Accepted Manuscript - Stony Brook Universityme.eng.stonybrook.edu/~chen/Downloads/IDETC2017-67024-TO_and… · In topology optimization, the design problem is recast as an optimal

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1 Copyright © 2017 by ASME

Proceedings of the ASME 2017 International Design Engineering Technical Conferences &

Computers and Information in Engineering Conference

IDETC2017

August 6-9, 2017, Cleveland, Ohio, USA

IDETC2017-67024

COMPUTATIONAL DESIGN AND ADDITIVE MANUFACTURING OF CONFORMAL

METASURFACES BY COMBINING TOPOLOGY OPTIMIZATION WITH RIEMANN

MAPPING THEOREM

Panagiotis Vogiatzis Department of Mechanical Engineering

State University of New York at Stony Brook Stony Brook, NY, 11794

Email: [email protected]

Ming Ma Department of Computer Science



Shikui Chen1 Department of Mechanical Engineering



Xianfeng David Gu Department of Computer Science



1 Address all correspondence to this author.

ABSTRACT In this paper, we present a computational framework for

computational design and additive manufacturing of spatial

free-form periodic metasurfaces. The proposed scheme rests on

the level-set based topology approach and the conformal

mapping theory. A 2D unit cell of metamaterial with tailored

effective properties is created using the level-set based topology

optimization method. The achieved unit cell is further mapped

to the 3D quad meshes on a free-form surface by applying the

conformal mapping method which can preserve the local shape

and angle when mapping the 2D design to a 3D surface. The

proposed level-set based optimization methods not only can act

as a motivator for design synthesis, but also can be seamlessly

hooked with additive manufacturing with no need of CAD

reconstructions. The proposed computational framework

provides a solution to increasing applications involving

innovative metamaterial designs on free-form surfaces in

different fields of interest. The performance of the proposed

scheme is illustrated through a benchmark example where a

negative-Poisson’s-ratio unit cell pattern is mapped to a 3D

human face and fabricated through additive manufacturing.

INTRODUCTION Metamaterials are generally defined as a group of artificial

materials which gain superior effective properties through the

inner structures rather than their composition. Metamaterials

have been considered as critical attributes and important

technology themes by a broad range of applications in

telecommunication, aerospace, defense, automotive and medical

devices, which may serve as building bricks for constructing

more advanced metadevices. Despite recent advances in

metamaterial research [1-4], conventional mechanical

metamaterial designs have been executed mainly in the regular

Cartesian space, e.g. in a 2D rectangular plane or a 3D cubical

space. With the development of additive manufacturing

technology, the time has come to think about achieving

metamaterials on a free-form surface, also termed conformal

metasurface. Such conformal metasurface can give rise to a new

range of applications, e.g. conformal cooling surfaces,

conformal sensors, artificial bone structures, etc.

There is increasing research on conformal structures and its

applications. Wang and Rosen [5] achieved a truss structure

conformed to a part’s shape using parametric modeling

techniques, and Chu et al. [6] proposed an additive

manufacturing method including conformal lattice with a unit

cell library of different mechanical properties. For radar-cross-

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section reduction, Jang et al. [7] proposed a conformal

metamaterial absorber achieving high absorption rates in

different incident angles, by utilizing three different unit cell

designs in different zones of a curved surface. In the field of

flexible electronics, Bernhard and Lewis’s team recently

implemented conformal printing of small electrical antennas

onto the surfaces of hemispherical glass substrates [8], which is

reported to improve the antenna performance by an order of

magnitude. The ability to rationally design and efficiently

realize artistic or functional metasurfaces are important in both

art and science. While research on designing the metamaterials

has drastically advanced recently, topology optimization of

conformal metasurfaces remains untapped. There is a distinct

lack of an appropriate method that can be utilized for

metasurface design.

A common practice in metamaterial design is to obtain a

2D or 3D unit cell, a representative volume element (RVE)

under periodic boundary conditions [9]. This constitutive

building unit can be later periodically assembled in two or three

directions, to build the desired metastructure (Fig. 1). It may

also be utilized as a building unit to form a rotational periodic

structure, such as a cylinder, by a straightforward

transformation from Cartesian to the cylindrical coordinate

system. Such an obtained geometry may possess different

characteristic behavior compared to the planar metastructure

[10]. Mapping a plane unit cell to a free-form surface is

nontrivial. A potential solution is conformal mapping, an angle-

preserving Riemann mapping that can preserve the local shape.

