Title Matlab code for a level set-based topology optimization method using a reaction diffusion equation Author(s) Otomori, Masaki; Yamada, Takayuki; Izui, Kazuhiro; Nishiwaki, Shinji Citation Structural and Multidisciplinary Optimization (2014), 51(5): 1159-1172 Issue Date 2014-11-19 URL http://hdl.handle.net/2433/201405 Right The final publication is available at Springer via http://dx.doi.org/10.1007/s00158-014-1190-z.; The full-text file will be made open to the public on 19 November 2015 in accordance with publisher's 'Terms and Conditions for Self- Archiving'.; この論文は出版社版でありません。引用の際 には出版社版をご確認ご利用ください。This is not the published version. Please cite only the published version. Type Journal Article Textversion author Kyoto University
Transcript
Title Matlab code for a level set-based topology optimizationmethod using a reaction diffusion equation
Citation Structural and Multidisciplinary Optimization (2014), 51(5):1159-1172
Issue Date 2014-11-19
URL http://hdl.handle.net/2433/201405
Right
The final publication is available at Springer viahttp://dx.doi.org/10.1007/s00158-014-1190-z.; The full-text filewill be made open to the public on 19 November 2015 inaccordance with publisher's 'Terms and Conditions for Self-Archiving'.; この論文は出版社版でありません。引用の際には出版社版をご確認ご利用ください。This is not thepublished version. Please cite only the published version.
Type Journal Article
Textversion author
Kyoto University
Structural and Multidisciplinary Optimization manuscript No.(will be inserted by the editor)
Matlab code for a level set-based topology optimizationmethod using a reaction diffusion equation
Abstract This paper presents a simple Matlab imple-mentation for a level set-based topology optimizationmethod in which the level set function is updated using
a reaction diffusion equation, which is different fromconventional level set-based approaches (Allaire et al.2002, 2004; Wang et al. 2003) that use the Hamilton-
Jacobi equation to update the level set function. Withthis method, the geometrical complexity of optimizedconfigurations can be easily controlled by appropriately
setting a regularization parameter. We explain the codein detail, and also the derivation of the topological deriva-tive that is used in the level set-based topology opti-
mization. Numerical results for stiffness maximizationproblems are provided to facilitate the reader’s under-standing. The presented code is intended for educa-
tional purposes only. This paper was inspired by pre-viously published papers presenting Matlab code for aSIMP method (Sigmund 2001; Andreassen et al. 2011),
a level set-based method (Challis 2010), and FreeFem++code for a structural optimization method (Allaire andPantz 2006). Readers can investigate results provided
by these different methods and discover the prominentaspects of each particular method. The code presentedhere can be downloaded from http://www.osdel.me.kyoto-
This paper presents Matlab code for a level set-basedtopology optimization method incorporating a reactiondiffusion equation for a compliance minimization prob-
lem. Concerning level set-based structural optimizationmethods, many have been proposed since the pioneer-ing works by Allaire et al. (2002, 2004) and Wang et al.
(2003). Programming code for a level set approach isfreely available: Scilab code (Allaire et al. 2004) can bedownloaded via http://www.cmap.polytechnique.fr/ al-
laire/levelset en.html. Also, Challis (2010) presented Mat-lab code for a discrete level set approach, which is avail-able in the manuscript. The Matlab code discussed in
this paper can be used as an aid to understand thesimilarities and differences between this level set-basedmethod and other topology optimization methods.
The level set method was first proposed by Osher
and Sethian (1988) as a method to implicitly representthe evolution of interfaces. In such analyses, the evolu-tion of interface boundaries is tracked by solving a so-
called Hamilton-Jacobi equation, using an appropriatevelocity normal to the interface of the moving bound-ary. In level set-based approaches, structural bound-
aries are represented by the zero iso-surface of the levelset function and the level set function is defined sothat the structural domain is represented wherever the
level set function has a positive value. With respectto boundary representation methods, we classify themethod presented here as a level set-based approach,
which was developed by several of the authors (Yamadaet al. 2010).
In this method, the level set function is updated bysolving a reaction-diffusion equation based on the topo-logical derivative of the objective functional. Therefore,
this method allows topological changes that generate
2 Masaki Otomori et al.
Fig. 1 Fixed design domain, D.
new boundaries during the optimization procedure andit does not require re-initialization of the level set func-tion, which is typically required in Hamilton-Jacobi-
based approaches to ensure accuracy when solving theHamilton-Jacobi partial differential equation. In addi-tion, our method enables qualitative control of the ge-
ometrical complexity of the optimized configurationsby using appropriate values when setting a regulariza-tion parameter. In the following, we provide a rigorous
derivation of the topological derivative for a mean com-pliance minimization problem. We note that althoughthe term representing the effect of boundary conditions
when a hole is created in the design domain was ignoredin previous research (Yamada et al. 2010), this effect isnow considered in the method presented here.
The rest of this paper is as follows. Section 2 de-scribes the formulation of the level set-based topology
optimization method using a reaction diffusion equa-tion and the formulation of an optimization problemfor a compliance minimization problem. Section 3 ex-
plains the Matlab code we use, and Section 4 providesnumerical examples and an extension of the code. Aconclusion is provided in Section 5, the Matlab code
is provided in Appendix A, and the derivation of thetopological derivative is provided in Appendix B.
