Eliminating occluded voids in additive manufacturing design via a projection-based topology optimization scheme

Journal Pre-proof Eliminating occluded voids in additive manufacturing design via a projection-based topology optimization scheme Andrew T. Gaynor, Terrence E. Johnson

PII:

S2214-8604(19)30713-4

DOI:

https://doi.org/10.1016/j.addma.2020.101149

Reference:

ADDMA 101149

To appear in:

Additive Manufacturing

Received Date:

5 June 2019

Revised Date:

21 February 2020

Accepted Date:

21 February 2020

Please cite this article as: Andrew T. Gaynor, Terrence E. Johnson, Eliminating occluded voids in additive manufacturing design via a projection-based topology optimization scheme, (2020), doi: https://doi.org/

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Eliminating occluded voids in additive manufacturing design via a projection-based topology optimization scheme Andrew T. Gaynora , Terrence E. Johnsonb

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a Materials Response and Design Branch Weapons and Materials Research Directorate The U.S. Army Research Laboratory B4600, 6300 Rodman Road Aberdeen Proving Ground, MD 21005, USA email: [email protected] b Multidisciplinary Design Optimization (A1C-9) Section Aerospace and Mechanical Engineering Group (A1C) Air and Missile Defense Sector (AMDS) The Johns Hopkins University Applied Physics Laboratory 11100 Johns Hopkins Road, Laurel, MD 20723-6099

Abstract

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Design for additive manufacturing (AM) requires knowledge of the constraints associated with your targeted AM process. One important design concern is the unintentional trapping of parasitic mass in occluded void geometries with either uncured or nonsolidified material, or in some cases, sacrificial support material. These occluded features create the need to physically alter the optimal topology to remove the material. In this work, a projection-based topology optimization design formulation is proposed to eliminate occluded void topological features in optimal AM designs. The algorithm is based on the combination and enhancement of two existing algorithms: a projection-based, overhang-constrained algorithm to design self-supporting structures in AM, and a void projection algorithm to design topologies through control of the void phase. The combined algorithm results in topologies with void regions that always possess an exit path to predefined outer surfaces – i.e. drainage pathways. Solutions are first demonstrated in two dimensions, with increasing design freedom allowed through algorithm enhancements. The algorithm is then adapted to 3D, adopting a multi-phase TO approach to not only regain control of the solid phase length scale, but also to drive toward superior performing topologies with minimal impact on the part performance. Keywords: topology optimization, parasitic mass, occluded void elimination, powder bed fusion, vat photopolymerization

1. Introduction

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down required post-processing.

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Designing for additive manufacturing (AM) is often nonintuitive, leading to instances where even an experienced design engineer will have limited intuition if the problem is altered even slightly. As such, it becomes increasingly necessary to harness powerful generative design methods, such as topology optimization (TO), to tackle these problems in a systematic fashion. In the world of AM, TO is quickly catching on as the go-to design methodology, with quite attractive performance-driven designs. TO is typically used in the early stages of design to produce efficient structures given prescribed loads and boundary conditions, targeted objectives and imposed constraints. However, TO designs from commercial codes and academic codes, alike, typically exhibit complex geometries and often require significant topological post-processing to meet manufacturing constraints. In this process, the designs often lose a significant amount of optimality, especially when postprocessing for traditional, more restrictive manufacturing approaches. As such, there is significant push, especially in the academic community, to incorporate increasing amounts of relevant design considerations in the TO algorithms in order to cut Preprint submitted to Additive Manufacturing

AM exhibits fewer, yet distinct, design limitations when compared with traditional manufacturing. While AM significantly widens the design space (i.e. geometric possibilities), manufacturing constraints and process limitations remain [1]. These manufacturing constraints include, but are not limited to: (1) minimum feature size, (2) maximum allowable feature distortion, (3) allowable overhanging features (often related to (2)) and (4) trapped parasitic mass via occluded void topologies. Minimum feature size can be directly tied to the print resolution: e.g. the bead width for extrusion-based AM, the sintered width in powder bed fusion, and the cured width in vat-based photo-polymeric AM. Several TO techniques have been proposed to achieve minimum feature size control [2, 3, 4, 5, 6], with advantages and disadvantages to each. The Heaviside Projection Method (HPM) framework [5], which has the advantage of separating the design space from the physical element density space, is utilized herein. Ever since the original paper addressing overhang control in additive manufacturing [7], there have been a large number of contributions exploring this design consideration. LanFebruary 21, 2020

