DESIGN AND FREEFORM FABRICATION OF DEPLOYABLE STRUCTURES WITH LATTICE SKINS

Uma Maheshwaraa, David Bourell, and Carolyn Conner Seepersad*

Mechanical Engineering Department 1 University Station, C2200

The University of Texas at Austin Austin, TX 78712

Abstract

Purpose

Frontier environments—such as battlefields, hostile territories, remote locations, or outer

space—drive the need for lightweight, deployable structures that can be stored in a compact

configuration and deployed quickly and easily in the field. In this paper, the concept of lattice

skins is introduced to enable the design, solid freeform fabrication (SFF), and deployment of

customizable structures with nearly arbitrary surface profile and lightweight multi-functionality.

Design/methodology/approach

Using Duraform FLEX® material in a selective laser sintering machine, large deployable

structures are fabricated in a nominal build chamber by decomposing them into smaller parts.

Before fabrication, lattice sub-skins are added strategically beneath the surface of the part. The

lattices provide elastic energy for folding and deploying the structure or constrain expansion

upon application of internal air pressure. Nearly arbitrary surface profiles are achievable and

internal space is preserved for subsequent usage.

Findings

A set of virtual and physical prototypes are presented, along with the computational

modeling approach used to design them. The prototypes provide proof of concept for lattice

* Corresponding Author. Email: ccseepersad@mail.utexas.edu. Phone: 512-471-1985.

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skins as a deployment mechanism in SFF and demonstrate the effect of lattice structures on

deployed shape.

Research implications

The research findings demonstrate not only the feasibility of a new deployment

mechanism—based on lattice skins—for deploying freeform structures, but also the potential

utility of SFF techniques for fabricating customized deployable structures.

Originality/value

A new lattice skin mechanism is introduced for deploying structures with nearly arbitrary

surface profiles and open, usable, internal space. Virtual and physical prototypes are introduced

for proof of concept, along with an optimization approach for automated design of these

structures.

Key Words: deployable structures, lattice structures, solid freeform fabrication, selective laser

sintering

Research Paper

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INTRODUCTION

Several mechanisms are available for compactly storing and deploying two-dimensional

surfaces and three-dimensional structures. Applications range from the deployment of satellite

booms and solar arrays to the construction of temporary shelters and retractable stadium roofs

with vast unsupported spans (Gantes, 2001; Pellegrino, 2001). Examples include tensegrity

structures comprised of a combination of exclusively compressive and tensile members, as

observed in structures such as the Georgia Dome in Atlanta, and pantographs comprised of bars

connected to one another by pivots at their center- and end-points and embodied in collapsing

laundry racks, temporary gates, and the popular children’s toy, the Hoberman sphere (Gantes,

2001; Pellegrino, 2001).

Conventional deployment mechanisms and fabrication techniques generally restrict the

geometry of a deployed structure to symmetric, polygonal or spherical shapes and make it

difficult to rapidly customize the geometry and functionality of the device. In contrast, a

freeform fabrication approach offers several potential advantages, including: (1) the capability of

fabricating complex, intricate internal and external geometries, which enable a host of

customized, embedded deployment mechanisms and parts with nearly arbitrary deployed

geometries, (2) rapid design and fabrication cycles, and (3) potentially lightweight,

multifunctional structures with embedded mechanisms that not only control deployment but also

provide additional functionality such as strength, stiffness, insulation, or blast amelioration.

Freeform deployment also has the potential to overcome build chamber size limitations by

building large parts in a condensed configuration and deploying them, post-build.

“Design for Freeform Deployment” methods and accompanying solid freeform fabrication

(SFF) techniques are established to enable the design, fabrication, and deployment of

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customizable structures with nearly arbitrary surface geometry and lightweight multi-

functionality. The first step towards this goal is to identify potential deployment mechanisms that

are suitable for freeform fabrication and support the desired capabilities. For this purpose, a

viable concept has been devised that uses lattice skins for deployment, as illustrated in Figure 1.