Figure 1. Planar metastructure with 4x4 unit cells.

In this paper, a computational framework is introduced for

topology optimization and additive manufacturing of

metamaterials on a 3D free-form surface, which rests on the

level-set based topology optimization approach and the

conformal mapping theory. A 2D metamaterial unit cell design

with tailored effective properties is first achieved using the

level-set based topology optimization scheme. After the free-

form object surface is subdivided into 3D quad mesh surfaces,

the conformal mapping from each 3D quad mesh surface to the

2D planar unit cell can be further calculated. Based on the

calculated conformal mapping, the metamaterial unit cell design

is aligned with the 3D quad mesh elements, to obtain the final

3D conformal metasurface, which acts as the input model for

additive manufacturing. The proposed framework (Fig. 2)

constitutes an automated process that begins with pre-specifying

the effective properties of the desired metamaterial design and

the 3D free-form surface and ends in a conformal metasurface

ready for 3D printing. It offers a direct coupling between

topology optimization and additive manufacturing and

alleviates the need for CAD reconstruction and further post-

processing work.

The paper is organized as follows: the background

regarding the level-set based topology optimization is presented

in the next Section, followed by an introduction to the theorem

and algorithms of conformal mapping. Later, the

implementation of the proposed 5-step procedure is presented,

accompanied by a demonstration example of mapping a 3D

human face with negative Poisson’s ratio metamaterial.

TOPOLOGY OPTIMIZATION Topology optimization [11] is a powerful computational

tool that has come to be an essential part of the design

innovation process. In topology optimization, the design

problem is recast as an optimal material distribution problem in

a specified design domain by optimizing an objective function

subject to different design constraints. During the last decades,

several topology optimization methods [12-14] have been

developed, including the Homogenization method [15, 16],

Solid Isotropic Material with Penalization (SIMP) [17],

Evolutionary [18], and Level-Set Methods [19-25]. In this

work, a level-set based topology optimization method is

employed for designing metamaterials. This alternative method

provides a clear description of the boundaries and a seamless

connection between the design and the fabrication process [26].

Figure 2. Computational framework for 3D free-form metasurfaces.

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In level-set methods, the boundary of the design is

implicitly embedded by a level-set function Φ with one higher

dimension, as expressed in Eq. (1). The level-set function is

regularized to be a signed distance function. According to the

sign of the level set function, the computational domain can be

divided into areas of material and void, as expressed in Fig. 3.

, 0,

, 0,

, 0, \

x t x material

x t x boundary

x t x D void

(1)

Figure 3. Schematic 3D level set function with the

corresponding 2D geometry.

As a geometric model, level set methods not only provide a

clear representation of the boundary, but also embed higher

order geometric information, such as the normal vectors or

curvatures. Such information can be utilized to create a

seamless connection between the design and the fabrication

process [26]. This is important in the topology-optimization-

driven design innovation, where the organic conceptual design

often needs post processing and a CAD reconstruction before it

can be manufactured. In level set methods, the external surfaces

(boundaries) of a 3D object are defined by the zero level of a

continuous 4D level set function. The embedded information

can be extracted for STL file generation and further

manipulation, which is more suitable for 3D printing. The

isosurface, formed by the boundaries of the design, can be

transformed to a triangle mesh using Delaunay triangulation

[27]. Each triangular facet has three vertices and a normal

vector n, implied by the change of the level set function’s sign.

The data of all the facets must be stored in a file with STL

(Stereolithography) format, a standard format widely

recognized by the 3D printers and CAD software.

Figure 4. Extraction of geometric information from the level

set model.

The structural optimization procedure and the boundary

evolution of the design are based on the Hamilton-Jacobi

equation [19, 28]:

,

, (x) 0x t

x tt

V , (2)

where (x)dx

dtV is a design velocity field on the boundary.

To obtain a feasible steady-state solution to Eq. (2), a shape

sensitivity analysis need be performed to construct the design

velocity field for the design evolution. The Courant-Friedrichs-

Lewy (CFL) condition must be satisfied to maintain the

numerical stability when solving the Hamilton-Jacobi PDE [28,

29].

Topology optimization is an iterative process starting from

an initial level-set function. In this paper, a Finite Element

Analysis (FEA) is carried out each iteration under the plane

stress assumption, and the effective properties are calculated

using the strain energy method [30, 31]. Then, the design

velocity field is constructed following the steepest descent

method [23, 28], and the Hamilton-Jacobi equation is solved

using the upwind finite difference approach [24, 28]. At the end

of each iteration, the design evolves and acts as an input for the

next iteration. The detailed flowchart of the optimization is

presented in Fig. 5.

