability of larger objects that do not fit into the printing envelop of a given printer. The build time for larger objects with higher build height significantly increases due to the increased number of layers. Layer geometries also play a prominent role in the manufacturability of an object. Contour plurality and non-uniform layer geometries [8, 9] resulting from curved free form shaped objects dramatically increase fabrication complexity in additive manufacturing processes. The curved free-form shaped objects sometimes demand sacrificial support material as well depending on the object geometry. All these factors demand more resources in 3D printing processes [8].

Splitting thin wall objects into discrete segments can solve the problem of limited print area. Also, a careful discretization of curved free-form shaped object into segments may result in uniform layer geometries of the object segments. For larger objects, Luo et al. [10] proposed an object partitioning approach where objects larger than the working volume of a 3D printer were partitioned into smaller parts so that each of the pars fit into the printing volume. Several partitioning criteria such as assemblability, joining interface, areas of high mechanical stress, object symmetry etc. were taken into account in the segmentation process. Vanek et al. [11] developed a mesh segmentation and packing approach where the 3D objects were first converted into shells and then segmented into smaller parts. It was shown that the overall support material and build time were reduced due to partitioning and packing the segments in the print envelop. Recently Cheng et al. [12] has demonstrated a skeleton based partitioning technique for large objects considering the integrity of meaningful parts. However, the object geometry and fabrication complexity have not been considered while partitioning the object model. Therefore, object segments obtained using these techniques may have inhomogeneous surface characteristics due to sharp change in curvature. The dissimilarity in the grain

(2)

AMM Nazmul Ahsan et al. / Procedia Manufacturing 26 (2018) 900–911 901

ScienceDirect

46th SME North American Manufacturing Research Conference, NAMRC 46, Texas, USA

Honeycomb pattern on thin wall object with grain based 3d printing

AMM Nazmul Ahsan, and Bashir Khoda, PhD*

Abstract

1.Introduction

The layer geometries are generated through slicing the digital model of the object. Materials are then deposited or cured inside these layers and stacked as it goes along the build direction. The succeeding layers Available online at www.sciencedirect.com

ScienceDirect

46th SME North American Manufacturing Research Conference, NAMRC 46, Texas, USA

Honeycomb pattern on thin wall object with grain based 3d printing

AMM Nazmul Ahsan, and Bashir Khoda, PhD*

Abstract

1. Introduction

The layer geometries are generated through slicing the digital model of the object. Materials are then deposited or cured inside these layers and stacked as it goes along the build direction. The succeeding layers

2 AMM Nazmul Ahsan and Bashir Khoda/ Procedia Manufacturing 00 (2018) 000–000 are supported by the previous layers during the whole

fabrication process. Depending upon the object geometry and the process plan, the object may require additional sacrificial support material to fabricate the overhangs.

To enhance the functionality and characteristics of 3D printed parts, research on novel structure design methods and techniques has received significant attention over the last few years. For instance, one common approach to reduce the deposition volume and material usage in this bi-modal layers pattern is hollowing the object and creating a scaled-down version inside it [1]. Another hollowing approach to reduce material wastage along with structural stability enhancement is a skin-frame structure design [2]

where frame structure is placed underneath the skin.

With the same objectives, Lu et al. [3] demonstrated another hollowing technique where the object interior was filled with voronoi honeycomb structure.

However, the hollowing techniques will introduce uncured trapped material inside the object for processes like powder bed fusion, vat polymerization, binder jetting etc. Removing the uncured material from the internal space can be tedious and time consuming and may often time require minor design modification to avoid trapped volume.

Hollowing the curved free form shaped thin wall objects can be done by placing repetitive patterns on the object surface and removing the interior of the patterns. These repetitive cutout patterns on thin wall objects are used for decorative purpose, easy uncured material removal, light weighting, or other functionality such as ventilation/air pass. The by- example pattern synthesis algorithm demonstrated by Dumas et al. [4] generates patterns as surface texture based on a given example pattern considering both appearance and structural properties of the object.