Beginning with a part of arbitrary surface profile and hollow interior, lattice sub-skins are added

beneath the surface of the structure during the design process. The part with integrated lattice

skins is fabricated using selective laser sintering (SLS) technology and a flexible, elastomer

material called Duraform® FLEX. If the structure is larger than the build chamber, it is

decomposed and fabricated as a collection of parts that are subsequently joined together. The

flexible structure can be folded manually for ease of storage and transport and then deployed in

the field via a combination of elastic strain energy and pneumatics. In this process, the lattice

structure serves several functions. First, during the folding step, it stores strain energy that can

be returned upon unfolding to help deploy the structure into its original configuration. Second,

in its deployed form, the lattice structure supports the surface of the flexible part to prevent

collapse and distortion of desired surfaces. Finally, if elastic energy is insufficient for deploying

a large part under its own weight, air pressure is applied within the lattice skin while the lattice

structure constrains the expansion of the part to prevent balloon-like inflation and preserve

desired surface profiles.

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Figure 1. Airfoil cross-section with lattice sub skin. Outer skin is translucent pink for visualization; inner

skin is solid pink; and lattice elements are gray. Lattice structures and cellular materials are particularly amenable to solid freeform

fabrication techniques, which provide flexibility for fabricating them as integral aspects of an

overall part. Lattice structures and cellular materials have been fabricated with a variety of SFF

technologies, including SLS (Stampfl, et al., 2004), direct metal deposition (Oruganti, et al.,

2004), selective laser melting (Brooks, et al., 2005), stereo/photolithography techniques (Wang,

2005; Zimbeck and Rice, 1999), and hybrid approaches that include SFF-based sacrificial molds

(Gervasi and Stahl, 2004; Stampfl, et al., 2004). They are used for applications that require

lightweight stiffness, compliance, impact absorption, heat exchange, catalysis, and other

properties (cf. (Evans, et al., 1999; Evans, et al., 2001; Gibson and Ashby, 1997) for an

overview). However, the use of lattice structures as deployment mechanisms has not been

investigated to date.

Lattice structures have several characteristics that make them particularly appropriate for

deployment applications. First, they are preferable to internal mechanism-like structures,

mentioned previously, because they are more flexible for shape control of relatively arbitrary,

freeform geometries; occupy very little space in the build chamber; and do not require small-

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scale pivots or joints that are difficult to fabricate. Second, they also offer a combination of low

relative density and high effective stiffness, providing high levels of rigidity for controlling the

deployment of a three-dimensional part without significantly impacting its apparent density for

consolidation and collapse (Hayes, et al., 2004). Third, the lattice skins do not occupy the

interior regions of the part, leaving the space available for other uses. In addition, the lattice skin

itself makes the deployed part suitable for multifunctional applications (Evans, et al., 2001;

Gibson and Ashby, 1997; Seepersad, et al., 2004; Seepersad, et al., 2005). For example, the

lattice skin could be filled with cooling fluids or air for heat exchange, insulation, or natural air

circulation; earth, foam, or other materials for impact absorption or blast protection; or explosive

materials for controlled detonation. Finally, lattice skins are conducive to portable deployment,

requiring only a portable air pump for deployment and a spray can of thermoset coating for

rigidization, if desired.

In the following sections, a method is presented for designing, fabricating, and deploying

structures with lattice skins, and preliminary results are presented.

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METHOD FOR DESIGN AND FABRICATION OF FREEFORM DEPLOYABLE

STRUCTURES WITH LATTICE SKINS

The general method for designing and fabricating deployable structures with lattice skins is

outlined in Figure 2. The starting point for the method is the 3D surface profile of the part to be

deployed.

Arbitrary 3D Surface

1. Add Lattice Elements

2. Optimize Lattice Structure

3. Decompose into Small Parts Virtually

4. Fabricate with Selective Laser Sintering (SLS)

5. Deploy

6. Stiffen

Fully Functional Part

Figure 2. A method for designing and fabricating deployable structures.