2 Formulation
2.1 Topology optimization
For structural optimization, optimization problems areformulated using a domain Ω filled with a material do-main, a void domain, and boundaries Γ , as follows:
infΩ
F [Ω] =
∫Ω
fd(x,u,∇u) dΩ +
∫Γ
fb(x,u,∇u) dΓ,
(1)
where fd and fb are arbitrary real functions defined for
domain Ω and boundary Γ , respectively. x represents apoint located in Ω and u represents the state variables.
In topology optimization, as Fig. 1 shows, the aboveoptimization problem is replaced with a material distri-
bution problem within a fixed design domain, D, usingthe characteristic function χΩ . The optimization prob-lem is then defined as
infχΩ
F [χΩ ] =
∫D
fd(x, χΩ) dΩ +
∫Γ
fb(x, χΩ) dΓ, (2)
where the χΩ is defined as
χΩ(x) =
1 ∀x ∈ Ω
0 ∀x ∈ D \Ω.(3)
In the above optimization problem, topological changes
such as an increase or decrease in the number of holesare allowed during the optimization procedure, in ad-dition to changes in the shape represented by outer or
inner boundaries. However, the continuity of the char-acteristic function is not guaranteed and the value ofthe characteristic function can be discontinuous at ev-
ery point. That is, since the characteristic function χΩ
is defined as a subset of a bounded Lebesgue space, L∞,which only assures integrability, the obtained solutions
can be discontinuous at every point in the fixed designdomain. The ill-posed nature of the optimization prob-lem is due to the lack of regularity of the admissible
shapes.
In other words, because increasing the number ofholes in a given design without changing its volume,by concomitantly decreasing their size, results in im-
proved values of the objective functional, the solutionof the optimization problem is rendered nonexistent.This is caused by the lack of closedness of the set of
feasible designs. To overcome this obstacle, the opti-mization problem must be regularized so that a solutioncan exist. Homogenization-based approaches (Bendsøe
and Kikuchi 1988; Allaire 2002) and fictitious densityapproaches such as the SIMP method (Bendsøe 1989)are two popular methods used to regularize optimiza-
tion problems.
Homogenization-based approaches use homogeniza-tion theory to relax the design domain, that is, enlargethe set of possible designs. On the other hand, ficti-
tious density approaches replace the characteristic func-tion with a fictitious material whose elasticity tensoris assumed to be a continuous function of the mate-
rial density. The SIMP (Solid Isotropic Material withPenalization) method is a type of fictitious density ap-proach that uses a continuous fictitious isotropic mate-
rial whose elasticity tensor is assumed to be a function
Matlab code for a level set-based topology optimization method using a reaction diffusion equation 3
of material density, penalized using an exponential pa-
rameter. In both the Homogenization-based approachesand fictitious density approaches, optimized configura-tions are represented as density distributions expressed
using continuous values that range from 0 to 1. There-fore, the obtained optimized configurations often in-clude grayscale areas where the density assumes an in-
termediate value between 0 and 1. Furthermore, theapplication of a boundary condition at the structuralboundaries is problematic in these approaches because
the structural boundaries are not clearly expressed.
To fundamentally overcome these problems, approachesbased on a level set method were proposed (Allaire etal. 2002, 2004; Wang et al. 2003), in which the bound-
aries of the optimal configuration are implicitly rep-resented using a level set function. However, becausethese particular level set-based methods are based on a
boundary advection concept, topological changes thatgenerate new boundaries during the optimization pro-cedure are not allowed, although the number of holes
in an existing configuration can be decreased. We notethat, for 3-dimensional problems, structural boundariescan easily merge to create holes in the fixed design
domain, but this is different from the generation ofholes inside the material domain, which is still disal-lowed. Therefore, although the above mechanism for
the generation of new holes makes the method muchmore flexible, conventional level set-based approachesthat use the Hamilton-Jacobi equation for advecting the
shape are subject to limitations of topological changeseven for 3-dimensional problems. As a result, the ini-tial configuration settings greatly affect the obtained
optimized configurations. To overcome this difficulty,Allaire et al. (2005) applied the bubble method (Es-chenauer et al. 1994) to a level set-based structural op-
timization method in which boundaries are updated us-ing the shape derivative, and holes are inserted basedon the value of the topological derivative. In Allaire
et al. (2005), hole nucleation based on the topologicalderivative is performed in several steps during the opti-mization procedure, as the level set function is updated
using the shape derivative.
2.2 Level set-based topology optimization using a
reaction diffusion equation
This subsection describes the level set-based topologyoptimization method applied in this paper. (Yamada et
al. 2010). In this method, the level set function is up-dated by solving a reaction-diffusion equation based onthe topological derivative of the objective functional.
Therefore, this method allows topological changes that
generate new boundaries during the optimization pro-
cedure, and re-initialization of the level set function isnot required.
As shown in Fig. 2, the structural boundaries in alevel set-based topology optimization method are im-plicitly represented using the iso-surface of the level set
function ϕ, as follows.
1 > ϕ(x) > 0 ∀x ∈ Ω \ ∂Ωϕ(x) = 0 ∀x ∈ ∂Ω
0 > ϕ(x) > −1 ∀x ∈ D \Ω.