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Figure 5. Flowchart of the level set based topology

optimization process.

CONFORMAL MAPPING Conformal mapping, also known as biholomorphic map, is

a transformation that can preserve the local angles [32]. In the

past few decades, conformal mapping has been widely

investigated and extensively applied in many engineering fields,

including global parameterization in computer graphics [33],

surface registration in computer vision [34], efficient routing in

sensor networks [35], and brain mapping in medical imaging

[36]. Stephenson [37] proposed a circle packing based method

to approximate conformal mappings between planar regions.

For a genus zero surface, Haker et al. [38] presented a

conformal mapping algorithm which maps the surface to a

sphere by solving a linear system. To map a multiply connected

mesh to the slit domain, Wang et al. [39] employed the

holomorphic 1-forms and applied this method to brain surface

parameterization.

More recently, Nasser [40] proposed a numerical

conformal mapping method for multiple connected regions onto

the second, third and four categories of Koebe's canonical slit

domains. Shi et al. [41] devised a novel conformal mapping

method to align anatomical features and reduce metric

distortion using Laplace-Beltrami eigenfunctions. Lately, Rorig

et al. [42] have employed a periodic conformal mapping

method for surfaces with cylinder topology which maps a

surface mesh to a cylinder or cone of revolution.

In this work, holomorphic differential 1-forms are utilized

to compute conformal mapping to transform the 2D

metamaterial design into a 3D free-form surface. Differential 1-

form based methods have been extensively employed in

numerous applications [43-47], such as vector fields

decomposition and surface parameterization. For

comprehensive reviews of a variety of conformal mapping

methods, readers are referred to [48, 49].

A conformal mapping between two surfaces preserves

angles. Suppose f:(M1,g1)(M2,g2) is a mapping between two

Riemann surfaces M1, M2 with Riemannian metrics g1, g2, and

the coordinate parameters of M1, M2 are (x,y), (u,v) respectively.

The pullback metric, induced by f, has the local representation:

*

2 2

, ,

, ,

T

u v u vf g g

x y x y

. (3)

The mapping f is called a conformal mapping if the

pullback metric induced by f satisfies:

* 2

2 1f g e g , (4)

where λ:M1ℝ is a function defined on M1, and e2λ is called the

conformal factor.

Two surfaces are conformally equivalent, if there exists a

conformal mapping between them. If two surfaces are

conformally equivalent, the conformal factor, which determines

Riemannian metric, can be employed to further differentiate

them. Let S be a genus zero surface with a single boundary and

four marked boundary points p1, p2, p3, p4 located counter-

clockwise. Such a surface is called a topological quadrilateral,

represented by Q(p1, p2, p3, p4). Following Riemann mapping

theorem, there exists a unique conformal mapping f, so it maps

Q(p1, p2, p3, p4) to a planar rectangle and f(p1)=0, f(p2)=1.

The ratio of height to width for the rectangle is called the

conformal module of Q(p1, p2, p3, p4). Conformal modules are

the complete invariant of the conformal structure. The concept

of the conformal module is illustrated in Fig. 6, where a human

face with four boundary corners, represented by a topological

quadrilateral, is conformally mapped to a planar rectangle.

Figure 6. Conformal module of a topological quadrilateral:

(left) The original face surface with four marked boundary

corners, (middle) the planar rectangle which is the

conformal mapping result, and (right) the face surface with

the checkerboard texture pulled back from the one placed

on the planar checkerboard.

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Holomorphic differential 1-form based conformal

mapping Given a Riemann surface S and a harmonic function

f:Sℝ. Suppose ω denotes a differential 1-form with the

representation fαdxα+gαdyα in the local coordinates (xα,,yα), and

fβdxβ+gβdyβ in the local coordinates (xβ,,yβ), so

a

a

x y

x x ff

ggx y

y y

. (5)

ω is a closed 1-form, if 0f g

y x

in each local

coordinate (xα,,yα). If a closed 1-form ω satisfies 0f g

x y

,

then ω is a harmonic 1-form. Hodge star operator * can

transform a differential 1-form ω to its conjugate *ω, which is

defined as:

* a a a ag dx f dy . (6)

A complex differential 1-form is a holomorphic 1-form if

and only if it can be decomposed into two harmonic 1-forms ω

and *ω which are conjugate to each other, and the holomorphic

1-form is written as:

1* . (7)

Suppose (S1,g) is a genus-zero surface with a single

boundary and Riemannian metric g. According to the conformal

mapping theory, the genus-zero surface S1 with a single

boundary can be conformally mapped to a planar rectangle Q1.