In Step 1, a lattice (sub-skin) structure is added beneath the surface of the part as a

deployment mechanism. The arrangement of the lattice structure is determined with a topology

optimization procedure or a standardized strategy. Two types of lattice sub-skins are described

in the next section: (1) an open lattice arrangement, as illustrated in Figure 3a, for directly

reinforcing the surface of the part, and (2) a closed lattice arrangement, as illustrated in Figure

3b, for pneumatic inflation.

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(a) Open lattice structure

(b) Closed lattice structure

Figure 3. An arbitrary 3D part with added lattice structure.

In Step 2, a finite element model of the part and lattice sub-skin is built to analyze the

stiffness and deflection profile of the part. If adjustments are needed, the dimensions and

density of the lattice structure are varied iteratively until the intended deflection is attained. This

step can be executed manually or with an optimization algorithm. For the lattice structure

illustrated in Figure 3a, the stiffness of the lattice structure should be such that it supports the

self- weight of the part with minimal deviation from the intended surface profile. If internal air

pressure is used to deploy a part with a closed lattice structure (Figure 3b), the lattice structure

should support its own self-weight and the self-weight of the part and constrain expansion as

internal air pressure is applied within the lattice skin. The ultimate objective is to achieve the

intended surface profile as closely as possible.

The deployment strategy is intended to be applied to large parts with volumes that are greater

than that of the SLS build chamber. Therefore, in Step 3 the structure is decomposed virtually

into parts that are smaller than the build chamber, as illustrated in Figure 4. The decomposed

parts (with integrated lattice structures) are fabricated in the SLS machine using Duraform® Flex

material in Step 4. This step also includes a post-processing phase in which parts are assembled

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by adhering them along built-in joints (e.g., butt, lap, or splice joints). Post-processing also

includes spraying the parts with an infiltrant to make them air-tight and folding them manually

into a compact, storable package. In Step 5, the prototyped part is deployed using internal air

pressure and/or any form of mechanical assistance (e.g., human-assisted unfolding). Since the

part is constructed of Duraform® FLEX, its skin is quite pliable—a useful feature for folding

and storing the part but not for maintaining a rigid structure. If a stiffer skin is desired, the part’s

skin can be coated with a thermoset polymer in Step 6. The outcome of the method is a fully

functional, deployable part.

Decomposed Virtually

Post Processed

Prototyped

Figure 4. Virtual decomposition of parts larger than the build chamber.

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LATTICE STRUCTURE

An integral part of the methodology for building deployable parts is the deployment

mechanism. The lattice structure provides elastic stiffness for supporting the surface of the part

and for constraining the expansion of a part during inflation. It can also store elastic strain

energy as a large structure is folded and return it as the part is unfolded to aid in the deployment

process. Two types of lattice structures are proposed, an open lattice structure (Figure 5a) and a

closed lattice structure for air deployable parts (Figure 5b).

(a) (b) Figure 5. Open lattice (a) for a self-supporting structure and closed lattice (b) for an air deployable part.

Open Lattice Structure

An open lattice structure is composed of lattice elements added to the inner surface of a part.

The arrangement of elements may be determined with a topology optimization routine or by

selecting a standard strategy. In the standard strategy illustrated in this paper, lattice elements

emanate from the inner surface of a part in an orientation normal to the surface. Their ends are

connected by similar lattice elements of same or different dimensions. Rows consisting of

numerous lattice elements are added to span the entire surface of the part. Finally the mesh of

lattice elements forming the lattice structure is obtained by connecting all the rows. Figure 6

shows the open lattice structure unit for a section of cylindrical surface.

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FRONT VIEW SIDE VIEW ISOMETRIC VIEW

Lattice element normal to surface Lattice interconnection

Figure 6. Open lattice unit.

The density and configuration of the lattice structure play a vital role in deploying the part.