(4)
The above level set function is used to represent theboundaries of the target structure; positive values rep-
resent the material domain, negative values representthe void domain, and zero represents the structuralboundaries. The level set function has upper and lower
limits imposed for the regularization term, used for reg-ularizing the optimization problem, which will be ex-plained in the next subsection. To avoid confusion con-
cerning the sign of the level set function, we note thateven though we define positive values of the level setfunction as corresponding to points in the material do-
main, as in (Wang et al. 2003; Yamada et al. 2010), aninverse definition where negative values correspond topoints in the material domain, as in (Allaire et al. 2002,
2004; Challis 2010), is equally valid. We also note thatthe level set function used in the method presented hereis not a signed distance function, such as is typically
used in many level set-based approaches. The optimiza-tion problem that minimizes an objective functional Funder an inequality constraint G is then formulated as
Fig. 2 Fixed design domain D and level set function ϕ.
4 Masaki Otomori et al.
follows, using the level set function ϕ defined above.
infχϕ
F [χϕ] =
∫D
fd(x, χϕ) dΩ +
∫Γ
fb(x, χϕ) dΓ (5)
subject to G[χϕ] =
∫D
g(x, χϕ) dΩ −Gmax 6 0 , (6)
where g is the density function and Gmax is the upper
limit value of the response∫Dg(x, χϕ) dΩ. The charac-
teristic function χϕ(ϕ) is now defined as
χϕ(ϕ) =
1 if ϕ > 0
0 if ϕ < 0 .(7)
The above optimization problem is now replaced with
an unconstrained optimization problem, using Lagrange’smethod of undetermined multipliers, as follows.
infϕ
F [χϕ, ϕ]= F + λG , (8)
where λ is the Lagrange multiplier. Based on the aboveformulation, the KKT (Karush-Kuhn-Tucker) conditionsof this optimization problem are described as
F ′ = 0 , λG = 0 , λ > 0 , G 6 0 , (9)
where the notation F ′ represents the derivative of the
Lagrangian F .Level set functions that satisfy the above KKT con-
ditions are candidate solutions of the level set function
that represent optimized configurations. However, it isnearly impossible to find such optimized solutions di-rectly, so the optimization problem is replaced with a
time evolution equation by introducing a fictitious timet. The level set function is then updated by solving thisequation, and an optimized configuration is ultimately
obtained, as explained below.
2.3 Time evolution equation
For the following formulation, which introduces ficti-tious time t, it is assumed that the variation of thelevel set function is proportional to the gradient of La-
grangian F , as follows.
∂ϕ
∂t= −KF ′ , (10)
where K > 0 is a coefficient of proportionality. The op-timization problem as formulated above is an ill-posed
problem, so it is regularized by adding a regularizationterm, as follows.
∂ϕ
∂t= −K(F ′ − τ∇2ϕ) , (11)
The above equation is a reaction diffusion equation,
and the diffusive term ∇2ϕ ensures the smoothness of
the level set function in the presented method. Further-
more, the value of the regularization parameter τ affectsthe degree of this diffusivity, with larger values of τ pro-viding increased diffusivity for the level set function.
Thus, an appropriately set value of τ can prevent thegeneration of structures that have excessive geometri-cal complexity, so that an optimized configuration that
has a desired degree of geometrical simplicity can beobtained.
Upper and lower limits are imposed on the levelset function so that the smoothing effect only operates
on points that are close to the structural boundaries.Therefore, this regularization term functions implicitlyas a kind of perimeter control. That is, the complex-
ity of the optimized configuration can be controlled byadjusting the value of the regularization parameter τ .For relatively small values of τ , relatively complex op-
timized configurations are obtained, and the converseis true. The details are provided in the numerical ex-amples. We note that adding the regularization term
to (10) to obtain (11) makes it currently difficult toguarantee that the objective functional monotonicallydecreases, although the advection velocity is rigorously
guaranteed to decrease in a descent direction in Allaireet al. (2004). We hope to address this issue in futureresearch.
Setting appropriate boundary conditions for (11),
the following equations are obtained.∂ϕ
∂t= −K
(F ′ − τ∇2ϕ
)in D
ϕ = 0 on ∂D.(12)
We note that although boundary ∂D is defined for val-ues of the level set function that are equal to zero, thisdoes not mean that ∂D represents structural bound-
aries. The use of this definition allows domains thatare close to the boundary of the design domain to re-main unaffected by boundary settings. Other boundary
conditions might work fine. However, we use the Dirich-let boundary condition ϕ = 0 on the boundary of thedesign domain for simplicity on implementation. The
optimized configuration can now be obtained by solv-ing the above time evolution problem.In this method,topological derivative dtF is used for the F ′ term.
2.4 Optimization problem
Figure 3 shows the fixed design domain D and bound-
ary conditions for a mean compliance minimization prob-lem. The displacement is fixed at boundary Γu and trac-tion ti is applied at boundary Γt. The objective is to find
the optimum layout of the design space that minimizes
Matlab code for a level set-based topology optimization method using a reaction diffusion equation 5
Fig. 3 Fixed design domain D and boundary conditions.
the mean compliance under a given volume constraint.
The optimization problem is defined as
infϕ
F =
∫Γ
tiui dΓ (13)
s.t. G =
∫Ω
dΩ − Vmax 6 0 (14)
div (Cijkluk,l) = 0 on Ω (15)
ui = ui in Γu (16)
ti = ti in Γt , (17)
where Vmax is the upper limit of the volume constraint,Cijkl is the elastic tensor, ui is the displacement andti = σijnj = Cijkluk,lnj is the traction. ui and ti are
constant values that represent the given displacementand traction, respectively.
