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MD-05-1288. 1 Seepersad et al

ROBUST DESIGN OF CELLULAR MATERIALS WITH TOPOLOGICAL AND DIMENSIONAL IMPERFECTIONS

Carolyn Conner Seepersad*

Mechanical Engineering Department The University of Texas at Austin

1 University Station , C2200 Austin, TX 78712

Janet K. Allen,** David L. McDowell, and Farrokh Mistree**

G.W. Woodruff School of Mechanical Engineering Georgia Institute of Technology

Atlanta, GA 30332-0405

ABSTRACT A paradigm shift is underway in which the classical materials selection approach in

engineering design is being replaced by the design of material structure and processing paths on

a hierarchy of length scales for multifunctional performance requirements. In this paper, the

focus is on designing mesoscopic material topology—the spatial arrangement of solid phases and

voids on length scales larger than microstructures but smaller than the characteristic dimensions

of an overall product. A robust topology design method is presented for designing materials on

mesoscopic scales by topologically and parametrically tailoring them to achieve properties that

are superior to those of standard or heuristic designs, customized for large-scale applications, and

less sensitive to imperfections in the material. Imperfections are observed regularly in cellular

material mesostructure and other classes of materials because of the stochastic influence of

feasible processing paths. The robust topology design method allows us to consider these

imperfections explicitly in a materials design process. As part of the method, guidelines are

established for modeling dimensional and topological imperfections, such as tolerances and

cracked cell walls, as deviations from intended material structure. Also, as part of the method,

* Corresponding Author. Email: [email protected]. Phone: (512) 471-1985. Fax: (512) 471-7682. ** Systems Realization Laboratory This is a revised version of Paper Number DETC2005/DAC-85061, published in the ASME Advances in Design Automation Conference.


robust topology design problems are formulated as compromise Decision Support Problems, and

local Taylor-series approximations and strategic experimentation techniques are established for

evaluating the impact of dimensional and topological imperfections, respectively, on material

properties. Key aspects of the approach are demonstrated by designing ordered, prismatic

cellular materials with customized elastic properties that are robust to dimensional tolerances and

topological imperfections.

KEYWORDS Robust design, topology design, cellular materials, material mesostructure, materials design NOMENCLATURE

Au Total area of a unit cell Cij Elastic constant Cij

H Homogenized elastic constant {d} Local displacement vector {D} Global displacement vector di

-, di+ Deviation variables

DSP Decision support problem E11/Es,

E22/Es Effective elastic compressive stiffness, in-plane principal directions

Es Elastic modulus of solid material {F} Vector of nodal loads G12/Es Effective elastic shear stiffness, in-plane transverse direction [k] Local stiffness matrix [K] Global stiffness matrix Le Length of element e MMA Method of moving asymptotes N Number of elements in a ground structure P Number of tailored elastic constants RD Set of nodes in a designed mesostructure Ri ith node in a ground structure RM Set of missing nodes in a realized mesostructure SVE Statistical volume element U Average strain energy vf Volume fraction Wk Weight for kth goal X Vector of design variables. In-plane thicknesses of elements in ground structure XL Lower bound for design variables XU Upper bound for design variables Z Deviation or objective function


ΔCij Range of elastic constant values due to dimensional imperfections ΔXi Range of dimensions for the in-plane thickness of element i ε Infinitesimal strain vector {ε0} Test strain field γi Probability that a node, Ri, is missing from the mesostructure γv Probability associated with Experiment v μCij Mean value of elastic constant σ Cauchy stress vector σCij Standard deviation of elastic constant values due to topological imperfections

1. FRAME OF REFERENCE The properties of materials are influenced by complex relationships with multi-scale material

structure and associated processing paths. Process-structure-property relationships are often cast

in terms of microstructural aspects of the material, including the arrangement of phases, grains,

and defects such as vacancies, dislocations, or cracks, but it is also important to investigate other

length scales. Larger mesostructural length scales,1 for example, are characteristic of the

prismatic cellular materials illustrated in Figure 1 and may take the form of cell dimensions,

shape, and arrangement and cell wall dimensions and connectivity—features that strongly

influence a wide range of desirable properties of these materials.

[INSERT FIGURE 1.]

Typically, the properties of cellular materials are designed by selecting a cellular topology

from a small set of standard topologies (e.g., square, triangular, hexagonal) and then adjusting its

relative density by modifying the cell wall thickness. Topology changes have a strong impact on

cellular material properties, but the small library of standard topologies limits our ability to reach

some regions of a property space, as illustrated in Figure 2 for in-plane, effective elastic

1 Mesoscopic length scales (on the order of tens to hundreds of micrometers in this research) are intermediate between microscopic length

scales, which apply to characteristics like gradients of chemical composition and microstructure (e.g., grain boundaries, dislocations, crystal structure), and macroscopic length scales, much greater than the characteristic lengths of heterogeneities, at which homogeneous continuum models are valid.


compressive and shear stiffnesses of several periodic cell structures.2 Using topology design

techniques, we can generate new cellular topologies with customized properties that are

unattainable with standard cellular topologies. Topology design facilitates modifying the form

of the cellular mesostructure—the size, shape, and connectivity of cell walls and the number,

shape, and arrangement of cell openings—rather than specifying these features a priori (cf. [1-3]

for recent reviews of topology optimization and [4] for seminal work in modern topology

optimization methods). Topology optimization methods have been applied for designing one-

and two-phase materials with customized elastic and thermoelastic properties [5-8] and for

designing cellular mesostructures for effective elastic moduli and conductivity [9,10].

[INSERT FIGURE 2.]

To date, design methods for cellular materials have been focused on identifying the optimal

density, dimensions, or topology of a cellular material. However, optimal solutions are elusive

for real materials with highly heterogeneous structures and morphologies that are limited and

stochastically influenced by feasible processing paths. For example, the prismatic periodic

cellular materials illustrated in Figure 1 are fabricated using a thermo-chemical extrusion process

developed by the Lightweight Structures Group at the Georgia Institute of Technology [11]. The

process affords significant freedom for tailoring in-plane topology with a few limitations on

minimum cell wall thickness (minimum of 50 micrometers), cell wall aspect ratios (maximum of

8:1), and relative density (maximum of approximately 30%). It also introduces imperfections in

the cellular mesostructure. Topological imperfections are associated with unintended variations

in cellular connectivity, such as cracked cell walls and missing cell wall joints. Dimensional

imperfections are associated with unintended variations in cellular dimensions such as tolerances

2 The plots in Figure 2 assume relative densities of 20% and doubly periodic structures. E11/Es, E22/Es represent effective elastic stiffness

for uniaxial loading in the in-plane principal directions. G12/Es is the effective elastic shear stiffness in the in-plane transverse direction.


on cell wall thickness. Other variations include shape imperfections such as curved or

corrugated cell walls, and material property imperfections such as porosity or retained oxides.

Porosity, shape variation, and missing cell walls have been shown to degrade properties such as

elastic moduli and compressive yield strength [12-14], and the impact varies with cell topology.

We need robust topology design methods for identifying cellular topologies and dimensions

with customized properties that are relatively insensitive to processing-induced imperfections.

The sensitivity of optimal topology to changes in prescribed loads has been investigated by

considering multiple loads (e.g., [15,16]), average performance under multiple loads [17],

reliability [18-20], or worst-case loads among a set of possible loads [21-23], and Sandgren and

Cameron [24] have considered the feasibility robustness of constraints with variations in loading

and material properties. However, these examples are representative of design for mean

performance or fail-safe or worst-case design, in which a structure is designed explicitly for

worst-case loading, rather than robust design, in which tradeoffs are sought between preferable

nominal performance values and minimal sensitivity of performance to uncontrolled variation.

Furthermore, variations in the topological structure itself, such as dimensional or topological

imperfections, have not been considered, partially because topology design was originally

focused on full-scale structures rather than materials. Similarly, robust design methods have

been established for improving the quality of products and processes by reducing their sensitivity

to variation [25-34], but they have been developed and demonstrated for applications with fixed

topology.

Our goal is to establish systematic design methods for tailoring material mesostructure to

provide robust properties for specific applications at higher length scales. In previous work, we

have presented a robust topology design method for designing material mesostructures with


properties that are robust to dimensional tolerances of the cell walls [10]. Here, we extend the

method to accommodate topological imperfections such as cracked cell walls and missing joints

and explore tradeoffs between dimensional and topological robustness.

2. A METHOD FOR ROBUST TOPOLOGY DESIGN OF CUSTOMIZED CELLULAR MESOSTRUCTURE Suppose that a large-scale load-bearing system requires lightweight cellular materials with

properties that are near the region of opportunity identified in Figure 2 and relatively robust or

insensitive to dimensional and topological imperfections. To meet these requirements, a method

is needed for designing novel cellular mesostructures—including the spatial arrangement,

connectivity, and dimensions of cells and cell walls—that provide robust, customized properties.