Zehnder et al. [5] proposed a decorative surface patterning technique formulated as energy minimization which creates ornamental curve networks on the object surface based on user-defined spline curves as the design primitives. Schumacher et al. [6] also presented an example pattern based cutout pattern/stencil generation technique taking aesthetics, stability, and material efﬁciency into account. The distribution and scaling of the patterns on object shell were determined solving an energy minimization problem. A stochastic search and a boosting algorithm proposed by Chen et al. [7] were used to cover an object surface with thin patterns termed as filigree.

This filigree synthesis was posed as a packing problem

with a focus on appearance. Besides providing some advantages, the thin wall objects with cutout patterns, however, suffers from fabrication complexity. For example, the overhang sections of the hollow cutout patterns frequently require increased amount of support structure from both outside and inside of the object. Again, support removal becomes another challenge because of the thin hollow sections of the object wall.

The print volume of the existing 3D printing machines is another important factor affecting the printability of larger objects that do not fit into the printing envelop of a given printer. The build time for larger objects with higher build height significantly increases due to the increased number of layers. Layer geometries also play a prominent role in the manufacturability of an object. Contour plurality and non-uniform layer geometries [8, 9] resulting from curved free form shaped objects dramatically increase fabrication complexity in additive manufacturing processes. The curved free-form shaped objects sometimes demand sacrificial support material as well depending on the object geometry. All these factors demand more resources in 3D printing processes [8].

Splitting thin wall objects into discrete segments can solve the problem of limited print area. Also, a careful discretization of curved free-form shaped object into segments may result in uniform layer geometries of the object segments. For larger objects, Luo et al. [10] proposed an object partitioning approach where objects larger than the working volume of a 3D printer were partitioned into smaller parts so that each of the pars fit into the printing volume. Several partitioning criteria such as assemblability, joining interface, areas of high mechanical stress, object symmetry etc. were taken into account in the segmentation process. Vanek et al.

[11] developed a mesh segmentation and packing approach where the 3D objects were first converted into shells and then segmented into smaller parts. It was shown that the overall support material and build time were reduced due to partitioning and packing the segments in the print envelop. Recently Cheng et al.

[12] has demonstrated a skeleton based partitioning technique for large objects considering the integrity of meaningful parts. However, the object geometry and fabrication complexity have not been considered while partitioning the object model. Therefore, object segments obtained using these techniques may have inhomogeneous surface characteristics due to sharp change in curvature. The dissimilarity in the grain

(3)

surface characteristics can contribute towards the amount of support volume and complex layer geometry with concave areas, which eventually can lead to higher build time, resources and cost [8]. Other 3D mesh segmentation techniques [13-16] proposed to segment the object intuitively into meaningful features which are not fully intended for 3D printing process.

Thus, partitioning methodology for 3D printing of objects needs to consider the geometry of the object segments in order to simultaneously reduce the fabrication complexity, support material, and build time. Besides, in case of printing hollow thin wall objects with cutout patterns, support material can be trapped inside the object and hollow patterns and it is often difficult to remove. Therefore, segmentation of both curved and hollow objects and then creating lattice shell segments by cutout patterns would reduce support material and accommodate the intended functionality.

Fig. 1. Framework of the proposed methodology.

In this paper, we propose a novel grain based mesh- model segmentation approach suitable for curved free form shaped thin wall objects. Facets on the mesh surface are clustered into homogeneous groups considering the attributes including facet normal, neighboring facet and locations, and spatial variation of functional response. The boundary defined by clustered facets is called grain here which is constructed with modified K-mean facet clustering algorithm. The grain geometry is optimized for maximum uniformity in their surface curvature to reduce their fabrication complexity, support volume and fabrication time. Once the grains are generated, they are optimally oriented on the base plane in order

to project the hexagonal honeycomb cells on them.

Since the grains are already generated based on the maximum uniformity in their surface curvature (flatness), this optimal orientation will ensure the least projection deviation of the honeycomb cells.

Removing the interior of the projected honeycomb cells will result in thin shell lattice grains. The thin wall lattice grains are manufactured with the optimal orientations and assembled together with glue. The overall framework of the proposed approach is demonstrated in Fig. 1.

2.Methodology

The object surface is discretized based on homogeneity in terms of surface curvature, neighboring facet and location, and spatial functional response variation. The generated homogeneous segments, which are termed as grains in this paper are then optimally oriented with respect to the base/build plane to tessellate with hexagonal honeycomb cells.