Using Lagrange’s method of undetermined multipli-
ers, the above optimization problem is replaced with anunconstrained optimization problem, using LagrangianF , and Lagrange multiplier ui and λ, as follows:
F =
∫Γ
tiui dΓ +
∫Ω
uidiv (Cijkluk,l) dΩ
+λ
(∫Ω
dΩ − Vmax
). (18)
The second term on the right-hand side in the aboveequation can be replaced using Green’s formula, as fol-lows: .
F =
∫Γ
tiui dΓ +
∫Γ
uiti dΓ −∫Ω
ui,jCijkluk,l dΩ
+λ
(∫Ω
dΩ − Vmax
). (19)
2.5 Topological derivative
Figure 4 shows the concept of the topological derivative,
which is a measure of the influence when a hole Ωε with
radius ϵ is created at a certain point in the domain Ω.
Γϵ represents the boundary of the created hole. Thetopological derivative dtF of objective functional F isdefined as
dtF := limε→0
(F + δF )− F
meas(Ω \Ωε)−meas(Ω). (20)
The topological derivative for Lagrangian (19) isgiven as follows.
dtF = limϵ→0
δF4πϵ3
3
= u0i,jAijklu
0k,l − λ , (21)
where the superscript 0 indicates the value without cre-ating holes, and Aijkl is defined as follows:
Aijkl =3(1− ν)
2(1 + ν)(7− 5ν)
[−(1− 14ν + 15ν2)E
(1− 2ν)2δijδkl
1
1+ 5E(δikδjl + δilδjk)
]. (22)
The details for the derivation of topological derivativeare provided in Appendix B.
3 Implementation
3.1 Reaction-diffusion equation
We now present a scheme for implementing the reaction-
diffusion equation (Yamada et al. 2010). As discussedabove, the level set function is updated using a reactiondiffusion equation, and the diffusive term ∇2ϕ ensuresthat the optimization has sufficient smoothness and nu-
merical stability.
First, we introduce a parameter C to normalize thesensitivities so that the value of τ can be chosen regard-
less of the particular problem being solved. We thenreplace (12) with dimensionless equations, as follows.
∂ϕ
∂t= −
(−CdtF − τ∇2ϕ
)in D
ϕ = 0 on ∂D.
(23)
Fig. 4 Concept of the topological derivative.
6 Masaki Otomori et al.
Here, K in (12) is set to 1. The effect of this term is
explained in Yamada et al. (2010). We note that theF ′ term in (12) is replaced by −CdtF . The opposite insign is due to the relationship between the definitions of
topological derivative and the sign of level set function.C is defined so that
C =
∫DdΩ∫
D| dtF | dΩ
. (24)
Using the finite difference approach, the equations in
(23) are discretized in the time direction as follows:ϕ(t+∆t)
∆t− τ∇2ϕ(t+∆t) = CdtF +
ϕ(t)
∆t
ϕ = 0 on ∂D,(25)
where∆t is the step for fictitious time t. Using the finite
element approach, the above equations are expressed inweak form as follows:
∫D
ϕ(t+∆t)∆t ϕdD +
∫D∇Tϕ(t+∆t)
(τ∇ϕ
)dD
=∫D
(CdtF + ϕ(t)
∆t
)ϕdD for ∀ϕ ∈ Φ
ϕ = 0 on ∂D,
(26)
where Φ is the functional space defined such that
Φ = ϕ(x)|ϕ(x) ∈ H1(D) with ϕ = 0 on ∂D. (27)
Discretizing (26) using the finite element method, weobtain the following:
T Φ(t+∆t) = Y
ϕ = 0 on ∂D,(28)
where Φ(t) is a vector that expresses the nodal valueof the level set function at time t. T and Y are nowdescribed as
T =N∪e=1
∫Ve
(1
∆tNTN+∇TNτ∇N
)dVe (29)
Y =N∪e=1
∫Ve
(CdtF +
ϕ(x, t)
∆t
)NdVe , (30)
where N is the number of elements and Ve is the vol-ume of an element.
∪Ne=1 represents the union set of
the elements, where e is the element number. N is the
interpolation function of the level set function.
After updating the level set function using (28), thelevel set function is replaced based on the following rule,
so that the upper and lower limit constraints of the levelset function in (4) are satisfied.
if ∥ϕ∥ > 1 then ϕ = sign(ϕ). (31)
3.2 Volume constraint
The volume constraint is dealt with using the aug-mented Lagrangian method, in which the Lagrange mul-
tiplier λ, is updated according to the following scheme.
λ =
∫DdtF dΩ∫D
dΩexp[p(
GGmax
+d)], (32)
where p and d are parameters that adjust the posi-
tion of the curve. This scheme can be considered as amodified version of the augmented Lagrangian method,in which Lagrange multiplier λ is updated as λi+1 =
λi + cG, where c is a penalty parameter. We note thatthe Maclaurin expansion of an exponential function isexpressed as ex = 1 + x + 1
2x2 + · · · , so the standard
updating scheme can be considered as a first-order ex-pansion of the above updating scheme when the volumeconstraint is active, with the value of pλi used as the
penalty parameter c.
When the constraint is sufficiently satisfied, thatis, when the value of G is very small, the value of λ
in (32) approaches 0. The sensitivity of the constraintfunctional then becomes relatively small compared withthat of the objective functional, in the sensitivity ex-
pression F ′ + λG′. In this way, the optimization is pri-marily affected by the value of the objective functional.On the other hand, when the constraint is far from be-
ing satisfied, the value of λ becomes very large, caus-ing the sensitivity of the constraint functional to be-come relatively large compared with that of the objec-
tive functional. When this occurs, the optimization isprimarily affected by the need to satisfy the constraintfunctional. Although other methods can be applied to
satisfy the volume fraction, we use this scheme for itssimplicity.