The ideal lattice element density depends not only on the geometry of the surface and the target

deflection profile but also on the stiffness of the Duraform® FLEX material. The laser power of

the SLS machine can be adjusted to alter the stiffness of the Duraform® FLEX material. If the

material is highly flexible, the lattice elements are more densely spaced and/or larger in

dimension to provide more support to the flexible surface. Denser spacing is also utilized to

support local surfaces with tighter curvature. As the density of lattice elements increases,

however, the self-weight of the structure also increases along with gravity-induced sagging and

deviations from the desired surface profile. A suitable compromise density must be chosen to

achieve minimal deflection in the structure. Also, the standard lattice configuration illustrated in

Figures 5a and 6 may be modified into more complex interconnecting truss structures—a

strategy that is explored for closed lattice skins in the next section.

For proof of concept studies, the lattice structure is added to the inner surface in an iterative

procedure. The inner surface of a 3D model is used as a reference to create the lattice structure in

SolidWorks (2005). In the standard configuration illustrated in Figures 5b and 6, a single lattice

element is created normal to the surface at a point. A linear array of lattice elements is added to

the interior of a cross sectional slice of the part. This procedure is then repeated along the depth

of the part to generate lattices over the entire inner surface. The free, inner ends of the lattice

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elements are interconnected by straight elements in a regular pattern. Iterative finite element

analysis (in ANSYS (2006) software) is used to determine the appropriate density of the lattice

structure for maintaining the surface profile of the final part. The initial cross-sectional

dimensions of the lattice elements are set to 2 mm x 2 mm, large enough to ensure satisfactory

resolution from the SLS machine. Based on surface deflection estimates from ANSYS, the lattice

element sizes and spacings are adjusted iteratively until the deflections are acceptable. The

iterative procedure can be guided by an optimization method, as demonstrated for the closed

lattice designs in the next section.

Closed Lattice Structure for Pneumatic Deployment

A closed lattice structure joins the outer surface of the part with a concentric, inner surface

and separates the two surfaces everywhere by a set distance. The purpose of the lattice structure

and concentric, inner skin is to maintain the desired outer surface profile of the part when air is

pumped into the skin between the parallel inner and outer surfaces. Without the lattice structure,

the flexible skin would expand naturally to a sphere-like shape, with maximum ratio of volume

to surface area. The lattice skin constrains the expansion so that the volume between the two

surfaces is maximized, which occurs when the skins are concentric and the lattice elements are

stretched to maximum length. As shown in Figure 7, lattice structures join the surfaces at

strategic points, according to the overall size and local curvature of a section.

The concentric inner skin is created with a surface offset procedure during the CAD

modeling of the part (cf. (Maekawa, 1999)). After the external surface of the part is rendered in

Solidworks (2005), an offset command is applied to create the surface of the concentric, inner

skin. The surfaces are separated by a fixed offset distance. Skin thickness values are applied to

the inner and outer skin surfaces, also using an offset command. The offset feature is preferred

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to scaling or copying to maintain concentric profiles and a set distance between skins. Finally,

the lattice configuration and dimensions are determined with a finite element-based lattice

optimization procedure, described in the next section, and the entire part is fabricated as an

integrated structure with SLS and Duraform® FLEX material.

Figure 7. A cross-section of a closed lattice structure for pneumatic deployment.

PROOF OF CONCEPT WITH VIRTUAL AND PHYSICAL PROTOTYPES

The feasibility and effectiveness of the proposed deployment strategy was evaluated with

virtual and physical prototypes of sample deployable structures in ANSYS. The prototyping

effort was conducted in several stages, as described sequentially in this section. First, material

properties and ANSYS finite element modeling capabilities were refined for improved accuracy

in predicting surface deflection profiles of deployed parts. Resulting material properties and

finite element modeling capabilities were used for iterative virtual prototyping, which

subsequently informed the creation of physical prototypes of open and closed lattice skins with

SLS and Duraform® FLEX material.