The interior of the honeycomb cells on each grain are removed and thus thin shell honeycomb lattice grains are generated. The detailed methodology is explained in the following sections.

2.1. Grain Based Segmentation 2.1.1. Surface Attributes Extraction

The input object model  represented as a mesh surface with spatial distribution of functional response consists of a set of triangular facets F{Fi}i1,2,,I

covering  , where I F is the total number of facets. Each facet F_i is uniquely identified with a unit normal vector ni and a set of face vertices V_iV as shown in Fig. 2, where V is the set of vertices of the mesh [17]. The performance application of the object will dictate the functional variation, for instance, thermal response including temperature distribution, thermal sensitivity etc., or structural response including load, stress, deformation distribution etc. along the object. In this paper, normalized color intensity over the facets ranging between 0 and 1 is used. The spatial distribution of the desired functional response magnitude is represented by color gradient map on the 3D object model (see Fig.

2(a)), where the color spectrum represents extreme values of the pseudo functional response.

In order to segment  into a finite set of manifold grains, each of the facets of is characterized with its 3D Mesh Model

Mesh surface characterization

Modified K-mean facet clustering algorithm Homogeneous object grains

Grain orientation optimization Optimal grain orientation

Honeycomb tessellation

Grain fabrication

Assembly

Functional variation map

location, normal vector, and color intensity of functional response. The facet centroids

I i

ci

C{ }1,2,, are used to identify the location of the corresponding facets and the normal vector is utilized to indicate the surface profile at that location. The facet color intensity values {f_i}_i₁_,₂_,_,_I indicate the level of functional response over the corresponding facets.

Thus, facet characteristics ( c_i, f_i,n_i) carry the corresponding surface attributes and cumulatively contribute towards the nature of the object surface for instance, surface concavity, curvature, need of sacrificial supports, structural parameters etc. More precisely, the direction and angle of the facet normals with respect to the build vector determine the surface quality [8] as well as the need of support structure [18].

A set of neighboring facets with similar color intensity values of function response corresponds to a homogeneous region on the object surface with consistent structural parameters. Therefore, the dissimilarity in the characteristics of a set of neighboring facets may also lead to incoherent surface regions which could introduce fabrication complexity and increase the amount of support volume. In the following section, the object facets are clustered into grains considering the homogeneity with respect to their characteristics discussed above.

2.1.2. Point Clustering and Homogeneous Grain Generation

The object surface is segmented into homogeneous grains such that each surface grain reduces the fabrication complexity and requires the least possible amount of support. If the difference in angles among the normal vectors of the neighboring facets in a grain is smaller, the segmented grain will be flatter requiring lesser amount of support material when built individually. Hence, the goal is to identify the homogeneous and coherent flatter regions and discretize them into manifold grains. A modified K- means clustering algorithm is used to group the facets belonging to the coherent regions. The modified K- means clustering algorithm combines angle between the normal vectors, functional response distribution, and centroid-to-centroid distance between the facets into the cost function to partition the object surface into total K number of grains. The set of the generated grains can be defined as

k k K k k k k

k F G G

G  |   }_,_{₁_,₂_,..., _}_and 

{  , where both k

and k indicate the grain number and kk. The grains are required to be manifold and there will be no common facet between any two grains. Hence, the

intersection between any two grains is an empty set )

( _.

Initially, a set of K number of facets F

S

S{ _k}_k1,2,_K  with corresponding centroids

K k

sck

SC{ }1,2, , normals SN{π_k}_k1,2,_K , and color intensities SI{k}k1,2,K are randomly selected as the seeds, which is shown in Fig. 2. Each of these seed facets will form a grain around it and partition the object surface. Thus, the total number of seeds, K will also correspond to total number of clusters/grains, which is a user defined parameter in our algorithm. The algorithm could sometime benefit from introducing a variable K. However, this would be computationally expensive and convergence will be difficult to achieve.

The angles between the normal vectors of all the non-seed facets {Tj}j1,2,...,IK F\S and the normal vector π_k of each seed facet S_kare determined using Equation (1) (see Fig. 2(b)).

K k

K I

k j

j k

j, cos^¹(n π ); 1,2,...,  and 1,2,...,



(1) where, j,k is the angle between the normal vector of jth non-seed facets (Tj) to the normal vector of seed facet Skand nj is the normal vector of facet Tj.

