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Spring 2003

Semester Project Report

Towards Finite Element-Based Thermal

Topological Design of Unit Cells for

Linear Cellular Alloys

ANSYS

MATLAB

prepared for

ME 6124

The Finite Element Method

Dr. Suresh Sitaraman

By

Hae-Jin ChoiMarco Gero Fernndez


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Tabe of Contents1. Overview 2

2. Objectives 2

3. Background Information 3

3.1 Linear Cellular Alloys 33.2 Topology Optimization 7

4. Purpose of this Investigation 8

5. Case Study Convectively Cooled Heat Sink for a Computer Chip 96. Modeling Details 11

6.1 Simplifying Assumptions 11

6.2 Finite Element Model for Heat Transfer Analysis 12

6.3 Modeling the LCAs in MATLAB

18

6.4 Modeling the LCAs in ANSYS

207. Results, Discussion, and Validation/Verification 21

7.1 The 2 x 2 LCA Design (MATLAB vs. ANSYS) 21

7.2 The 8 x 8 LCA Design (MATLAB vs. Finite Difference) 257.3 The 1 x 1 LCA Design (MATLAB vs. Hand Calculations) 27

8. Closure 27

9. Problems Encounteredand Learning 2810. References 29

11. Appendices 30

11.1 Appendix A Derivation of Finite Element used for MATLAB

HeatTransfer Analysis

11.2 Appendix B Hand Calculation for a 1 x 1 Cell11.2 Appendix C Final Presentation Slides

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1. Overview

The objective of this project was to facilitate the design of Linear Cellular Alloys

(LCAs), as developed by the Lightweight Structures Group at Georgia Tech, through

robust topology design techniques (i.e., a method combing topology optimization and

robust design that is currently being developed in the Systems Realization Laboratory,also at Georgia Tech). Since topology design is very computationally expensive and

highly dependent on the ability to explore a given design space effectively, analysis

routines, used to judge the performance of the structures changing morphology arecritical. The primary obstacle to effective and efficient design space exploration is the

speed with which the algorithms used to conduct the analysis can be invoked. This is turn

is highly dependent on the ability to interface different software tools successfully andupdate changing inputs/outputs automatically. Previous implementations have relied on

the coupled use of iSIGHT from Engineous Software for optimization and ANSYS

fromANSYS Incorporated for analysis. Unfortunately, passing any information among

heterogeneous and potentially distributed software tools, though novel, introduces a

significant lag and greatly increases the amount of time required per iteration.Considering that optimization can necessitate on the order of hundreds of runs, such

inherent lags can be detrimental.

In order to address this need for quick and efficient design space exploration, a 99

line MATLAB

code developed by Ole Sigmund from the Department of SolidMechanics at the Technical University of Denmark [1] has been modified for topology

design and analysis based on structural considerations. Since a key strength of LCAs istheir performance with regard to multifunctional criteria (i.e., structural and thermal), this

is not sufficient. The intent of this report is to provide an overview of the implementation

of thermal Finite Element Analysis in MATLAB

, as required for the expedition ofmulti-objective topology design. The example considered here is an air-cooled heat sink

for a computer chip, as introduced in Section 5.

2. Objectives

The objectives addressed in this project are as follows:

1) Develop thermal analysis for LCAs that interfaces with existing structural

analysis routines in MATLAB

where topologies are modeled through the use oftwo node frame elements with four degrees of freedom per node (i.e., 2displacements, 1 rotational, and 1 temperature).

2) Develop a Finite Element for modeling convection that is compatible with the

frame elements currently employed for structural optimization.

3) Consider laminar flow for modeling convectively cooled heat sinks andpotentially extend to turbulent flow for modeling combustor liners.

4) Compare and contrast results obtained using the developed MATLAB

algorithms with results obtained using ANSYS and via hand calculations

Deliverables include:

MATLAB Thermal Analysis Code Analysis of sample heat sink designs using the developed code

Validation of MATLAB analysis code using ANSYS

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3. Background Information

3.1 Topology Design

Topology design involves simultaneously adjusting both the external shape and the

number and shape of internal boundaries for a given 2D or 3D domain and associatedboundary conditions and design objectives [2,3]. Using topological design techniques,

vastly different topologies can be obtained from an arbitrary, initial domain. By

adjusting the topology of a structure, important properties like compliance, stiffness,strength, eigenfrequencies, convective coefficients, and other properties sensitive to

material arrangement can be tailored. It is possible to distribute material strategically,

resulting in lightweight structures with desirable properties. Increasingly, manufacturingprocesses, like additive fabrication and processing of cellular materials, are emerging that

facilitate the fabrication of structures with nearly arbitrary topologies. Thus, topological

design is a timely topic.Topology Design, as implemented here, is defined as the sum total of methods used to

address the question: How can material be distributed efficiently in a given design regionto tailor properties that are sensitive to material distribution (e.g., compliance, stiffness,

strength, convection, etc.)? It is important to note that nothing is known about structureor shape a priori in topology optimization. In fact, it is during the course of topology

optimization that the shape and number of discontinuities (i.e., voids) are determined. A

typical topology design approach, as proposed by Carolyn Conner Seepersad in her PhDproposal involves the following steps:

Step1 - Establish design requirements, objectives, and domain. This step was completed

in the previous discussion.Step 2 - Divide domain into finite elements. In Figure 1, the design domain (in this case

a simply supported cantilever beam with a load applied at one end) is divided into 2-D

planar finite elements.

Step 3 - Assign density variable to each finite element (i).

Step 4 - Modify density variables according to solution (optimization) algorithm. Smalldensity values for an element imply that the element is empty (i.e., part of a hole). Large

density values imply solid material.

Step 5 - Calculate effective properties of structure.A. Select penalization power, p>3. The penalization power penalizes

intermediate densities and encourages convergence to regions of solid (full

density) and void (minimum density).B. Calculate effective properties in each element. For example, a stiffness

matrix (K) for an element becomes: Ki = ip

KsolidC. Calculate effective properties for the structure.

Step 6 - Return to Step 4 until convergence is achieved.

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The computational model for topology design used in this investigation stems from a

99 line MATLAB

code for compliance minimization of statically loaded structures,developed by Ole Sigmund from the Department of Solid Mechanics at the Technical

University of Denmark [1]. The code was intended for engineering education and

contains both a mesh independency filter and a finite element code. It is based on a

number of simplifying assumptions aimed at reducing code complexity. For example, thedesign domain is modeled as a rectangle and is discretized using square finite elements,

as indicated in Figure 1. Element and node numbering proceeds on a column-by-column

basis, starting in the upper left corner. The aspect ratio of the structure to be optimized isdetermined by the number of horizontal (nelx) and vertical (nely) elements as specified

by the user.

The chosen implementation of topology optimization within this algorithm is basedon the power law approach or SIMP approach (Solid Isotropic Material with

Penalization), where properties are assumed constant within each element and design

variables are the element relative densities. The objective here is to minimizecompliance. The basic structure of a topology optimization problem formulated as such

is the following:

Finite

Element

Mesh

Finite

Element

Mesh

1 2

m xn

1 2

m xnFigure 1 Dividing the Cantilever Beam Design Domain into

Finite Elements (Courtesy of CCS)

Minimize:1

( ) ( )U KU u k uN

T p

e e o

e

c x x=

= = T e

Subject to:( )

o

V xf

V=

KU = F

min0 1x x<

U and F are the global displacement and force vectors, respectively, and K is the

global stiffness matrix; ue and ke correspond to the element; xe is the vector of designvariables bounded with respect to relative density by xmin (non-zero to avoid singularity)

and xmax; N = m x n is the number of elements within the discretized design space; p isthe penalization power; V(x) is the material volume; Vo is the design domain volume; fis

the prescribed volume fraction.

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Mesh-Independency Filtering

The purpose in including a mesh-independency filter within the topologyoptimization algorithm is to avoid disjointed solid regions (i.e., chess/checker board

patterns). From a practical consideration, applying such a filter improves the likelihood

of manufacturability and can be used to control the minimum feature size. The parameter

used to controll the degree of filtering is rmin. In filtering the mesh fordisjointed/unconnected elements an area equal to a square with side lengths of two times

rmin around each element is searched. Setting minr 1< essentially deactivates the filter.

Finite Element Code

The finite element code implemented within this MATLAB routine takes advantage

of the sparse matrix capability within MATLAB

and as such is quite efficient, avoidingwhat would otherwise be a significant amount of computational overhead associated with

required matrix inversion calls. Elements are modeled as four node quadrilateralelements having a total of two degrees of freedom (horizontal and vertical displacement)

per node (i.e., 8 dof per element). Supports are modeled through the elimination of fixed

degrees of freedom from the linear equations. Elemental stiffness matrices are calculatedbased on an analytic 8 x 8 matrix for a square bi-linear 4-node element. The user has the

ability to adjust both the Youngs Modulus E and the Poissons Ratio .Note: For more information please refer to [1].

In order to make this code more useful for multi-objective topology optimization a

number of modifications were made, the most important among them being (1) theimplementation of frame elements with four degrees of freedom per node (i.e., 2

displacements, 1 rotational, and 1 temperature) in lieu of the more limited square cell

finite elements originally included and (2) the incorporation of a more powerful andconsistent optimization scheme. While the first was motivated by the need for increased

flexibility in modeling topology, the second requires a more detailed explanation.Topology Optimization for the design of Linear Cellular Alloys (LCAs) is a highly

non-linear optimization problem. It involves either multiple nonlinear objectives or

multiple non-linear constraints, and very large numbers of continuous design variables.

The role of optimization algorithms in topology design is to aid in converging on a

particular solution, or rather, driving the process of topology optimization towardsachieving a structure that closely represents the posed requirements without violating any

constraints. As stated, topology design is characterized by high degrees of non-linearity

that involve either multiple nonlinear objectives or multiple non-linear constraints, andvery large numbers of continuous design variables. A first step towards effective

structural synthesis is the determination of stable optimization routines, capable of

effectively handling large numbers of variables and yielding consistent results. Previousresearch was focused on comparing a number of different gradient-based algorithms with

regard to performance and stability, among them the Method of Moving Asymptotes

(MMA) [4,5], Sequential Quadratic Programming (SQP) [6,7], and Sequential LinearProgramming (SLP) [8,9]. The clear indication of the study was that MMA had the most

consistent performance among the three and the most favorable convergence behavior.

Developed by Krister Svanberg, this non-linear programming method focuses on an

iterative process in which strictly convex approximations in the form of sub problems are

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generated and solved [10,11]. The generation of these approximations in the form of sub

problems is accomplished through the adjustment of asymptotes in a manner that canboth stabilize and speed up the convergence of the more general process. MMA was

developed explicitly for use in structural optimization and is based on a special type of

convex approximation. Consequently, MMA is capable of handling (1) general

requirements of non-linear programming problems (e.g., the ability to handle all types ofconstraints, provided only that derivatives of constraint functions can be calculated either

analytically or numerically, etc.), as well as (2) requirements that are particular to

structural optimization (e.g., costly function evaluations and large numbers of variables).Finally, MMA is designed to remain stable, capable of generating a sequence of

improved feasible, or quasi-feasible solutions, and be implementable with considerable

ease. The general form of the non-linear programming problem addressed in an MMAproblem is

Minimize:2

1

( ) ( 0.5 )n

o o i i i

i

if x a z c y d y=

+ + +

Subject to: ( ) 0i i if x a z y , i m1,...,= min max

j j jx x x , 1,...,j n=

, y , i m0z 0i 1,...,=

Thus, since MMA was developed explicitly for the type of problem underconsideration it is ideally suited for the dynamic and highly computationally intensive

nature of topology optimization. Gradient information is required. Although numerical

approximations are often easier to obtain than analytical expressions they are quite costlyand can significantly reduce algorithmic performance, especially considering the large

number of variables that lies at the core of topology optimization. In this particular

application analytical expressions for required gradients were implemented in order tominimize the adverse impact of numerical approximations on computation time. Another

benefit of MMA is the inherent ability to influence convergence. Moving the asymptotes

away from the current iteration point, for example, speeds up convergence for monotone

or slow progression. Moving the asymptotes closer to the current iteration point, on theother hand, has the effect of stabilizing the process in the case of unnecessary

oscillations. This is a key advantage in a process with the objective of distinguishing asclearly as possible between solid and void regions (i.e., treating a variable that is modeled

continuously, despite being Boolean by nature) and stands in marked contrast to the

tendency of SLP, for example, to push the values of parameters to either extremum (i.e.,due to linearization of objective function and constraints.

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3.2 Linear Cellular Alloys

Linear Cellular Alloys (see Figure 2) are metallic cellular materials with a constant

cross section, fabricated through a process developed by the Lightweight Structures

Group at Georgia Tech. The process combines extrusion of ceramic slurry, composed of

metal oxides and water through a die, allowing for the achievement of quasi-arbitrarytwo-dimensional cellular topologies. Extrusion of the ceramic is followed by exposure to

thermal and chemical treatments that cure the composites. The inherent advantage in

producing materials using this process is the ability to tailor properties of the resultingstructure such as the effective moduli of elasticity and conductivity by altering the

topologies of the cells. Structures may be composed of either periodically repeating unit

cells or functionally graded, non-uniform cells of various topologies. LCAs areparticularly suitable for multifunctional applications that require not only structural

performance but also lightweight thermal or energy absorption capabilities

Accordingly, LCAs have potential for use in applications such as actively cooledsupersonic aircraft skins or engine combustor liners [12].

Figure 2 -- Square-Cell Linear Cellular Alloy [13]

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4. Purpose of this Investigation

Typically, topology design and optimization involve the general steps outlined in

Figure 3. As indicated, every change in geometry requires renewed analysis to evaluate

system performance with regard to desired objectives. Considering that such changes in

geometry also require the recalculation of temperature dependent (i.e., inlet, outlet, andbulk) properties such as fluid viscosity , convective coefficient h, Prandtl NumberPr,Reynolds NumberRe, Hydraulic DiameterDh, etc. and the reevaluation of potentially

huge stiffness matrices computational expense is considerable. This is especially true

when a number of different software applications are involved.

The overarching objective in this project is to investigate the potential of conducting

optimization alongside analysis exclusively within MATLAB, in order to avoid the

computational overhead associated with relying on distributed computing. Relying oniSIGHT, for example, results in significant delays due to reading and writing numbers

from/to input/output files, that are amplified further by the large number of design

variables required for topology optimization). Porting FEA analysis required for the

topology optimization routine from ANSYS, as used in conjunction with iSIGHT, toMATLAB

native code eliminates the required use of iSIGHT to wrap the

optimization procedure. Having self-standing code has the additional advantage of

reducing the time required for completing optimization runs by an order of magnitude

or more. This task is accomplished through an adaptation of the 99 line topologyoptimization algorithm, developed by Ole Sigmund and extended by Carolyn Conner

Seepersad, as described in Section 3.1.

Discretize Domain

and ID Appropriate

Variables,Objectives, and

B.C.s.

Assign Element

Material Properties

Build and Solve

Finite Element

Model for SpecificBoundary

Conditions

Calculate Values

of Objectives

Based on FEResults

Update Element

Material Properties

to Improve

Objective Function

Figure 3 General Steps for Topology Design and Optimization (Courtesy of CCS)

This investigation is concerned with developing, validating, and smoothly integrating

thermal analysis capabilities into the modified MATLAB

code in order to make quickexploration of multi-objective design spaces possible. With this in mind, a motivating

example, necessitating the consideration of multiple objectives is introduced in the next

section.

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5. Case Study Convectively Cooled Heat Sink for a Computer Chip

The example LCA application considered in this report is that of a convectively

cooled heat sink for a computer chip. A sample schematic of the structure is given in

Figure 4. The general requirements for a CPU heat sink are that it 1) remove enough heat

from the chip so as to ensure steady state operation and 2) withstand the relatively highcompressive forces exerted by clamps used to attached the heat sink to the chip as tightly

as possible, see Figure 5. With this in mind, it is important to note that constant

temperature at the chip interface is assumed in this investigation. Although, it may seemmore intuitive to model constant heat flux instead, the idea is to design a heat sink that is

capable of removing enough heat to keep the chip below 1) its maximum operating

temperature or 2) its melting temperature (in the case of potential over-clocking)

Figure 4 Compact, Forced Convection Heat

Exchanger with Graded Rectangular

LCAs [12]

W

H

D

w1 w2 w3 wNh. . .

h2

h1

hNv

th

tv

Heat

Source

Tsource

Air Flow, T in

.

.

.

x

y

z

Figure 5 Steps Involved in CPU/Heat

Sink Assembly

The basic structure of the problem being addressed in this report (see Figure 4) is thefollowing :

Given:

1) Overall dimensions of the structure (width (W) x depth (D) x height (H)).

2) A potential LCA heat sink design, as defined by the number of cells in the

horizontal and vertical directions (i.e., Nh and Nv), respective Cell Heights (Ch)and Cell Widths (Cw), Wall Thicknesses in the x-direction (Wt-x), and Wall

Thicknesses in the y-direction (Wt-y).

3) A heat source (i.e., the CPU), maintained constant temperature Ts, tightly clamped

to the top face of the heat sink.

4) Insulated boundaries (i.e., zero heat flux) at the right, bottom, and right side faces.

5) Air flow rate and initial bulk temperature (Tbulk), Tin.

6) Conditions are representative of fully developed laminar flow.

Determine:

1) The steady state temperature distribution in the structure

2) The overall rate of steady state heat transferQ total

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The values of design variables, used in this investigation are as follows:

W = 25 mm

D = 75 mm

H = 25 mm

Nh = variesNv = varies

Ch = varies

Cv = variesWt-x = varies

Wt-y = varies

350sT K=

300in

T K=

kgvaries

s=

W363

m-Ksk =

The boundary conditions and coordinate system used for the thermal finite element

analysis are provideD in Figure 6.

Z

Figure 6 - FEA Boundary Conditions

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6. Modeling Details

6.1. Simplifying Assumptions

In implementing thermal analysis for multi-objective topology design, a number of

simplifying assumptions were made. Some were a direct consequence of the requirement

to interface with the existing MATLAB

code. Others resulted from modelingconsiderations. Each is explained in detail as follows:

Type of Finite Element UsedIn order to add modeling/analysis capability of the two-dimensional cross section

(determined as a result of topology design with regard to structural considerations) in thethird dimension, a fundamental requirement was the reliance on previously determined

information. Since structure in topology design is determined through the successive

determination of solid and void areas, the most important piece of information is thestatus (i.e., activity/inactivity) of nodes.

The nodes in question comprise the endpoints of frame elements that determine the

constant cross-section of the LCAs. Each node has four degrees of freedom (i.e., 2displacements, 1 rotational, and 1 temperature). The information derived from the

structural considerations is a two dimensional grid of active nodes, as determined by the

active frame elements. The challenge now becomes effectively adding a third dimension,

used to analyze the convective properties of the structure. Since the cross section ismodeled in terms of one-dimensional elements, this translates to modeling thermal

characteristics in the third dimension using two-dimensional elements. Although a three

dimensional element might be better suited to model the thermal behavior of a three-dimensional structure, this would require the specification of two-dimensional elements

for modeling the cross section. This is indicated in Figure 11. In essence, the element

developed in Section 6.2 is a plate element, capturing two-dimensional geometry and

variation of DOFs and having constant behavior in the third.

Modeling ConsiderationsModeling three-dimensional geometry in terms of a combination of one-dimensional

and two-dimensional elements has several consequences, the most significant of which

can be explained by elaborating on the nature of plate elements. As plate elements aretwo-dimensional analogues to beams they do not capture all the effects associated with

the three dimensional geometries they are meant to model. In our case, conduction

throughout the thickness of the plate element was not modeled explicitly. In order todifferentiate between elements of different thickness the thermal conduction coefficient

on an area basis was multiplied by the thickness of the structure.

In order to model the three-dimensional structures effectively, several otherassumptions were made.

Negligible pressure drop at the inlet

Negligible heat conduction along the direction of extrusion Constant surface temperature, equal to average duct wall temperature (because

temperature difference of adjacent duct walls is very small)

Fluid temperature difference between inlet and outlet is very small

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6.2 Finite Element Model for Heat Transfer Analysis

In this section, the linear (four node) rectangular element, employed to model the

thermal characteristics of LCAs along the direction of extrusion (in MATLAB), is

presented in light of the simplifying assumptions discussed in Section 7.1. An overview

of the elements derivation is provided alongside governing equations, stiffness matrices,convective coefficients, etc. The reader is referred to Appendix A for mathematical

details.

Thermal analysis as implemented in MATLAB consists of two parts. The first isaimed at modeling the thermal behavior of the structural material, the second at modeling

the changing properties of the fluid flowing through structural voids. The finite element

approach employed in this investigation (and mapped out in Figure 7) is somewhatunique in so far that sections of the structure, rather than the structure as a whole, are

analyzed one at a time. This is akin to the way in which finite difference codes work.

The inherent advantage is that there is no need to calculate immense stiffness matrices,reducing computational complexity. Additionally, this procedure facilitates the

determination of changing properties of the convective fluid and the incorporation ofthese to the thermal analysis of the structure. In essence, this is what a coupled

ANSYS/FLOTRAN analysis accomplishes.

Fluid

Rectangular

DuctFluid

Rectangular

Duct

W

H

D

w 1 w 2 w 3 w Nh. . .

h2

h1

hNv

th

tv

Heat

Source

T source

A ir Fl ow , T in

.

.

.

x

y

z

Plane element

In FEA

Fluid Cell

Element

Plane element

In FEA

Fluid Cell

Element

Figure 7 - Modeling Approach for Thermal Analysis of LCA Performance

With this in mind, a brief overview of the linear (four node) rectangular and fluidelements, developed for the purposes of this investigation, follows.

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1 2

34

a

b

tt

Linear (Four Node) Rectangular Finite

Element

Linear (Four Node) Rectangular Finite

Element

The finite element used for LCA thermal

analysis is shown in Figure 8. As indicated,

geometrically, the element is defined by length

(a) and width (b). Each of the four nodes (1

The finite element used for LCA thermal

analysis is shown in Figure 8. As indicated,

geometrically, the element is defined by length

(a) and width (b). Each of the four nodes (1, 2,3, and 4) has one degree of freedom

temperature. The element also takes into

consideration internal heat generation asindicated by the black arrows, as well as

convection effects on either face. This is

indicated by the light blue arrows pointing awayfrom either face in Figure 8. Figure 8 - Linear (Four Node)

Rectangular Thermal ElemenThe governing equation for steady state heat

transfer in plane systems is given by

( ), x Tf x y k ky Tx y y

= [1]

where T is the temperature (inoK), kx and ky are the thermal conductivities of the

material (in W m-1

oK

-1) along the xand y directions respectively, and f is the internal

heat generation per unit volume (in W m-3

). For convective boundary conditions, the

natural boundary conditions are a balance of energy transfer across the boundary due to

conduction and/or convection (i.e., Newtons Law of Cooling) [14]:

( ) x x y yT T

k n k n T T x x

+ + nq= [2]

where b is the convective conductance (or the convective heat transfer coefficient) (in

W m-2

oK

-1), T is the ambient temperature of the (ambient) surrounding fluid, and q is

the specified heat flow. The first two terms account for heat transfer by conduction, thethird for heat transfer by convection; the term on the right hand side accounts for the

specified heat flux, if any [14].

n

The system illustrated in Figure 4 is thus described by the following relationship:

{ } { } { }e e e e

K H T F P + = + e

[3]

whereKe

is the stiffness of the element, He

describes the influence the convection oneach node, T

eis the temperature of the element, F

eis the internal heat generation of the

element, and Pe

is used to define the convection on the top and bottom faces of the

element.

The interpolations functions for a linear (four node) rectangular element used here are:

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1 1 1y

a b

=

[4]

2 1y

a b

=

[5]

3

x y

a b

=

[6]

4 1y

a b

=

y

[7]

The elemental stiffness matrix (for an isotropic material with xk k= ) is defined by:

0 0

b aj ji i

ijK k dxdyx x y y

= +

dy

dy

[8]

The convective coefficients on an elemental basis are determined through:

( )0 0

b a

ij i j i T BH T dxdy = + [9]

Internal heat generation is accounted for by:

[ ]0 0

b a

nF wq dx= [10]

Convection on the top (T) and bottom (B) surfaces is given by

[ ]0 0 0 0

b a b a

T T B BP w T dxdy w T dx = + [11]

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Using the interpolation functions for a linear (four node) rectangular element, the

stiffness matrix (for an isotropic material with xk ky= ) may be determined to be:

2 22

2 22

[ ]6 2 2

2

2 22

a b a b a b a b

b a b a b a b a

a b a b a b a b

b a b a b a b akK

a b a b a b a b

b a b a b a b a

a b a b a b a b

b a b a b a b a

+ +

+ +

=

+ +

+ +

[12]

Similarly, the matrix of convective coefficients computed to be:

( )

1 1 11

2 4 2

1 11

2 2[ ]

1 1 191

4 2 2

1 1 11

2 4 2

T BabH

+ =

1

4 [13]

The following matrix gives heat generation on a nodal basis:

1

1[ ]

14

1

nabqF

=

[14]

Finally, the effects of convective boundary conditions are accounted for through:

[ ] ( )

1

1

14

1

T T B B

abP T T

= +

[15]

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Fluid Element

The fluid element, detailed in Figure 9, is used to calculate the conductivity of thefluid, viscosity, Reynolds Number, Prandtl Number, Nusselt Number, convective

coefficients, and hydraulic diameter of the fluid passed through the structural voids, using

the inlet temperature and mass flow rate. The convective coefficient h is stored in the

fluid cell slot and used as both T and B in thePmatrix. The following equations areused to calculate the convective coefficients.

(1.329 ) / 2.8896.163/(1 )ratioNu e = + (Laminar Flow) [16]

where ratio is the aspect ratio of the duct.

0.5 2/ 3

( /8) Re Pr

1.07 12.7( / 8) (Pr 1)

frNu

fr

=

+ (Turbulent Flow) [17]

where fr is the friction factor.

h

Nu kh

D

= [18]

Nodal temperatures for each of the four elements making up a rectangular duct arethen calculated using the FEA formulation of the Linear (Four Node) Rectangular Finite

Element. An average of the four nodal temperatures pertaining to each element is taken

and assigned as the surface temperature. This information, in turn, is then used to

calculate the exit temperature Tout and the total heat transfer rate by the fluid within each

cell Q , respectively by

/

( ) phA mC

out surface surface inT T T T e= [19]

(p out inQ mC T T = ) [20]

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Finally, the calculated Tout is assigned as Tin of the next layer and the process

repeated.

Calculate

Viscosity,h, k,

Pr, Re, h, Dh

GivenTin, mass flow rate

Get

Tout? Q?

Calculate

Viscosity,h, k,

Pr, Re, h, Dh

GivenTin, mass flow rate

Get

Tout? Q?

CalculatingFilm Coefficient and

Bulk Temperature

1. Given Tin, mass flow rate

2. Calculate

k, viscosity, Re, Pr, Nu, h, Dh

(Hydraulic Diameter)

3. Solve for node temperatures using

FEA formulation (previous slide)

Plug h and Tin into FEA matrix

4. Assign average node temperature as

surface temperature and calculate

Tout and Heat transfer rate by fluidwithin each cell.

5. Assign Tout as Tin of next layer

Figure 9 - Overview of Fluid Element Calculations

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6.3 Modeling the LCAs in MATLAB

The routine used to conduct thermal analysis (i.e., model

and solve) for LCA structures in MATLAB

is shown in

Figure 10. The thermal module in MATLAB

starts with

specifying input parameters, such as node numbers in the x, y,and z directions, inlet temperature and mass flow rate of the

fluid, and thermal conductivity of the structural material. Theprogram then forms nodes and elements for the LCA structure

based on these input parameters. These nodes and elements

are first formed in the x-y plane and extruded along the z-axis

(i.e., the direction of extrusion). This is done in order toensure consistency of structural and thermal analyses. It is

thus possible to model any type of cross sectional geometry as

long as it is composed of rectangular (shown in Figure 11) ortriangular (not shown) cells.

The node numbering scheme, implemented within theMATLAB code, is illustrated at the hand of a simple 2 by 2

LCA design in Figure 11. The thermal model is fully

compatible with the structural model. In fact, it is formed viaextrusion along the z-axis, as indicated in the previous

paragraph. Although, this example shows only a single slice

of the structure along the z-direction, an arbitrary number may be used, as required for a

particular analysis. As indicated, the thermal element is a rectangular plane while thestructural element is a one-dimensional line. Nevertheless, thermal elements are

numbered in the same pattern as nodes corresponding to the structural elements, so as not

to duplicate global node and element numbers in the z direction. More specifically, node

Specify inputparameters

Formulate Nodesand Elements

Figure 11 An example of node numbering scheme in structural model

and equivalent thermal model

Structural Model

18

93

2

1

6

4

5

7

8

17

16

15

14

13

Thermal Model

12

11

10

zx

y 9

8

7

6

5

4

3

2

1

Calculate fluid exittemperature and Qin each fluid cell

No

Yes

Display Results

Sum the total Q

LastLayer?

Solve for nodalTemperatures

Formulate GlobalStiffness, H, P, andF matrices for eachla er

Calculate FluidProperties for eachlayer

Define BoundaryCondition

Figure 10 Flow chartof Matlab FEA Routine

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numbering for structural elements follows the pattern indicated in the left half of Figure

11. The equivalent thermal model then builds on the numbering scheme of the structuralmodel as shown in the right half of the same figure. Consequently, material properties

such as conductivity, thickness, etc., assigned to the structural model, can be directly

inherited by the thermal model. This is especially important since the thickness of any

given element is likely to change based on the results of the preceding optimization loopand is thus consistently applied to both structural and thermal models.

After formulating nodes and elements, boundary conditions are defined by fixing the

attributes of the corresponding nodes. The temperature attribute, for example has twoproperties one being the temperature and the other being its status (fixed or not).

Temperatures that are imposed as a boundary conditions are thus marked as fixed. Much

the same is true for heat flux, the other possible boundary condition for thermal analysisas implemented here.

Once the boundary conditions are imposed the program begins the process of solving

the structure. As mentioned previously, this is done on a layer-by-layer basis (along thedirection of extrusion). For each layer, the property calculation module first calculates

fluid (air) properties (e.g., film coefficient, Re, Pr, k, kinematic viscosity, Nu (for laminaror turbulent flow), etc.), given inputs of air inlet temperature and mass flow rate. Globalstiffness, H, P, and F matrices are then formulated for each layer. Since each of these

matrices is just for the layer in question, sizes are considerably smaller than global

stiffness matrices corresponding to the entire structure. Since the solver makes use of the

sparse matrix capability of MATLAB (which is significantly faster than ordinary matrixoperations involving matrices with large numbers of zeros), computational efficiency is

aided further. The output of the solver is the numerical value of temperatures for all

nodes corresponding to the given boundary condition.Calculated nodal temperatures are then used to calculate other fluid properties,

specifically the outlet temperature and heat transfer rate. The outlet temperature of thefluid is stored alongside other fluid cell properties and used to assign the outlet

temperature of the current cell as the inlet temperature of the interfacing cell within the

next layer. Heat transfer rates for each fluid cell are also stored as fluid cell properties, tobe summed over all slices at the conclusion of the analysis routine to determine the total

heat transfer rate by the fluid (corresponding to the total heat drawn from the CPU in the

example considered here). At the conclusion of the thermal analysis, the results (i.e.,node temperatures, graphically represented using color swatches, a depiction of cross

sectional LCA geometry, maximum and minimum temperature readings, and total heat

transfer rate by the fluid) are displayed in a window.

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6.4 Modeling the LCAs in ANSYS

The main purpose of analyzing the sample LCA designs discussed in Section 7 using

ANSYS was to verify and validate the performance of the MATLAB code, developed inorder to address the needs enumerated in Section 2. Unlike the MATLAB model, no

restrictions were placed on the geometry/nodal configurations of the elements used here.In order to obtain the most accurate results possible, both a two-dimensional and three-dimensional analysis were conducted. The elements, chosen to model the LCA

geometries in ANSYS were the following:

PLANE 55 2D 4-Node Thermal Solid SOLID 70 3D 8-Node Thermal Solid

These elements were chosen for their capability of modeling both the thermalproperties we were concerned with in this investigation as well as structural properties

potentially considered for any further validation of our robust topology design efforts in

the future. Since the geometry of the LCAs is fairly simple and structural and thermalgradients do not change drastically for the applications taken into consideration here, no

higher order elements (e.g., PLANE 77 2D 8-Node Thermal Solid, SOLID 90 3D 20-

Node Thermal Solid) were considered.

LCA geometries were initially modeled in three dimensions. After establishing thatthe use of average temperature and film coefficient values produced good quality results

for thermal analysis, however, ANSYS models were simplified to two dimensions.

Convergence of results (i.e., temperature distributions) was then established throughmesh refinement for the two dimensional models. It is quite likely that the upper limit

(128,000 elements) of ANSYS computational capabilities would have been breeched, had

the use of average fluid properties not proven accurate. Although finer granularities with

respect to the two-dimensional cross section would not have been required, as indicatedby the results shown in Figure 16, this effect is likely to have been offset by the increasein granularity required along the direction of extrusion, mandated by gradients and

elemental aspect ratio considerations. This line of reasoning is supported by the

MATLAB results given in Section 8.1, indicating that increasing the number of elementsused to model a given cross section is not quite as important as increasing the number of

slices used to analyze a structure.

In both the two and three-dimensional cases, ANSYS free mesh features consistentlyproduced large numbers of bad elements with poor aspect ratios. This is due to the

irregular geometry that results from the extremely thin walls, characteristic of LCAs.

Meshes were thus created manually by dividing the cross sectional geometry into myriad

rectangular elements. Having provided a brief overview of modeling considerations,results are discussed in Section 8.

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7. Results, Discussion, and Validation/Verification

7.1 The 2 X 12 LCA Design (MATLAB vs. ANSYS)

MATLAB FEA Results

Figure 12 Input Specifications, Boundary Conditions, and LCA Cross-Sectional Geometry for the 2 x 12

LCA Design

Geometry of 2 x12 LCAGiven Specification

Material: Copper (k = 363 W/m K)

Fluid: Air Width : 0.025 m (even width for each cell) Height: 0.025 m (0.01365, 0.00701 for each

cell)

Wall Thickness: 0.001 m Depth: 0.075 m Inlet Air Temperature = 273.15 K Inlet Air Velocity = 2.97 m/s

Boundary Condition

Heat Source Temperature at Top = 373.15 K Complete Insulation at side and bottom walls

MATLAB FEA results are shown in Figure 13. The numbers of Layers in the z-

direction were varied from 20 to 90 layers in increments of 10. As shown in the figure,

maximum temperature, minimum temperature, total heat transfer rate by the fluid, andaverage film coefficient (since film coefficients are calculated for each fluid cell) are

provided as alongside visual color swatches representing the temperature distribution

throughout the structure. The left side of each figure is the inlet and the right side theoutlet. The temperature of the top face of the results shown was fixed at the given

temperature of 373.15 K.

Figure 13 MATLAB Results for the 2 x 12 LCA Design

9080

706050

403020

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ANSYS Results

ANSYS FEA results are shown in Figure 14. After establishing that the use ofaverage fluid temperature and film coefficient values gave acceptable results, the LCA

structures were modeled in two dimensions using the PLANE 55 (4-Node Thermal Solid)

element. Mesh refinement was conducted by increasing the number of elements used to

model geometrical features of the LCA cross section. The total number of elementsvaried from 179 at the lowest resolution to 10425 at the highest resolution.

As shown in the figure, maximum and minimum temperatures are provided alongside

a visual presentation of the temperature distribution throughout the structure. The left,bottom, and right sides of each cross section were insulated, while the top face was held

at a constant temperature of 373.15 K.

Figure 14 - ANSYS Results for the 2 x 12 LCA

1042566723753

1668417179

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Comparison Minimum Temperature of MATLAB FEA with ANSYS Result

As shown in Figures 15 and 16, the results obtained for the minimum temperature

within the 2 x 12 LCA design in both in MATLAB and ANSYS are very close.

MATLAB FEA results converged to 360.18 K while ANSYS results converges to

361.95K. The difference between the two results is attributable to differences in the filmcoefficient and bulk temperatures used for computations. Specifically, film coefficients

and air bulk temperatures were dynamically calculated and updated in MATLAB FEA

from slice to slice, capturing the variation of the fluid properties along the length of theLCA. In the ANSYS analysis, o the other hand, film coefficients and air bulk

temperatures were fixed at their average value as determined through the use of our

MATLAB code. For this reason, it is reasonable to assume that MATLAB FEA resultsare more accurate than ANSYS results.