When the volume fraction of an initial guess is greater
than the maximum allowable volume fraction Vmax, thevolume constraint is relaxed according to the followingformula, to stabilize the convergence.
G=
∫D
χϕdΩ−Vmax−(V0−Vmax)max
(0, 1− i
nvol
)60 ,
(33)
where i is the current iteration number and V0 is thevolume fraction of the initial guess. The first term rep-
resents the volume of the configuration at the current it-eration. The third term in the right-hand side of (33) isadded to the primal volume constraint to relax the up-
per limit of the volume constraint so that the constraintis gradually tightened during nvol iterations. After nvol
iterations, the constraint functional (33) represents the
original volume constraint.
Matlab code for a level set-based topology optimization method using a reaction diffusion equation 7
4 Matlab code
The simple Matlab code for the mean compliance mini-mization problem is provided in Appendix A. The pro-
gram consists of four parts: parameter definitions (lines3–13), finite element analysis preparation (lines 14–40),loads and boundary settings (lines 41–53), and the main
loop (lines 54–88). Figure 5 shows the design domainand boundary conditions of the problem. Displacementis fixed at the left boundary and a downward force is
applied at the center of the right boundary. The designdomain is discretized using a rectangular mesh. Thenumbers of elements are nelx in the horizontal direc-tion and nely in the vertical direction.
The Matlab code is evoked with the following call:
levelset88(nelx,nely,Vmax,tau),
where Vmax is the upper limit of the volume fractionand tau is the regularization parameter τ .
4.1 Parameter definition: lines 3–13
Explanations for the various parameters are given inTable 1. E0, Emin, and nu are parameters used for analy-
sis, Young’s modulus in the material and void domains,and Poisson’s ratio, respectively. nvol corresponds tothe iteration number for the volume constraint, nvol in
(33). When the initial volume is greater than the up-per limit of the volume constraint Vmax, the volumeconstraint is relaxed, and then it is gradually tight-
ened during the iterations prescribed by nvol. dt, d,and p are optimization parameters. dt corresponds tothe fictitious time step ∆t in (29) and (30). d and p
are parameters used in the updating scheme appliedto Lagrange multiplier λ for the volume constraint inthe augmented Lagrangian method formulated in (32).
Fig. 5 Fixed design domain and boundary conditions of de-sign model 1.
Table 1 Parameter definition
parameter meaning
nelx∗ number of elements in x direction
nely∗ number of elements in y direction
Vmax∗ maximum allowable volume
tau∗ regularization parameter
E0 Young’s modulus in material domain
Emin Young’s modulus in void domain
nu Poisson’s ratio
nvol iteration number for volume constraint
dt step size for fictitious time t
d, p parameter for augmented Lagrangian method
∗: set according to the user’s function call.
phi and str represent the level set function ϕ, and thematerial distribution, respectively.
4.2 Preparation of finite element analysis: lines 14–40
These lines of code define the preparation for solvingthe displacement field and reaction diffusion equation,
and computing the topological derivative. This code fol-lows the same procedure for computing the global stiff-ness matrix as that of Andreassen et al. (2013), where
assembly of the global stiffness matrix is efficiently per-formed using the sparse function in Matlab. This pro-cedure is advantageous from the standpoint of compu-
tation time because it avoids the use of for loops. Lines16–19 define the components of the element stiffnessmatrix KE for the displacement field and these are as-
sembled to create the global stiffness matrix K at eachiteration.
Due to the similarity in the formulation of the topo-
logical derivative u0i,jAijklu
0k,l defined in (22), and the
mutual strain energy density ui,jKijkluk,l (Nishiwakiet al. 1998; Howell 2001), the topological derivative can
be computed using the same procedure as that for com-puting the mutual strain energy density, by replacingYoung’s modulus E and Poisson’s ratio ν as follows:
E → 4A22
A1 + 2A2, (34)
ν → A1
A1 + 2A2, (35)
where
A1 = − 3(1− ν)(1− 14ν + 15ν2)
2(1 + ν)(7− 5ν)(1− 2ν)2E , (36)
A2 =15E(1− ν)
2(1 + ν)(7− 5ν). (37)
The detailed derivation of this relationship is providedin Appendix C. a1 and a2 in the code correspond to
A1 and A2 in the above equation, respectively, and A
8 Masaki Otomori et al.
corresponds to Aijkl. We note that when the given dis-
placement at boundary Γu is set to u = 0 as in Fig. 5,the adjoint problem (46)-(48) is equivalent to the stateproblem (15)-(17). Therefore the optimization problem
becomes self-adjoint problem and ui,j = ui,j .
nodenrs is a matrix that indicates the node number
for all elements and edofVec is a vector representingthe first DOF index for all elements. The i-th columnof edofMat consists of the node number of the i-th el-
ement. iK and jK are index vectors used when assem-bling the global stiffness matrix using the element stiff-ness matricies. Details of this procedure are given inAndreassen et al. (2011).