The robust topology design method outlined in Figure 3 has been devised for this purpose [35].

As described in Section 2.1, the first step in the method is to establish a robust topology design

space by representing the design space of possible topologies, characterizing dimensional and

topological imperfections or noise factors, and identifying an accompanying set of design

parameters. After the design space is composed, a mathematical model is formulated for the

multiobjective decision to be solved, as described in Section 2.2. Then, as described in Section

2.3, a simulation infrastructure is created for evaluating the properties of alternative cellular

mesostructures. Finally, as discussed in Section 2.4, the robust topology design problem is

solved using optimization procedures to identify preferred alternatives efficiently, and the results

are validated.

[INSERT FIGURE 3.]


2.1 Phase I: Formulate a Robust Topology Design Space

The first phase of the robust topology design method involves constructing a design space for

representing the cellular mesostructure, modifying it for properties of interest, and modeling

associated dimensional and topological imperfections. The design space for robust design of

doubly periodic 2D cellular mesostructures is represented by the set of design parameters

summarized in Table 1 and described in this section.

[INSERT TABLE 1.]

2.1.1 Properties of Interest

As listed in Table 1, the properties of interest include a constraint on the volume fraction, vf,

of solid cell wall material and targets for the elastic constants, Cij. The elastic constants are

components of the tensor of elastic constants, C, that describes the macroscopic behavior of the

material in response to applied stress according to the constitutive equation for a homogeneous,

linearly elastic material [36]:

{ } [ ]{ }Cσ ε= (1)

A cellular material with three mutually orthogonal planes of symmetry is orthotropic. When two

of the principal axes of the orthotropic cellular material are aligned with planes of symmetry and

a state of plane strain is assumed parallel to the plane of the axes, the constitutive law can be

expressed in 2D as [36]:

11 1 11 12 1

22 2 12 22 2

12 3 33 3

00

0 0

C CC C

C

σ σ εσ σ εσ σ ε

⎧ ⎫ ⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥⎩ ⎭ ⎩ ⎭ ⎣ ⎦ ⎩ ⎭

(2)


where 1 11

2 22

3 12

ε εε εε ε

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

where Cij represents the four independent elastic constants. For cellular materials, the constants

are ‘effective’ elastic constants because they are measured as a fraction of the corresponding

properties for a fully dense piece of material. For this example, the independent elastic constants

are related to the effective elastic compressive stiffness in the in-plane principal directions and

the effective elastic shear stiffness in the in-plane transverse direction, as plotted in Figure 2, as

follows:

11 11 / sC E E= (3)

22 22 / sC E E= (4)

33 122 / sC G E= (5)

In addition to the nominal values of the independent elastic constants, the properties of interest

for this example include the mean value of each elastic constant, μCij, the range of values for

each elastic constant due to dimensional imperfections, ijCΔ , and the standard deviation of each

elastic constant due to topological imperfections, Cijσ . Models for each of these properties are

described in Section 2.3.

2.1.2 Topology Representation and Modification of a Doubly Periodic Cellular Mesostructure

with a Ground Structure

To establish the topology design domain, we assume that the 2D cellular mesostructure is

doubly periodic in a basic unit cell (i.e., the smallest repetitive unit of the cellular mesostructure),

as illustrated in Figure 4. To customize the properties of interest, the topology of a unit cell is

represented and modified using a discrete topology design approach based on ground structures


(cf. [2,37,38] for relevant reviews and [39] for an introduction). As shown in Figure 4B, the

topology design space for a single unit cell is modeled as a ground structure, consisting of a grid

of regularly spaced nodes that are connected with frame finite elements with six degrees of

freedom (cf. [40]).3 Doubly periodic boundary conditions are applied to the unit cell to simulate

the effect of replication of the unit cell in both 2D principal directions. In the ground structure of

Figure 4B, the unit cell is divided into four quadrants by two planes of symmetry that are aligned

with the vertical and horizontal principal axes of orthotropy. Within each quadrant, every pair of

nodes is connected with a frame finite element. The entire ground structure in Figure 4B has 25

nodes and 132 finite elements, and it is assumed to occupy a square domain with an area of 1

cm2.4 The ground structure in Figure 4B is chosen because it is sufficiently dense to include

candidate topologies that satisfy targets for nominal elastic constants for this example; whereas

coarser ground structures may restrict the topology design space too much. Because topology

design results can depend on the initial ground structure, we consider a more complex, 81-node

ground structure, as illustrated in Figure 5B, for validation and comparison with results from the

coarser ground structure in Figures 5A and 4B.

[INSERT FIGURE 4.] [INSERT FIGURE 5.]

In the ground structure, a design variable, Xi, is assigned to the in-plane thickness of each

finite element5 in a single quadrant of the ground structure.6 Consistent with the ground

structure approach, the design variables vary between an upper bound, XU, on the order of 1000

3 Frame finite elements are a superposition of 1D beam and bar finite elements. Frame elements are used to account for transverse loads

and bending in cell walls, in addition to axial deformation. Both mechanisms are observed in prismatic cellular materials subject to in-plane loading and elastic deformation.

4 The area of the unit cell is chosen to correspond to typical size ranges for unit cells fabricated with the thermo-chemical extrusion process described in Section 1.

5 All elements are assigned a unit depth in the out-of-plane direction. Therefore, the in-plane thickness, Xi, of element i is equivalent to its cross-sectional area.

6 Orthotropy implies that changes made in one quadrant are mirrored symmetrically to the other three quadrants.


um and a lower bound, XL, with an extremely small but positive magnitude on the order of 0.01

μm. After the optimization algorithm converges, the elements have different in-plane

thicknesses, as depicted in Figure 4C with lines proportional to the thicknesses. As shown in

Figure 4D, elements with in-plane thicknesses near the lower bound typically are removed in a

post-processing step and are not depicted in the final design. Finally, a doubly periodic cellular

mesostructure may be depicted by repetition of the designed unit cell, as shown in Figure 4E.

2.1.3 Characterizing Dimensional and Topological Variation

As noted in Section 1, a realized (as-fabricated) cellular mesostructure is likely to differ

stochastically from an intended (as-designed) cellular mesostructure due to processing-induced

imperfections. In this work, we consider two types of imperfections: dimensional variation in

the in-plane thickness of each cell wall and topological imperfections in the form of missing cell

walls or joints.

Dimensional variation is modeled as a range of potential dimensions, ΔXe, a function of the

nominal in-plane thickness, eX , of a cell wall or element as follows:

21e e eX X X ααΔ = − (6)

where α1 and α2 are constants with values of 0.502 and 1.085, respectively, and eX is measured

in μm. The model is fit via nonlinear regression to manufacturing observations of tolerance

values that are approximately 15 μm for a 50 μm wall thickness—the minimum realizable cell

wall thickness, XMinMfg, in accordance with present thermochemical, extrusion-based processing

capabilities—and gradually approach 10% tolerances for larger dimensions (e.g.,

100 μmeXΔ ≅ for eX = 1000 μm). The model has several desirable properties, including first

and second order continuity or smoothness to aid convergence during the optimization process


and a guarantee that ΔXe < eX . These properties are valid for all bounded values of eX

(specifically, in this example, 0.01 μm < eX < 1000 μm).7 Since the model is a function of cell

wall thickness, it is scale-dependent—a feature that impacts the results presented in Section 3.

[INSERT FIGURE 6]

Topological imperfections are modeled as the probability that a joint is missing from the

realized (or fabricated) mesostructure, as illustrated by example in Figure 6. Since each joint in

the mesostructure corresponds to a node in the ground structure, the topological variation is

modeled as the probability, γi, that any specific node, Ri, in a ground structure is missing or

randomly defective in a realized mesostructure:

( )M Di i iP R Rγ = ∈ ∈R R (7)

where RD is the set of nodes in the initial ground structure, and RM is the set of missing nodes.

Although a single unit cell is adequate for evaluating the elastic constants of undamaged

mesostructures or mesostructures with uniform dimensional imperfections, a larger design

domain is desirable for evaluating the impact of topological imperfections. The design

domain—also known as a statistical volume element (SVE) or window size—for analyzing

topological imperfections consists of nine identical unit cells, arranged in a 3x3 matrix. If

topological imperfections were analyzed with an SVE of a single unit cell, then any random

topological imperfection in the parent unit cell would be assumed to repeat periodically in all of

the surrounding unit cells in the mesostructure. By using a larger SVE for analyzing topological

imperfections, this assumption is relaxed, and a less periodic and more realistic distribution of

imperfections is permitted.