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Finite Element Modeling in ANSYS

Finite element analysis of Duraform® FLEX parts needs to account for the elastomeric

behavior of the material with relatively large Poisson’s ratio and large displacements under

relatively small applied loads. The behavior of a Duraform® FLEX part was simulated using

finite element analysis models in ANSYS, and the embedded nonlinear analysis capabilities in

ANSYS were utilized to investigate large deformations in the parts. Before deployable

structures were analyzed, the ANSYS models were calibrated with some simple experimental

tests. A cantilever beam test was performed on a Duraform® FLEX block of dimensions 93.5

mm x 32.5 mm x 3.5 mm. As shown in Figure 8, the boundary conditions included fixed

deflection and rotation at one end of the specimen and a set of increasing loads applied to the

opposite end. Data for deflection as a function of applied load were recorded in Table 1.

93.5 mm LOAD

DEFLECTION

3.5 mm

X

Y

Figure 8. Schematic diagram for cantilever beam test.

Table 1. Tip displacement in y-direction for cantilever beam test. Load (N) DEFLECTION (mm)

0.16785 24.61 0.355231 30.22 0.462616 30.98 0.576859 35.12

0.852239 37.08

A corresponding three-dimensional finite element model of the specimen was developed in

ANSYS, using the appropriate material properties listed in Table 2. Both large deformation and

small displacement analyses were conducted, and the results were compared with experimental

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data, as shown in Figure 9. For both analyses, a SOLID45 element was used with element size

of 0.15 mm per edge. In small displacement analysis, the load was applied in a single step based

on the assumption that part geometry remained unchanged. When the large deformation analysis

was enabled in ANSYS, the load was applied incrementally in a series of steps. After each step,

ANSYS updated the geometry to reflect the simulated deformations. The number of load steps

for the large deformation analysis was set to 200.

Table 2. Material properties for ANSYS analysis. Property Value

Tensile Modulus 3.8 MPa Density 486.1 kg/m3

Poisson Ratio 0.45

As shown in Figure 9, the large deformation model matches the experimental data much

more closely than the small displacement model over a broader range of displacements. The

small displacement model is very accurate for small displacements, but increasingly

overestimates deflection as the magnitude increases. As shown in Figure 9, the force-deflection

curve for the small displacement model is linear, as opposed to the nonlinear relationship

observed experimentally. The large deformation model is relatively accurate over the entire

range of displacements and typically approximates the experimentally measured deflection

within 10%. With the low magnitude of Duraform® FLEX’s elastic modulus, large deflections

are expected and may occur under self-weight for large structures.

The simple experiment confirms our capability of analyzing the force-deflection behavior of

simple structures composed of Duraform® FLEX material, using large deformation analysis in

ANSYS. The next step is to demonstrate how this capability can be used to analyze and design

fully deployable structures.

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Figure 9. Comparison of deflection values between experiments and ANSYS.

Virtual and Physical Prototypes of Open Lattice Skin Structures

Two different arch prototypes were used to study the capability of an open lattice skin to

maintain the required surface profile upon deployment. The first prototype, illustrated in Figures

10 and 11, was a single arch with overall dimensions of 30 cm (length) x 8 cm (height) x 0.3 cm

(thickness of skin). Prototypes of the arch were fabricated with and without the underlying lattice

structure, with identical dimensions for the length, height, and thickness of the outer skin. Both

prototypes were fabricated in an identical arched shape using an SLS machine with a laser power

of 15W. As shown in Figure 10, the arch collapses under its own weight without the underlying

lattice structure. The center of the lattice prototype deflects vertically by 2.1 cm under its own

weight. As shown in Figure 11, ANSYS models of the structure predict a deflection of 2.0 cm,

within 5% of the actual experimental value. In the ANSYS model, the effect of self weight was

simulated by adding gravitational body forces and specifying the measured density of the

Duraform® FLEX material. Boundary conditions included constraints on horizontal and vertical

deflection on one end of the arch and constraints on vertical deflection on the other end.

Material properties were reported in Table 2.