Fig. 2. Facet characteristics and relation between seed and non- seed facets. The facets in white color are the initial random seeds.

Also, the color intensity differences {j,k} between all the non-seed facets ({fj}\SI)_and each seed facet (_k) are determined. Here,

|

,k | k j

j  f

 is the color intensity difference between the j^th non-seed facets (Tj) and the k^th

0 1

nj

πk

k^thseed )(Sk

j^thfacet(Tj)

k

j,

cj

sck

(a) (b)

(4)

surface characteristics can contribute towards the amount of support volume and complex layer geometry with concave areas, which eventually can lead to higher build time, resources and cost [8]. Other 3D mesh segmentation techniques [13-16] proposed to segment the object intuitively into meaningful features which are not fully intended for 3D printing process.

Thus, partitioning methodology for 3D printing of objects needs to consider the geometry of the object segments in order to simultaneously reduce the fabrication complexity, support material, and build time. Besides, in case of printing hollow thin wall objects with cutout patterns, support material can be trapped inside the object and hollow patterns and it is often difficult to remove. Therefore, segmentation of both curved and hollow objects and then creating lattice shell segments by cutout patterns would reduce support material and accommodate the intended functionality.

Fig. 1. Framework of the proposed methodology.

In this paper, we propose a novel grain based mesh- model segmentation approach suitable for curved free form shaped thin wall objects. Facets on the mesh surface are clustered into homogeneous groups considering the attributes including facet normal, neighboring facet and locations, and spatial variation of functional response. The boundary defined by clustered facets is called grain here which is constructed with modified K-mean facet clustering algorithm. The grain geometry is optimized for maximum uniformity in their surface curvature to reduce their fabrication complexity, support volume and fabrication time. Once the grains are generated, they are optimally oriented on the base plane in order

to project the hexagonal honeycomb cells on them.

Since the grains are already generated based on the maximum uniformity in their surface curvature (flatness), this optimal orientation will ensure the least projection deviation of the honeycomb cells.

Removing the interior of the projected honeycomb cells will result in thin shell lattice grains. The thin wall lattice grains are manufactured with the optimal orientations and assembled together with glue. The overall framework of the proposed approach is demonstrated in Fig. 1.

2. Methodology

The object surface is discretized based on homogeneity in terms of surface curvature, neighboring facet and location, and spatial functional response variation. The generated homogeneous segments, which are termed as grains in this paper are then optimally oriented with respect to the base/build plane to tessellate with hexagonal honeycomb cells.

The interior of the honeycomb cells on each grain are removed and thus thin shell honeycomb lattice grains are generated. The detailed methodology is explained in the following sections.

2.1. Grain Based Segmentation 2.1.1. Surface Attributes Extraction

The input object model  represented as a mesh surface with spatial distribution of functional response consists of a set of triangular facets F{Fi}i1,2,,I

covering  , where I F is the total number of facets. Each facet F_i is uniquely identified with a unit normal vector ni and a set of face vertices V_iV as shown in Fig. 2, where V is the set of vertices of the mesh [17]. The performance application of the object will dictate the functional variation, for instance, thermal response including temperature distribution, thermal sensitivity etc., or structural response including load, stress, deformation distribution etc. along the object. In this paper, normalized color intensity over the facets ranging between 0 and 1 is used. The spatial distribution of the desired functional response magnitude is represented by color gradient map on the 3D object model (see Fig.

2(a)), where the color spectrum represents extreme values of the pseudo functional response.

In order to segment  into a finite set of manifold grains, each of the facets of is characterized with its 3D Mesh Model

Mesh surface characterization

Modified K-mean facet clustering algorithm Homogeneous object grains

Grain orientation optimization Optimal grain orientation

Honeycomb tessellation

Grain fabrication

Assembly

Functional variation map

location, normal vector, and color intensity of functional response. The facet centroids

I i

ci

C{ }1,2,, are used to identify the location of the corresponding facets and the normal vector is utilized to indicate the surface profile at that location. The facet color intensity values {f_i}_i₁_,₂_,_,_I indicate the level of functional response over the corresponding facets.

