2088
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
2 NUMERICAL OPTIMIZATION ALGORITHM
An iterative algorithm is used to solve the optimization problem
numerically. First a topology is created and second its load-
displacement behavior is determined using a finite-elemente
analysis. In the next step, the results of the finite-element
analysis are interpreted and transmitted to a topology
optimization algorithm, which creates a new improved
topology. Afterwards step two is performed again.
The SIMP-Method (Solid Isotropic Material with
Penelization) after Sigmund (2001) is used as topology
optimization algorithm. The algorithm is based on the idea, that
the material of the optimized structure already exists in the
design domain
, but is not optimally distributed. Therefore,
the material is equally distributed in the design domain
at the
beginning of the optimization process. The material distribution
changes during the optimization process and the material
compacts in areas where it is needed to achieve the optimization
task.
The design domain
is descretized with finite elements.
The material parameters are specified individually for each
element depending on the material distribution. The virtual
density
at a point
a
has to be between 0 and 1, see Equation 1.
Regarding a geotechnical application, for example a
foundation made of concrete, a finite-element with
(a)
= 0 is a
soil element and with
(a)
= 1 is a concrete element.
Using the SIMP-Method, the objective function is the
minimization of the compliance of the structure in the design
domain
. Thus, the stiffness of the structure is maximized. The
compliance of the structure can be expressed using the internal
energy of the system. The internal energy
c
of an elastic
material is defined by Equation 2.
In Equation 2
U
is the global deformation tensor,
K
the
global stiffness matrix and
x
the tensor of design parameters.
The virtual density
(a)
matches the design parameters of
x
i
of
the tensor
x
at point
a.
The minimization of the compliance is restricted by two
constraints. The first constraint ensures that the observed system
is in equilibrium an every step of the optimization process.
Using the finite-element method, this constraint is ensured by
Equation 3.
F
is the global tensor of the external forces.
The second constraint ensures that the volume of the
material distributed in the design domain remains constant
during the optimization process, see Equation 4.
V
is the
volume of the structure.
Additionally, the design parameters
x
i
are limited by an
upper and an lower bound, such that the optimized material
parameters lie within to the physically possible range.
The optimization task for minimal compliance design can be
written using Equation 5.
U
e
is the element deformation tensor,
K
e
the element stiffness matrix,
e
the element virtual density,
the volume fraction,
V
the structure volume and
V
0
is the
volume of the design domain. The values of the material
distribution are limited by
x
min
to avoid singularities during the
finite-element analysis. Using the algorithm for geotechnical
application, this limit is not necessary because the stiffness of
an element belongs to the soils stiffness at
x
i
= 0 and cannot
tend to zero.
The penalty
p
controls the material change-over to ensure
complete material change for example from soil to concrete, see
Figure 4.
The improved topology in every iteration step is determined
using the method of optimal criteria, see Equation 6 (Bendsøe
1995). A positive move-limit
m
and a numerical damping
coefficient
= 0.50 are introduced, see Bendsøe (1995). The
move-limit
m
limits the change of the topology in each iteration
step. The sensitivity of the objective function is expressed in
Equation 7. Using the Lagrangian multiplier
,
B
e
is defined in
Equation 8.
(1)
(2)
(3)
(4)
(5)
material
1
material
no 0
)(
a
KUU c
T
)(
x
F KU
V
mat
d1
1
0
) (
)(
:
subject to
:min
min
0
x
x
V V
F KU
UKU
KU U xc
N
e
e e
T
e
p
e
T
Figure 2. Shape optimization: a) main topology, b) variation 1: anchor position, c) variation 2: anchor inclination, d) variation 3: anchor length
Figure 3. Dimension optimization