7 To preserve and enhance the effectiveness of gradient-based optimization algorithms for solving a robust topology design problem, the

variation model in Equation (6) is extrapolated beyond the manufacturable range of cell wall thicknesses, [XMinMfg, XU], to the bounds established for the topology design process, [XL, XU], with XL assumed to be arbitrarily small (i.e., 0 < XL < XU).


Each node in the SVE is assigned equal probability, γi, and only one node is assumed to be

missing at a time although it is possible to consider two or more simultaneously missing nodes.

It is also possible to model similar probabilities for missing individual elements or cell walls, but

only nodes and joints are considered here.

2.2 Phase II: Formulate a Robust Topology Design Problem

After the design space is defined in Phase I, the robust topology design problem is

formulated in Phase II as a compromise Decision Support Problem—a mathematical model of

the multiobjective decision to be solved [41]. The compromise Decision Support Problem (DSP)

is a hybrid multiobjective construct that incorporates concepts from both traditional

mathematical programming and goal programming [41]. It is used to determine the values of

design variables that satisfy a set of constraints and bounds and achieve a set of conflicting,

multifunctional goals as closely as possible. The system descriptors, namely, system and

deviation variables, system constraints, system goals, bounds and the objective function are

described in detail elsewhere [41]. Here, we apply it for robust design of periodic 2D cellular

mesostructure. We use the compromise DSP, instead of the deterministic, single-objective,

nonlinear programming formulations that are typically used for topology design, because the

compromise DSP has several features that facilitate robust topology design. Those features

include the capability of accommodating uncontrolled variation in design variables, constraints,

and goals and the capability of balancing the multiple objectives associated with meeting targets

for multiple material properties and simultaneously minimizing variation in each of those

properties.

The compromise DSP is presented in Figure 7 for the present example and its associated

design parameters (Table 1). Accordingly, the design variables are the in-plane thickness, X, of


each element in the ground structure. An upper bound, XU, and a lower bound, XL are placed on

each design variable in Equation (12), and a maximum limit, vf-limit, is placed on the volume

fraction, vf, in Equation (8). The constraint limits, design variable bounds, and other constants

are summarized in Table 2 for two examples that are described in Section 3.

[INSERT FIGURE 7]

[INSERT TABLE 2.]

The goals are formulated in Equations (9) through (11) in the compromise DSP and include

meeting targets for the mean value of each elastic constant, μCij, and minimizing the variation in

elastic constant values due to dimensional variation, ΔCij, and topological variation, σCij.

Separate goals are included for elastic constant variation due to dimensional and topological

imperfections so that the impact of the two types of imperfections can be assessed and

minimized separately. As in goal programming [42], the goals are formulated in terms of

achieving target values, μCij-target, ΔCij-target, and σCij-target, for each goal. Deviation variables, di-

and di+, measure the extent to which each goal target value is under- or over-achieved,

respectively. Each goal formulation is normalized by μCij-target to ensure that the deviation

variables range from 0 to 1. Restrictions are included in Equation (13) to limit the deviation

variables to positive values and to ensure that only one deviation variable per goal is positively

valued at any specific point in the design space [41]. For example, if μCij-target and μCij assume

values of 0.5 and 0.25, respectively, in Equation (9), the target value is underachieved rather than

overachieved. Accordingly, di+, the overachievement deviation variable, is assigned a value of

0, and di-, the underachievement deviation variable, is assigned a value of 0.5, thereby satisfying

Equations (9) and (13).


The objective function is expressed as a linear, weighted combination of the deviation

variables for each goal, as formulated in Equations (14), (15), and (16) for non-robust design,

robust design with dimensional variation, and robust design with dimensional and topological

variation, respectively. Depending on the scenario, one of the objective functions is selected and

minimized with the aid of an optimization algorithm. During the solution process, the focus is

on identifying values for element thicknesses that satisfy the design variable bounds and

constraints and achieve the chosen set of goals as closely as possible, as measured by the

objective function value. Both weights and goal target values can be adjusted to generate

families of solutions that embody a variety of tradeoffs between nominal performance and

robustness to dimensional and/or topological imperfections.

2.3 Phase III: Establish a Simulation Infrastructure for Solving the Robust Topology Design

Problem

After a robust topology design problem is formulated as a compromise DSP, a simulation

infrastructure is established in Phase III of the robust topology design method. The simulation

infrastructure includes models for evaluating the nominal elastic constant values and models for

evaluating the variation in elastic constant values due to dimensional and topological

imperfections.

2.3.1 Analysis Models for Evaluating Elastic Constants

The analysis model for this example is a finite element-based homogenization approach that

is used to obtain the macroscopic (continuum) constitutive properties of the material, expressed

as the elastic constants in Equation (2), in terms of its doubly periodic cellular mesostructure.

The approach is similar to that utilized by Sigmund [5,6] and Neves and coauthors [43]. The

homogenization approach is applied to a representative volume element (RVE) that statistically


represents the mesoscopic heterogeneities of the material. In this case, the RVE is a periodically

repeating unit cell of the 2D cellular material, as illustrated in Figure 4. If the RVE or unit cell is

represented in Equation (2) by an equivalent homogeneous, linearly elastic solid characterized by

a homogenized tensor of elastic constants [CH], the homogenized elastic constants can be

calculated using energy considerations. Specifically, the elastic strain energy of a unit cell

characterized by an homogenized upper bound tensor of elastic constants [CH] subjected to a test

strain field {εo} is equivalent to the average elastic energy integrated over the mesostructure (unit

cell) volume subjected to an equivalent test strain field and doubly periodic boundary conditions,

i.e.,

{ } { } { } { }( ) { } { }( )0 0 0 01 12 2 u

TT HAu

C C dAA

ε ε ε δε ε δε⎡ ⎤ ⎡ ⎤⎣ ⎦⎣ ⎦ = + +∫ (17)

where [C] is the local tensor of elastic constants at each point in the mesostructure, Au is the area

of the unit cell, { }ε is the local strain in the mesostructure, and { } { } { }0δε ε ε= − is the local

strain perturbation from uniform test strain at each point in the mesostructure [44]. By subjecting

the unit cell to each of three uniform test strain fields, corresponding to {ε0}1 = {1,0,0}, {ε0}2 =

{0,1,0}, and {ε0}3 = {0,0,1}, all the elements of [CH] can be calculated. To facilitate evaluation

of the right side of Equation (17) for complex mesoscopic topologies, the unit cell is discretized

into frame finite elements according to the ground structure shown in Figure 4B, and the induced

strain is calculated using standard finite element equations and boundary information pertaining

to each of these uniform strain fields via

[ ]{ } { }0i

iK D F ε= (18)


for a unit cell, where { }0

iF ε is the vector of nodal loads that induce the initial strain field {ε0}i,

[K] is the global stiffness matrix compiled from N element stiffness matrices [ke], and {Di} is the

vector of global displacements. The strain energy of a finite element can be calculated as

follows [45]:

{ } [ ]{ }12

Te e e eU d k d= (19)

where {de} is the vector of displacements and rotations associated with frame element e and [ke]

is the stiffness matrix for element e. The stiffness matrix for a frame element may be obtained

from standard finite element textbooks [40]. The average strain energy integrated over the

mesostructure volume can be approximated based on finite element results, i.e.,

{ } [ ]{ }1

12

NT

e e eeu

U d k dA =

= ∑ (20)

Making use of the property of unit applied test strains with Equations (17)-(20), the

homogenized elastic constants can be calculated based on finite element results, i.e.,

{ } [ ]{ }1

1N TH i jij e e e

e u

C d k dA=

⎛ ⎞⎡ ⎤ = ⎜ ⎟⎣ ⎦

⎝ ⎠∑

(21)

where {dei} is the vector of displacements associated with element e due to induced strain field

{ε0}i. To obtain these displacement vectors, the ground structure finite element model is

subjected to each of the three test strains discussed previously, and periodic boundary conditions

are applied to the unit cell to simulate the periodic nature of the cellular material (cf. [46]).

Finally, it is important to calculate the remaining response in the compromise DSP—the

portion of a unit cell occupied by solid material. The volume fraction can be calculated as


1

1 N

f e eeU

v X LA =

= ∑ (22)

where AU is the area of the entire unit cell domain, eX is the nominal in-plane thickness of

element e, and Le is the length of element e. The results of Equation (22) are used in Equation

(8) in the compromise DSP of Figure 7.

The accuracy of the frame finite element model for simulating the mean values of elastic

constants for cellular materials has been confirmed by comparing results calculated with the

finite element model with theoretical results reported by Hayes and coauthors [47] for standard

unit cell topologies. The results agree with an error of less than 7% for the elastic constants, C11

and C22, for standard mixed triangular and square cell topologies8 [35].