Thus, facet characteristics (c_i,f_i,n_i ) carry the corresponding surface attributes and cumulatively contribute towards the nature of the object surface for instance, surface concavity, curvature, need of sacrificial supports, structural parameters etc. More precisely, the direction and angle of the facet normals with respect to the build vector determine the surface quality [8] as well as the need of support structure [18].

A set of neighboring facets with similar color intensity values of function response corresponds to a homogeneous region on the object surface with consistent structural parameters. Therefore, the dissimilarity in the characteristics of a set of neighboring facets may also lead to incoherent surface regions which could introduce fabrication complexity and increase the amount of support volume. In the following section, the object facets are clustered into grains considering the homogeneity with respect to their characteristics discussed above.

2.1.2. Point Clustering and Homogeneous Grain Generation

The object surface is segmented into homogeneous grains such that each surface grain reduces the fabrication complexity and requires the least possible amount of support. If the difference in angles among the normal vectors of the neighboring facets in a grain is smaller, the segmented grain will be flatter requiring lesser amount of support material when built individually. Hence, the goal is to identify the homogeneous and coherent flatter regions and discretize them into manifold grains. A modified K- means clustering algorithm is used to group the facets belonging to the coherent regions. The modified K- means clustering algorithm combines angle between the normal vectors, functional response distribution, and centroid-to-centroid distance between the facets into the cost function to partition the object surface into total K number of grains. The set of the generated grains can be defined as

k k K k k k k

k F G G

G  |   }_,_{₁_,₂_,..., _}_and

{  , where both k

and k indicate the grain number and kk. The grains are required to be manifold and there will be no common facet between any two grains. Hence, the

intersection between any two grains is an empty set )

( _.

Initially, a set of K number of facets F

S

S{ _k}_k1,2,_K  with corresponding centroids

K k

sck

SC{ }1,2, , normals SN{π_k}_k1,2,_K , and color intensities SI{k}k1,2,K are randomly selected as the seeds, which is shown in Fig. 2. Each of these seed facets will form a grain around it and partition the object surface. Thus, the total number of seeds, K will also correspond to total number of clusters/grains, which is a user defined parameter in our algorithm. The algorithm could sometime benefit from introducing a variable K. However, this would be computationally expensive and convergence will be difficult to achieve.

The angles between the normal vectors of all the non-seed facets {Tj}j1,2,...,IK F\S and the normal vector π_k of each seed facet S_kare determined using Equation (1) (see Fig. 2(b)).

K k

K I

k j

j k

j, cos^¹(n π); 1,2,...,  and 1,2,...,



(1) where, j,k is the angle between the normal vector of jth non-seed facets (Tj) to the normal vector of seed facet Skand nj is the normal vector of facet Tj.

Fig. 2. Facet characteristics and relation between seed and non- seed facets. The facets in white color are the initial random seeds.

Also, the color intensity differences {j,k} between all the non-seed facets ({fj}\SI)_and each seed facet (_k) are determined. Here,

|

,k | k j

j  f

 is the color intensity difference between the j^th non-seed facets (Tj) and the k^th

0 1

nj

πk

k^thseed )(Sk

j^thfacet(Tj)

k

j,

cj

sck

(a) (b)

(5)

seed facet (S_k) . _j,_k , _j,_k , and d_j_,_k are used to determine an aggregate cost function value j,k for each facet Tj with respect to seed k using Equation (2), where dj,kis the distance between the centroid of facet Tj and seed facets S_k. The values of j,k, j,k, and d_j_,_kare normalized with their extreme values.

k j d w w

w_a ^norm_j_k _r ^norm_j_k _d ^norm_j_k

k

j_,  _,  _,   _, ; ,

 (2)

Here, w_a, w_r , and w_d are the associated weights which represent the relative importance of normal vector angle, intensities of functional response, and distance, respectively. Summation of all these weights are unity. For simplicity, w_aw_rw_d 1/3 are used in this paper. However, different set of weightage can be used considering the part characteristics, its functionality, expected performance, manufacturing capabilities, geometry, and other related attributes.

The aggregate cost function value j,k in Equation (2) represents the degree of similarity of facet Tj to the seed facet k with respect to their characteristics including normal vector angle, intensity of functional response, and location. The smaller the value of j,k,

the higher the similarity is. Therefore, the non-seed facets {Tj} are clustered into K number of groups by assigning the j^th facet to k^th cluster when

} { min1,2,..., ,

,k k K jk

j 

  _ . Such grouping ensures that a facet is clustered to the seed, which has the maximum similarity to that facet compared to the other facets.