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The second prototype—illustrated in Figures 12-15 is a double arch with overall dimensions

of 30 cm (length) x 7 cm (height) x 0.3 cm (thickness of skin). The purpose of constructing a

second prototype was to investigate a more arbitrary surface profile with convexities and

concavities in the surface. As illustrated in the figures, prototypes of the double arch were also

fabricated with and without the lattice structure, with all other features identical. As shown in

Figures 12 and 15, the prototype without the lattice structure collapsed under its own weight to a

single, concave arch. As shown in Figures 13 and 14, the prototype with the lattice structure

retained its shape. The center of the prototype deflected vertically by 1.1 cm under its own

weight. ANSYS models of the structure also predicted a deflection of 1.1 cm, matching the

actual experimental value within the level of precision of the measurements. Boundary

conditions and material properties were identical to those of the single arch model. A larger

prototype of the single arch was also created with dimensions of 90 cm (length) x 30 cm (height)

x 0.9 cm (thickness of skin). The deflection in the centre of the prototype was measured to be

19.8 cm due to its own weight. ANSYS models of the prototype predicted a deflection of 18.7

cm, within 5% of the actual experimental value.

These virtual and physical prototypes indicate that open lattice skins effectively prevent

excessive deformation and collapse of Duraform® FLEX parts and significantly reduce deviation

from intended surface profile. The overall deflection can be reduced even further if the

configuration and density of the lattice skins are refined by coupling the finite element analysis

with an optimization routine. This technique is demonstrated for the closed lattice structure in

the following section.

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Figure 10. Arch prototype with and without lattice structure.

Figure 11. ANSYS model of the arch withstanding its own weight. Colors represent displacement

contours with blue contours on the order of 2 cm for an arch with a span of 30 cm and a height of 8 cm.

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Figure 12. ANSYS result for the double arch without lattice

structure, showing deflection contours under self-weight. Blue contours represent maximum deflection, on the order of

25.0 cm, which occurs where the convex arches collapse.

Figure 13. ANSYS result showing the double arch with lattice structure maintaining its profile. Dotted lines

represent the original, undeformed shape.

Figure 14. ANSYS result of the double arch with lattice

structure showing the deflection contours. Blue contours represent maximum vertical deflection, on the order of 1.1

cm.

Figure 15. Physical prototype of the concave arch with

and without the lattice structure.

Virtual Prototypes of Closed Lattice Skin Structures

An airfoil cross-section was used to study the capability of a closed lattice structure to

maintain a desired surface profile upon pneumatic inflation and deployment. As illustrated in

Figure 16, the airfoil cross-section was specified with overall dimensions of 540 mm (in-plane

width) x 270 mm (in-plane height) x 20 mm (out-of-plane depth) and an inner and outer skin

thickness of 3 mm. Several different lattice structure configurations were investigated in a series

of virtual prototypes, all of which were analyzed for displacement under self-weight using large

deformation analysis in ANSYS. A 2D cross-section of the airfoil was extracted for analysis.

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Self-weight was modeled with body forces and material density specified according to Table 2.

To simulate pneumatic inflation, an internal air pressure of 700 N/m2 (gauge) was applied

between the inner skin and the outer skin. Vertical (y-direction) displacement was constrained at

the base of the model to simulate the support of a table, as illustrated in Figure 16. Horizontal (x-

direction) displacement was constrainted at a single point on the bottom surface of the model.

The lattice elements were modeled using BEAM3 elements in ANSYS, and the skin was

modeled with PLANE183 elements.

To investigate the behaviour of the airfoil part with pneumatic inflation, several different

lattice element configurations were evaluated. For the first virtual prototype, a radial

configuration of elements was borrowed from the open lattice structure, as illustrated in Figure

17a. In initial experiments, bulging was observed in the outer skin between lattice elements,

especially in sections of the model where the skin profile was relatively straight, as illustrated in

Figure 18. To minimize these bulges, the intersections of each lattice element with the inner and

outer skins were chamfered so that the thickness of the element tapers to the middle instead of

remaining flat, as illustrated in Figure 18. Chamfering was incorporated in the first virtual

prototype, as illustrated in Figure 17a, resulting in enhanced ability to maintain a desired outer

surface profile. Even with the incorporation of chamfered elements, however, the deviation from

the desired surface profile was unsatisfactorily large, with a maximum deflection of nearly 36

mm. Large deflections were attributed to insufficient stiffness in the radially configured lattice

structure, particularly in sections with relatively high curvature.