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gelaar and colleagues contributed a number of design algorithms in this area, most notably a novel adjoint sensitivity approach for fast calculation of derivative information [8, 9]. This approach was recently adapted to the framework proposed by Gaynor and Guest [10, 11]. Langelaar also harnessed the same underlying approach to simultaneously design component and support structures [12], and to design for component and support topology, along with build orientation [9]. An alternative approach from the same research group looks at harnessing a front propagation filter to also guarantee selfsupporting structures [13]. Aside from these two main contributors, a number of other intriguing algorithms have been proposed [1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 20, 23, 24]. The overhang algorithm framework harnessed herein builds upon that found in Behrou et al. [10] and Johnson and Gaynor [11]. For additional details on the basics of the overhang projection framework, please see Gaynor and Guest [25]. Behrou et al [10] and Johnson and Gaynor [11] provide details on a recent adjouint speedup and adaptation to the 3D context. Occluded voids – the design feature focus of this paper – trap loose powder in powder-bed fusion processes, or trap liquid photo-polymer in vat-based steriolithography processes, potentially adding significant parasitic mass to the structure (unless it is somehow removable post-print). Interestingly, in an extreme case, Reddy K et al. [26] showed that trapped material can account for as much as 74% of the total material needed to make a part. Fig. 1 illustrates the laser powder-bed fusion process with trapped powder within a part – a cross-section shown here. As the fabrication piston moves down, metal powder is rolled onto the piston head. A laser beam then melts the powder in the configuration of a prescribed cross-section shape (horizontal plane). The process repeats until a final structure is produced. In steriolithography, a vat-based photo-polymer process, a similar phenomena occurs, where uncured resin is trapped inside solid features. Additionally, in extrusion-based AM processes, closed cavity geometries that trap support material can also result in parasitic mass, as support material is impossible to remove if there is no access from the outside. Several material density-based TO methods have been proposed for avoiding occluded voids in design. Liu et al. [27] developed a multi-physics TO-based methodology to remove occluded voids from structures. The method, coined the virtual temperature method (VTM), sets void elements as highly conductive, solid elements as thermal insulators and the domain boundaries as heat sinks. Li et al. [28] expanded Liu’s [27] methodology to include internal heat sinks and showed that 2D and 3D TO solutions can be created with occluded voids for passage of physical structures such as pipes. Harzheim and Graf [29] used TopShape to design topologically optimized cast parts. Their proposed approach for removing occluded voids eliminates material from the outside-in. Zhou et al. [30] proposed manufacturing and extrusion constraints within a TO scheme to remove occluded voids and enforce constant crosssections. The method introduces uni- and multi-directional constraints that reduce element density from the inside-out or from a so called “growth interface” to the boundaries of the design domain. By extension, Lu and Chen [31] expanded the method-

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Figure 1: Diagram of Selective Laser Sintering Process showing trapped material in a part (2D slice shown here).

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ology developed by Zhou [30] to search in additional directions (4 or more) in various orientations. Gersborg and Andreasen [32] developed a methodology to produce castable designs (design free of concave surfaces and occluded voids) by using a Heaviside design parameterization in a specified casting direction. Guo et al. [1] proposed using Moving Morphable Voids (MMV) to eliminate occluded voids by imposing a restriction so that printable features exist only at the boundaries of the design domain. Finally, Zhou and Zhang proposed a side constraint scheme within a level set TO framework where voids are essentially tied to the edge of the domain – this guarantees a void pathway to any void within the physical domain [33]. In contrast to the above schemes, the approach presented in this manuscript is generally less restrictive on the design space and harnesses a previously developed framework for overhang projection [7, 25] with a slight modification for the riddance of occluded voids, and an adjoint speedup for sensitivity calculations [11, 10]. As such, the presented approach does not require additional constraints, but does require the projection scheme to proceed in a directional layer-by-layer manner. Thus, if the framework for overhang projection is already implemented [11, 10], then the removal of occluded voids can be implemented within this framework with minimal additional coding effort. The remaining paper is outlined as follows: Sections 1.1 and 1.2 will briefly explain the void projection method (VPM) and overhang projection method (OPM) schemes, respectively. Section 2 combines the VPM with the OPM to create an algorithm capable of tackling the identified occluded void problem. Section 3 briefly describes the adjoint approach employed to calculate sensitivities. Section 4 presents the optimization problem formulation. Section 5 displays initial results for single di-

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rection two dimensional void growth optimization. Significant enhancements to open up the design space but still eliminate the occurrence of occluded voids are seen in Section 6. Extension to 3D is explored in Section 7, demonstrating how void elimination in a 3D context is a much less restrictive design rule. Finally, there is a discussion of the pros and cons, algorithmic nuances and some final thoughts in the Discussion Section 8.

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1.1. Void Projection Method (VPM) At a fundamental level, the void projection method (VPM) [3, 34, 35] inverts the design problem such that the algorithm is determining where the void should exist instead of determining where the solid should exist, as is the case in typical solid projection TO – i.e. project void (ρe = 0) as opposed to solid (ρe = 1). As stated previously, this work uses the Heaviside Projection Method (HPM) [5] to achieve minimum length scale control. Figure 2 illustrates HPM mapping in three dimensions. In this diagram, the design variables are chosen to exist at the nodes of finite element mesh. Figure 2a demonstrates the design variable projection to elemental density space within a radial distance of rmin , defining the minimum allowable feature size. Alternatively, one can view the projection from the elemental perspective, as seen in Figure 2b, and define the neighborhood set, denoted as N e , of design variables within rmin of the element centroid. Design variables within N e either project solid or void information to the element of interest, e. More specifically, the weighted average, µe , of the magnitudes of the design variables within N e is calculated and is either linearly or nonlinearly projected to the element of interest. Linear projection is effectively the same as linear density filtering and is discussed in Burns and Tortorelli [36]. In HPM, the weight averaged design variables are passed through a regularized (continuous) Heaviside function to produce an element density, ρe . The added non-linearity allows for a crisper definition of the feature boundary – i.e. less intermediate-density elements. Solid projection is first defined for completeness. The relative element densities for solid phase projection is defined as:    1, if µe (φ) > 0 e e ρsolid = H(µ (φ)) =  (1)  0, if µe (φ) = 0

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(a) Nodal perspective: A design variable projects solid or void phase to elements within a radial distance defined by rmin

where H represents the Heaviside function, µe is the weighted average of the design variables, φ, and ρesolid is the associated element density. As shown in [5], a discrete Heaviside operator can be made continuous (i.e. regularized) as shown in the following:

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(b) Elemental perspective: An element receives projection information from design variables within its neighborhood set.

e

ρesolid = 1 − e−βµ (φ) +

µe (φ) −βφmax e φmax

Figure 2: Minimum length scale imposed through a projection of design variables to physical space.