This conformal mapping algorithm starts with the computation

of harmonic 1-form basis ωi and its conjugate *ωi, where *

denotes the Hodge star operator. Then, the holomorphic 1-form

basis is computed as ωi+i*ωi. With the integration of

holomorphic 1-forms along the path in the primary domain 1S ,

the conformal mapping 1 1:f S Q is obtained. The detailed

conformal mapping algorithm is illustrated in Table 1.

Table 1. Algorithm 1: Conformal Mapping.

Algorithm 1: Conformal Mapping

Input: A triangle mesh 1S of genus zero with a single boundary.

Output: A conformal mapping 1 1:f S Q , where 1Q is a planar

rectangle.

1: Compute a harmonic 1-form basis of 1 1 2, { , }S .

2: For a harmonic 1-form i , compute its conjugate * i .

3: Compute the holomorphic 1-form base by 1*i i .

4: Repeat step 2 through step 3 until all the harmonic 1-forms i

in the set are processed.

5: Slice 1S along the cut graph of 1S to obtain a fundamental

domain 1S .

6: Select a root vertex 0 1S , and integrate the holomorphic 1-

form i along the path i from 0 to arbitrary vertex i during

breadth-first traversal to obtain ( )i

if

.

7: The conformal mapping is given by 1 1:f S Q after above

integration is completed.

Aligning unit cells with quadrilaterals in 3D quad

mesh surface

The algorithm for aligning unit cells with quadrilaterals in

3D quad mesh surface mainly employs the conformal mapping

between the 3D surface with a 2D planar rectangle and the

Barycentric interpolation method. Based on the conformal

mapping result, each triangle mesh unit cell C is firstly placed

into the quadrilateral in planar quad mesh rectangle Q2, and

then transformed and aligned with the quadrilateral in 3D quad

mesh surface S2 using Barycentric interpolation. The final

designed surface for 3D printing comprises a set of connected

triangle mesh unit cells. The detailed algorithm is given in Table

2.

Table 2. Algorithm 2: Alignment of Unit Cells with

Quadrilaterals.

Algorithm 2: Alignment of Unit Cells with Quadrilaterals

Input: A 3D quad mesh surface 2S of genus zero with a single

boundary, a planar quad mesh rectangle 2Q , and a triangle mesh

unit cell C with its facet parallel to yz-plane and x coordinate

starting from 0.

Output: A 3D surface input model 3S for 3D printing comprising

a set of connected triangle mesh unit cells.

1: Create a new copy of the triangle mesh unit cell C .

2: Project each vertex i of C on the yz-plane, and save its x

coordinate value as height ih . The resulting unit cell is denoted as

prC .

3: Scale and place prC into a quadrilateral in planar quad mesh

rectangle 2Q .

4: Using Barycentric interpolation method, compute the new

position of each vertex i of prC , which will be placed into the

corresponding quadrilateral of 3D quad mesh surface 2S .

5: Lift each vertex of prC on the quadrilateral of 3D quad mesh

surface 2S by the height ih along the vertex normal direction to

recover its 3D unit cell structure. The resulting unit cell is then

represented by lftC .

6: Repeat step 1 through step 5 until all the quadrilaterals of 3D

quad mesh surface 2S are aligned with the triangle mesh unit cell

lftC .

7: The 3D surface input model 3S for 3D printing is given by such

a set of connected triangle mesh unit cells lftC .

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NUMERICAL IMPLEMENTATION AND

DEMONSTRATION EXAMPLE In this Section, the proposed method will be illustrated

through a demonstration example. The schematic representation

of the computational method is shown in Fig. 2. The proposed

computational framework comprises five steps that can be

summarized as follows, and the algorithm is presented in Table

3.

Design a metamaterial unit cell model: Designing the

metamaterial using a level-set based topology optimization

method. The designed unit cell serves as a constitutive element

for mapping onto the surface model.

Conformally map the 3D triangle mesh surface S1 to a 2D

triangle mesh rectangle Q1: Under Riemann mapping theorem,

for any topological quadrilateral surface S1, there exists a

conformal mapping which maps S1 to a planar rectangle Q1.