For the second virtual prototype, cross trusses were added to increase stiffness in strategic,

high-stress locations, as illustrated in Figure 17b. As a result, deviation from desired surface

profile (i.e., vertical deflection) was reduced to 23.5 mm as shown in Figure 17b. Although the

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cross elements reduced the overall deflection by approximately 33%, relative to the first virtual

prototype, the deflection remained unsatisfactorily large. Since the configuration and dimensions

of the lattice elements were chosen arbitrarily without engineering guidelines or informed

iterations, an optimization approach was devised to improve the performance of the third virtual

prototype.

Figure 16. Airfoil model showing constraints and lattice elements.

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(a) Maximum deflection without cross truss = 35.59 mm. Yellow contours indicate maximum deflection.

(b) Maximum deflection with cross truss = 23.59 mm. Yellow contours indicate maximum deflection.

Figure 17. ANSYS results for airfoil cross-section with two different configurations (a) without cross truss (b) with cross truss lattice elements.

Figure 18. Flat section of closed lattice skin with and without chamfered lattice elements. Lower figures

illustrate the deformed (black solid) structure after inflation versus the undeformed, desired surface profile (wireframe sketches).

The third virtual prototype was derived from a topology optimization procedure. The

topology optimization procedure was used to identify lattice element configurations and

dimensions with minimal overall deflection. The procedure was based on a ground structure

topology optimization approach (cf. (Dorn, et al., 1964; Kirsch, 1989; Ohsaki and Swan, 2002;

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Topping, 1984)) which begins with a very dense initial configuration of lattice elements, as

illustrated in Figure 16. Each lattice element is assigned a variable in-plane thickness that is

adjusted by an optimization algorithm. (As part of the 2D analysis, all elements were assumed to

have unit out-of-plane depths.) During the optimization process, the in-plane thickness variables

are assigned reasonable upper bounds (e.g., 1 cm) and extremely small lower bounds (e.g., 1E-4

cm). Elements with post-optimization thicknesses near the lower bound are removed from the

final lattice structure under the justified assumption that they contribute very little to the stiffness

of the overall lattice structure. A volume fraction constraint serves as an upper bound on the

total amount of material utilized in the lattice structure.

For this application, iSIGHT (2003) design integration and optimization software was

coupled with ANSYS to execute the topology optimization. Each lattice element was modeled

with a BEAM3 (1D) element, to make it easier for iSIGHT to change its in-plane thickness. The

in-plane thickness of each BEAM3 element was adjusted by the iSIGHT software for each

iteration of the optimization algorithm, by adjusting the appropriate BEAM3 element property in

the input APDL file before executing ANSYS in batch mode. To accommodate the intersections

of cross truss elements with independently variable thicknesses, as shown in Figure 16, each of

the two intersecting elements was divided into two BEAM3 elements with coupled in-plane

thicknesses. PLANE 183 elements were utilized for the inner and outer skins, and their

dimensions were not adjusted during the optimization process. In this case, only cross and radial

lattice elements in the left arc of the structure were modified. The dimensions of the remaining

lattice elements were retained from previous analyses.

The optimization problem was formulated as shown in Figure 19. An iSIGHT-based

optimization sequence—a genetic algorithm followed by a sequential quadratic programming

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algorithm—was used to explore the design space. The optimization algorithms adjusted the in-

plane element thicknesses of 29 lattice elements to minimize the maximum displacement of the

outer surface while satisfying bounds on the in-plane element thicknesses and the overall volume

fraction. Thicknesses were decreased for elements with relatively small effects on the overall

objective, thereby allowing remaining elements to become thicker without violating the volume

fraction constraint. After the algorithm converged to a solution, thin elements (with thicknesses

less than 1 mm in this case) were removed from the lattice structure.