2.3.2 Variability Assessment Models for Evaluating the Impact of Dimensional and Topological

Imperfections

The simulation infrastructure includes not only analysis models but also variability

assessment models for evaluating the impact of dimensional and topological imperfections on

elastic constant values. Specifically, the parameters in Equations (17)-(22) represent nominal

values, but the compromise DSP formulation in Figure 7 requires an estimate of mean elastic

constant values, μCij , ranges of elastic constant values, ijCΔ , induced by variation in control

factor values, ΔX , and standard deviations of elastic constant values, σCij, associated with

topological imperfections. Nominal or mean elastic constant values are evaluated at nominal

values of the vector of design variables, X:

( )Cij ijCμ = X (23)

8 Some additional approximation and round-off error is associated with the extremely small magnitude of C33 for square cells.


where the elastic constant values are obtained from the homogenized tensor of elastic constants

[CH] in Equation (21). A Taylor series expansion is used to evaluate elastic constant ranges

associated with dimensional imperfections in the form of tolerances for in-plane cell wall

thickness.9 It is applied as follows:

1

N

e

ijij e

e

CC X

X=

=∂

Δ Δ∂∑

(24)

To evaluate Equation (24), the partial derivative of an elastic constant is calculated for unit cell

boundary conditions and constant prescribed displacements:

{ } [ ]{ }Tij ei je e

e e

C kd d

X X∂ ∂

=∂ ∂

(25)

Here, dei is the portion of the global displacement vector associated with element e and

prescribed test strain {ε0}i , and ke is the stiffness matrix for element e. The Taylor series

expansion is an efficient approach for estimating the ranges of elastic constant values, ijCΔ ,

because it requires fewer system evaluations per iteration than Monte Carlo analysis or

statistically designed experiments. This characteristic is especially important for robust topology

design with large numbers of design variables and non-negligible computational times for system

analysis. Furthermore, the Taylor series analysis is efficient because the partial derivative

required for Equation (24) is already calculated analytically as input for the gradient-based

optimization algorithm. The accuracy of the Taylor series-based model has been verified by

comparison with a worst-case analysis, with errors of less than 3% [35].

The impact of topological imperfections on cellular material properties is evaluated with a

series of experiments to simulate missing nodes. As discussed in Section 2.1.3, we assume that

9 The Taylor series approach is also known as worst-case analysis, a term introduced by Parkinson and coauthors [28], because fluctuations are assumed to occur simultaneously in a worst-case combination. It is most accurate for small tolerances and weak or negligible interactions among the factors that fluctuate, and it is based on the assumption that tolerance ranges, rather than statistical distributions, are assigned to relevant factors.


any individual node, Ri, in the initial ground structure is available for inclusion in the designed

mesostructure, but may be missing randomly from the realized mesostructure with a small

probability, γi. Also, we utilize an SVE of 9 unit cells for evaluating the impact of topological

imperfections. If the set of D nodes in the SVE is expressed as RD:

{ }1 2, ,...,DDR R R=R (26)

then a sample space, Sj, can be defined of possible combinations, Rj, of D nodes, selected j at a

time:

{ }: , , j j j D j j j D≡ ⊆ = ≤S R R R R (27)

For the present case, there are 25 nodes in the initial ground structure for a single unit cell, as

illustrated in Figure 4B, and 169 nodes in the initial ground structure for the SVE. If we assume

that any single node may be missing randomly from the initial ground structure for the SVE, j

may be less than D by a magnitude of one (i.e., j=168). Therefore, the sample space of nodes,

Sj=168, includes 169 permutations, Rj=168, or possible combinations of the 169 nodes, selected 168

at a time, namely:

( ) ( ) ( )2 3 169 1 3 169 1 2 168, ,..., , , ,..., ,..., , ,...,R R R R R R R R R (28)

where R1 is the first node, R2 is the second node, and so on. Therefore, a total of V=170

experiments are conducted to simulate topological variation in the initial ground structure of the

SVE (i.e., 169 experiments for missing nodes and 1 for the intact ground structure). In the present

case, the orthotropic and periodic properties of the SVE can be used to reduce the number of

experiments. 10 In the first experiment, the effective elastic properties of the intact ground

10 Orthotropic symmetry implies that modifications in the designed quadrant are mirrored immediately to the other three quadrants during

the topology design process. Therefore, removing a node in one quadrant is equivalent, in its effect on material properties, of removing any of its three symmetric nodes. Similarly, periodicity implies that each unit cell in the SVE is identical in the designed mesostructure. If only one node is removed from the SVE at a time, removing a node from one unit cell in the SVE is equivalent, in its effect on material properties, of removing the same node from any other unit cell.


structure of the SVE are evaluated with the finite element model described in Section 2.3.1. In

the second experiment, the first node is removed from the ground structure of the SVE, and all of

the elements attached to the node are removed from the finite element model. For the third

experiment, the first node and its corresponding elements are replaced; the second node and its

corresponding elements are removed; and so on until all of the experiments are completed.

Effective elastic properties are calculated for each modified SVE ground structure with the finite

element model described in Section 2.3.1, modified appropriately for the missing node and

elements. Doubly periodic boundary conditions are applied to the SVE for the analysis.

Accordingly, it is assumed that the imperfections that appear in a single SVE are repeated

periodically in the surrounding SVEs.

Based on the experimental data, the standard deviation of an elastic constant is calculated as

follows:

( )( )22

1

V

Cij v ij v Cijv

Cσ γ μ=

= −∑ X (29)

where Xv is the vector of design variables for permutation or Experiment v,11 and γv is the

probability associated with Experiment v. We maintain two separate measures of elastic constant

variation—σCij and ijCΔ —so that tradeoffs can be explored between nominal elastic constant

values and robustness to dimensional and topological variation independently. Equation (29) is

used in Equation (11) in the compromise DSP of Figure 7 and completes the formulation of the

variability assessment model and simulation infrastructure.

11 The vector of design variables changes for each experiment because a different node is removed in each experiment along with the

elements that are connected to it.


2.4 Solve the Robust Topology Design Problem

In Phase IV, the compromise DSP is solved using the simulation infrastructure and the

Method of Moving Asymptotes (MMA) algorithm [48]—a gradient-based nonlinear

programming algorithm. The resulting unit cell design is post-processed as described in Section

2.1.2 and validated with several techniques described in [35]. Further details of the solution

process are provided in [35].

The formulation of the robust topology design method is now complete. In the following

section, we present the results from two example applications of the method.

3. EXAMPLES OF CUSTOMIZED, ROBUST CELLULAR MESOSTRUCTURES To demonstrate the effectiveness of the robust topology design method, we apply it for two

cases of periodic 2D cellular mesostructure design:

(1) A preliminary case with targeted effective elastic compressive stiffness in both principal

directions, with corresponding elastic constants, C11 and C22. The goal target values and other

parameters are listed in the last column of Table 2.

(2) A primary case with targeted effective elastic compressive stiffness in both principal

directions and targeted effective elastic shear stiffness. The corresponding elastic constants

are C11, C22, and C33. The goal target values and other parameters are listed in the first column

of Table 2 and are intended to address the region of opportunity identified in Figure 2.12

For the preliminary case, we already know that the optimal topology is a square cell with

orthogonal walls aligned with the principal directions. Because we know the optimal solution, it

is a useful case for verifying the effectiveness of the robust topology design method for

generating periodic 2D cellular mesostructures with specific properties and for distinguishing

12 Note the relationship between C33 and G12 recorded in Equation (5).


between robust and nonrobust solutions. The primary example is intended to demonstrate that

the robust topology design method is effective for generating periodic 2D cellular mesostructures

with customized properties that cannot be obtained with standard cellular topologies and for

identifying robust and non-robust variants of the topology.

3.1 Cellular Mesostructures with Dimensional Robustness

We begin by designing cellular mesostructures that achieve target values for elastic constants

as closely as possible with minimum sensitivity to dimensional variation. The dimensionally

robust designs are obtained by solving the compromise DSP in Figure 7 with Equation (15) as

the objective function, the 5x5 node ground structure in Figure 4B, and the design parameter

values recorded in Table 2. The goals in Equation (15) are weighted equally.

For comparison purposes, non-robust designs are obtained by solving the compromise DSP

in Figure 7 with Equation (14) as the objective function and all other factors equivalent. When

Equation (14) is the objective function, variation in the elastic constant values is not considered,

and only nominal elastic constant values are considered.

Dimensionally robust cellular mesostructures for the two example cases are reported in the

left columns of Tables 3 and 4 for the preliminary and primary cases, respectively. Non-robust

results are reported in the right columns of Tables 3 and 4. The diagrams depict the designed

unit cell of the cellular mesostructure after post-processing. For ease of visualization, the

designed unit cells are periodically repeated to depict a segment of material. Dimensions are

labeled for the cell walls of each unit cell.13 Values for all goals are presented in the bottom

rows of Tables 3 and 4. For non-robust designs the ranges of elastic constant values, ΔCij,

associated with dimensional variation are calculated and reported for comparison purposes, even

13 Recall that the unit cells have orthotropic symmetry; therefore, only one quadrant of cell wall dimensions is labeled for each unit cell.


though they are not considered in the objective function during the non-robust design process.

