COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Topology Optimization of Structures: Applications in the Simulation
and Design of Cellular Materials
H. C. Rodrigues *
Mechanical Engineering Department, Instituto Superior Tecnico, Technical University of Lisbon, 1049-001 Lisboa,
Portugal
Email: [email protected]
Abstract: In this work a review of different formulations for topology design of structures and materials is
presented. Using the concept of material distribution introduced in 1988 by Bendsøe and Kikuchi, different
treatments have been developed in the last decade not only for the design of structural components but also to
applications in material design namely of cellular, composite and/or piezoelectric materials and to problems in
biomechanics, wave propagation , adaptive structures, fluid flow etc. In this paper we review topology design
formulations developed for periodic material design and extend them to the simultaneous design of the structure and
the material used in its production. Examples of application are presented for elasticity and heat transfer transient
problems.
Keywords: cellular, composite, materials, optimization, topology.
INTRODUCTION
The concept of optimum material allocation was introduced in 1988 by Bendsøe and Kikuchi [1] in order to compute
optimum structural topologies. The goal was then to obtain the global shape and topology of the stiffest structure, for
prescribe load(s), and made of a given isotropic material. A homogenization approach was used in order to achieve
this goal since orthotropic materials were needed to expand the space of possible solutions. This approach was
proved to work well, and has been widely studied and applied to different criteria in mechanical design, such as
stress, buckling loads, natural frequencies, displacement, and applied to problems in biomechanics, material science,
electronics, fluid-flow, wave propagation etc (see [2-4] and references therein). Lately these models have been
extended to multidisciplinary and multi-physics problems [5].
In this paper it is presented some applications and methodologies recently developed for periodic materials (cellular
or composite) design and for the simultaneous design of the mechanical component and the material used in its
construction.
HOMOGENIZED PROPERTIES AND DESIGN PARAMETRIZATION
Let us consider a structure made of a cellular (or composite) material with a microstructure obtained locally by the
periodic repetition of a characteristic (or unit) cell (see Fig. 1). For this unit cell let “d” be its characteristic
dimension and y the position variable. Also let “D” and x be the characteristic dimension of the structure and its
position variable. Let us consider that d D = ε → 0 i.e. the micro geometry dimension is much smaller than the
dimension of the structure. Consequently we can identify two scales in the domain, a “macro” (or global) scale with
variable x and a “micro” (or local) scale with variable y, related by y = x ε .
Furthermore consider that the structure has a linear behavior due to the external applied actions. Under these hypotheses
homogenization models permit the mechanical characterization of an equivalent locally homogeneous material through
the study of the cellular material(s) characteristic cell(s). Note that the local periodic repetition does not exclude the
possibility of different characteristic cell at different macro location, i.e. at different x points in Ω. For a detailed
description of the homogenization models applied to different problems see e.g. Lions [6] and Sanchez-Palência [7].
— 125 —
D
Ω
Y
d
X
Y
Figure 1: Structure and characteristic unit cell
In the case of linear elasticity we will consider cellular materials with characteristic material cell made of linear
elastic material (base material). This unit cell domain will be assumed parallelepiped (2D or 3D) with the volume
fraction ρ given as the quotient of the volume (or area) occupied with material ¥ with the total volume (area) of the
unit cell Y .
Under these conditions the homogenized mechanical properties (superscript “H ”) are given by (see e.g. [8]):
H
ijkm
E
1
=
Y
⎛
∂χ pkm
∫¥ E pqrs ⎜⎜ δ pk δ qm − ∂ yq
⎝
⎞ ⎛
∂χ rij ⎞
⎟⎟ ⎜ δ ri δ sj −
⎟ dY
∂ ys ⎠
⎠ ⎝
(1)
where Epqrs identifies the components of the base material mechanical properties tensor, δ ij is Kronecker delta and
the “micro” displacements χ km (k, m = 1, 2 in 2D applications) are the solutions of a set of equilibrium problems at
the unit cell level,
∫ Eijpq
ゥ
∂χ pkm ∂ vi
∂ vi
dY = ∫ Eijkm
dY , ∀v ∈ V
∂ yq ∂ y j
∂ yj
(2)
V = { v : v , Y-Periodic} . Note that all these equations are defined at the material unit cell “micro” domain Y.
Assume now a base material(s) not homogeneous in the micro domain Y, the design parameterization is introduced
through the micro density field γ (y )= ( 0,1] with the base material mechanical properties dependence on γ given as
Eijkl = γ ( y ) E ijkl0 , with the power p interpreted in the sense of the SIMP method (see e.g. [3, 9]). Typically the
p
exponent p is set equal to 3 or 4.