Next, lines 30 through 40 compute the global ma-trices NTN, ∇NT∇N for the reaction-diffusion equa-
tion. NN and NNdif respectively correspond to NTNand ∇NT∇N defined in (29). Although the number ofvariables defined for each node is different for the dis-
placement field (two variables: u, v) and the level setfunction (one variable: ϕ), the procedure for comput-ing NN and NNdif is same as that for computing global
stiffness matrix K.
4.3 Loads and boundary settings: lines 41–53
Lines 42–48 define the boundary conditions for the dis-
placement field. Line 45 defines the load vector thatis applied at the center of right-hand boundary. Thelength of the line where the load is applied is 1/16th
that of the entire length of the right-hand boundary.Line 49 defines the stiffness matrix T shown in (29)for the reaction-diffusion equation. The load vector Y
shown in (30) for the reaction-diffusion equation is de-fined in the main loop, since it includes the level setfunction that varies during optimization. Lines 49–53
define the boundary conditions for the reaction-diffusionequation.
4.4 Optimization loop: lines 54–88
The optimization loop consists of four parts: finite ele-ment analysis and sensitivity computation, convergencecheck, computation of Lagrange multiplier λ for the vol-
ume constraint, and the level set function update.
4.4.1 Finite element analysis and sensitivity
computation
sK is a vector that is created by reshaping the elementstiffness matrix, and global stiffness matrix K is con-structed using the sparse function. SED and TD are
the strain energy density and topological derivative,
respectively. TD describes the value of the topological
derivative in each element and these values are mappedonto nodes. TDN is the value of the topological deriva-tive used on the nodes. In line 65 and 66, the objective
functional, objective, and the volume of the currentconfiguration, vol, are calculated. Lines 67–70 print theoptimization results and plot the material distribution
str.
4.4.2 Convergence check
Lines 72–75 are for the convergence check of the algo-
rithm. The convergence check is performed after nvoliterations. The optimization terminates if both of thefollowing conditions are satisfied: the volume is within
0.005 of the required value, Vmax, and the five previousvalues of the objective functional differ by less than 1%.
4.4.3 Updating the Lagrange multiplier for the volume
constraint
ex corresponds the second and third terms in (33), and
indicates the relaxed upper limit of the volume con-straint so that the constraint will be gradually tight-ened during nvol iterations. lambda corresponds to the
Lagrange multiplier λ defined in (32). Line 79 computesthe normalization parameter C defined in (24). And g2
in line 80 expresses the nodal topological derivative in
vector notation.
4.4.4 Updating the level set function
Matrix Y is computed based on the Lagrange multiplierlambda (line 82) and the level set function phi obtainedduring the previous iteration. The level set function phi
is then updated by solving the reaction-diffusion equa-tion (line 83). The updated level set function may notsatisfy the upper and lower limit constraints of the level
set function. To ensure that these constraints are sat-isfied, the level set function is replaced based on therule described in (31) (line 84 in the code). Nodal level
set function phin is mapped onto the elements. An up-dated structure is obtained using the elemental levelset function phie. The process then returns to the first
step of the optimization loop.
5 Numerical examples
5.1 Mesh independency
First, we examine three cases whose degree of discretiza-tion is subject to the following mesh parameters: 80×64,
160×128, and 320×256. The regularization parameter
Matlab code for a level set-based topology optimization method using a reaction diffusion equation 9
τ is set to 2×10−4 and a volume fraction upper limit of
0.5 is used in all cases. The Matlab code calls for thesethree cases are respectively:levelset88(80,64,0.5,2e-4),
Figure 6 shows the optimized configurations for each
case. The results indicate that the optimized configu-ration has minimal dependency on the mesh discretiza-tion.
5.2 Effect of the regularization parameter τ
Next, we examine three cases in which different degreesof diffusivity are imposed by setting parameter τ to thefollowing values: τ = 2×10−4, 6×10−4, and 1× 10−2. A
mesh discretization of 320× 256 and a volume fractionupper limit of 0.5 is used in all cases. Figure 7 showsthe optimized configurations for each case. The results
show that relatively complex optimized configurationsare obtained when τ is set to smaller values, and viceversa. We note that in stiffness maximization problems
for cantilever structures, a structure with an infinitenumber of infinitely thin members is known to be op-timal. Therefore, as the regularization parameter is set
to smaller values, the optimized configuration becomesmore complex and correspondingly better values of theobjective functional are obtained. The results indicate
that essentially any desired degree of geometrical com-plexity in the optimized configuration can be obtained,by setting the regularization parameter τ to an appro-
priate value.
5.3 Other boundary conditions
Last, we offer an additional example in which different
boundary conditions are used. Figure 8 shows the anal-ysis domain and boundary conditions. A load is appliedat the bottom center of the analysis domain. Vertical
and horizontal displacements at the bottom left cornerof the analysis domain are fixed, as is the horizontaldisplacement at the bottom right corner of the analy-
sis domain. The design domain is symmetric along they-axis and a roller constraint is imposed on the bound-ary at the bottom right corner, so only the right half
of the design domain is considered in the optimizationproblem.
To enable different load and boundary conditions
for this optimization problem, lines 44, 45, 49 and 50of the code are changed, as follows.
Line 44:
Fig. 8 Fixed design domain and boundary conditions of de-sign model 2
F(2*(nely+1):2*(nely+1):2*(nely+1)*
(round(nelx/32)+1),1) = 1;
Line 45:
fixeddofs = [1:2:2*(nely+1) 2*(nely+1)*
(nelx+1-round(nelx/32)):2*(nely+1):2*
(nely+1)*(nelx+1)];
line 49, 50:fixeddofs phi = [];
Figure 9 shows the optimization results obtained usingtwo different settings for the regularization parameter
τ . A mesh discretization of 120× 120 (half model) anda volume fraction upper limit of 0.5 is used in bothcases. We can confirm that the optimization obtained
appropriate optimized configurations.