All of the designs reported in this section have a 20% volume fraction, vf, of solid cell wall

material.

[INSERT TABLE 3.]

[INSERT TABLE 4.]

Resulting mesostructures are different for each case and for each level of robustness.

Although each design has a 20% volume fraction of solid material, the connectivity and

configuration of the cell walls and the number of voids per unit area are different for each design.

For the preliminary case, the observed rectangular grid patterns in Table 3 are expected

outcomes because they maximize effective elastic stiffness in the principal in-plane directions—

the two components of the constitutive tensor targeted in this example. However, the rectangular

cell designs of Table 3 have very poor effective elastic shear stiffness. When effective elastic

shear stiffness is considered for the primary case, diagonal elements are present in the final

topology (as illustrated in Table 4) to increase the shear stiffness of the design. It is interesting

to observe that the cell topologies for the primary case (Table 4) are significantly different from

any of the standard cell topologies discussed in the literature for prismatic cellular materials

(e.g., square, triangular, hexagonal, kagome, etc.; cf. [47]). A novel cellular topology is expected

for this example because the standard cell topologies cannot meet the combination of effective

elastic stiffness targets specified for this material. This is an example of materials design in

which material structure is tailored to achieve a desired set of properties that are unattainable

with available material assets.

A comparison of the robust and non-robust topologies for each example yields important

insights into the effectiveness of the robust topology design method. For each example, the


robust and non-robust topologies have similar mean elastic constants, μCij, but the range of

elastic constant values induced by cell wall tolerances, ΔCij, is up to 40% higher for non-robust

designs. This disparity provides evidence for the relative insensitivity of robust design

performance to control factor variation as well as the effectiveness of the robust topology design

method in generating relatively robust topologies. The fact that robust and non-robust designs

for each case exhibit similar mean elastic constant values indicates that the designs represent

alternative local minima to the materials design problem posed in Figure 7. In fact, by adjusting

weights, starting points, and other convergence parameters for the non-robust designs, it is

possible to obtain additional topologically distinct local minima, as reported in [35].

Dimensional robustness introduces scale-related effects because of the scale-dependent

tolerance function in Equation (6). In Tables 3 and 4, the phenomenon is embodied in robust

designs that have simpler topologies with fewer elements and voids per unit cell, on average,

than the non-robust, standard topologies. The robustness of a cellular structure with respect to

cell wall tolerances is largely a function of the number of cell walls per unit area in the

mesostructure. In many cases, as with the rectangular grid designs in Table 3, it is possible to

achieve identical or nearly identical nominal performance with either large numbers of thin cell

walls (i.e., small-scale topology) or small numbers of relatively thick cell walls (i.e., large-scale

topology). The latter category of designs yields lower overall performance variation if the ratio

of tolerances to nominal cell wall thickness decreases with increasing thickness. However, if

tolerances were strictly proportional to thicknesses, then the induced performance variation

would be equivalent for the two designs in Table 3 and other similar designs. However, in most

cases, tolerances are not necessarily proportional to nominal dimensions; instead, they may be

decreasing as a percentage of the nominal dimension as it increases. This is especially true when


tolerances are more difficult to maintain for smaller dimensions, as in this example. This

behavior is embodied in the tolerance function in Equation (6), which is increasing and concave

in nominal values of cell wall thickness so that the ratio of tolerances to nominal thickness is

monotonically decreasing over the region of interest.

3.2 Cellular Mesostructures with Dimensional and Topological Robustness

Next, we design cellular mesostructures that achieve target values for elastic constants as

closely as possible with minimum sensitivity to dimensional and topological variation. The

dimensionally and topologically robust designs are obtained by solving the compromise DSP in

Figure 7 with Equation (16) as the objective function and the design parameter values recorded

in Table 2. The results are recorded in the middle columns of Tables 5 and 6 for the preliminary

and primary cases, respectively, alongside results for dimensionally robust and non-robust

designs. The new piece of information is the standard deviation of each elastic constant, σcij,

which quantifies the spread in elastic constant values associated with potential topological

imperfections, as described in Section 2.3.2.

[INSERT TABLE 5.]

[INSERT TABLE 6.]

For both the preliminary and the primary cases, the designs in each table are variations of a

similar underlying pattern. In Table 5, the primary geometric difference between the designs is

that they accomplish the same elastic constant goals with different numbers of elements—i.e., a

few thick orthogonal elements, a large number of thinner orthogonal elements, or something in

between. Similarly, the designs in Table 6 are variations of an underlying diamond pattern, with

additional redundant elements incorporated in the more complex topologies. The mean values of

elastic constants are nearly identical for all of the designs in Table 5 and similar for the designs


in Table 6. This fact implies that the designs represent multiple local minima—alternative

topologies with different scales but similar on-target performance—for the topology design

problem.

Significant differences between the designs are observed in the associated elastic constant

ranges and standard deviations. Local minima with an abundance of thinner elements impose

higher elastic constant ranges, ΔCij, relative to more efficient designs with fewer thicker elements

because dimensional tolerances tend to be relatively high (as a percentage of element thickness)

for thinner elements. Therefore, the impact of tolerances on elastic constant ranges increases

with the number of elements per unit cell area in a final topology, and simple topologies are

preferred for robust design for dimensional variation. This is reflected in the elastic constant

ranges, which are smaller for simpler topologies with fewer elements and voids for a given

domain.

A different trend is observed when one considers topological noise and its impact on elastic

constant variation, namely, the standard deviations of elastic constant values, σCij, reported in the

tables. In this case, the simpler topologies have much higher standard deviations than the more

complex topologies. This conclusion is intuitively related to the mechanics of the problem.

When experiments are conducted to simulate the impact of topological defects on elastic

constants, the possibility is considered of missing each node in turn, and standard deviations are

derived from the resulting experimental values of elastic constants. In relatively simple

topologies such as the dimensionally robust topologies in the left columns of Tables 5 and 6,

only a few elements are available for providing stiffness or carrying structural loads. If one or

more of the elements fail, there are few ‘back-up’ elements to provide some measure of stiffness.

In more complex topologies such as the non-robust designs in the right columns of Tables 5 and


6, there are many more elements. The failure of any single element or node has a much smaller

impact on the stiffness of the overall structure. In fact, this effect is so strong that if dimensional

variation were not considered, the non-robust topologies in Tables 5 and 6 would be the

dominant designs, offering on-target nominal performance and minimal deviation due to

topological noise.

From the designs in Tables 5 and 6, we observe a tradeoff between robustness to topological

noise and robustness to dimensional variation, with designs performing well with respect to one

criterion performing poorly with respect to the other. It is possible to discern a family of

designs, embodying tradeoffs between robustness to dimensional variation and robustness to

topological variation. If relatively large weight is placed on the impact of dimensional variation,

the left-most design is preferred in each table. Conversely, if relatively large weight is placed on

the impact of topological variation, the preferred design is the right-most design. For

intermediate weights or for relatively equal weights on topologically- and dimensionally-induced

performance variation, intermediate designs are preferred such as the dimensionally and

topologically robust designs reported in the center columns of Tables 5 and 6. Visually, it is

noticeable that the sensitivity to topological variation of the dimensionally and topologically

robust designs (in the middle columns) is reduced by introducing additional, redundant elements.

These additional elements reduce the impact of random removal of a node or element on elastic

constant values.

3.3 Impact of the Initial Design Space on Final Designs

Both the density of the initial ground structure (i.e., number of nodes and elements) and the

number of unit cells considered during the design process can have a significant impact on the

topological nature and associated properties of the designed cellular mesostructures. To


investigate the impact of the density of the initial ground structure, we compared results from the

5x5 node initial ground structure with those from a denser 9x9 node initial ground structure,

illustrated in Figure 5. For both the preliminary and primary design cases, we found that the

dimensionally robust designs remained topologically consistent for larger ground structures,

confirming that the 5x5 node initial ground structure is dense enough to support cellular

mesostructures with the targeted nominal elastic properties (i.e., μCij) [35]. We observed that

topologically robust designs become more complex with increasing ground structure density;

they tend to incorporate as many redundant, ‘back-up’ elements as allowed by the initial ground

structure, in order to minimize the impact of potential topological imperfections. Dimensional

robustness and manufacturability tend to decline with increasing topological complexity;

therefore, for manufacturability or other purposes, a designer may wish to limit the initial ground

structure density to the minimum density required for supporting designs with targeted nominal

properties.