With this type of interpolation, at points where γ (y) is zero one has no stiffness, so we have a void, and at points
where γ (y) is equal to one has full base material. Thus using such a micro density field we have a complete
characterization of the cellular material unit cell geometry (see Bendsøe, Sigmund [3] for a comprehensive
discussion on different types of design parameterizations in topology optimization).
Substituting this design parameterization, the homogenized elastic properties (1) are given as,
H
ijkm
E
⎛
∂χ pkm ⎞ ⎛
1
∂χ rij ⎞
p 0
= ∫ γ E pqrs ⎜ δ pk δ qm −
⎟ ⎜δ δ −
⎟ dY
⎜
Y Y
∂ yq ⎟⎠ ⎝ ri sj ∂ ys ⎠
⎝
(3)
with similar changes for the state equations (2).
In the case of heat transfer, we will assume a composite material with a characteristic (unit) cell made of two
isotropic linear conducting materials, one with a high conductivity (black) and the other with a low conductivity
(white). The high-conducting material has mass density ρ1, specific heat per unit mass c1 and conductivity
coefficient k1. The lower conducting one has properties ρ2, c2 and k2 respectively. We will use a similar design
parameterization as for the cellular material, thus inside each cell the microstructure is defined by a density function
γ (y) describing the conductivity inside each cell as
— 126 —
Κ ε ( x , y ) = ( γ p (k1 − k2 ) + k2 ) Ι ,
(4)
where I is the identity tensor.
With this interpolation the components of the homogenized conductivity are given as
K ijH ( x ) =
1
Y
∫ (γ
Y
p
(k1 − k2 ) + k2 ) ⎡⎣ ei + ∇ yφ (i ) ( x , y ) ⎤⎦ ⋅ ⎡⎣e j + ∇ yφ ( j ) ( x , y ) ⎤⎦ dY ,
(5)
where the Y-periodic functions φ ( i ) , i = 1, 2 (for two-dimensional problems) are solutions of the following cell
problems:
div y ⎡⎣(γ p (k1 − k2 ) + k2 ) ( ei + ∇ yφ (i ) ( x , y ) ) ⎤⎦ = 0,
i = 1, 2 .
(6)
The homogenized heat capacity is computed as,
( ρc)
H
=
1
Y
∫ ⎡⎣γ ( ρ c − ρ c ) + ρ c ⎤⎦ dY
1 1
2 2
(7)
2 2
Y
.
See Guedes and Kikuchi [8] for a comprehensive explanation on the computational modeling of the homogenization
problem(s).
For the subsequent use we note here that the homogenized properties are, at fixed macroscopic point x, functionals
of γ, thus one can now design the cellular or composite material microstructure through the proper choice of γ (y)
CELLULAR MATERIAL OPTIMIZATION
1. Formulation of the problem Consider a cellular material characterized by the periodic repetition of a unit cell
(see Fig. 2). Let γ (y) identify the design variable i.e. a material density that permits the description of the micro
structural topology, as described in the previous section.
σyy
γ=0
σxy
σxx
γ ∈ ( 0,1)
γ =1
Figure 2: Cellular material design
Formulating the problem in terms of stresses the optimization problem considered here corresponds to finding the
stiffest microstructure for the given stresses (loads). In terms of material science terminology (see, e.g., [10]) we
then seek the (single scale) microstructure that attains an energy bound defined as the minimization of the
complementary energy for the given averaged stress fields, subjected to a upper bound constraint on the cellular
material volume fraction ρ:
min
(E )
]
γ ( y )∈[ 0,1
γ dY ≤ ρ
∫
H −1
ijkl
Aijkl ;
Aijkl = ⎡⎣λ1σ ij1σ kl1 + ... + λM σ ijM σ klM ⎤⎦ or Aijkl = ∫ σ ij ( x )σ kl ( x ) dx
Ω
(8)
Y
In this problem statement λ1 ,..., λM are weighting factors expressing the relative importance of the given stress
tensors σ 1 ,..., σ M . The second definition of Aijkl signifies a spatial average over a domain of interest for some
given stress field and the homogenized elasticity tensor E H definition is given in Eq. (3). The reader is referred to
[11, 12] where a complete description of this model and its computational approximation is presented.
— 127 —
2. Numerical Examples This first example considers two load cases corresponding to uniaxial equal stress fields in
two perpendicular directions, respectively (see Fig. 3). The weighting factor λ varies from zero to one. For λ = 0 we
are at the single load situation σ22 = 1 and for λ = 1 we have a single load with σ11 =1.