(2)

where β is the regularization parameter. Defining the aggressiveness of the regularized Heaviside function, if β is large, Eq. 2 has a high degree of nonlinearity, producing crisp topological boundaries in the transition from solid to void (minimal intermediate density elements). Note that in Eq. 2, φmax = 1. Conversely, in the VPM the design algorithm controls the length scale of void phase, and the length scale control of the solid phase is lost. The VPM is defined as: 3

ρevoid

   0, if µe (φ) > 0 e = 1 − H(µ (φ)) =   1, if µe (φ) = 0

by setting the thresholding Heaviside parameter, βT , to a sufficiently high value. Additionally, it is found that including a small penalization term, ηT , helps with convergence by driving the φ in the “wedge” neighborhood set to 0 or 1. ηT = 1.5 for all example problems herein. The threshold, T , varies between 0 and 1 and is chosen such that the support condition is correctly enforced. For the OPM, the user must designate the boundary on which the build plate exists. This equates to setting ρ s = 1 along the bottom edge of the design domain. Once this is established, the calculation of ρ s and φ proceed in a layer-by-layer fashion, as previously mentioned. This layer-by-layer “directionally dependent” approach is fundamental to the proposed algorithm in Section 2.

(3)

Thus, the void projection is simply one minus the solid projection and is represented here: e

ρevoid = e−βµ (φ) −

µe (φ) −βφmax e φmax

(4)

More specifically, in the VPM, µe > 0 produces ρevoid = 0, while µe = 0 produces ρevoid equal to one as β goes to ∞. This is the inverse of solid projection logic. Guest [35] showed that applying the VPM alone often produces solutions with rounded holes instead of sharp corner topologies common from solid projection methods.

1.3. Adjusted Overhang Projection Neighborhood Set for 2D

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1.2. Overhang Projection Method (OPM)

The three dimensional OPM support neighborhood set, NS remains the same as in the manuscript by Johnson and Gaynor [11]. This approach for overhang control “looks” one layer of φ down in the design space to determine the support metric, ρ s . In two dimensions, a one layer down support neighborhood set (NS ) only captures three material placement points, φ [10], in the neighborhood set. While conceptually simple, the one layer approach in 2D often results in non-monotonically decreasing optimization progression. To counteract this phenomena, the authors have gone back to an approach taken in previous papers [7, 25], in which NS included a multi-layered wedge region below the point of concern. In these papers, the NS wedge extended at least two layers of φ below the design point of concern. A multi-layered NS will result in overlapping of neighborhood sets in the build direction (y direction in 2D as seen in Fig. 3), significantly reducing the nonlinear behavior and ensuring smooth convergence. The diagram in Fig. 3 demonstrates how this support neighborhood set, NS , is defined for one particular material placement variable located at the yellow dot. The scheme “looks” two material placement layers below in the blue triangular region to form the NS . To define the overhang angle, the spacing of the independent design variables ψ and dependent material placement variables φ (locations of ψ and φ are coincident) is varied by adjusting the spacing variables δ x and δy . The δ overhang angle from horizontal is defined as tan−1 ( δyx ). Fig. 3 demonstrates the support neighborhood set definition on top of an unstructured triangular finite element mesh to emphasize that the design space and material placement space are independent from the underlying finite element mesh. It should be noted, however, that all examples in this manuscript use quadrilateral elements for 2D problems and a hexahedral elements for 3D problems – this is done for simplicity and because these element types generally result in smoother topologies in post-processing efforts. The threshold value, T , in Eq. 6 is set depending on number of nodes below. For the case seen in Fig. 3, the nominal T value is set to 28 for internal points. Importantly, the nominal T is then shifted so that ρS equals roughly 0.93 at the threshold (rule of thumb) – see Johnson and Gaynor [11] for more details. Note

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φi = ψi ρis

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The overhang projection method (OPM) guarantees entirely self-supporting structures, thus removing the need for sacrificial support material. The OPM scheme is actualized by three variables: ψ, ρs and φ. The independent design variable vector, ψ indicates whether material is desired at a given location, while the dependent variable vector, φ – referred to here as the “material placement variable” – determines whether material will be projected to the element density space at a given location, i. φi is defined as: (5)

= HT (φ) =

  tanh (βT T ) + tanh βT (µis (φηT ) − T ) tanh (βT T ) + tanh (βT (1 − T ))

(6)

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This dependent variable vector is then either projected to the physical space through solid projection (Eq. 2) or through void projection (Eq.4). Ultimately, to convert the independent design variable, ψ, to the dependent material placement variable, φ, one must proceed in a systematic fashion, calculating φ layerby-layer via Eq. 5. Further diving into Eq. 5, the variable vector, ρs – designated the “support indicator” – is a function of φ in the support neighborhood set, NS , and indicates whether a point is adequately supported. NSi for a particular φi is shown in Fig. 3, indicated as the φ captured in the blue wedge. The support indicator, ρis , for a particular point, i, is calculated as:

In Eq. 6, the average of the material placement variables, φ, for a particular point, i, is denoted as µis , and is defined in the following equation: P µis

j∈NSi

= P

φ j ws

j∈NSi

ws

,

(7)

where w s is the uniform support region weighting function (w s = 1), and j is a counter of φ in the support neighborhood set, NSi . This average is passed through Eq. 6 to calculate the “support indicator,” ρ s . The threshold value, T , is enforced 4

(a) ψ design variable field

Figure 3: Two layer overhang projection method mapping scheme. The blue Neighborhood set, NSi is illustrated for the yellow design point of concern.

(b) φ standard void projection

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that the T for a design point on the edge of the space is different than the T for an internal design point, as a varying number of φ points are captured.