The number of unit cells in the initial design domain—also known as the statistical volume

element (SVE) or window size—impacts our ability to design for randomly distributed

topological imperfections. The implication of SVE size is that defects are assumed to be

periodically repeated in each of the surrounding SVEs in the material. In this case, the statistical

volume element (SVE) for analyzing topological imperfections is assumed to be a 3x3 matrix of

9 unit cells. By adopting SVEs that are larger than the unit cell, we consider defects that are

more randomly distributed throughout the material. Larger SVEs (cf. [12]) would enable us to

consider less periodic distributions of topological defects and their impact on designed

mesostructures, but it would increase the computational complexity of the robust topology

design process.


4. CLOSURE In this paper, the mesostructures of periodic 2D cellular materials are designed with

customized structural elastic properties that are robust to dimensional variations and topological

imperfections such as missing cell walls or joints. For the examples in Section 3, three

categories of cellular mesostructures are generated: (1) designs with structural elastic properties

that are robust to dimensional and topological variation, (2) designs with structural elastic

properties that are robust to dimensional variation only, and (3) benchmark non-robust designs

for which variation is not considered. When the robust designs are compared with benchmark,

non-robust topology designs, the effectiveness of the robust topology design methods is evident

in both the performance and the structure of the resulting designs. Dimensionally robust

topology designs tend to have nearly identical levels of nominal performance, much lower levels

of performance variation, and much simpler topologies than their non-robust counterparts. The

simpler topologies reduce the build-up of tolerance effects on performance variation, and they

also tend to be easier to manufacture. On the other hand, the more complex, non-robust

topologies tend to be less sensitive to topological variation because element removal has a

smaller impact on a complex topology with large numbers of redundant elements. When both

dimensional and topological variation are considered, the robust topology design method yields

topologies that offer a compromise between the simpler topologies with superior robustness to

dimensional variation and the more complex, non-robust topologies with low levels of

robustness to dimensional variation and higher levels of robustness to topological noise.

The periodic 2D cellular mesostructures are designed with the robust topology design method

presented in this paper. Its effectiveness stems from four constituent phases that have been

devised to address many of the challenges of integrating robust design, topology design, and


multiobjective decision support techniques for materials design applications. In the first phase,

topological and dimensional variation are modeled, respectively, as sets of potential

permutations or subsets of an intact ground structure and as tolerance ranges with special

characteristics that make them suitable for robust topology design. In the second phase, the

robust topology design problem is formulated as a compromise DSP, a flexible decision model

that facilitates exploration of families of solutions that embody a spectrum of tradeoffs between

nominal performance and robustness to topological and dimensional imperfections. In the third

phase, the impact of dimensional and topological variation is assessed via Taylor series-based

techniques and strategic experiments in potential topological permutations, respectively. Finally,

in the fourth phase, robust topology design problems are solved with gradient-based optimization

algorithms, and the results are validated.

The method has potential to be used in industrial applications. Customized mesostructures

provide lightweight multifunctionality for ultralight load-bearing combined with energy

absorption, heat transfer, and other properties. It is possible to fabricate parts and materials with

these customized mesostructures using additive fabrication techniques, such as selective laser

sintering, in addition to the thermo-chemical extrusion process discussed in this paper. The

method facilitates mesostructure design for specific industrial applications, with minimal

sensitivity to the lack of reproducibility often associated with these fabrication techniques. The

method may be extended to address non-structural performance criteria and non-periodic,

functionally-graded topology for these industrial applications.

The method and example also have important materials design implications. Although

imperfections are known to impact the performance of cellular materials significantly, no work

has been done on designing the material mesostructure to minimize their impact on overall


structural performance. The robust topology design method has been shown to be effective not

only for minimizing the sensitivity of material mesostructures to dimensional and topological

variation but also for adjusting the complexity or simplicity of the resulting topologies. This

feature is useful for customizing materials for applications such as catalysis that require complex

structures or for considerations such as manufacturability that require simplicity. Furthermore, in

this example, we use the robust topology design method to identify a new standard cellular

topology that meets requirements that are beyond the scope of other standard cellular topologies,

as illustrated in Figure 2. The method can be used to design additional cellular topologies for

specific requirements, including robustness considerations. Furthermore, due to the

multiobjective nature of the underlying decision support, the method facilitates the search for

compromise solutions rather than solutions that are predominantly single objective in nature. In

summary, the method is representative of a systematic approach to materials design that is

requirements-driven, exploratory, structured, and focused on the robust design of products and

materials that are relatively insensitive to commonly encountered variations and imperfections.

ACKNOWLEDGMENTS Financial support from an AFOSR MURI (1606U81), NSF DMI-0085136 and DMI-

0407627, and the University of Texas at Austin is gratefully acknowledged. During her graduate

study at Georgia Tech, Carolyn Conner Seepersad was sponsored by a National Science

Foundation Graduate Fellowship and a Hertz Foundation Fellowship. We are grateful to Krister

Svanburg of the Royal Institute of Technology in Stockholm, Sweden, for supplying a MATLAB

version of his MMA algorithm.


REFERENCES

1. Eschenauer, H. A. and N. Olhoff, 2001, “Topology Optimization of Continuum Structures: A Review,” Applied Mechanics Reviews, Vol. 54, No. 4, pp. 331-389.

2. Ohsaki, M. and C. C. Swan, 2002, "Topology and Geometry Optimization of Trusses and Frames," Recent Advances in Optimal Structural Design (S. A. Burns, Ed.), American Society of Civil Engineers, Reston, VA.

3. Soto, C., 2001, “Structural Topology Optimization: From Minimizing Compliance to Maximizing Energy Absorption,” International Journal of Vehicle Design, Vol. 25, No. 1/2, pp. 142-160.

4. Bendsoe, M. P. and N. Kikuchi, 1988, “Generating Optimal Topologies in Structural Design Using a Homogenization Method,” Computer Methods in Applied Mechanical Engineering, Vol. 71, pp. 197-224.

5. Sigmund, O., 1994, “Materials with Prescribed Constitutive Parameters: An Inverse Homogenization Problem,” International Journal of Solids and Structures, Vol. 31, pp. 2313-2329.

6. Sigmund, O., 1995, “Tailoring Materials with Prescribed Elastic Properties,” Mechanics of Materials, Vol. 20, pp. 351-368.

7. Sigmund, O., 2000, “A New Class of Extremal Composites,” Journal of the Mechanics and Physics of Solids, Vol. 48, No. 2, pp. 397-428.

8. Sigmund, O. and S. Torquato, 1997, “Design of Materials with Extreme Thermal Expansion Using a Three-Phase Topology Optimization Method,” Journal of the Mechanics and Physics of Solids, Vol. 45, No. 6, pp. 1037-1067.

9. Hyun, S. and S. Torquato, 2002, “Optimal and Manufacturable Two-Dimensional, Kagome-Like Cellular Solids,” Journal of Materials Research, Vol. 17, No. 1, pp. 137-144.

10. Seepersad, C. C., J. K. Allen, D. L. McDowell and F. Mistree, 2003, "Robust Topological Design of Cellular Materials," ASME Advances in Design Automation, Chicago, IL. Paper No. DETC2003/DAC-48772.

11. Cochran, J. K., K. J. Lee, D. L. McDowell and T. H. Sanders, 2000, "Low Density Monolithic Honeycombs by Thermal Chemical Processing," Proceedings of the 4th Conference on Aerospace Materials, Processes, and Environmental Technology, Huntsville, AL.

12. Wang, A. and D. L. McDowell, 2004, “Effects of Defects on In-Plane Properties of Periodic Metal Honeycombs,” International Journal of Mechanical Sciences, Vol. 45, No. 11, pp. 1799-1813.

13. Silva, M. J., W. C. Hayes and L. J. Gibson, 1995, “The Effects of Non-Periodic Microstructure on the Elastic Properties of Two-Dimensional Cellular Solids,” International Journal of Mechanical Sciences, Vol. 37, No. 11, pp. 1161-1177.

14. Bocchini, G. F., 1986, “The Influence of Porosity on the Characteristics of Sintered Materials,” The International Journal of Powder Metallurgy, Vol. 22, No. 3, pp. 185-202.

15. Diaz, A. and M. P. Bendsoe, 1992, “Shape Optimization of Structures for Multiple Loading Situations Using a Homogenization Method,” Structural Optimization, Vol. 4, pp. 17-22.


16. Diaz, A., R. Lipton and C. A. Soto, 1995, “A New Formulation of the Problem of Optimum Reinforcement of Reissner-Mindlin Plates,” Computer Methods in Applied Mechanics and Engineering, Vol. 123, pp. 121-139.

17. Christiansen, S., M. Patriksson and L. Wynter, 2001, “Stochastic Bilevel Programming in Structural Optimization,” Structural and Multidisciplinary Optimization, Vol. 21, pp. 361-371.