Table 1 presents the weighted energy values attained by the optimal microstructures and the respective optimal
bound for different λ values and ρ = 0.5 (50% volume fraction). The same results, comparing the energy value
attained by optimal microstructures and the optimal analytical bound, are graphically shown in Fig. 4.
λ
(1−λ)
Figure 3: Load/stress cases
Table 1: Weighted Energy Values Comparison
λ
Analytical Bound
Computational
Dif. (%)
0
1,000
1,000
0,0
0,1
1,300
1,369
5,3
0,2
1,407
1,479
5,1
0,3
1,465
1,530
4,5
0,4
1,490
1,579
6,0
0,5
1,500
1,584
5,6
0,6
1,490
1,579
6,0
0,7
1,460
1,530
4,8
0,8
1,408
1,479
5,1
0,9
1,301
1,369
5,3
1
1,000
1,000
0,0
In Fig. 5 some microstructure designs are shown for different λ values. Note that the single length scale design
problem is non-convex with several local optimal solutions. Also there exist several optimal microstructures for the
same volume fraction and load values. Fig. 5(c) illustrates this, showing an alternative design obtained with a
different computational initial design point in the case of ρ = 0.5, λ = 0.5 (compare with Fig. 5(b)).
Fig. 5(d) shows a 3D result. It is an optimal microstructure for a single case uniaxial load (for example with λ = 0 or 1).
The second example considers nineteen load cases corresponding to equal shear stress fields each one rotated θ = 5º
from the previous one (see Fig. 6(a)). The weighting factor is equal to all stress fields. The optimal microstructure is
presented graphically in Figs. 6(b) and 6(c). Due to the high number of load cases with different orientation the
optimal cellular material has an almost isotropic equivalent behavior as expected.
The interested reader is referred to references [11-14] where this type of modeling and the respective computational
approximation are discussed and demonstrated with several application examples.
— 128 —
Figure 4: Graphical comparison of weighted energy values
(a) ρ = 0.5, λ = 0.3
(b) ρ = 0.5, λ = 0.5
(c) ρ = 0.5, λ = 0.5
(d) 3D microstructure for uniaxial load
Figure 5: Optimal microstructures
(a) Design Domain and loads
(b) Optimal Characteristic Cell
(c) Cellular Material
Figure 6: Design Domain and Optimal Material
— 129 —
HIERARCHICAL MODEL FOR SIMULTANEOUS OPTIMIZATION OF MATERIAL AND
STRUCTURE
The hierarchical material design model can be illustrated based on Fig. 1. The design domain Ω is defined for the
macro level (global) where the goal is to find an optimal structure topology for given “load(s)” and “support”
conditions. Conversely the design domain Y is defined for the micro level (local) where the aim is to find the
optimal design of the material unit cell from which the structure is manufactured.
Thus at each point x in Ω, the hierarchical design model computes not only the volume fraction ρ of the cellular
material but also the material characteristic cell geometry defined by the “micro” density γ (x , y) function of y in Y .
Thus a two scale material distribution problem (macro/global and micro/local) has been identified and a “density”
field governs and defines each one, ρ(x) and γ (x, y), respectively.
The local problem solution is the topology for a single microstructure (material base cell) that is assumed to be
periodically repeated inside a small neighborhood of each point x∈Ω. This periodicity is only local so the unit cell
topology may vary with the macroscopic variable x although smooth transitions must be assumed. This feature is
illustrated in Fig. 1 where two different unit cell designs are shown at different x points in Ω
In the following we will describe two applications of this methodology. One in structural design with a compliance
based objective and another in design for a thermal application.
1. Structural Compliance Optimization Assuming a multiple load measure (with weighing factors r) for structural
compliance as objective function, the density fields ρ(x) and γ (x, y), are solution of the optimization problem,
⎧⎪ M
⎛
⎞ ⎫⎪
min ⎨∑ α r ⎜ ∫ bir uir dΩ + ∫ tir uir dΓ ⎟ ⎬ with:
ρ ,γ
⎟⎪
⎪⎩ r =1 ⎜⎝ Ω
Γtr
⎠⎭
M
∑α
r
=1
(9)
r =1
Subjected to:
∫ ρ ( x ) dΩ ≤ V ,
∀x ∈ Ω
(10)
1
γ ( x, y ) d Y , ∀x ∈ Ω
Y Y∫
(11)
Ω
ρ( x) =
0 < γ min ≤ γ ( x, y ) ≤ 1, ∀x ∈ Ω and ∀y ∈ Y(x)
(12)
Where the displacement field u r solves the elastostatic problem, for the applied body and boundary forces p r and
tr ,
∫ E (γ ) ε (u )ε
H
ijkl
r
ij
kl
Ω
(v r )dΩ − ∫ bir vir dΩ − ∫ tir vir dΓ = 0, ∀v r admissible, r = 1, ..., M
(13)
Γtr
Ω
H
with Eijkl
(γ ) given by Eqs. (2, 3).