(c) φ for VOPM

(d) ρ standard void projection

2. Combined formulation: the Void Overhang Projection Method (VOPM)

(e) ρ for VOPM

Figure 4: Design variable mapping for standard void projection and for void overhang projection method (VOPM)

Drawing upon the logic of the solid overhang constrained algorithm, the result of coupling the VPM with the OPM is to produce structures with designed “void pathways” which nucleate at the structure’s boundaries and stop within, or penetrate through the structure. The presence of void pathways ensures that unsolidified material will not be trapped by the structure. The combined formulation – deemed the void overhang projection method (VOPM) – simply involves sequentially implementing the OPM and the VPM. First the design variables, ψ are passed through the OPM to determine the material placement variables, φ. Subsequently, the material placement variables are passed through the VPM to project void information to the physical density space, ρ, as specified in Eq. 4. This can be summed up in the following equation:

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projects to the physical space as before and creates the topology in Fig. 4e, which allows for void pathways to the bottom surface, as desired. Note that in this case, the material in the blue void space would be able to flow downward at an angle of 45 deg or greater from horizontal. Under the combined VOPM algorithm, instead of a “supported” solid feature growing from the build plate in the +y , a “supported” void region must grow from the designated side in the +y direction. This manufacturing constraint turns out to be rather restrictive, completely eliminating the creation of “holes” or occluded voids seen in the VPM and OPM. Hence, each void region must have an uninterrupted pathway to the designated surface. In this example, an overhang angle of 45 degrees is chosen for convenience. The engineer should choose an angle that corresponds with reliable powder and resin flow under gravitational forces. Obviously, the closer the OPM angle gets to 0 (horizontal), the more the design space is relaxed. However, in practice, small angles (less than roughly tan−1 ( 13 )) can cause issues due to the skewed design point spacing (δ x >> δy in Fig. 3).

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ρevoid = ρevoid (φ(ψ)) = 1 − H(µe (φ(ψ)))

(8)

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For solutions to typical VPM and OPM problems, readers are referred to previous publications on the matter: VPM see [3, 34, 35], OPM see [25, 11, 10]. In this section, we demonstrate the sequential projection scheme through a simple example. Fig. 4a presents a test design variable field, ψ, where red indicates a value of 1 and blue indicates a value of 0. In this projection example, the design variable field is passed through both the VPM and the VOPM to highlight the difference. In the case of the VPM, the φ material placement variables are equal to the ψ design variables (Fig. 4a), and the projected topology is seen to have rounded void space corners (Fig.4d). Alternatively, in the VOPM case with the OPM projection from bottom to top with a defined angle of 45 deg, the φ material placement variables are quite different from the VPM case. Notably, the two square regions of ψ = 1 are projected (i.e. filtered) to be φ = 0, as there is no pathway to the bottom surface, where all voids must nucleate. Also, the “L-like” ψ region is projected to create a vertical ψ region with an angled region attached to the side at a 45 deg angle, as defined – see Fig. 4c. This φ field

3. Sensitivity Analysis The objective and constraint function sensitivities are similar to those found in Johnson and Gaynor’s 3D overhang projection paper [11]. A more thorough derivation of the self-supporting adjoint approach adapted to the HPM context from that seen in Langelaar’s papers [8, 37] is found in a recent paper [10]. This adjoint approach allows for order-of-magnitude speedup in the sensitivity calculation by eliminating the need to directly calcu∂φ term. It is emphasized that the adjoint approach also late the ∂ψ simplifies implementation and significantly reduces memory requirements, allowing for better scalability (previously [25] re5

∂φ quired storage of the ∂ψ term, which is on the order of number of design variables (ndv) by ndv in size). To obtain the sensitivities of the OPM, one must first calculate the sensitivities with respect to the material placement variables, φ. The sensitivity of an arbitrary function, C, w.r.t. φ is calculated according to:

∂C X ∂C ∂ρe = ∂φ e∈Ω ∂ρe ∂φ

4. Problem Formulation All design cases in this manuscript solve the well-known minimum compliance (maximum stiffness) with maximum volume constraint problem [38], as seen in the following formulation: min

f (ψ) = FT d

subject to:

K(ψ)d = F X ρe (ψ)ve ≤ V

ψ

(9)

Next, the calculated derivative w.r.t. the material placement variables are passed through an adjoint approach to obtain the derivatives w.r.t. the design variables, ψ. As demonstrated in ∂φ term is eliminated, replaced [11, 10], the calculation of the ∂ψ instead with an equivalent sensitivity through an adjoint ap∂C proach. This ∂ψ calculation is seen in the following equation: j

e∈Ω

0 ≤ ψi ≤ ψmax

where the objective function, f , is the structural compliance, FT d. The compliance and maximum volume constraint, P e e e∈Ω ρ v ≤ V, are a function of the elemental density vector, ρ, which is ultimately a function of the independent design variable vector, ψ. Kd = F is an equilibrium constraint – where F is the global load vector and K is the global stiffness matrix. Solving Kd = F yields, d, the nodal displacement vector. v is the vector of elemental volumes, and V is the maximum allowable volume of material. Herein, the RAMP material penalization scheme [39] is employed, as seen here: ! ρe Ke (16) Ke = ρmin + 1 + η(1 − ρe ) 0

(10)

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where the vectors λ j are adjoint operators (multipliers). For convenience, a new function, m, ˘ is defined as: m ˘ j = φi = ψi ρis

∀i∈Ω

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! ˘j ∂m ˘ j+1 ∂m ∂m ˘j ∂C ∂C = + λ j+1 = λj ∂ψ j ∂φ j ∂φ j ∂ψ j ∂ψ j

(15)

(11)

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Defining a function in this way also helps draw parallels to the aforementioned approach by Langelaar [8, 37]. The subscript j on the function m ˘ indicates the “layers” of the design variable Cartesian grid, where j = 1 is the lowest layer (the build plate for solid OPM). Layer j = ni is the top layer of the design variable grid. Equation 10 shows that each adjoint operator depends on the adjoint operators associated with the layer above. Therefore, the sensitivity calculation begins at the top layer of the design variable grid and proceeds sequentially downward, until reaching the bottom layer. The following equation provides the multiplier at the top layer: λTni =

