Bendsøe (1989) presented a new topology optimisation method based on the
homogenisation method. In this new approach Bendsøe (1989) used the rel-
ative densities directly as design variables. He called it the direct approach.
Unlike the preceding homogenisation method, in the direct approach there is
(a) microcells with square holes
(b) layered materials
Figure 2.11 The final results obtained by the homogenisation method for SCB problem.
there is no microstructures in use. Instead it is assumed that the structure
is made of an artificial material whose elasticity constants are changing by
its density. For this reason the approach is also referred to as ‘the artificial
material model’ by some authors (e.g. Hassani and Hinton 1998c). Later on
the name SIMP standing for ‘Solid Isotropic Microstructures with Penalisa-
tion’ was selected by Rozvany et al. (1992) for this approach. The same term
SIMP was also used by Bendsøe and Sigmund (1999) with ‘M’ standing for
‘Material’.
2.5.1
Material model
The relationship between the elasticity tensor and the density of the base
material is commonly referred to as material interpolation scheme (Bendsøe
and Sigmund 1999). In his original paper Bendsøe (1989) used the so-called
power-law approach as material interpolation scheme. The power-law inter-
polation scheme can be written as
Eijkl(ρ) = [ρ(ξ)]pE¯ijkl, ξ ∈ Ω (2.23)
where Eijklis the interpolated stiffness tensor which replaces the homogenised
stiffness tensor in the homogenisation method; ¯Eijkl stands for elasticity con-
stants of the base material and ρ(ξ) is the relative density function with
0 ≤ ρ(ξ) ≤ 1. ξ indicates the location and Ω is the design domain. The
parameter p is a penalisation factor which penalises the intermediate density
and ρ = 0) topology.
Using an isotropic base material, unlike homogenisation approach, this
material model yields an isotropic interpolated material. The resulted topol-
ogy is thus an ISE topology. However note that in (2.23) the material changes
continuously from void to solid and hence the resulted topology is not a bi-
nary ISE but rather a relaxed one. Applying high penalty factors, the re-
sulted topology will be more close to a binary ISE topology. On the other
hand setting p = 1 in (2.23) the optimisation problem will change to a vari-
able thickness sheet problem (Bendsøe 1989). For comparison, the resulted
elasticity constants of SIMP material model with p = 2 and p = 3 are de-
picted in Figure 2.12 along with results of homogenised microcells with square
holes. In this graph, the base material has modulus of elasticity of 0.91 and
Poisson’s ratio of 0.3.
Figure 2.12 The SIMP material model with penalty values of p = 2 and p = 3 compared with microcells with square holes.
(1989) mentioned the fictitious material properties in the SIMP material
model and stated that the homogenisation method is preferred. Nevertheless
the SIMP approach superseded the original homogenisation method shortly
after introduction. Later Bendsøe and Sigmund (1999) proposed a physical
interpretation of the so-called artificial material model. According to Bendsøe
and Sigmund (1999) the power-law material model can correspond to a real
physical microstructural model providing
p ≥ max 2 1 − ν, 4 1 + ν , in 2D (2.24) p ≥ max 15 1 − ν 7 − 5ν, 3(1 − ν) 2(1 − 2ν) , in 3D (2.25)
with ν denoting the Poisson’s ratio of the base material.
Note that the power-law interpolation scheme will result in singular stiff-
ness for ρ = 0. In order to avoid singularity, a soft material should be used
instead of void. This can be achieved by increasing the lower bound of ρ
from 0 to a small positive number ρ. The box constraints on relative density
in the SIMP method thus becomes 0 < ρ≤ ρ ≤ 1.
2.5.2
Deriving optimality criteria
The solution procedure for the SIMP method is similar to that of the ho-
pliance design problem takes the form
min
u,ρ c(ρ) = f Tu
such that K(ρ)u = f ,
ρi− 1 ≤ 0, i = 1, . . . , N ρ − ρi ≤ 0, i = 1, . . . , N N X i=1 (ρiVi) − ¯V ≤ 0 (2.26)
The Lagrangian functional for the above equation can be expressed as
L= fTu + ¯uT(Ku − f )+ N X i=1 λui(ρi− 1) + λli(ρ− ρi) + Λ XN i=1 (ρiVi) − ¯V (2.27)
Stationarity of L with respect to ρi implies that
∂c ∂ρi
+ λui − λli+ ViΛ = 0, ∀i = 1, . . . , n (2.28)
Similar to homogenisation method, if the parameter Bi is defined as
Bi = −∂c ∂ρi ViΛ (2.29)
the following update scheme for ρ can be proposed ρK+1i = max{(1 − ζ)ρK i , ρ} if ρKi (BKi )η ≤ max{(1 − ζ)ρKi , ρ} min{(1 + ζ)ρK i , 1} if ρKi (BKi )η ≥ min{(1 + ζ)ρKi , 1} ρKi (BKi )η otherwise (2.30)
The partial derivatives of the mean compliance with respect to ρ in (2.29)
can be easily calculated using the power-law equation (2.23) in (2.14)
∂c ∂ρi
= −pρp−1i uTi Kiui, i = 1, . . . , N (2.31)
Like homogenisation method, the Lagrange multiplier of volume constraint
Λ need to be calculated in an inner loop in each iteration.
The algorithm of the SIMP method has been reviewed in Figure 2.13. A
99-line code in Matlab for the SIMP method has been published by Sigmund
(2001).
1: Discretise the problem’s domain.
2: Select initial values of densities. A uniform distribution is a good starting point.
3: repeat
4: Perform FE analysis and calculate the objective function.
5: repeat
6: Update the design variables using the update scheme (eq. 2.30).
7: Update Lagrangian multiplier of volume constraint.
8: until volume constraint becomes active
9: until convergency criteria are met
10: print the results
11: end
2.5.3
Numerical example
The short cantilever beam problem depicted in Figure 2.10 is solved using the
SIMP method to demonstrate the application of this method. Using (2.24),
the minimum penalty factor for a two dimensional case with the Poisson’s
ratio of 0.3 can be calculated as p = 3. Here this minimum value is adopted.
The lower limit of densities is chosen as ρ = 0.001. The move limit and the
damping factor in (2.30) are selected as ζ = 0.1 and η = 0.5 respectively.
The final topology and the evolution of the objective function are illustrated
in Figure 2.14.
Figure 2.14 The final results obtained by the SIMP method for SCB problem. The graph on the right side shows the values of objective function in each iteration.
The initial objective function was 338.72 corresponding to the initially
uniform distribution of material. This value reduces to 54.89 after optimi-
sation. Note that because of the power-law material model, the objective
function values in the SIMP method are not comparable to those obtained
the SIMP method for this problem is illustrated in Figure 2.15.
Figure 2.15 The topologies obtained by the SIMP method for the SCB prob- lem at different iterations. The darkness of the elements in these grey-scale images corresponds to the value of their relative densities.