This optimization problem can be equivalently restated as (see e.g. [3, 15-16] for a full discussion of this aspect):
max
∫
ρ (x )
0 ≤ ρ (x ) ≤1
ρ (x )dΩ ≤V
⎡1
M
⎛ r r
⎞⎤
1
M
r
r
r
⎢
⎜
⎟⎥
Φ
ρ
Ω
α
Ω
s
min
,
u
,...,
u
d
p
u
d
t
u
d
×
−
⋅
+
⋅
) ∑ ⎜∫
∫ (
∫k
u r ∈U ⎢ 2
⎟⎥
r
=
1
Ω
Ω
ΓT
r =1,..., P ⎣
⎝
⎠⎦
(14)
Ω
Φ ( ρ , u1 ,..., u P ) =
max
∫
M
∑α
γ ( y)
r =1
0≤γ ( y) ≤1
γ ( y)dY = ρ (x )
r
H
⎡ Eijkl
(γ ) ε ij ( u r ) ε kl ( u r )⎤⎦
⎣
(15)
Y
The previous formulation clearly shows the global-local character of the design uncoupling it into a global problem
for ρ(x) (14) and the local problem on γ (y) (15), linked through constraint (11).
— 130 —
Note that the local optimization problem (15) is simply a strain based formulation of the material optimization
problem (8) presented in the previous section. Thus one is faced with a series of material optimization problems (one
for each location “x”) where now the strain (stress) fields are not prescribed (as in Eq. (8)) but instead solution of the
global equilibrium problem (13), i.e. induced by the respective boundary and body loads.
Furthermore the uncoupling observed in the problem statement (14-15) is especially appropriate for the use of
parallel processing techniques, where each of the local problems can be solved in a different processor, with
important savings on computational time and efforts. A complete description of this hierarchical model and
discussion of the respective computational implementation can be found in references [15, 16].
2. Numerical Example This example considers a supported beam subjected to a multiple load condition as shown in
Fig. 7. The problem was solved with 588 finite elements for the global problem while the local problem was solved
with a 20x20x20 mesh. The volume constraint considered is 35% of the total volume of the structure. It is assumed a
base material E0ijkl linear and isotropic. The material properties are 210 GPa for Young’s Modulus and 0.3 for
Poisson ratio (see [16] for a complete description on the computational modeling for this example). The final global
material distribution and some selected cellular material microstructures are presented in Fig. 8. In this figure, and
the following ones, darker areas denote higher cellular material volume fraction ρ, lighter areas lower volume
fraction values.
Figure 7: Hierarchical optimization-design domain and loads
This type of hierarchical model as been studied also in the context of the remodeling behavior observed in natural
cellular materials such as bone (see e.g. [17]).
3. Thermal Transient Problems A similar hierarchical modelling can be developed for thermal problems. Let us
consider a structure occupying the domain Ω (see Fig. 1) but now let us assume, not a cellular material but instead a
composite material (see previous section “HOMOGENIZED PROPERTIES AND DESIGN PARAMETRIZATION”). The thermal transient problem is stated as,
∂θ
⎧
H
⎪qi ( x , t ) = K ij ( x) ∂x
⎪
j
⎨
⎪ ∂qi ( x , t ) = ( ρ C ) H θ&( x, t )
⎪⎩ ∂xi
(16)
in Ω × (0, T ]
with the initial and boundary conditions,
⎧ θ ( x, t ) = θˆ( x , t )
on ∂Ωθ × ( 0, T ]
⎪
⎪qi ( x, t ) ni = qˆ ( x, t ) on ∂Ω q × ( 0, T ]
⎨
a
⎪ qi ( x, t ) ni = h(θ ( x, t ) − θ ( x, t )) on ∂Ω h × ( 0, T ]
⎪ θ ( x, 0) = θˆ ( x ) on Ω
0
⎩
— 131 —
(17)
Figure 8: Optimal solution
In the previous problem statement, θ(x) is the temperature field, KH the homogenized (equivalent) conductivity
tensor (5), (ρ C)H the homogenized heat capacity (7). The boundary and initial conditions are defined by; the
prescribed temperature θˆ on the ∂Ωθ boundary, the prescribed heat flux q̂ through ∂Ωq , natural or forced
convection with the pair ⎡⎣ h, θ a ( x , t ) ⎤⎦ on ∂Ωh and initial temperature field θˆ0 ( x ) . Vector n is the outward unit
normal to the domain boundary.