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While the occluded void elimination scheme will also work with the more typical SIMP penalization scheme [38, 40], RAMP is found to produce more stable convergence behavior and superior performing topologies. Specifically, RAMP produces in non-zero derivative values at ρe = 0 while SIMP yields zero-valued derivates at ρe = 0 – the non-zero derivatives help provide constant derivative information, even if elemental densities goes to zero. Similar benefits are seen in Ha and Guest [41] and Guest [42]. In Eq. 16, ρmin is set to ρmin = 0.001 to prevent singularity issues in the system solve. The penalty parameter, η, is set sufficiently high such that intermediate values of ρe are deterred through penalized. The Method of Moving Asymptotes (MMA) optimizer [43], was employed to tackle the defined problem in Eq. 15. The MMA algorithm is slightly altered such that the asymptote increase parameter is set to 1.15 instead of the typical 1.2 and the asymptote decrease parameter was set to 0.6 instead of the typical 0.7. The initial asymptote value was set to 0.5/(max(β, βT ) + 1). These parameters produced smooth convergence behavior.

∂C ∂φni

(12)

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For completeness, the derivative of m ˘ with respect to material placement variable, φ, is calculated as:

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∂m ˘ j+1 ∂HT (φ) =ψ ∂φ j ∂φ j

(13)

Finally, the derivative of m ˘ w.r.t. the design variables, ψ, takes on the form: ∂m ˘j = ρs ∂ψ j

5. Two-dimensional Algorithm Results (14)

A two-dimensional example is solved here to demonstrate the algorithm and to help define the meaning of the overhang angle in the void projection context. As defined in Fig. 5a, a pinned-pinned beam with an angled load on bottom center is used. The design domain here is defined by L = 60, H = 10 with 600 by 100 element mesh discretization. In this example,

Derivatives are not given here for the void projection scheme, as they are previously published [35]. Additionally, derivatives for the RAMP material interpolation scheme are straight forward. 6

L

L 2

H=

Param rmin T η

L 6

F

(a) Problem Definition

Max opt iters β βT rS F E V

(b) Void growth at max of 63 deg from horizontal

Param Value 3.2*elemsize see Sec. 1.3 10 for 100 iters 20 for 400 iters 500 15 12 two layers below 1.0 1.0 50%

Table 1: 2D parameter values

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(c) Void growth at max of 45 deg from horizontal

(d) Void growth at max of 27 deg from horizontal Figure 5: Void projection at various angles. Pinned-pinned beam with angled point load.

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Figure 6: Bow tie Problem Definition

the void grows from the bottom, with the allowable angle of “void growth” varied. In Figures Figures 5b to 5d, the angle of void growth decreases from 63 degrees, to 45 degrees, to 27 degrees to horizontal. As can been seen in the angled member attached to the point load, the allowable void growth becomes less restrictive as the angle of growth decreases. Hence, 27 degrees is much less restrictive on the design space than 63 degrees.

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6. Algorithm Extensions and Enhancements

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The above demonstration produces solutions which adhere to the imposed “void growth” constraint. However, the constraint of requiring the void grow from only one surface and in one direction is both restrictive on the design space, and is also not necessarily realistic restriction for the occluded void design issue. The above algorithm can be extended through straightforward schemes to better approximate the manufacturing limitations seen in AM. The following section demonstrates how the underlying “void growth” algorithm can be extended to allow the void to exit to side regions and in more than one direction, opening up the design space fairly significantly. A “bow tie” test problem is devised to demonstrate these extensions. As seen in Fig. 6, a point load is applied in the downward direction on a rectangular design domain with edges fixed on the right and left. The design domain here is defined by L = 15, H = 5 with a 300 by 100 element mesh discretization. Symmetry is not used in this case, instead optimizing for the entire region. Free-form topology optimization produces the design seen in Fig. 7. For reference, the free-form objective function value is f = 9.57. As can be seen in the “bow tie” topology, there are four occluded pores trapping blue void material. Clearly, this parasitic material serves no structural purpose and should be eliminated (released) if possible. Four examples demonstrate how the parasitic material can be eliminated, with an increasingly relaxed design space through allowing void growth from (a) bottom up (BU), (b) bottom-up plus sides (BU+s), (c) bottom-up plus top-down (BU+TD), and (d) bottom-up plus top-down plus sides (BU+TD+s).

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Inspecting the formulation, the void growth angle is perhaps a bit arbitrary at first glance. However, experienced AM professionals will attest that when draining material from internal voids, the topology of the internal void regions can still trap material, even if there is an exit pathway. This is especially a problem when a winding internal pathway has no exit on the other end, eliminating the possibility of using pressurized air to blow out the material. The proposed algorithm, by comparison, always guarantees a “downhill” exit pathway for any internal void regions. This is a major advantage, as a part can simply be vibrated in a single orientation and the powder or resin should exit to the bottom through gravity alone. An additional advantage comes from the designer’s ability to specify the minimum diameter of the exit pathways (drainage holes). More viscous materials may require larger pathways, while “easy flowing” low viscosity materials may allow for much smaller pathways. For the two dimensional VOPM approach, it is useful to define some parameter values (found to work well). In addition to the parameters seen below in Table 1, the element type of each beam was a 4-node quadrilateral. The most critical parameters were β and βT – the values in Table 1 were chosen such that OPM logic was correctly imposed and so that convergence was fast and targeted quality local minima.