18. Maute, K. and D. M. Frangopol, 2003, “Reliability-Based Design of MEMS Mechanisms by Topology Optimization,” Computers and Structures, Vol. 81, pp. 813-824.

19. Thampan, C. K. P. V. and C. S. Krishnamoorthy, 2001, “System Reliability-Based Configuration Optimization of Trusses,” Journal of Structural Engineering, Vol. 127, No. 8, pp. 947-956.

20. Bae, K.-R., S. Wang and K. K. Choi, 2002, "Reliability-Based Topology Optimization," 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Atlanta, GA. Paper No: AIAA-2002-5542.

21. Ben-Tal, A. and A. Nemirovski, 1997, “Robust Truss Topology Design via Semidefinite Programming,” SIAM Journal of Optimization, Vol. 7, No. 4, pp. 991-1016.

22. Cherkaev, A. and E. Cherkaeva, 1999, "Optimal Design for Uncertain Loading Conditions," Homogenization (V. Berdichevsky, V. Jikov and G. Papanicolaou, Eds.), World Scientific, pp. 193-213.

23. Kocvara, M., J. Zowe and A. Nemirovski, 2000, “Cascading--An Approach to Robust Material Optimization,” Computers and Structures, Vol. 76, pp. 431-442.

24. Sandgren, E. and T. M. Cameron, 2002, “Robust Design Optimization of Structures through Consideration of Variation,” Computers and Structures, Vol. 80, pp. 1605-1613.

25. Taguchi, G., 1986, Introduction to Quality Engineering, Asian Productivity Organization, UNIPUB, White Plains, NY.

26. Taguchi, G. and D. Clausing, 1990, “Robust Quality,” Harvard Business Review, Vol. Jan/Feb, pp. 65-75.

27. Phadke, M. S., 1989, Quality Engineering Using Robust Design, Prentice Hall, Englewood Cliffs, NJ.

28. Parkinson, A., C. Sorensen and N. Pourhassan, 1993, “A General Approach for Robust Optimal Design,” ASME Journal of Mechanical Design, Vol. 115, No. 1, pp. 74-80.

29. Chen, W., J. K. Allen, K.-L. Tsui and F. Mistree, 1996, “A Procedure for Robust Design: Minimizing Variations Caused by Noise Factors and Control Factors,” ASME Journal of Mechanical Design, Vol. 118, No. 4, pp. 478-485.

30. Otto, K. N. and E. K. Antonsson, 1993, “Extensions to the Taguchi Method of Product Design,” ASME Journal of Mechanical Design, Vol. 115, No. 1, pp. 5-13.

31. Ramakrishnan, B. and S. S. Rao, 1996, “A General Loss Function Based Optimization Procedure for Robust Design,” Engineering Optimization, Vol. 25, No. 4, pp. 255-276.

32. Michelena, N. F. and A. M. Agogino, 1994, “Formal Solution of N-Type Robust Parameter Design Problems with Stochastic Noise Factors,” ASME Journal of Mechanical Design, Vol. 116, No. 2, pp. 501-507.

33. Sundaresan, S., K. Ishii and D. R. Houser, 1995, “A Robust Optimization Procedure with Variations on Design Variables and Constraints,” Engineering Optimization, Vol. 24, pp. 101-117.

34. Yu, J. and K. Ishii, 1998, “Design for Robustness Based on Manufacturing Variation Patterns,” ASME Journal of Mechanical Design, Vol. 120, No. 2, pp. 196-202.


35. Seepersad, C. C., 2004, "A Robust Topological Preliminary Design Exploration Method with Materials Design Applications," PhD Dissertation, G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA.

36. Malvern, L. E., 1969, Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Upper Saddle River, NJ.

37. Kirsch, U., 1989, “Optimal Topologies of Structures,” Applied Mechanics Reviews, Vol. 42, No. 8, pp. 223-239.

38. Topping, B. H. V., 1984, “Shape Optimization of Skeletal Structures: A Review,” Journal of Structural Engineering, Vol. 109, No. 8, pp. 1933-1951.

39. Dorn, W. S., R. E. Gomory and H. J. Greenberg, 1964, “Automatic Design of Optimal Structures,” Journal de Mecanique, Vol. 3, pp. 25-52.

40. Reddy, J. N., 1993, An Introduction to the Finite Element Method, 2nd Ed., McGraw-Hill, Boston.

41. Mistree, F., O. F. Hughes and B. A. Bras, 1993, "The Compromise Decision Support Problem and the Adaptive Linear Programming Algorithm," Structural Optimization: Status and Promise (M. P. Kamat, Ed.), AIAA, Washington, D.C., pp. 247-286.

42. Charnes, A. and W. W. Cooper, 1961, Management Models and Industrial Applications of Linear Programming, John Wiley & Sons, New York, NY.

43. Neves, M. M., H. Rodrigues and J. M. Guedes, 2000, “Optimal Design of Periodic Linear Elastic Microstructures,” Computers and Structures, Vol. 76, No. 1-3, pp. 421-429.

44. Nemat-Nasser, S. and M. Hori, 1999, Micromechanics: Overall Properties of Heterogeneous Materials, 2nd Ed., Elsevier, Amsterdam.

45. Cook, R. D., D. S. Malkus and M. E. Plesha, 1989, Concepts and Applications of Finite Element Analysis, 3rd Ed., John Wiley and Sons, New York.

46. van der Sluis, O., P. J. G. Schreurs, W. A. M. Brekelmans and H. E. H. Meijer, 2000, “Overall Behavior of Heterogeneous Elastoviscoplastic Materials: Effect of Microstructural Modeling,” Mechanics of Materials, Vol. 32, pp. 449-462.

47. Hayes, A. M., A. Wang, B. M. Dempsey and D. L. McDowell, 2004, “Mechanics of Linear Cellular Alloys,” Mechanics of Materials, Vol. 36, pp. 691-713.

48. Svanberg, K., 1987, “The Method of Moving Asymptotes--A New Method for Structural Optimization,” International Journal for Numerical Methods in Engineering, Vol. 24, pp. 359-373.


Figure Titles

Figure 1. Examples of ordered, prismatic cellular materials.

Figure 2. Effective elastic properties of standard periodic cellular topologies.

Figure 3. Outline of the robust topology design method [35].

Figure 4. An initial ground structure for a cellular mesostructure (A) and for a representative unit

cell (B). A designed unit cell after topology design (C) and after post-processing (D). A

designed cellular mesostructure (E) comprised of a doubly periodic pattern of designed

unit cells (D).

Figure 5. Course (A) and fine (B) initial ground structures for cellular mesostructure design.

Figure 6. An example of a topological imperfection in a cellular mesostructure.

Figure 7. Decision Support Problem for robust topology design of 2D periodic cellular

mesostructure with topological and dimensional variation.


Table Titles

Table 1. Summary of design parameters for design of periodic 2D cellular mesostructure.

Table 2. Design variable bounds, constraint limits, and goal target values for the compromise

DSP in Figure 7.

Table 3. Robust vs. non-robust periodic cellular mesostructure for effective stiffness in both

principal directions (C11 and C22), considering dimensional variation only.


principal directions (C11, C22) and in shear (C33), considering dimensional variation only.


principal directions, considering topological and dimensional variation.

Table 6. Robust vs. non-robust periodic cellular mesostructure for effective stiffness in principal

directions and shear, considering topological and dimensional variation.


Figure 1


Figure 2

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.02 0.04 0.06 0.08 0.1 0.12

X1

X2

X1

X2

X1

X2

X1

X2

X1

X2

X1

X2

X1

X2

X1

X2

X1

X2

X1

X2

Effe

ctiv

e El

astic

She

ar S

tiffn

ess

In-P

lane

Tra

nsve

rse

Dire

ctio

n, G

12/E

s

Effective Elastic Compressive StiffnessIn-Plane Principal Directions, E11/Es or E22/Es

An Area ofOpportunity

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.02 0.04 0.06 0.08 0.1 0.12

X1

X2

X1

X2

X1

X2

X1

X2

X1

X2

X1

X2

X1

X2

X1

X2

X1

X2

X1

X2

Effe

ctiv

e El

astic

She

ar S

tiffn

ess

In-P

lane

Tra

nsve

rse

Dire

ctio

n, G

12/E

s

Effective Elastic Compressive StiffnessIn-Plane Principal Directions, E11/Es or E22/Es