Construct a conformal mapping between a 3D quad mesh

surface S2 and a 2D quad mesh rectangle Q2: With the

conformal mapping between two triangle meshes, a conformal

mapping between a 3D quad mesh surface S2 and a 2D quad

mesh rectangle Q2 is achieved utilizing the Barycentric

interpolation method.

Align each triangle mesh of the unit cell C with the

quadrilateral in the 3D quad mesh surface S2: To this end, an

algorithm has been created, so the resulting connected unit cells

form a 3D surface input model S3 for 3D printing.

3D Print the surface model S3: All the unit cells that

compose the model S3 are assembled into one piece, and the

data are stored in an STL file for 3D printing.

Figure 7. Notation of surfaces throughout the process: 3D

triangle mesh surface S1, 3D Quad mesh surface S2, 3D

surface model S3, 2D triangle mesh rectangle Q1, 2D quad

mesh rectangle Q2.

Table 3. Algorithm 3: A Novel Framework for Conformal

Metasurfaces.

Algorithm 3: A Novel Framework for Conformal Metasurfaces

Input: A triangle mesh surface 1S of genus zero with a single

boundary, and target properties of the desired metamaterial design.

Output: A 3D printed surface 3S .

1: Design a metamaterial unit cell with the desired effective

properties.

2: Compute a conformal mapping from the triangle mesh surface

1S to a planar rectangle 1Q using Algorithm 1.

3: Based on triangle mesh surface 1S and planar rectangle 1Q ,

calculate the corresponding 3D quad mesh surface 2S and 2D

quad mesh rectangle 2Q using Barycentric interpolation method.

4: Align each copy of triangle mesh unit cell C with the

quadrilateral in the 3D quad mesh surface 2S using Algorithm 2,

so a 3D surface input model 3S for 3D printing comprising a set of

connected triangle mesh unit cells is obtained.

5: 3D print the surface model 3S .

Designing the metamaterial through level-set based

topology optimization methods

After the effective properties of the desired metamaterial

have been specified, the optimization process, detailed earlier,

is adopted. The result of the topology optimization process is a

unit cell design which will act as an input for the conformal

mapping. In this work, a 2D negative Poisson’s ratio

metamaterial design (or auxetics) is selected for optimization

and will finally be mapped on a 3D face. The characteristic

behavior of negative-Poisson’s-ratio (NPR) materials is that, in

contrast to ordinary materials, when compressed NPR materials

contract in the perpendicular direction.

One way to formulate the problem for auxetic metamaterial

design is to express it as a least square optimization problem,

minimizing the difference between the effective properties of

the design Eff

ijklC and the design objectives *

ijklC for the entries of

the elasticity matrix for orthotropic materials:

Minimize: 2

, , , 1

1

2

dEff

ijkl ijkl ijkl

i j k l

J w C C

(8)

Subject to:

0

,

T

ij ijkl kl ij ijkl klD D

u C v H d C v H d

V V

(9)

where d denotes the dimension of the problem (2D or 3D), and

wijkl is the weighting factor for each elasticity tensor. H(ϕ) is the

Heaviside function, D the computational domain, V(Ω) is the

volume of the unit cell, and V* is the volume target.

The volume target of the design has been selected to be

40%, and the targeted effective properties are set as

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C*1111=0.2GPa, C*

2222=0.2GPa, and C*1122=-0.08GPa, driving

to a design with Poisson’s ratio ν=-0.4. The weight factors for

Eq. (8) are w1111=0.5, w2222=0.5, and w1122=5, and the

constitutive material’s Young’s modulus is E=0.91GPa. The

void is represented by a dummy material with Evoid=10-6GPa, to

avoid singularity in finite element analysis.

The design evolution history with the objective function

and the effective properties of the unit cell can be found in Fig.

8 and 9, respectively. In Table 4, the unit cell design has been

assembled to a 4x4 metastructure, and a prototype has been

fabricated. The effective elastic properties indicate a

metastructure with Poisson’s ratio ν=-0.38. More details about

constructing the design velocity field and implementing the

negative Poisson’s ratio problem can be found in [31], where

results have been numerically verified and experimentally

validated.