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Figure 7: Bow tie test case free-form solution, f = 9.63

Figure 10: Bow tie problem: Optimal topology solution allowing void from bottom and sides (BU+s), f = 13.72

can now escape to the left or right edges as well. Running the design problem from Fig. 6 with the ρ s fixed as in Fig. 9 produces the topology seen in Fig. 10 – i.e. case (b) “BU+s.” The resulting “BU+s” topology is both visually closer to the free-form topology solution and achieves a better compliance value than for the ”BU” situation ( f = 13.72 vs. previous f = 15.49), although not as good as the free-form solution ( f = 9.63). A plot with all compliance values for the bow tie problem are seen in Fig. 15. Here, the penalized compliance values are normalized by the free-form penalized compliance and multiplied by 100. Of course, requiring voids to exit to the bottom is not that reasonable of an approach to eliminating internal voids, since voids can clearly exit to the top in addition to the bottom (or any surface). Hence, a slightly enhanced, further relaxed approach is proposed whereby void growth is allowed from both the top and bottom simultaneously. This is conceptually similar to a multi-axis machining topology optimization approach proposed by Langelaar [44], in which material is removed from multiple direction simultaneously – i.e. material is removed from outside-in, eliminating the occurrence of internal voids. In this approach, the various material removal directions are combined via an efficient cumulative summation. The effect of the proposed algorithm herein would be conceptually equivalent if the overhang angle were set to 0 deg. Multi-directional void pathways are achieved by mapping the design variables, ψ, to two sets of void placement variables, φ1 and φ2 , as calculated for a particular point, i:

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Figure 8: Bow tie problem: Optimal topology solution allowing void from bottom (BU), f = 15.49

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As demonstrated in Sec. 5 on the angled load problem, the bow tie problem is now solved with void growth allowed from the bottom only – i.e. case (a) “BU.” All design problems in this section are executed with a minimum allowable void growth angle of 27 degrees from horizontal. As seen in Fig. 8, allowing void growth from the bottom only severely restricts the achievable topology, and drastically impacts the compliance value ( f = 15.49 as compared with f = 9.59 of the free-form case). The first enhancement – or relaxation of the design space – comes through simply allowing voids to nucleate on the sides of the domain. This is achieved through setting ρ s on side regions equal to 1, indicating that a void is “supported” and can nucleate from a side. This is illustrated in Fig. 9, where a yellow inner circle indicates a ρ s set to 1. For this enhancement, there is still just one neighborhood set. Hence, the voids can nucleate and grow from the bottom face of the domain at greater or equal to the prescribed angle, and also from the sides under the same angle restriction. This opens up the design space a fair amount, as voids are not required to have a direct path to the bottom edge of the design domain, but

φi1 (ψ) = ψi ρis1

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φi2 (ψ) = ψi ρis2

(17)

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where φi1 indicates a void growing from the bottom-up, while φ2 indicates a void growing from the top-down. As opposed to the previous enhancement, this enhancement requires determining and storing an additional support neighborhood set, NS , for the φ2 projection calculation. Once the design space is mapped to the material placement space, the two material placement variable vectors must be combined before they can be projected to the physical space, ρ. The combination is seen in the following equation: φ=

1 (φ1 (ψ)) + φ2 (ψ)) 2

(18)

Note that this is an average of the two void placement variables. While the calculated φ in Eq. 18 will be 1/2 when one of the

Figure 9: Design point setup for void growth from bottom-up and sides (BU+s). Inner circle colored as yellow indicates ρ s is set to 1 for these design points.

8

(a) solution

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Figure 11: Design point setup for void growth from bottom-up and top-down (BU+TD). Inner circle colored as yellow indicates ρS 1 is set to 1 for these design points. Outer circle colored as red indicates ρS 2 is set to 1 for these design points

(b) ψ field

φi = 1, and will equal 1 when φ1 = φ2 = 1. Since the proposed approach uses a relatively high β value in the HPM, both of these situations result in nearly the same projection to the density space. It should be noted that one cannot replace Eq. 18 with a multiplicative scheme, as this would require both φ1 and φ2 to be 1 for void material to be projected. In contrast, the proposed scheme should project void material whenever either φ1 or φ2 is 1. This could be achieved by a max function, or through the project of the average, as is proposed here (the average φ is projected to to the physical space, ρ). Figure 11 demonstrates how the ρS variables are specified for this new multi-directional void growth scheme. ρS 1 is set to 1 on the bottom edge of the design domain, while ρS 2 is set to 1 on the top of the design domain. In Figure 11, a yellow inner circle indicates ρS 1 = 1 while a red outer circle indicates ρS 2 = 1. Applying the multi-directional mapping scheme (BU+TD) to the test problem from Fig. 7, we achieve a slightly altered “bow tie” optimal topology, displayed in Fig. 12. The mapping process is demonstrated in Fig. 12: starting with design variable vector, ψ, and mapping to φ1 and φ2 , and the combined φ. While the number of material placement variables doubles, the number of design variables remains constant. In general, the multi-directional VOPM scheme results in similar convergence behavior to that of the single-direction VOPM. Clearly, the algorithm is attempting to drive the topology as close to the free-form “X” solution as possible, but can only achieve a variable thickness beam under the two direction void growth scheme. Finally, to open up the design space to an even fuller extent, the multi-φ approach (Eq. 18) is expanded to allow for voids to nucleate from the side as well (case (d) “BU+TD+s”). A schematic of the design domain is seen in Fig. 13. Here, the ρ s1 are set to one on the bottom and both sides – indicated with the yellow inner circle. Likewise, the ρ s2 are set to one on the top and both sides – indicated with a red outer circle. Applying this scheme to the same bow tie test problem, one obtains the topology seen in Fig. 14, where the void is allowed

(d) φ2 field

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(c) φ1 field

(e) φ field

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Figure 12: Bow tie problem: Optimal topology solution allowing void from bottom-up and top-down (BU+TD), f = 14.95

Figure 13: Design point setup for void growth from top-down and bottom-up, including side growth nucleation (BU+TD+s). Inner circles colored as yellow indicates ρ s1 is set to 1 for these design points. Outer circles colored as red indicates ρ s2 is set to 1 for these design points.

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Figure 14: Bow tie problem: Optimal topology solution allowing void from bottom, top and sides (BU+TD+s), f = 10.91

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Figure 15: Bow tie problem: Comparison of normalized penalized compliance values for various void exit directions

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to nucleate on both sides in addition to the bottom and top, opening up the design space relatively significantly, and allowing the optimization to approach the free-form solution (Fig. 7).

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Plotting all normalized penalized objective function values for the bow tie problem, it can be seen how allowing additional void growth directions relaxes the design space and lets the solutions approach the free-form TO solution. Here, the most relaxed case (d) “BU+TD+s” achieved a penalized compliance only 13% higher than the free-form solution, while the most restricted case (a) “BU” achieved a compliance 61% higher than the free-form solution.

Figure 16: Convergence plot and design evolution for BU+TD+s

Lastly, a demonstration of the convergence behavior is included in Fig. 16 (shown here for the BU+TD+s case). As can be seen in the objective function vs. iteration plot, the convergence is generally very smooth, with only a bit of undesirable oscillatory behavior in the beginning, where the structure attaches and detaches from itself. This behavior is seen in the design evolution of the first 10 iterations or so (see figs. 16b to 16f). Clearly the structure always adheres to the imposed constraint, and quickly converges to a high quality local minima. 10

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Figure 17: 3D torsion design definition

7. Three-dimensional Algorithm Results

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Two dimensional design cases are great to demonstrate proof of concept, although the algorithm restricts the design space rather significantly, even when voids are allowed to grow from multiple directions and from side regions. Therefore, the VOPM algorithm is demonstrated in 3D, where the added dimension allows for a much closer match between free-form TO solutions and the proposed internal void elimination TO scheme. First, it is shown that free-form TO solutions may exhibit large occluded void topological features, where parasitic mass would be trapped. Second, the 2D algorithm is implemented in 3D and applied to the same problem. It is shown that visually similar solutions may be obtained with minimal loss of optimality in comparison to free-form. While there are a number of potential test problems, we showcase the algorithm’s capabilities on a 3D torsion problem, as defined in Figure 17. The design problem here has relative dimensions of L = 64 and H = 32 discretized with elements nel x = 64, nely = 32, and nelz = 32. A patch of nodes on the “wall” are fixed in x, y and z: here the patch is size 12 by 12, equating to 13 by 13 patch of fixed nodes centered on the lefthand face. On a similar patch on the right-hand face, a torsion load is applied to the perimeter. The magnitude of the line load on each perimeter line is 1.0 on each node, except for the corner nodes, where the magnitude is 0.5. The free-form reference case is solved (Fig. 18) and the compliance is normalized to a value of 100 in the penalized compliance plot in Figure 22. As can be seen in Fig. 18b, there exists a large ellipsoid-like void region. This void region of the design is completely closed, meaning none of the unsolidified material would naturally drain. To rectify the issue, the algorithm outlined above, and demonstrated in a 2D sense, is adapted to a 3D context. All concepts of the VOPM are directly translatable. Additionally, and importantly, for the 3D design case, a multi-phase TO scheme is employed to design both the void phase and the solid phase, regaining control of the solid phase minimum length scale and helping the optimizer obtain better local minima – the optimization problem’s nonlinearity is alleviated through introduction

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(a) Free-form solution solid topology

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(b) Free-form solution void topology

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Figure 18: 3D Torsion free-form solution

11

Param rmin T η Max opt iters βvoid βsolid βT rS F E V

Param Value 3.2*elemsize see Johnson [11] 10 for 100 iters 20 for 400 iters 500 17 17 15 two layer below 1 1 50%

(a) Solid solution, thresholding out low density elements.

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Table 2: 3D parameter values

1 (ρvoid (ψvoid ) + ρsolid (ψsolid )) 2

(19)

(b) Cut view, showing transparent void phase exiting to bottom surface

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of a second phase. Here, the minimum length scale of the void phase guarantees the minimum drainage hole diameter for powder or resin. As noted by Guest [35], the multiphase algorithm simply requires doubling the number of design variables, such that there is a set of independent design variables associated with solid phase and another set associated with the void phase. In this manuscript, the location of the design points for both phases are coincident. The element density calculation for the multi-phase formulation takes on the following form:

The calculation of ρsolid proceeds in the standard HPM manner, without the intermediate “material placement” dependent variables, φ:

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Figure 19: 3D torsion problem: void from bottom up (BU)

ρesolid = ρesolid (ψsolid ) = H(µe (ψsolid ))

(20)

points. The overhang angle for the void growth is again set to 27 degrees – this value tends to work well, as in 2D. As can be seen in the solution in Fig. 19a, the topology is extremely similar to that seen in free-form solution (Fig. 18). The only difference is two small drainage holes on the bottom face of the design domain, as seen in the cut view in Fig. 19b. Due to the imposed allowable void growth angle, an inclined exit path allows for easy material removal. It is noted that the existence of two holes is a byproduct of void growth angle – i.e. at a more extreme defined angle, the optimal topology would likely only require one drainage hole. Looking at the normalized performance (compliance) of the topologies in plot in Fig. 22, it’s noted that the penalized compliance value of 102.69 is only slightly larger than that of the free-form solution ( f = 100). This 2.7% increase is minimal in comparison to the drastic performance decrease when imposing the internal void elimination scheme to problems in 2D. Clearly, the constraint is much less restrictive in a 3D context. As was demonstrated in two dimensions, the introduction of multi-directional void growth allows for further design relaxation. The torsion test problem is now solved with the design scheme which allows void growth from the top and the bottom. As in 2D, the problem employs the multiple φ approach as seen in Eq. 18. The solution in Fig. 20 resembles that of the single

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The void projection scheme proceeds in the same VOPM manner as before: ρevoid = ρevoid (φ(ψvoid )) = 1 − H(µe (φ(ψvoid )))