Figure 19. Optimization problem formulation.

Minimize Maximum displacement, δ, of any node on the outer surface

Find Thickness, ti, of each lattice element, i Subject to Bounds on the design variables, 0.1tmax < ti < tmax, where tmax is the maximum

allowable element thickness (3mm)

An upper bound on volume fraction, 0.5ν ≤ , where i i

i max

l tl t

ν =∑ and il is the

length of each lattice element.

The final optimal configuration is illustrated in Figures 20 and 21. This configuration boasts

a maximum overall displacement of 10.16 mm which is 3% of the overall height (270 mm) of the

model and less than 30% of the deflection of the first virtual prototype. Thirteen lattice elements

were removed from the original configuration (Figure 16) which contained 29 lattice elements.

The optimal configuration had a volume fraction of 0.219. The total mass of the lattice elements

was reduced from 3.521g to 0.7713g as a result of the optimization process. In Figure 21, the

optimal configuration is illustrated along with the thicknesses of the remaining elements.

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Figure 20. Airfoil with the optimal configuration showing a maximum deflection of 10.16 mm (in red).

Figure 21. Optimal airfoil configuration with the corresponding lattice element thickness values.

CLOSURE

A method has been presented for designing and fabricating deployable structures, with lattice

structures as the deployment mechanism. The deployable structures are fabricated from an

elastomeric Duraform® FLEX material using selective laser sintering (SLS). The flexible,

elastomeric material allows the structures to be folded and compactly stored. For achieving a

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target surface profile during deployment, lattice structures are utilized to reinforce the surfaces of

flexible structures when they are unfolded and to constrain their expansion when they are

pneumatically deployed. Virtual prototypes provide proof of concept for some preliminary

designs with open and closed lattice structures, and the behavior of physical SLS prototypes

corresponds closely to that of virtual models of open lattice structures.

Ongoing work involves physically prototyping the closed lattice skin parts with pneumatic

deployment. Prototyping the closed lattice deployable structures involves several challenges,

such as sealing the part with a polyurethane infiltrant to withstand air pressure during

deployment and spray coating with a thermoset polymer for stiffness after deployment. Current

plans call for manually spraying the thermoset polymer, but it may be possible to encapsulate

pockets of thermoset polymer that harden when exposed to elevated air pressures or temperatures

during deployment (cf. (Tsunoda and Senbokuya, 2002)). Also, an important aspect of the

deployment methodology involves prototyping parts that are larger than the SLS build chamber.

Currently, large structures are decomposed into smaller parts for fabrication in a nominal build

chamber. This decomposition is a manual process, based on designer intuition and trial-and-

error, but a more systematic and automated process needs to be implemented (cf. (Ilinkin, et al.,

2002)). Decomposed parts are currently joined with butt or splice joints, but alternatives, such as

tongue and groove joints, should be explored for improved air-tightness and surface smoothness.

Also, ongoing work involves developing strategies for virtually collapsing (folding, rolling,

wadding) the parts in CAD and post-processing the parts to return them to their intended

configurations. Related work is reported on a methodology for generating 3D conformal textiles

(Bingham, et al., 2006), but deployable structures introduce a new set of challenges associated

with collapsing continuous surfaces rather than interlinked matrices of fabric. Finally, it would

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be interesting to explore auxiliary functions for the gaps in the closed lattice skins. Example

applications include trapped air for insulation; natural air circulation for convective cooling; and

sand, earth, or other materials to fill the gaps for blast protection or even radiation protection in

habitable space structures.

ACKNOWLEDGMENTS

The authors gratefully acknowledge financial support from the University of Texas at Austin.

The authors owe a special thanks to the undergraduate students who participated in this project:

Brian Nowotny, Trevor Page, Catherine Tradd and Brandon Walther.

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