An Area ofOpportunity


Figure 3

Phase I. Formulate Robust Topology Design Space

Robust Design Space

StructureVariables, Factors

BehaviorProperties

RequirementsConstraintsGoals

Topology Representation

u=uo

t=to

u=uo

t=to

Characterization of Variation

Product/Processx y

zNoise Factors

ResponsesControl Factors

Overall Design Requirements

Phase II. Formulate Robust Topology Design Problem

The Compromise DSP

GivenFindSatisfyMinimize

Robust Design Specifications

Phase IV. Solve Robust Topology Design ProblemSimulation

InfrastructureSearch & Optimization

Algorithms

Phase III. Establish Simulation Infrastructure

Simulation Infrastructure

Variability AssessmentAnalysis Models

Phase I. Formulate Robust Topology Design Space

Robust Design Space


BehaviorProperties


Robust Design Space


BehaviorProperties



u=uo

t=to

u=uo

t=to


u=uo

t=to

u=uo

t=to


Product/Processx y

zNoise Factors



Product/Processx y

zNoise Factors

ResponsesControl Factors Product/

Processx y

zNoise Factors


Overall Design Requirements

Phase II. Formulate Robust Topology Design Problem

The Compromise DSP

GivenFindSatisfyMinimize

Robust Design Specifications


Infrastructure


InfrastructureSearch & Optimization

Algorithms








Figure 4

-

A

B C D

E

-

A

B C D

E


Figure 5

A

B

1

2

1

2


Figure 6

Intact Ground StructureFor a Unit Cell

Designed Unit Cell Realized Unit Cell with 1 Missing Node


Figure 7

Given Robust topology design space (Sect. 2.1) Simulation infrastructure (Sect. 2.3) Targets, bounds, weights (Table 2) Find

iX In-plane element thickness i = 1, …, N N = # elements

di-, di

+ Deviation Variables i = 1, …, 3P P = # tailored elastic constants Satisfy Constraint -limitf fv v≤ Eq. (8), cf. Eq. (22) Goals Mean value of elastic constant

k k

Cij Cij target

Cij target Cij target

d d 1μ μ

μ μ− + −

− −

+ − = = k = 1, …, P Eq. (9), cf. Eq. (23)

Range of elastic constant (due to dimensional variation)

ij-target

Cij-target Cij-target

ij

k k

C Cd d

μ μ− +

Δ Δ+ − = k = (P+1), …, 2P Eq. (10), cf. Eq. (24)

Standard deviation of elastic constant (due to topological imperfections)

Cij-target Cij-target

Cij-targetk k

Cij d dσ σ

μ μ− ++ − = k = (2P+1),…, 3P Eq. (11), cf. Eq. (29)

Bounds , ,i L i i UX X X≤ ≤ i = 1,…, N Eq. (12)

0i id d− +• = ; 0,i id d− + ≥ i = 1,…, 3P Eq. (13) Minimize Non-robust Design:

( )k k k

P

k=1Z W d d− += +∑ , k

P

k=1W = 1∑ P = # tailored elastic constants Eq. (14)

Robust Design for Dimensional Variation

( )1

k k k

2P

kZ W d d− +

=

= +∑ , k

2P

k=1W = 1∑ Eq. (15)

Robust Design for Dimensional and Topological Variation

( )1

k k k

3P

kZ W d d− +

=

= +∑ , k

3P

k=1W = 1∑ Eq. (16)


Table 1

Fixed Factors • Initial ground structure (Fig. 4) and boundary conditions Sources of Variation • Dimensional imperfections (i.e., tolerances for in-plane thickness of each cell wall)

• Topological imperfections (i.e., randomly missing cell walls or joints) Design Variables • X, Vector of in-plane thicknesses of elements in ground structure Properties • vf, Volume fraction

• Cij, Elastic constant • μCij, Mean or nominal value of elastic constant • ΔCij, Elastic constant variation due to dimensional imperfections • σCij, Elastic constant standard deviation due to topological imperfections


Table 2

Primary Example Effective Elastic Stiffness in Principal Directions and

in Shear

Preliminary Example Effective Elastic

Stiffness in Principal Directions Only

vf-limit 0.2 0.2 μC11-target 0.035 0.1 μC22-target 0.09 0.1 μC33-target 0.045 NA ΔCij-target, 0 0 σCij-targett 0 0 XU 500 μm 1000 μm XL 0.01 μm 0.01 μm Wi Example-Specific,

Noted in Section 3 Example-Specific, Noted in Section 3


Table 3


Non-Robust Design

Material

Material

Unit Cell

1 cm

1 cm

1 cm

1 cm

0.05 cmcell walls

1 cm

1 cm

1 cm

1 cm

0.05 cmcell walls

Unit Cell

1 cm

1 cm

1 cm

1 cm

0.02 cmcell walls

1 cm

1 cm

1 cm

1 cm

0.02 cmcell walls

Design Performance Design Performance

μC11 = 0.10 μC22 = 0.10

ΔC11 = 0.015 ΔC22 = 0.015

μC11 = 0.10 μC22 = 0.10

ΔC11 = 0.021* ΔC22 = 0.021*

* (40% higher than robust design)


Table 4


Non-Robust Design

Material

Material

Unit Cell

1 cm

1 cm

1 cm

1 cm

0.032 cmcell wall

0.045 cm, cell wall

1 cm

1 cm

1 cm

1 cm

cell wall

1 cm

1 cm

1 cm

1 cm

0.032 cmcell wall

0.045 cm, cell wall

1 cm

1 cm

1 cm

1 cm

cell wall

Unit Cell

1 cm

1 cm

1 cm

1 cm

0.015 cm

0.025 cm0.013 cm

0.025 cm

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

0.015 cm

0.025 cm0.013 cm

0.025 cm

1 cm

1 cm

1 cm

1 cm

Design Performance Design Performance

μC11 = 0.032 μC22 = 0.096 μC33 = 0.043

ΔC11 = 0.0052 ΔC22 = 0.017

ΔC33 = 0.0072

μC11 = 0.029 μC22 = 0.080 μC33 = 0.036

ΔC11 = 0.0084 ΔC22 = 0.022 ΔC33 = 0.011


Table 5


Robust Design for Dimensional and Topological Variation

Non-Robust Design And Robust Design for Topological Variation

Material

Material

Material

Unit Cell

1 cm

1 cm

1 cm

1 cm

0.05 cmcell walls

1 cm

1 cm

1 cm

1 cm

0.05 cmcell walls

Unit Cell

1 cm

1 cm

1 cm

1 cm

0.03 cm

0.02 cm0.03 cm

0.02 cm

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

0.03 cm

0.02 cm0.03 cm

0.02 cm

1 cm

1 cm

1 cm

1 cm

Unit Cell

1 cm1

cm1 cm1

cm

0.02 cmcell walls

1 cm1

cm1 cm1

cm

0.02 cmcell walls

Design Performance Design Performance Design Performance

μC11 = 0.10 μC22 = 0.10

ΔC11 = 0.015

ΔC22 = 0.015 σC11 = 0.0071 σC22 = 0.0071

μC11 = 0.10 μC22 = 0.10 ΔC11 = 0.02

ΔC22 = 0.02 σC11 = 0.0068 σC22 = 0.0068

μC11 = 0.10 μC22 = 0.10

ΔC11 = 0.021

ΔC22 = 0.021 σC11 = 0.0053 σC22 = 0.0053


Table 6


Robust Design for Dimensional and Topological

Variation

Non-Robust Design

Material

Material

Material

Unit Cell

1 cm

1 cm

1 cm

1 cm

0.032 cmcell wall

0.045 cm, cell wall

1 cm

1 cm

1 cm

1 cm

cell wall

1 cm

1 cm

1 cm

1 cm

0.032 cmcell wall

0.045 cm, cell wall

1 cm

1 cm

1 cm

1 cm

cell wall

Unit Cell

1 cm

1 cm

1 cm

1 cm

0.034 cm

0.025 cm0.01 cm

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

0.034 cm

0.025 cm0.01 cm

1 cm

1 cm

1 cm

1 cm

Unit Cell

1 cm

1 cm

1 cm

1 cm

0.015 cm

0.025 cm0.013 cm

0.025 cm

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

0.015 cm

0.025 cm0.013 cm

0.025 cm

1 cm

1 cm

1 cm

1 cm

Design Performance Design Performance Design Performance μC11 = 0.032 μC22 = 0.096 μC33 = 0.043

ΔC11 = 0.0052 ΔC22 = 0.017

ΔC33 = 0.0072 σC11 = 0.0017 σC22 = 0.0049 σC33 = 0.0020

μC11 = 0.025 μC22 = 0.093 μC33 = 0.029

ΔC11 = 0.0054 ΔC22 = 0.017

ΔC33 = 0.0064 σC11 = 0.0009 σC22 = 0.0048 σC33 = 0.0011

μC11 = 0.029 μC22 = 0.080 μC33 = 0.036

ΔC11 = 0.0084 ΔC22 = 0.022

ΔC33 = 0.011 σC11 = 0.0008 σC22 = 0.0021 σC33 = 0.0008

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