(21)

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Aside from the multi-phase scheme, the parameter values and optimization strategy are similar to that seen for 2D and are tabulated in Table 2. The torsion example problem is demonstrated with two levels of restriction on the void growth. For all design cases, a buffer layer of design variables is added outside the bounds of the physical mesh. Here, we add four extra design points in the x, y, and z direction (i.e. two extra on each side). The extra design points allow relaxation of the design space, resulting in improved convergence of the algorithm without any significant computational burden. Physically this allows for void nucleation away from the boundary of the physical mesh – lessening the compliance impact of new void nucleation, and improving optimization behavior. In the first instance in Fig. 19, the problem is solved allowing void growth from bottom-up (BU) only – i.e. the 3D equivalent to the typical 2D mapping (ρ s on bottom of domain set to 1), except with the aforementioned buffer zone of extra design 12

(a) Solid solution, thresholding out low density elements.

(b) Cut view, showing transparent void phase exiting to bottom and top surface

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(b) Cut view, showing transparent void phase exiting to bottom and top surface

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(a) Solid solution, thresholding out low density elements.

Figure 21: 3D torsion problem: void from bottom-up and top-down (BU+TD) with alternative initial guess and standard MMA parameters

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Figure 20: 3D torsion problem: void from bottom-up and top-down (BU+TD)

in any case, the achieved compliance in all three instances is within 5% of the free-form solution.

direction (Fig. 19), but possesses void exit pathways to both the bottom and the top. As seen in Fig. 20b, the void has 2 clear exit pathways to the top and two clear exit pathways to the bottom. While one might expect the addition of more design freedom to lower the objective function in relation to the single-direction void growth scheme, the opposite is true here – Fig. 22 reports a value of f = 104.67, as opposed to the previously achieved f = 102.69. While this is not expected, it makes logical sense, as the algorithm may not be able to discern whether or not to use an exit boundary (sensitivity is symmetric). To test the theory that the algorithm found an inferior local min, the initial guess is perturbed slightly and the MMA parameters are relaxed to their default values. Thus, the initial ψ is set to ψnew = ψoriginal − δ/2 + δR(ndv), where δ is some small value (here δ = 0.08), and R indicates a random number vector the length of the number of design variables (ndv). The design from the alternative initial guess yields a normalized performance of f = 102.74, which is almost identical to the performance achieved in the BU design case. Additionally, it should be noted that the design in this case is no longer symmetric. Thus is can be concluded that the slight perturbation in initial guess and the relaxation of the MMA parameters allowed the solution to progress to a better local minima. Other design cases may not be as suseptible to initial guesses – although

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8. Discussion

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The proposed VOPM algorithm for eliminating the occurrence of parasitic occluded void material in TO solutions involves combining two previously developed algorithms for (1) overhang control in additive manufacturing (OPM) and (2) void design control (VPM). This algorithm is a clear alternative to the virtual temperature method (VTM). Inspecting the solutions from VTM, it is often seen that there are generally more holes created in the domain, as the algorithm chooses to route the “thermally loaded” void region to the closest available heat sink border. Hence, it is unlikely that the VTM algorithm would allow for long exit pathways of the void phase. In the proposed VOPM algorithm, that inherent problem is eliminated. Additionally, it is asserted that the VOPM algorithm herein has the extra advantage of only requiring the structural simulation. However, there are clearly some disadvantages of the proposed scheme, including the inability of the algorithm to allow for certain serpentine void pathways. Two dimensional test problems clearly demonstrate the algorithm’s ability to design for occluded void elimination. These 13

Acknowledgments Penalized Compliance, f

100

The authors would like to thank Krister Svanberg allowing use of the MMA optimizer code.

80 9. Abbreviations and variables 60

AM - Additive Manufacturing HPM - Heaviside Projection Method OPM - Overhang Projection Method TO - Topology Optimization VOPM - Void Overhang Projection Method VPM - Void Projection Method ψ - design variable φ - material placement variable ρS - support indicator variable ρe - elemental density variable µe - elemental weighted average of material placement variables

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µS - weighted average of material placement variables in support wedge T - Heaviside threshold value β - Heaviside projection method curvature parameter βT - Thresholding Heaviside curvature parameter

Figure 22: Comparison of penalized compliance values for the 3D torsion problem

problems also demonstrate the influence of certain parameters, such as the void “overhang” angle. Since this design consideration is rather restrictive on the design space, extensions are made to allow for (1) void nucleation from the side of the domain (i.e. not just from build plate), and (2) void growth from multiple directions. These enhancements are seen to allow the design to closer approach the free-form solution. It should be noted that multiple drainage directions is an open topic in AM, as it becomes increasingly difficult to remove powder or resin with increased drainage topology complexity. With the relaxation of the design space comes increased nonlinearity in the projection (filtering) schemes – thus one must be careful in the formulation to eliminate nonlinearity wherever possible.

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The algorithm is then demonstrated in three dimensions, where the design problem is fundamentally much less constrained than in 2D. This relaxation is due to the fact that void growth can move up and around solid features, whereas in 2D, if void is to exist at the top of the domain, it must cut a path there, eliminating any solid design features in its way. The 3D torsion design case presented here represents a situation in which material should exist on the boundaries of the design domain. Clearly, the efficacy of the algorithm is to figure out where to “poke” drainage holes in order to allow for the large internal void region. Finally, the authors note that the proposed algorithm has potential for use beyond direct manufacture AM. While the methodology was developed for removing occluded voids in AM structures, it could potentially also be used to generate TO structures for manufacture via casting, especially 3D printed sand molds. Essentially, one could design and print the negative and cast the positive, with the negative guaranteed to flow out due to the imposed constraint. The extension is natural and promising. 14

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