6 Conclusions
This paper presented Matlab code for a level set-based
topology optimization method that uses a reaction dif-fusion equation to update the level set function. A rig-orous derivation of the topological derivative for a com-
pliance minimization problem was provided. Althoughthe term representing the effect of boundary conditionswhen a hole is created in the design domain was ig-
nored in previous research (Yamada et al. 2010), thiseffect was considered in the method presented here.
We hope that dissemination of this code will enable
readers to more easily understand the operation of thistopology optimization method and also allow them tocompare results obtained when using different topol-
ogy optimization methods. We believe that this codeis very compact, comprehensible, and computationallyefficient. It employs the loop vectorization and memory
preallocation detailed in Andreassen et al. (2011).
10 Masaki Otomori et al.
Mesh
80-64
(a)
Mesh
160-128
(b)
Mesh
320-258
(c)
Fig. 6 Optimized configurations for design model 1 obtained using different meshes: (a) 80×64; (b) 160×128; (c) 320×256.
tau
1e-4
(a)
tau
3e-4
(b)
tau
3e-3
(c)
Fig. 7 Optimized configurations for design model 1 obtained using different regularization parameter τ settings: (a) 2×10−4;(b) 6× 10−4; (c) 1× 10−2.
tau
(a)
tau
(b)
Fig. 9 Optimized configurations for design model 2 obtained using different values for the regularization parameter τ : (a)2× 10−4; (b) 6× 10−4.
Acknowledgements The authors would like to express theirgratitude to Prof. Toshiro Matsumoto and Prof. Hiroshi Isakarifor their helpful discussions concerning topological derivativesand encouragement of this work.
References
Allaire G (2002) Shape optimization by the homogeniza-tion method. Vol. 146 of Applied Mathematical Sciences,Springer-Verlag, New York.
Allaire G, Jouve F, Toader AM (2002) A level-set methodfor shape optimization. C. R. Acad. Sci. Paris, Ser. I,334:1125-1130
Allaire G, Jouve F, Toader AM (2004) Structural optimiza-tion using sensitivity analysis and a level-set method.J. Comp. Phys., 194:363-393
Allaire G, de Gournay F, Jouve F, Toader AM (2005) Struc-tural optimization using topological and shape sensitivityvia a level set method. Control and Cybernetics, 34:59-80
Matlab code for a level set-based topology optimization method using a reaction diffusion equation 11
Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sig-mund O (2011) Efficient topology optimization in MAT-LAB using 88 lines of code. Struct Multidiscipl Optim,43:1-16
Bendsøe MP, Kikuchi N (1988) Generating optimal topolo-gies in structural design using a homogenization method.Comput Methods Appl Mech Eng 71;197-224
Bendsøe MP (1988) Optimal shape design as a material dis-tribution problem. Struct Multidiscipl Optim, 1;193-202
Challis VJ (2010) A discrete level-set topology optimizationcode written in Matlab. Struct Multidisc Optim 41;453-464
Eschenauer HA, Kobelev VV, Schumacher A (1994) Bubblemethod for topology and shape optimization of struc-tures. Struct Multidisc Optim 8;42-51
Howell HH (2001) Compliant Mechanisms. John Wiley &Sons, Inc.
Guzina BB, Bonnet M (2004) Topological derivative for theinverse scattering of elastic waves. The Quarterly Journalof Mechanics and Applied Mathematics 57(2);161-179
Lurie AI, Belyaev A (2005) Theory of elasticity. Springer,New York
Nishiwaki S, Frecker MI, Min S, Kikuchi N (1988) Topologyoptimization of compliant mechanisms using the homog-enization method. Int. J. of Numerical Methods in Engi-neering 42;535-559
Osher SJ, Sethian JA (1988) Fronts propagating with cur-vature dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79;12-49
Sigmund O (2001) A 99 line topology optimization code writ-ten in Matlab. Struct Multidisc Optim 21;120-127
Wang MY, Wang X, Guo D (2003) A level set method forstructural topology optimization. Comput Methods ApplMech Eng, 192:227-246
Yamada T, Izui K, Nishiwaki S, Takezawa A (2010) A topol-ogy optimization method based on the level set methodincorporating a fictitious interface energy. Comput Meth-ods Appl Mech Eng, 199:2876-2891
12 Masaki Otomori et al.
A Matlab code
1 % Matlab code for topology optimization using a reaction diffusion equation
2 function [str,phi] = levelset88(nelx,nely,Vmax,tau)
The boundary value problem for a created hole can be de-scribed as
div (Cijkl (uk,l + δuk,l)) = 0 in Ω \Ωε (38)
ui + δui = ui on Γu (39)
ti + δti = ti on Γt (40)
ti + δti = 0 on Γϵ . (41)
We note that the boundary conditions expressed in (39) and(40) indicate that the displacement and traction after creatinga hole must satisfy the original displacement and tractionconstraints, respectively. (41) represents the free surface onthe boundary of the created hole.