Thus, every facet cluster eventually forms a grain on the mesh surface resulting in K number of grains

K k

Gk} ₁_,₂_,...,

{  (see Fig. 3(b)).

Fig. 3. (a) Initial random seeds and (b) generated grains from the initial seeds.

Fig. 4. Seed update procedure: (a) k^th grain with its centroid g_k, (b) angle determination between the facet normal and imaginary vector connecting the facet centroid to grain centroid, and (c) updated seed.

Once all the grains are generated at the end of first iteration as shown in Fig. 3(b), the centroid facets of the grains become the new seeds for the next iteration.

In order to determine the centroid facets of the grains, the area centroid pints {g_k}_k_₁_,₂_,...,_K of the grains are calculated. If a grain centroid gk lies on the grain

surface, the facet containing the grain centroid will be the centroid facet of that grain. However, if the grain centroid falls outside of the grain surface as shown in Fig. 4(a), the grain centroid facet is determined as follows. In each grain, a set of facet imaginary vectors connecting the centroids of the facets of that grain to

(a) (b)

Initial Seeds

Grains {Gk}₁_,...,K

the grain centroid is created. The facet corresponding to the minimum angle between the facet normal and facet imaginary vector will be considered as the centroid facet of the grain (see Fig. 4). Similarly, the centroid facets of all the grains are determined and the seed facets are updated for the next iteration. Since each seed is updated to a new facet of the corresponding grain, the updated seed will possess the characteristics including centroid, color intensity, and normal vector of that new facet.

Using the updated seeds, a set of new grains is formed at the end of the second iteration. This entire process is repeated until no update happens for the present seeds, which indicates the convergence of the algorithm. Therefore, after convergence, the resulting clusters {Gk_₁_,₂_,...,KF} will consist of facets with similar characteristics making homogeneous grains as

shown in Fig. 5(f). The entire clustering and grain formation process is illustrated in Fig. 5.

The boundary contours of the generated grains usually become geometrically jagged because of the nature of the triangular mesh. These jagged edges can undermine the printability and assembly of the grains due to their resolution. To resolve this issue a weighted boundary smoothing technique adopted from our previous work [19] is implemented in this paper. A jagged zone between two neighboring grains is first generated by constructing two extreme profile curves connecting the edges of the boundary facets. The facets inside the jagged zone can be defined as internal and external to the grain. By using their ratio, weighted points are inserted on the common facet line segments between two neighbor grains. Connecting the weighted points will generate a piecewise linear curve with smooth edge grain boundary as shown in Fig. 6(a).

Fig. 5. Facet clustering and object grain formation: (a) object mesh surface, (b) initial random seeds, (c) grain formation using the initial seeds, (d) seed updating, (e) grain formation using updated seeds, (f) final grains.

2.2. Optimal orientation for grains

In order to generate hollow thin shell lattice grains, each grain of the object surface is tessellated with shape conforming hexagonal honeycomb cells. The

grain boundary contour is first projected on the base plane and is tessellated with 2D hexagonal honeycomb cells. After tessellation, the 2D honeycomb cells are then projected backed on the grain surface yielding 3D shape conforming honeycomb lattice. Due to the

Seeds

(c)

(a) (b)

1^stIteration

1^stIteration Seeds

Clusters

2^ndIteration 2^ndIteration n^thIteration

Seed updating and re-clustering

Updated Seeds

1^stIteration

clustersNew Updated Seeds for 2^nditeration

(e) (f)

(d)

(6)

seed facet (S_k) . _j,_k , _j,_k , and d_j_,_k are used to determine an aggregate cost function value j,k for each facet Tj with respect to seed k using Equation (2), where dj,kis the distance between the centroid of facet Tj and seed facets S_k. The values of j,k, j,k, and d_j_,_kare normalized with their extreme values.

k j d w w

w_a ^norm_j_k _r ^norm_j_k _d ^norm_j_k

k

j_,  _,  _,   _, ; ,

 (2)

Here, w_a, w_r , and w_d are the associated weights which represent the relative importance of normal vector angle, intensities of functional response, and distance, respectively. Summation of all these weights are unity. For simplicity, w_aw_rw_d1/3 are used in this paper. However, different set of weightage can be used considering the part characteristics, its functionality, expected performance, manufacturing capabilities, geometry, and other related attributes.