Following the material description present earlier let us assume a locally periodic material whose characteristic cell
is made of two base materials; one high-conducting (material 1) and the other low-conducting (material 2). The
distribution of each material within the unit cell will be defined by the variable γ identifying the local volume
fraction of material 1. In the limit in the unit cell regions where γ equal to one, material 1 is present while in regions
where γ equals zero one has material 2 (see interpolation formula (4)).
The objective is to identify locally the composite material and its distribution that minimizes the difference between
the temperature distribution θ (x, T), at a prescribed time T, and a desired target distribution θ tg(x) i.e.
Min
ρ ,γ ,θ
2
1
tg
θ
x
T
−
θ
x
dX
,
(
)
(
)
(
)
2 Ω∫
(18)
such that,
ρ ( x) =
1
Y
∫ γ ( x, y ) dY
Yx
0 ≤ γ (x,y ) ≤ 1
(19)
∫ ρ ( x ) dX ≤ V
Ω
— 132 —
In this sense the present formulation is an extension of the one presented by S. Turteltaub [18] where a simpler law
of mixtures was utilized. A complete derivation of the necessary conditions for the optimization problem (18-19), its
separation into local and global problems and a discussion of the computational model developed and its
implementation is presented in [19].
4. Numerical Example The structural domain Ω is defined as a 3 × 2 rectangular domain. The boundary and initial
conditions are obtained from the following data: the initial temperature field is uniform θ 0 = 100 , the flux is
constant along part of the lower boundary for all t ∈ ( 0, T ] , all the rest of the boundary has a convection condition
(see Fig. 9). The objective is to have a uniform temperature θ tg = 75 0C in the domain, at time Τ = 10.
Note that for the problem conditions the target uniform temperature distribution is unattainable, so the objective is to
get as close as possible in a L2 norm sense.
The materials properties used are the following: For material 1 one has k1=1 and (ρ1c1) =10, for material 2 the
properties are k2=10−3 k1 and (ρ2c2) =1 (see [18, 19] for further details). The volume upper bound is set equal to 83%
and the initial design is uniform with initial ρ uniform and set to 0.83. The macro domain finite element model has a
mesh with 60×40 four node isoparametric elements and the micro domain (unit cell) has a 30×30 mesh.
In Fig. 10 the initial solution and its temperature distribution is shown. Fig. 11 depicts the “optimal” structure, as
well as the material cell geometry at selected locations (macro elements), and the resultant temperature distribution.
The difference between the target temperature and the initial structure and final structure temperatures is depicted in
Figs. 12(a) and 12(b), respectively. Note here that at the optimal solution the “error” between actual temperature
field and temperature target is concentrated close to the boundary as anticipated.
Figure 9: Thermal Problem
(a) Initial Design
(b) Temperature Distribution at Initial Design
Figure10: Initial Design and Temperature Distribution
— 133 —
(a) Final Design
(b) Temperature Distribution at Final Design
Figure 11: Final Design and Temperature Distribution
(a) Initial Design θ(x,Τ )− θ tg (Fobj = 1637)
(b) Final Design θ(x,Τ ) − θ tg (Fobj = 197)
Figure 12: Temperature Difference at Initial and Final Designs
— 134 —
FINAL COMMENTS
Some problems and the respective computational models were presented in the application of topology design
techniques for the optimization of cellular/composite materials and the simultaneous optimization of material and
structure. The proposed approaches for the simultaneous design of material and structure are based on the
hierarchical design model concept, presented for elasto-static applications and thermal transient problems.
The hierarchical model described, which uncouples the problem into local and global sub-problems, has several
advantages. Formulating and solving separately the material optimization problem permits the easy introduction of
material specific design constraints, such as manufacturing ones. Also even though computationally expensive the
uncoupling in a local and global problem permits the use if parallel processing methodologies with significant gains
in terms if computational cost and time (see e.g. [16, 20]). Lately it has been observed an even greater interest in this
area of structural optimization with applications to design materials and structures for impact control, vibration and
noise attenuation and interesting extensions to multidisciplinary engineering applications (see e.g. [4] and references
therein).
Acknowledgements
The results presented in this communication derive from a very gratifying collaboration with Profs. M. P. Bendsøe
(DTU, DK), S. Turteltaub (Univ. Delft, NL), Mr. P. Coelho (UNL, PT) and Profs. J. M. Guedes P.R. Fernandes, J.
Folgado of IST-UTL. I would like to acknowledge these contributions and thank these researchers for the very
interesting collaboration and discussions. I would like to acknowledge also the support given by Portuguese Science
Foundation (FCT), through several programs, and by NATO-RTA to the research work reported here.
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— 136 —