At the end of the optimization, the unit cell design is

implicitly described by a level-set function Φ. In the

demonstration, the obtained metamaterial is 2D, which can be

further extruded to obtain a 3D structure (2.5D). If a 3D

topology optimization is followed, the result would be a 3D

metamaterial with specific effective properties. Either way, the

design must be translated into a triangle mesh so the

communication between all the steps can be possible [50]. The

design, now, is a set of triangular facets described by three

vertices for the points and one normal vector, each.

Figure 8. Design evolution during topology optimization.

Figure 9. Target properties C* and effective properties CEff

during topology optimization.

Table 4. 4x4 periodically assembled unit cells, and

corresponding elasticity matrix.

Unit Cell 4x4 metastructure

4x4 printed prototype Elasticity matrix

0.200 0.076 0

0.076 0.200 0

0 0 0.038

EffC

3D printing of the conformal metasurface

The original unit cell design consists of a triangle mesh of

373,416 triangles. The number of the triangles has remained

unchanged during the one-to-one mapping of the unit cell to

each quadrilateral. After the successful conformal mapping of

the unit cell onto the surface through algorithms 1 and 2, the

final design comprises a group of cells from the same original

unit cell. Each cell has a varied design and locates at a different

position on the surface. All the individual meshes must go

through a Boolean operation that unites them into a single

layout. The unified design, described by the vertices and normal

vector for each triangular facet is stored in an STL file for

additive manufacturing. For convenience, the triangle mesh of

the unified design has been refined to become manageable for

3D printing. Part of the design can be seen with the original

triangle mesh in Fig. 10, and after the refinement in Fig. 11.

The process has been applied to a 3D human face design,

obtained through 3D scanning, with two quad mesh size. In the

first example (Fig. 12), the surface has been discretized into a

15x14 quad mesh (original number of triangles: 78,000,000,

after refinement: 3,200,000), and the design was fabricated

using a state-of-the-art 3D printer (Objet260 Connex, Stratasys

Ltd) with a flexible digital material (40% VeroWhite and 60%

TangoPlus). The second example (Fig. 13) has a denser mesh

(31x28) (original number of triangles: 320,000,000, after

refinement: 2,000,000), and it has been fabricated using a

desktop 3D printer with ABS plastic. The size of the unit cell of

the current examples is, for a better demonstration, relatively

large. For more demanding applications, a high-resolution 3D

printer enables the realization of conformal metastructures with

a smaller in size metamaterial unit cell, as a constitutive

element.

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Figure 10. Part of the design with the original triangle

mesh.

Figure 11. Part of the design with refined triangle mesh.

Figure 12. First human face example with 15x14 quad

mesh: a) 3D triangle mesh surface of human face S1, b) 3D

quad mesh surface S2, c) unified 3D surface model S3, and d)

3D printed prototype.

Figure 13. Second human face example with 31x28 quad

mesh: a) 3D triangle mesh surface of human face S1, b) 3D

quad mesh surface S2, c) unified 3D surface model S3, and d)

3D printed prototype.

CONCLUSION In this paper, we present a computational framework to

address key issue in computational design and additive

manufacturing of spatial free-form periodic metasurfaces,

including how to create a metamaterial design with tailored

effective properties through level-set based topology

optimization method; how to map the design to a 3D free-form

surface with the local shape preserved by synthesizing the

information from the geometric level set model and conformal

mapping; and how to generate the conformal metasurface with

consideration of the file format for additive manufacturing. The

proposed method is validated through a demonstration example

of a metasurface with NPR unit cells. An issue we found in our

study is that some unit cells are highly distorted compared to

the original design at the spots where the surface curvatures are

high. The effective mechanical properties of such unit cells

might differ greatly from the desired properties. This problem is

alleviated by subdividing the surface with smaller quad mesh Q,

which can better preserve their original shapes of the unit cells.

Future applications may include customizable designs based on

a wide database of metamaterials, ready to be utilized in

complicated designs to achieve the desired performance.

Potential applications of the proposed method to designing

artistic and functional structures are to be explored soon.

ACKNOWLEDGMENTS This research was supported by the National Science

Foundation (CMMI1462270), the Region 2 University

Transportation Research Center (UTRC), the unrestricted grant

from Ford Research & Advanced Engineering, and the start-up

funds from the State University of New York at Stony Brook.

The authors thank Mr. Cui Xiao for his contribution at the early

stage of this research.

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Accepted Manuscript - Stony Brook Universityme.eng.stonybrook.edu/~chen/Downloads/IDETC2017-67024-TO_and… · In topology optimization, the design problem is recast as an optimal

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