The aggregate cost function value j,k in Equation (2) represents the degree of similarity of facet Tj to the seed facet k with respect to their characteristics including normal vector angle, intensity of functional response, and location. The smaller the value of j,k,

the higher the similarity is. Therefore, the non-seed facets {Tj} are clustered into K number of groups by assigning the j^th facet to k^th cluster when

} { min1,2,..., ,

,k k K jk

j 

  _ . Such grouping ensures that a facet is clustered to the seed, which has the maximum similarity to that facet compared to the other facets.

Thus, every facet cluster eventually forms a grain on the mesh surface resulting in K number of grains

K k

Gk} ₁_,₂_,...,

{  (see Fig. 3(b)).

Fig. 3. (a) Initial random seeds and (b) generated grains from the initial seeds.

Fig. 4. Seed update procedure: (a) k^th grain with its centroid g_k, (b) angle determination between the facet normal and imaginary vector connecting the facet centroid to grain centroid, and (c) updated seed.

Once all the grains are generated at the end of first iteration as shown in Fig. 3(b), the centroid facets of the grains become the new seeds for the next iteration.

In order to determine the centroid facets of the grains, the area centroid pints {g_k}_k_₁_,₂_,...,_K of the grains are calculated. If a grain centroid gk lies on the grain

surface, the facet containing the grain centroid will be the centroid facet of that grain. However, if the grain centroid falls outside of the grain surface as shown in Fig. 4(a), the grain centroid facet is determined as follows. In each grain, a set of facet imaginary vectors connecting the centroids of the facets of that grain to

(a) (b)

Initial Seeds

Grains {Gk}₁_,...,K

the grain centroid is created. The facet corresponding to the minimum angle between the facet normal and facet imaginary vector will be considered as the centroid facet of the grain (see Fig. 4). Similarly, the centroid facets of all the grains are determined and the seed facets are updated for the next iteration. Since each seed is updated to a new facet of the corresponding grain, the updated seed will possess the characteristics including centroid, color intensity, and normal vector of that new facet.

Using the updated seeds, a set of new grains is formed at the end of the second iteration. This entire process is repeated until no update happens for the present seeds, which indicates the convergence of the algorithm. Therefore, after convergence, the resulting clusters {Gk_₁_,₂_,...,K F} will consist of facets with similar characteristics making homogeneous grains as

shown in Fig. 5(f). The entire clustering and grain formation process is illustrated in Fig. 5.

The boundary contours of the generated grains usually become geometrically jagged because of the nature of the triangular mesh. These jagged edges can undermine the printability and assembly of the grains due to their resolution. To resolve this issue a weighted boundary smoothing technique adopted from our previous work [19] is implemented in this paper.

A jagged zone between two neighboring grains is first generated by constructing two extreme profile curves connecting the edges of the boundary facets. The facets inside the jagged zone can be defined as internal and external to the grain. By using their ratio, weighted points are inserted on the common facet line segments between two neighbor grains. Connecting the weighted points will generate a piecewise linear curve with smooth edge grain boundary as shown in Fig.

6(a).

Fig. 5. Facet clustering and object grain formation: (a) object mesh surface, (b) initial random seeds, (c) grain formation using the initial seeds, (d) seed updating, (e) grain formation using updated seeds, (f) final grains.

2.2. Optimal orientation for grains

In order to generate hollow thin shell lattice grains, each grain of the object surface is tessellated with shape conforming hexagonal honeycomb cells. The

grain boundary contour is first projected on the base plane and is tessellated with 2D hexagonal honeycomb cells. After tessellation, the 2D honeycomb cells are then projected backed on the grain surface yielding 3D shape conforming honeycomb lattice. Due to the

Seeds

(c)

(a) (b)

1^stIteration

1^stIteration Seeds

Clusters

2^ndIteration 2^ndIteration n^thIteration

Seed updating and re-clustering

Updated Seeds

1^stIteration

clustersNew Updated Seeds for 2^nditeration

(e) (f)

(d)