The Lagrangian that includes the created hole is given as
F + δF =
∫Γu∪Γt
(tiui + tiδui + δtiui) dΓ
+
∫Γu∪Γt
ui(ti + δti) dΓ +
∫Γε
ui(ti + δti) dΓ
−∫Ω\Ωε
ui,jCijkl(uk,l + δuk,l) dΩ
+λ
(∫Ω\Ωε
dΩ − Vmax
). (42)
When the boundary condition (41) is inserted into (42), thethird term on the right-hand side becomes zero. Subtractingthe Lagrangian (19) from the Lagrangian (42), the variationδF of the Lagrangian then becomes
δF =
∫Γu∪Γt
(tiδui + δtiui) dΓ +
∫Γu∪Γt
uiδti dΓ
−∫Ω\Ωε
ui,jCijklδuk,l dΩ +
∫Ωε
ui,jCijkluk,l dΩ
−λ
∫Ωε
dΩ. (43)
The third term on the right-hand side in the above equationcan be replaced using Green’s formula, as follows:∫
Ω\Ωε
ui,jCijklδuk,l dΩ =
∫Γu∪Γt
tiδui dΓ (44)
+
∫Γε
Cijklui,jnlδuk dΓ −∫Ω\Ωε
div (Cijkluk,l) δui dΩ,
where ti is the derivative of Lagrange multiplier ui in thenormal direction and nl is the normal vector at boundary Γϵ.Substituting the above equation into (43) and considering theconditions, δui = 0 on Γu and δti = 0 on Γt, the variation ofthe Lagrangian is then given as
δF =
∫Ω\Ωε
div (Cijkluk,l) δui dΩ +
∫Γt
(ti − ti)δui dΓ
+
∫Γu
(ui + ui)δti dΓ+
∫Ωε
ui,jCijkluk,l dΩ
−∫Γε
Cijklui,jnlδuk dΓ−λ
∫Ωε
dΩ. (45)
The adjoint equation is now defined so that the integrals ofthe term that includes δui or δti are canceled out, as follows:
div (Cijkluk,l) = 0 in Ω (46)
ui = −ui on Γu (47)
ti = ti on Γt . (48)
The variation of the Lagrangian can be obtained as follows,using the adjoint variable obtained by solving the above ad-joint field.
δF =
∫Ωε
ui,jCijkluk,ldΩ−∫Γε
Cijklui,jnlδukdΓ−λ
∫Ωε
dΩ.
(49)
The first term on the right-hand side of the above equationrepresents state variable ui and adjoint variable ui, whichare known values. On the other hand, the value of δui in thesecond term is unknown. We note that this second term wasignored in previous research (Yamada et al. 2010), so that themethod did not consider the effect of boundary condition Γϵ
that arises when a hole is created in the design domain.By subtracting the boundary value problem (15)-(17) from
the boundary value problem (38)-(41), we obtain the follow-ing boundary value problem by which the unknown value δuis governed.
div (Cijklδuk,l) = 0 in Ω \Ωε (50)
δui = 0 on Γu (51)
δti = 0 on Γt (52)
δti = −ti on Γϵ . (53)
In the above problem, since the radius of the hole ϵ is suf-ficiently small, the effect of boundaries Γu and Γt can beignored for solving δui on Γϵ. δui is then a solution of thefollowing boundary value problem.
div (Cijklδuk,l) = 0 in Ω \Ωε (54)
δti = −σ0ijnj +O(ϵ) on Γϵ , (55)
where the superscript 0 indicates the value without creatingholes, and σij = Cijkluk,l. The solution of this boundaryvalue problem in sphere Γϵ is given (Guzina and Bonnet 2004;Lurie and Belyaev 2005) as:
δui = −ε
µ
(4− 5µ
7− 5µσ0ijnj−
3− 5µ
4(7− 5µ)σ0jjnj+O(ε)
). (56)
The variation of Lagrangian F is then given by:
δF =4πε3
3
3(1− ν)
2(1 + ν)(7− 5ν)
[−(1− 14ν + 15ν2)E
(1− 2ν)2δijδkl
1
1+ 5E(δikδjl + δilδjk)
]u0i,ju
0k,l−λ
, (57)
where δij is Kronecker’s delta function. Finally, the topolog-ical derivative of the Lagrangian is given as
dtF = limϵ→0
δF4πϵ3
3
= u0i,jAijklu
0k,l−λ , (58)
where Aijkl is defined as follows:
Aijkl =3(1− ν)
2(1 + ν)(7− 5ν)
[−(1− 14ν + 15ν2)E
(1− 2ν)2δijδkl
1
1+ 5E(δikδjl + δilδjk)
]. (59)
Matlab code for a level set-based topology optimization method using a reaction diffusion equation 15
C Topological derivative implementation
First, for simplicity, we reformulate the tensor Aijkl defined in (22) as follows:
Aijkl := A1δijδkl +A2(δikδjl + δilδjk) . (60)
Namely,
A1 = −3(1− ν)(1− 14ν + 15ν2)
2(1 + ν)(7− 5ν)(1− 2ν)2E, A2 =
15E(1− ν)
2(1 + ν)(7− 5ν). (61)
The first term on the right-hand side in (21) can then be given as follows:
A1+2A2. Compare this with the following formulation of the mutual strain energy density:
ui,jEijkluk,l =E
(1− ν)2
[ϵ11 ϵ22 τ12
] 1 ν 0ν 1 00 0 1−ν
2
ϵ11ϵ22τ12
. (68)
The topological derivative can be therefore computed by substituting A1, A2 into E, ν in the procedure for computing themutual strain energy density, as follows:
