1. A Semi-Analytical Unit Cell Synthesis Method for
Design of Metamaterials with Targeted Nonlinear
Deformation Response
Shanyun Gao
Department of Mechanical Engineering
Clemson University
November, 2016
Advisor: Dr. Gang Li
Committee Members: Dr. Gang Li, Dr. Georges Fadel, Dr. Lonny Thompson
0

2. 1
Outline
 Introduction
 Metamaterial and design methods
 Analytical Functions of EFGs’ Large Deformation Behavior
 Non-dimensional load / deformation parameters
 Elemental Functional Geometry (EFG) large deformation behavior
 Analytical deformation solutions of EFGs combined in parallel / series
 Semi-Analytical Unit Cell Synthesis Method
 EFG selection and combination to construct metamaterial unit cell
 Unit cell size optimization
 Case study of designing unit cells to match a targeted nonlinear deformation curve
 Summary

3. 2
 Metamaterials(Sihvola, 2007)
 Macroscopic composite of periodic micro-structures
 Engineered to satisfy prescribed requirements
 Unit cell (UC) is the smallest repeatable structure
 Exotic Properties
Introduction
 Fluidic property(Guest, Prevost, 2007) (e.g. permeable materials in fluid transport)
 Thermal property(Sigmund, Torquato,1997) (e.g. extreme thermal expansion)
 Electromagnetic property(Smith, Pendry, Wiltshire, 2004) (e.g. negative index of refraction)
 Mechanical property(Milton, 1992) (e.g. negative Poisson’s ratio: Auxetic material)
Auxetic material and its application in footwear upper

4. 3
Introduction
 Metamaterial Design Methods
Topology Optimization
 Iterative process of distributing a certain amount of material within a design domain
 Seeks a material layout that can satisfy the objective function which subjects to constraints
 Problem setup includes an objective function and a set of constraints
 Metamaterial gain its exotic property from the unit cell structure
 Topology Optimization(Bendsoe, Sigmund, 2013) is the most predominant method
Initial design Optimal UC Optimal metamaterial

5. 4
Introduction
 Topology Optimization Methods
 Limitations of Topology Optimization
 Geometric nonlinearity has not been addressed
 Difficult to achieve a targeted nonlinear deformation behavior
 Unit cell aspect ratio is not considered as a design variable
 Homogenization Method(Bendsoe, Kikuchi, 1988)
 Evolutionary Structural Optimization Method (ESO) (Xie, Steven, 1997)
 Solid Isotropic Material with Penalization Method (SIMP) (Bendsoe, Sigmund, 1999)
 Level Set Method (LSM) (Osher, Sethian, 1988)
UC variables in square void
 Unit Cell Synthesis Method[Satterfield, Kulkarni]

6. 5
Introduction
 Unit Cell Synthesis Method
 Pros and Cons
 Matching predetermined nonlinear deformation response
 No quantitative solution of EFG’s large deformation behavior
 Relying heavily on nonlinear FEA, computational costly, Design workflow
 EFG: Elemental Functional Geometry
UC geometry
UC tessellation to metamaterial

7. 6
Objectives and Research map
 Objectives
 Develop a systematic approach to obtain analytical force-displacement functions of
EFGs subjected to large deformations
 Utilize the above force-displacement relations in the “Unit Cell Synthesis” method to
design metamaterial with prescribed nonlinear deformation response

8. 7
Outline
 Introduction
 Metamaterial design method overview
 Analytical Functions of EFGs’ Large Deformation Behavior
 Non-dimensional load / deformation parameters
 Elemental Functional Geometry (EFG) large deformation behavior
 Analytical deformation solutions of EFGs combined in parallel / series
 Semi-Analytical Unit Cell Synthesis Method
 EFG selection and combination to construct metamaterial unit cells
 Unit cell size optimization
 Case study of designing unit cells to match a targeted nonlinear deformation curve
 Summary

9. Analytical Functions of EFGs’ Large Deformation Behavior
 Cantilever beam (“Canti”)
𝐹 : force at free end
𝐿 : beam length
𝐸 : Young’s modulus
𝐼 : moment of inertia
𝑦 : vertical deformation
8
2
2
FL
EI
  Non-dimensional load parameter(Bisshopp, 1945)
Non-dimensional deformation parameter
 Solution is not trivial or handy, elliptical integral must be
evaluated numerically
 Beam is assumed to be thin and long, i.e. large aspect ratio
 Metamaterial unit cell design needs small aspect ratio structural
entities
y
L
 
0
0
0
2 0
sin sin
d 

 
 

0
0
0
1 sin
2 sin sin
d  

  
 
 Elliptical integral
2
(3 )
6
F x L x
y
EI
 
 deformation-force relation

10. Analytical Functions of EFGs’ Large Deformation Behavior
9
 Cantilever beam with small aspect ratios
 Solution is inaccurate when load gets larger
 Non-dimensional deformation and aspect
ratio has a negative correlation
L
h
  aspect ratio
2 2
2 2
3
6 6
( )
2
2
12
o
FL FL F L F
hEI Eh h Eh
E
      

26 nF
Eh
  
 Large deformation behavior with varying aspect ratio
 Reduce the aspect ratio term’s impact on
the overall large deformation characteristic
 Deduct a certain amount of exponent over
aspect ratio
 Determine the value of n

11. 10
Analytical Functions of EFGs’ Large Deformation Behavior
 Cantilever beam with small aspect ratios
26 nF
Eh
  
 
Optimize n
 
25 10
1 1
min : iji j
f   
  
2
0.1
2
FL
EI




Round-off
 Optimize parameter n such that curves from different aspect ratios converge
Error value with respect to exponent value

12. Analytical Functions of EFGs’ Large Deformation Behavior
 Cantilever beam with small aspect ratios
 Fit the deformation curve using polynomial fitting to gain analytical solution
3
0
5 30i
ii
a  
   
 Conclusion: cantilever beam’s large deformation response is obtained, and
expressed as a polynomial
 Note: solution is effective within a certain aspect ratio range
While the original load parameter works for larger aspect ratios
11

13. Analytical Functions of EFGs’ Large Deformation Behavior
 Circular Beam Pulled Up (“CirUp”)
12
2
o
FR
EI
 
2
0.03
FR
EI




y
R
 
R
h
 
base non-dimensional load parameter
optimal non-dimensional load parameter, n=0.03
non-dimensional deformation parameter
aspect ratio
base load parameter optimal load parameter analytical vs FEA

14. Analytical Functions of EFGs’ Large Deformation Behavior
 Circular Beam Pushed Down (“CirDown”)
13
2
o
FR
EI
 
2 0.03
FR
EI




y
R
 
R
h
 
base non-dimensional load parameter
optimal non-dimensional load parameter, n=-0.03
non-dimensional deformation parameter
aspect ratio
base load parameter optimal load parameter analytical vs FEA

15. Analytical Functions of EFGs’ Large Deformation Behavior
 Fixed-Fixed Beam (“FFB”)
14
2
FR
EI
 
2
1.5
FR
EI




y
R
 
L
h
 
base non-dimensional load parameter
optimal non-dimensional load parameter
non-dimensional deformation parameter
aspect ratio
base load parameter optimal load parameter analytical vs FEA

16. EFG Combination
15
 Spring Systems and Their Effective Stiffness / Compliance
springs in parallel springs in series
, 1 2
,
1 2
1
1 1
eff p
eff p
k k k
C
C C
 


,
1 2
, 1 2
1
1 1eff s
eff s
k
k k
C C C


 
effective stiffness
effective compliance
effective stiffness
effective compliance

17. EFG Combination
16
 EFGs in Parallel
 EFG’s Compliance Expression
( )f  ( )y L f c F  
( ) ( )
( )
dy F df c F
C F L
dF dF

   y
L
c



deformation
characteristic length
constant
Note: compliance C is a function of force, not a constant

18. EFG Combination
17
 EFGs in Parallel
 Given analytical expression of FFB and CirDown deformation behavior:
1
1 1eff
FFB CirDown
C
C C


How the total force is distributed
on each EFG is unknown, hence
compliances are unknown
?
( ) ( )FFB FFB FFB CirDown CirDown CirDowny g F y g F 
( ) ( )
total FFB CirDown
eff FFB FFB CirDown CirDown
F F F
y g F g F
 
 
 Solve equation system below to calculate the structure’s deformation

19. EFG Combination
18
 EFGs in Parallel
 FEA validation
 Effectiveness is verified
(a) (b)
(c) (d)
 Aspect ratios:
(a) CirDown: 20; FFB: 35
(b) CirDown: 25; FFB: 30
(c) CirDown: 30; FFB: 30
(d) CirDown: 40; FFB: 30

20. EFG Combination
19
 EFGs in Series
1 2 2sin costotaly y L y     
 Moment has a part to play in deformation behavior
 Angle of rotation is required to calculate total deformation
 Expression of deformation in terms of force and moment is required
 Analytical solution of angle of rotation is required
 Two cantilever beams in series
Beam loading condition
Total deformation decomposition Total deformation is decomposed into three parts
, 1 2eff sC C C 
( ) ( )
( )
dy F df c F
C F L
dF dF

  

21. EFG Combination
20
 EFGs in Series
2
0.1
2
F
FL
EI



 2
M
ML
EI
 
 Non-dimensional force and moment parameters
DeformationAngle of ration

22. EFG Combination
21
 EFGs in Series
 Polynomial fitting to obtain analytical solutions of deformation, and angle of rotation
2 2 3 2
00 10 01 20 11 02 30 21
2 3
12 03
( , )F M F M F F M M F F M
F M M
a a a a a a a a
a a
           
  
              
   
2 2 3 2
00 10 01 20 11 02 30 21
2 3
12 03
( , )F M F M F F M M F F M
F M M
b b b b b b b b
b b
           
  
              
   

23. EFG Combination
22
 EFGs in Series
1 2 2sin costotaly y L y     
 Total deformation
1 2 2 2 2 2( , ) sin ( , ) ( ,0) cos ( , )totaly y F FL L F FL y F F FL     
2
1 2 1
1 2 1 0.1
1
( , ) ( , )
2 2
FL FL L
y F FL L
EI EI



 

2
2
2 2 0.1
2
( ,0) ( ,0)
2
FL
y F L
EI


 

 Effectiveness is verified
FEA validation results

24. 23
Outline
 Introduction
 Metamaterial design method overview
 Analytical Functions of EFGs’ Large Deformation Behavior
 Non-dimensional load / deformation parameters
 Elemental Functional Geometry (EFG) large deformation behavior
 Analytical deformation solutions of EFGs combined in parallel / series
 Semi-Analytical Unit Cell Synthesis Method
 EFG selection and combination to construct metamaterial unit cells
 Unit cell size optimization
 Case study of designing unit cells to match a targeted nonlinear deformation curve
 Summary

25. 24
Semi-Analytical Unit Cell Synthesis Method
 EFG Selection and UC Synthesis
Elemental Structural Geometry (ESG)
 Support / rigid connection
 High stiffness

26. 25
Semi-Analytical Unit Cell Synthesis Method
 UC Synthesis and Size Optimization
 UC Size Parameters (for size optimization)
 UC Conceptual Tessellation

27. 26
Semi-Analytical Unit Cell Synthesis Method
 Case Study
 Design Objective: Targeted Deformation Response
100strain
H

 
2
1
min : : ( )
N t
i ii
strain error f  
  
 Strain definition
 Objective function
t
i i are target strain and effective
strain at load step i

28. 27
Semi-Analytical Unit Cell Synthesis Method
 Size Optimization
 UC Structure and Size Parameters
 Unit Cell Size Optimization
 Software: Matlab & modeFRONTIER
 3 design variables
 DoE: Uniform Latin Hypercube (ULH)
 Optimization: NSGA-II algorithm
 24 DoE * 100 generations = 2,400 designs
Optimization workflow (modeFRONTIER)

29. 28
Semi-Analytical Unit Cell Synthesis Method
 Optimization Process (converging)
t3 strain-error t2 g
 Optimization Results
4
4 10strain error 
  
 Constraint for the objective:
the unit cell design is
deemed feasible when the
optimizer is able to generate
sufficient design points.
Design summary chart
 CPU time: less than 5
minutes

30. 29
Semi-Analytical Unit Cell Synthesis Method
 Optimal “Canti” UC Design
Optimal Design
Optimal design curve vs target curve FEA validation

31. 30
Semi-Analytical Unit Cell Synthesis Method
 “CantiCirUp” UC Design
 EFG Selection
 EFG Combination
 ESG to Form UC
 Add an EFG in series
makes the bulk
material softer

32. 31
Semi-Analytical Unit Cell Synthesis Method
 “CantiCirUp” UC Design
 UC Loading Condition
 UC Size Parameters  Conceptual UC Tessellation

33. 32
Semi-Analytical Unit Cell Synthesis Method
 Unit Cell Size Optimization
 Software: Matlab & modeFRONTIER
 5 size parameters
 DoE: Uniform Latin Hypercube (ULH)
 Optimization: NSGA-II algorithm
 50 DoE * 100 generations = 5,000 designs
 Optimization Process
g R
t3 t4

34. 33
Semi-Analytical Unit Cell Synthesis Method
 “CantiCirUp” UC Design
Design Summary
 Optimization Results
4
4 10strain error 
  
 Constraint for the objective:
 Nearly 50% out of 5,000 design
points satisfy the objective and
all the constraints
 CPU time: 30 minutes

35. 34
Semi-Analytical Unit Cell Synthesis Method
 “CantiCirUP” Optimal UC Design
Optimal design curve vs target curve FEA validation

36. 35
Outline
 Introduction
 Metamaterial design method overview
 Analytical Functions of EFGs’ Large Deformation Behavior
 Non-dimensional load / deformation parameters
 Elemental Functional Geometry (EFG) large deformation behavior
 Analytical deformation solutions of EFGs combined in parallel / series
 Semi-Analytical Unit Cell Synthesis Method
 EFG selection and combination to construct metamaterial unit cells
 Unit cell size optimization
 Case study of designing unit cells to match a targeted nonlinear deformation curve
 Summary

37. 36
Summary
 Developed a systematic approach to obtain analytical deformation-
load functions of EFGs subjected to large deformations
 Four polynomial functions are obtained for deformation-load relations
for three EFGs with four loading conditions
 Analytical solutions are obtained for deformation behavior of multiple
EFGs connected in series or parallel
 Utilized the above expressions and developed Semi-Analytical Unit
Cell Synthesis Method to design metamaterial UC topology, to match
predetermined nonlinear deformation response, in a much more efficient
way

38. 37
Future Work
 Include more EFGs to expand the repository
 Incorporate different loading conditions and deformation in multiple
directions for EFGs
 Consider more design aspects such as stress profile,
manufacturability, implement multi-objective optimization process
 Develop algorithms to enable design process automation, and optimal
design selection to maximize the potential of EFGs’ analytical solution

39. 38
Thank you
Thank you