M A T L A B M e s h C o n v e r g e n c e

3 5 9 . 9 5

3 6 0

3 6 0 . 0 5

3 6 0 . 1

3 6 0 . 1 5

3 6 0 . 2

2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0

N u m b e r o f L a y e r s ( L C A L e n g t h )

A N S Y S M e s h C o n v e r g e n c e

3 6 1 .8

3 6 1 .8 53 6 1 .9

3 6 1 .9 5

3 6 2

3 6 2 .0 5

3 6 2 .1

3 6 2 .1 5

3 6 2 .2

1 7 9 4 1 7 1 6 6 8 3 7 5 3 6 6 7 2 1 0 4 2 5

N u m b e r o f E l e m e n t s

Figure 16 - ANSYS 2D Mesh Convergence with Respect to the Minimum Temperature within the

Structure (2 x 12 LCA)

Figure 15 - MATLAB Mesh Convergence along Direction of Extrusion with Respect to theMinimum Temperature within the Structure (2 x 12 LCA)

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Comparisons the total heat transfer rate of FEA in MATLAB and Finite Difference

Analysis

In order to validate the results of the total heat transfer rate computed by MATLAB FEA,

these were compared to Finite Difference results for the same scenario (i.e., geometry,input specification, and boundary conditions) [12]. As shown in Figure 17, the

MATLAB FEA results converged to a value of 106.1 W. The total heat transfer rate

documented by Seepersad, et al., however, is 94.96 W. Although we are not certain, it islikely that this difference is due to a simplification of cell wall widths as having constant

thicknesses of 1 mm for walls in both the in x and y direction. This stands in marked

contrast to the thicknesses shown on the right-hand side of Figure 17 that were usedwithin the Finite Difference code. Overall comparison with Finite Difference code

results for the 8 x 8 structure, described in Section 8.1, which featured constant wall

thicknesses showed, close agreement. In fact, taking into consideration that theexperimental setup is likely to indicate higher heat transfer rates due to the inability to

reproduce perfectly insulated boundaries, it seems that the MATLAB FEA is in closeragreement with theoretical results.

MATLAB Mesh Convergence

105.7

105.8

105.9

106

106.1

106.2

20 30 40 50 60 70 80 90

Number of Layers (LCA Length)

T

otalHeatTransfer

(W)

Figure 17 MATLAB Mesh Convergence along Direction of Extrusion with Respect to the Total

Heat Transfer Rate (2 x 12 LCA)

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7.2 The 8 X 8 LCA Design (MATLAB vs. Finite Difference)

MATLAB Results

In order to validate the results of the MATLAB FEA code using experimental data,

the identical conditions (see Figure 18) cited for the experiments documented by

Seepersad, et al. [12] were simulated. Case study 1 consisted of 6 different setups withtop temperatures of 305, 312.5, 315, 319, 323, and 325 K and Reynolds number 1200

respectively. The second case study was conducted at Reynolds number 891 with top

temperatures of 305, 312.5, 315, 317.5, 319, 323, and 325 K respectively. A sampleresult of MATLAB FEA analysis for a top temperature as 327 K and air velocity 8.8513

m/sec, corresponding to Re=891, is given in Figure 19.

Given Specification

Material: Copper (k = 363 W/m K) Fluid: Air Width: 0.0138 m (even width for each cell) Height: 0.0138m (even height for each cell)

Depth: 0.0508 m Wall Thickness: 0.0012 m Inlet Air Temperature = 273.15 K Inlet Air Velocity 1 = 11.92 m/s (Re = 1200) Inlet Air Velocity 2 = 8.8513 m/s (Re = 891)

Boundary Condition

Heat Source Temperature at Top = 305~327K Complete Insulation at side and bottom walls

Figure 19 - MATLAB Results for the 8 x 8 LCA

Figure 18 Input Specifications, Boundary Conditions, and LCA Cross-Sectional Geometry for the 8 x 8

LCA Design

Geometry of 8 x 8 LCA

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Comparisons the total heat transfer rate of FEA in MATLAB with Experimental Data

As can clearly be seen in Figures 20 and 21, the total heat transfer rate calculated for

each Heater Temperature (i.e., the top surface temperature) tested, using the MATLAB

FEA code is very close to experimental data corresponding to both Reynolds number

1200 and 891. In fact, the total heat transfers between 312.5 K and 315 K are almostidentical for both FEA and the experiments. The difference of the total heat transfer rate

at higher top temperatures can be explained by the fact that it is virtually impossible to

implement completely insulated boundary conditions assumed during MATLAB FEAanalysis (at the left, right and bottom walls of structure) in physical setups.

Consequently, slightly higher heat transfer is to be expected, as shown in both cases.

Total Heat Transfer vs. Heater Temperature

(Re = 1200)

0

5

10

15

20

25

300 305 310 315 320 325 330

Heater Temperature (K)

TotalHeatTransferRa

Matlab Analysis Experimental Data

00

Total Heat Transfer vs Heater (Re=891)

0

5

10

15

20

25

310 315 320 325 330

Heater Tempe rature (K)

TotalHeatTransferRat

Matlab Analysis Experiment

00

Figure 20 Comparison of MATLAB FEA and Experimental Results for Total Heat Transfer at

RE = 12

Figure 20 Comparison of MATLAB FEA and Experimental Results for Total Heat Transfer at

RE = 12

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7.3 The 1 x 1 LCA Design (MATLAB vs. Hand Calculations)

Hand calculations for a simple 1 x 1

channel indicated that MATLAB code is in

agreement with theoretical results. This

should come as no surprise since theMATLAB algorithm essentially comprises

a computational instantiations of the

element derived in Appendix A. Any slightdifferences in the numerical results obtained

for larger structures can thus be attributed to

round off error. For detailed handcalculations, please refer to Appendix B.

8. Closure

Finite Element Thermal analysis hasthus been developed and successfully

deployed in MATLAB. Results obtainedusing these analysis modules have been

validated against (1) theoretical results (i.e.,

hand calculations for a simple rectangularduct), (2) numerical results using both finite

difference and finite element code, and (3)

experimental results. Assumptions,underlying this development were also

verified satisfactorily. Specifically, it was determined that (1) heat conduction along theLCA length at layer interface is indeed negligible, (2) constant surface temperature

(Internal Pipe Flow) holds, and (3) approximating the Inlet Temperature as Bulk

Temperature is valid. Furthermore, the ability to calculate Fluid Heat Transfer Rate(Internal Pipe Flow) was added to current analysis capabilities. This is a fundamental

step towards successful multi-objective topology design.

Figure 21 MATLAB FEA Results for a 1 x 1

Cell

Having developed thermal analysis capabilities in MATLAB, future work will consistof extending the applicability of fluid cell analysis to different geometries. In essence,

this effort will focus on developing an algorithm to (1) determine the extent of void

regions and (2) calculate the corresponding hydraulic diameters, thus allowing us to

determine the nature of the flow in question (i.e., laminar or turbulent), when given massflow rate and inlet temperature. Thermal analysis will also have to be integrated with

preexisting structural analysis modules in order to run the multi-objective optimization

routines, required for robust topology design.

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9. Challengesand Thoughts on Learning

Challenges

The main challenge in this investigation was to develop a thermal analysis module

that interfaced directly with existing structural analysis codes in MATLAB. Since

structural analysis is currently based on one-dimensional frame elements (chosen toincrease flexibility in modeling two-dimensional topologies) this limited our flexibility in

answering this need. Since elements are continuously removed in topology optimization,

a key requirement was to use existing node locations within a given two-dimensionalcross section and simply model the corresponding elements with regard to their thermal

characteristics along the direction of extrusion.

Other challenges included reducing the computational time required for conductingthermal analysis as much as possible. This aim was accomplished by slicing the structure

along the direction of extrusion and solving for the thermal performance of the structure

on a layer-by-layer basis. Our investigation showed that even the lowest number of slicesgave results that were extremely good. Additionally, we found that improvements in the

quality of results to be obtained by increasing the number of elements used to model agiven LCA cross section beyond the minimum were negligible. Slicing the structure anddetermining thermal characteristics in a sequential manner thus proved to greatly reduce

computational intensity, in part through global stiffness matrix reduction.

With respect to modeling the given LCA designs in ANSYS, the prime difficulty

turned out to be obtaining elements with good aspect ratios. Any of the automaticmeshing features within ANSYS proved incapable of dealing with the rather thin

geometries of the LCAs effectively. Consequently, each mesh had to be created

manually, a task that was complicated by the fact that edges of inner and outer geometricfeatures did not match up. Creating a suitable mesh thus turned out to involve quite a bit

of artistic finesse.Although we initially attempted to model the fluid in ANSYS using FLOTRAN,

further investigation showed that applying average values of bulk temperatures and

convective coefficients, as obtained through our MATLAB code gave us results that wereaccurate enough. Using such constant values, further made it possible to rely on two

dimensional ANSYS models. Although several geometries considered were also

modeled using symmetry considerations, this avenue of investigation was no pursuedonce the models were reduced to two dimensions. Considering that LCA cross sections

are not necessarily symmetric and may be functionally graded in more than a one

direction and using two-dimensional analysis (relying on average fluid properties)seemed more appropriate, at least for the case at hand.

Learning

As a whole, we feel that this project was an excellent vehicle for gaining a morefundamental understanding of the Finite Element Method that complemented the

theoretical foundation we were given in class rather well. Developing thermal analysis

code in MATLAB required us to draw on underlying theory in deriving our thermal andfluid elements as well as take into consideration computational aspects, fundamental to

the implementation of FEA. We developed a solid understanding of ANSYS in terms of

its execution as well as an appreciation of its rather complicated underpinnings.

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10. References

1. Sigmund, O., 2001, A 99 Line Topology Optimization Code Written in Matlab,

Structural Multidisciplinary Optimization, Vol. 21, pp. 120-127.

2. Eschenauer, H. A. and N. Olhoff, 2001, Topology Optimization of Continuum

Structures: A Review,Applied Mechanics Reviews, Vol. 54, No. 4, pp. 331-389.3. Rozvany, G. I. N., 2001, Aims, Scope, Methods, History, and Unified Terminology

of Computer-Aided Topology Optimization in Structural Mechanics, Structural and

Multidisciplinary Optimization, Vol. 21, pp. 90-108.4. Sigmund, O., 2001, Design of Multiphysics Actuators Using Topology Optimization

-- Part I: One-Material Structures, Computer Methods in Applied Mechanics and

Engineering, Vol. 190, No. 49-50, pp. 6577-6604.5. Neves, M. N., O. Sigmund and M. P. Bendsoe, 2002, Topology Optimization of

Periodic Microstructures with a Penalization of Highly Localized Buckling Modes,

International Journal for Numerical Methods in Engineering, Vol. 54, pp. 809-834.6. 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 AppliedMechanics and Engineering, Vol. 123, pp. 121-139.7. Saxena, A. and G. K. Ananthasuresh, 1999, "Towards the Design of Compliant

Continuum Topologies with Geometric Nonlinearity," ASME Advances in Design

Automation, Las Vegas, NV, ASME DETC99/DAC-8578.

8. Frecker, M. I., G. K. Ananthasuresh, S. Nishiwaki, N. Kikuchi and S. Kota, 1997,Topological Synthesis of Compliant Mechanisms Using Multi-Criteria

Optimization,ASME Journal of Mechanical Design, Vol. 119, No. 2, pp. 238-245.

9. Sigmund, O. and S. Torquato, 1997, Design of Materials with Extreme ThermalExpansion Using a Three-Phase Topology Optimization Method, Journal of the

Mechanics and Physics of Solids, Vol. 45, No. 6, pp. 1037-1067.10. Svanberg, K., 1987, The Method of Moving Asymptotes - A New Method for

Structural Optimization, International Journal for Numerical Methods in

Engineering, Vol. 24, pp. 359-373.11. Svanberg, K., 1999, MMA and Some Modeling Aspects, Optimization and

Systems Theory, KTH.

12. Seepersad, C. C., B. M. Dempsey, J. K. Allen, F. Mistree and D. L. McDowell, 2002,"Design of Multifunctional Honeycomb Materials," 9th AIAA/ISSMO Symposium on

Multidisciplinary Analysis and Optimization, Atlanta, GA, AIAA, Paper Number

AIAA-2002-5626.13. Hayes, A. M., A. Wang, B. M. Dempsey and D. L. McDowell, 2001, "Mechanics of

Linear Cellular Alloys," Proceedings of IMECE, International Mechanical

Engineering Congress and Exposition, New York, NY.

14. Reddy, J. N., 1993, An Introduction to the Finite Element Method, McGraw-Hill,Boston, MA.

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11. Appendices

11.1 Appendix A Derivation of Finite Element used for MATLAB

Heat Transfer

Analysis

11.2 Appendix B Hand Calculation for a 1 x 1 Cell

11.2 Appendix C Final Presentation Slides

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Appendx ADerivation of Finite Element used for MATLAB

LCA Heat Transfer Analysis

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Appendx BHand Calculation for a 1 x 1 Cell

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Appendx CSemester Project Presentation Slides

hc.fm.final.report

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