Using BESO method to optimize the shape and reinforcement of the underground openings

Abstract

In excavation design, optimizing the reinforcement and finding the optimal shape of the opening are two significant challenges. Both of these problems can be viewed as searching for the optimum distribution of material in the design domain. In structural design, the state-of-the-art topology optimization techniques have been successfully used to deal with such problems. One of these techniques, known as bi-directional evolutionary structural optimization (BESO), is employed here to improve the shape and reinforcement designs of underground openings. The BESO algorithm is extended to simultaneously optimize the shape of the opening and the topology of the reinforcement. The validity of the proposed approach is tested through a simple example.

Full text

1 INTRODUCTION
During the last two decades topology optimization
has attracted considerable attention and the tech-
niques in this field have been improved signifi-
cantly. Several physical problems have been tackled
in this context. However the application of topology
optimization tools in geotechnical problems has not
been studied thoroughly. In spite of the great poten-
tial in this class of problems there are only a few
published works in this area. Among these works
some attempted to optimize the shape of under-
ground openings (Ren et al. 2005, Ghabraie et al.
2007) while others tried to optimize the topology of
reinforcement around a tunnel with a predefined
shape (Liu et al. 2008, Yin et al. 2000, Yin & Yang
2000a,b). Both of these optimizations can lead to
considerable savings in excavation designs. In this
paper, attempts have been made to optimize the
shape of the opening and the topology of the sur-
rounding reinforcement simultaneously. This cou-
pling can improve the solutions leading to greater
savings. It will be shown that the sensitivities of
these two optimization problems only differ in con-
stant values. Hence the two optimization problems
can be solved together with almost no extra compu-
tational effort.
The BESO method was proposed in late 90s
(Querin et al. 1998, Yang et al. 1999) as an im-
proved version of the ESO method which was origi-
nally introduced in early 90s by Xie and Steven (Xie
& Steven 1993, 1997). The ESO method improves
the design by gradually removing the inefficient
elements. In the BESO method, on the other hand, a
bi-directional evolutionary strategy is applied which
also allows the strengthening of the efficient parts by
adding material. The efficiency of elements can be
calculated by sensitivity analysis of the considered
objective function or can be assigned intuitively (Li
et al. 1999).
In this paper the BESO method is used for solv-
ing both problems of shape optimization of the
opening and topology optimization of reinforce-
ments. These two problems can both be modeled as
two-phase material distribution problems. For shape
optimization the material is changing between solids
and voids. In reinforcement optimization, on the
other hand, the material can be switched between
original rock and reinforced rock.
In the original BESO inefficient elements are
completely eliminated from the mesh. Such topology
optimization techniques are sometimes referred to as
hard kill methods as oppose to soft kill methods. In
hard kill methods only the non-void elements will
remain in the mesh and so the finite element analysis
can be performed faster. However in these methods
the sensitivity of void elements cannot be calculated
directly from the analysis results and should be ex-
trapolated from the surrounding solid elements. In
this paper a soft kill BESO has been adopted where
the void elements are represented by a very soft ma-
terial. In this manner the sensitivities of voids are di-
rectly calculable.
Using BESO method to optimize the shape and reinforcement of the
underground openings
K. Ghabraie, Y.M. Xie & X. Huang
School of Civil, Environmental and Chemical Engineering, RMIT University, Melbourne, Australia
ABSTRACT:
In excavation design, optimizin
g the reinforcement and finding the optimal shape of the ope
n-
ing are two significant challenges. Both of these problems can be viewed as searching for the optimum distri-
bution of material in the design domain. In structural design, the state-of-the-art topology optimization tech-
niques have been successfully used to deal with such problems. One of these techniques, known as bi-
directional evolutionary structural optimization (BESO), is employed here to improve the shape and rein-
forcement designs of underground openings. The BESO algorithm is extended to simultaneously optimize the
shape of the opening and the topology of the reinforcement. The validity of the proposed approach is tested
through a simple example.

2 MATERIAL MODEL
Although the geomechanical materials are naturally
inhomogeneous, non-linear, anisotropic, and inelas-
tic (Jing 2003), in excavation design in rocks, mod-
eling them as an isotropic, homogeneous material
and assuming linear elastic behavior can be instruc-
tive and sometimes can predict the real behavior
with acceptable accuracy (Brady & Brown 2004). In
fact the simplified linear elastic material model is
still the most common material model used in ge-
omechanics (Jing 2003). Moreover the results of a
linear analysis can be used as a first-order approxi-
mation of non-linear cases. Most of previous works
(Ren et al. 2005, Liu et al. 2008, Yin et al. 2000, Yin
& Yang 2000a,b) which applied the topology opti-
mization techniques in excavation design have
adopted the linear elastic material model. For sim-
plicity and in order to produce comparable results
with previous works and verifying the current ap-
proach, the linear elastic material model has been
used here. The optimized results can be used in
cases where the linear elastic material behavior can
be assumed. Even in elasto-plastic media the propos-
ing optimization technique can provide the starting
guess design.
The homogeneity assumption is valid in cases of
intact rocks and highly weathered rocks. In case of
massive rocks with few major discontinuities, the
overall behavior of the rock mass is highly influ-
enced by the discontinuities.
In underground excavation it rarely happens that
the ground material can adequately resist the conse-
quences of stress relief. The use of supports is thus
usually unavoidable. A common technique to sup-
port the underground excavations is through rock
bolts and grouting. Using rock bolt systems the rock
mass can effectively be reinforced only where it is
not strong enough. To simplify the numerical model,
homogenized properties can be used to model the
behavior of the reinforced parts of rock mass (Ber-
naud 1995). In this paper, in line with previous pub-
lications in this context (Liu et al. 2008, Yin et al.
2000, Yin & Yang 2000a,b), a linear elastic behavior
has been assumed for reinforced rock. Further dis-
cussion on validity of this type of analysis can be
found in (Yin et al. 2000). The modulus of elasticity
of host rock and reinforced rock are represented by
O
E
and
R
E
respectively. As mentioned before the
void areas are considered to be made of a very weak
material. The modulus of elasticity of this weak ma-
terial is represented by
V
E
and it is assumed that
OV
EE 001.0=. It is also assumed that all these ma-
terials have same Poisson's ratio equal to 0.3.
3 OBJECTIVE FUNCTION AND PROBLEM
STATEMENT
Consider a simple design case depicted in Figure 1.
In this figure,
Γ
represents the boundary of the
opening. The minimum dimensions, shown in the
figure, can be due to some design restrictions. The
placement, orientation and the length of rock bolts
has been depicted by solid line segments in this fig-
ure. The dark shaded area
Ω
with the outer bound-
ary of
Ω
∂
and inner boundary of
Γ
is the reinforced
area of the design. Having found this reinforced area
one can choose the proper location and length of the
reinforcing bars and vice versa. The simultaneous
shape and reinforcement optimization can be viewed
as finding the optimal
Ω
when both its inner and
outer boundaries
Γ
and
Ω
∂
are changing.
Figure 1. A simple design case.
In this paper the mean compliance has been selected
as objective function which is the most common ob-
jective function used in topology optimization prob-
lems. Considering a volume constraint on rein-
forcement material and restricting the size of the
opening, the problem of concern can be expressed as
VV
RR
T
xxx
VV
VV
c
n
=
≤
=
,s.t.
min
,,,
21
uf
K
(1)
where
n
xxx ,,,
21
K
are design variables,
c
is the
mean compliance, f is the nodal force vector, u
stands for nodal displacement vector, and
RVR
VVV ,,
and
V
V are reinforcement and void volume and their
corresponding limits respectively. Minimizing com-
pliance will be equivalent to maximizing the stiff-
ness of structure. The problem thus will be finding
the stiffest design with prescribed opening size and
predefined upper limit for the volume of reinforce-
ment material.
For problems with constant load (where load is
not a function of design variables) sensitivities of
mean compliance can be easily calculated via adjoint

method (Bendsøe & Sigmund 2004) or direct differ-
entiation (Tanskanen 2002) as
u
K
ui
T
ixx
c∂
∂
−=
∂
∂
(2)
where
K
stands for stiffness matrix and
i
x is the i-
th design variable. The stiffness matrix in element
level can be related to design variables by
O
i
Oi
ii
E
xE
xKK )(
)( = (3)
where
E
is the elasticity modulus of the element
i
which is a function of the element's design variable.
O
i
K is the stiffness matrix of element
i
as if it was
made of original rock.
In order to maintain the topology of the hole for
shape optimization, the boundary of the hole should
be determined and only the boundary elements
should be allowed to change. In this paper it is as-
sumed that there is a shotcrete lining around the
opening with material properties similar to that of
reinforced rock. In this manner, in the shape optimi-
zation of the opening, the material can only switch
between void and reinforced rock. In the reinforce-
ment optimization, on the other hand the two mate-
rial phases are original rock and reinforced rock.
4 MATERIAL INTERPOLATION SCHEME
For a general two-phase material case, the interpo-
lated modulus of elasticity can be defined as
(
)
121
)( EExExE −+=
(4)
where
1
E
and
2
E
are Young's moduli of the two
materials. The value of
0
=
x results in
1
EE =
and
thus represents the first material. Similarly the sec-
ond material can be represented by 1
=
x. Using
Equation 4 and 3 in Equation 2, the latter can be re-
written as
i
O
i
T
i
O
i
EEE
x
c
uKu
12
−
−=
∂
∂
(5)
where
i
u
indicates local nodal displacements at the
element level for the i-th element. The change in ob-
jective function due to a change in an element can
then be approximated as
i
i
x
c
xc ∂
∂
∆=∆ (6)
If the material of an element changes, one can calcu-
late the approximate change in objective function by
substituting corresponding
i
x
∆
value in Equation 6.
4.1
Shape optimization of the tunnel
In shape optimization of the opening the two phases
of the material are void and reinforced rock so Equa-
tions 4 and 6 can be rewritten as
(
)
VRV
EExExE
−+=
)( (7)
and
i
O
i
T
i
O
VR
i
E
EE
xc
uKu
−
∆−=∆
(8)
respectively. Note that in Equation 7 the void and
the reinforced rock phases are represented by values
of 0 and 1 for x respectively. Now for an element
changing from void to reinforced rock
(
1+=−=∆
VR
i
xxx ) one can write
V, ∈
−
−=∆ i
E
EE
c
ii
T
i
V
VR
uKu (9)
with V standing for the set of the numbers of cur-
rently void elements. Note that in Equation 9 the i-th
element is void so
V
ii
KK =.
Similarly for an element changing from rein-
forced rock to void
R, ∈
−
=∆ i
E
EE
c
ii
T
i
R
VR
uKu (10)
where R stands for the set of the numbers of cur-
rently reinforced elements.
Based on Equations 9 and 10 we define the fol-
lowing sensitivity numbers for shape optimization of
the opening



∈− ∈−
=R,)(
V,)(
iEEE
iEEE
ii
T
i
VRR ii
T
i
VRV
S
uKu uKu
α
(11)
Here the sensitivity number is defined as the change
in compliance multiplied by the square of the
Young's modulus. This definition prevents infinite
sensitivity numbers for the case of 0=
V
E (hard
kill). Considering this definition, the reinforced ele-
ments with the lowest sensitivity numbers are the
least efficient elements and should be change to
voids while the void elements with the highest sensi-
tivity numbers are the most efficient ones and should
be switched to reinforced rock.
4.2 Reinforcement optimization
In topology optimization of reinforcements, the two
material phases are original and reinforced rock.
The sensitivity numbers can thus be easily obtained
by replacing
V
E
by
O
E
into Equation 11:



∈− ∈−
=R,)(
O,)(
iEEE
iEEE
ii
T
i
ORR ii
T
i
ORO
R
uKu uKu
α
(12)

Here O represents the set of the numbers of original
rock elements.
Implementing Equation 12, the reinforced ele-
ments with the lowest sensitivity numbers are the
least efficient elements and should be changed to
original rock. On the other hand the rock elements
with the highest sensitivity numbers are the most ef-
ficient ones and should be reinforced.
Sensitivity numbers defined in Equations 11 and
12 only differ in constant coefficients and both can
be obtained by multiplying the strain energy of the
elements by the calculated coefficients. Thus the
computational time to solve these two problems is
nearly same as that of a single optimization problem.
5 FILTERING SENSITIVITIES
It is known that some topology optimization meth-
ods, including the BESO method, are prone to nu-
merical instabilities such as the formation of check-
erboard patterns and mesh dependency (Sigmund &
Petersson 1998). One of the simplest approaches
known to be capable of overcoming these two insta-
bilities is filtering the sensitivities (Sigmund & Pe-
tersson 1998, Li et al. 2001, Huang & Xie 2007). In
filtering technique a new sensitivity number is calcu-
lated based on the sensitivity numbers of the element
itself and its surrounding elements. The following
filtering scheme has been used in this paper to calcu-
late the filtered sensitivity numbers
∑
∑
=
=
=
n
jij
n
jijj
i
H
H
1
1
ˆ
α
α
(13)
where
i
α
ˆ is the filtered sensitivity number of the i-th
element, n is the number of elements and
{
}
ijfij
drH −= ,0max (14)
Here
f
r is the filtering radius and
ij
d is the distance
between the centers of the i-th and the j-th elements.
The Equation 13 is actually a weighted average
which results in greater values in elements near the
areas of high sensitivity and vice versa. Using this
filtering scheme will result in stable results and
smoother topologies.
6 BESO PROCEDURE
The BESO procedure iteratively switches elements
between different materials (and voids) based on
their sensitivity numbers. If in the initial design the
materials' volumes are not within the constraints in
Equation 1, then these volumes should be adjusted
gradually to meet the constraints. This can be
achieved by controlling the number of switches be-
tween different materials. Huang & Xie (2007) pro-
posed an algorithm for gradually adjusting the mate-
rials' volumes. However, if one starts from a feasible
design there is no need to change the volume. In this
case the number of adding elements should be equal
to the number of removing elements in order to keep
the volume constant. In the examples solved here a
feasible initial design is used. This reduces the com-
plexity of the algorithm and eases the verification of
the results.
At every iteration a number of elements will
switch between reinforcements and voids to opti-
mize the shape of the opening based on Equation 11.
Then some other switches will be applied between
normal and reinforced rocks to optimize the topol-
ogy of reinforcement's distribution based on Equa-
tion 12. By restricting the program to switch only a
few elements each time, one can prevent sudden
changes to the design. The maximum number of
switches between different elements at each iteration
is referred to as move limit. Using larger move limits
one can obtain faster convergence but may lose
some optimum points. With a small move limit, the
evolution of the objective function should show a
relatively monotonic trend with a steep descent at
the initial iterations reaching a flat line at the end in-
dicating convergency. Getting such evolution trend
one can ensure that the optimization procedure is
working well.
To keep up with the shotcrete lining the elements
on the boundary of the hole should be changed to
shotcrete elements after each update in the hole's
shape. Therefore the number of shotcrete elements
might change during optimization while the total
volume of the reinforced rock and the shotcrete lin-
ing is constrained. In order to satisfy this volume
constraint, in reinforcement optimization the number
of reinforcing and weakening elements should be ad-
justed.
7 EXAMPLES
A simple example has been considered to verify the
proposed BESO algorithm. The relative values of
moduli of elasticity of reinforced rock, original rock,
and void elements have been considered as
10000:3000:3 respectively. It is assumed that the
tunnel is long and straight enough to validate plane
strain assumption. The outer boundaries of the de-
sign domain have been considered as non-designable
rock elements in order to prevent reinforcing of far
fields. Because the discretized domain is very large
in compare to the size of the opening, changes in the
opening's shape will not have a considerable effect
on the overall compliance. The objective function is
thus limited to the compliance of designable domain
only. The filtering radius is considered equal to
twice of the elements' size. The move limit has been
limited to five elements. It is also assumed that the

tunnel should have a flat floor. To fulfill this re-
quirement a layer of non-designable reinforced rock
has been considered at the bottom of the opening.
The initial guess design together with non-
designable elements has been depicted in Figure 2.
The minimum size of the opening is 2.4m×1.6m.
This area is restricted to void elements by setting a
rectangular area of non-designable voids. The size of
the opening is 7.92m
2
. The upper limit for the vol-
ume of the reinforcement material is chosen equal to
14.8m
2
. The infinite domain has been replaced by a
large finite domain of size 20m×20m surrounding
the opening. Because of symmetry only half of the
design domain has been considered in finite element
analysis with proper symmetry constraints. A typical
2D mesh consisting of 50×100 equally sized quadri-
lateral 4-node elements has been used to discretize
the half model.
Figure 2. An initial guess design illustrating the design domain,
non-designable elements, loading, and restraints.
The tunnel is considered under biaxial stresses. To
model the stress conditions uniform distributed loads
with consistent magnitudes have been applied on
top, right and left sides and the bottom is restrained
against vertical displacement (Fig. 2).
Three cases with different values of horizontal to
vertical stress ratio (
λ
) has been considered. Figure
3 shows the final topologies for
4.0
=
λ
, 7.0
=
λ
,
and 2.1
=
λ
. It can be seen that the final shape of the
opening and the final topology of reinforcements
change dramatically with the applied load ratio. The
aspect ratio of the optimum opening shapes show a
correlation with the applied load ratios which is also
reported in (Ren et al. 2005) and (Ghabraie et al.
2007). The evolutions of the objective functions
have been depicted in Figure 4. In all cases the ob-
jective function changes almost monotonically and
smoothly. The initial and the final values of the ob-
jective function are reported in Table 1.
Table 1. The initial and final objective function's values for
the three load cases.
________________________________________________
Case Initial value Final value Difference
________________________________________________
λ=0.4 13.72 11.72 14.57%
λ=0.7 14.73 13.27 9.90%
λ=1.2 22.19 21.01 5.33%
________________________________________________
8
CONCLUSION
The topology optimization of reinforcement around
an underground opening in rock mass and shape op-
timization of the opening itself have been solved si-
multaneously. Among different topology optimiza-
tion methods the BESO method has been chosen due
to its clear topology results and its fast convergence.
The binary nature of the BESO method makes it
suitable for solving shape optimization problems.
However unlike the regular BESO, in this paper a
soft kill approach has been followed and a weak ma-
terial has been used to model void elements. Mean
compliance has been considered as objective func-
tion for the optimization procedures together with
constraints on maximum volume of reinforcements
and on the size of the opening.
The problem then reduced to two two-phase ma-
terial distribution problems. The first problem repre-
sents the shape optimization of the opening where
the material is changing between reinforced rock and
void. The second one relates to the reinforcement
optimization where the two material phases are
original and reinforced rock. The sensitivities of the
objective function with respect to the design vari-
ables have been calculated for these problems. Two
different sensitivity numbers have then been defined
based on the calculated sensitivities. It has been
shown that the two sensitivity numbers only differ in
some constant coefficients. Hence the two optimiza-
tion problems can be solved using nearly same com-
putational effort as required by a single problem.
A shotcrete lining has been assumed around the
opening with mechanical properties similar to that of
reinforced rock. A filtering scheme has been used to
prevent numerical instabilities such as checkerboard
patterns. The filtering approach also smoothes inter-
material boundaries, resulting in a topology free of
jagged edges.
The proposed approach has been verified by solv-
ing a simple example. The evolution of the objective
function shows a smooth, relatively monotonic and
converging curve. The proposed method can be used
to improve the design of underground excavations in
linear elastic and homogeneous rocks. It can also be
used to provide initial designs for excavations in
elasto-plastic media.

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(a) λ=0.4 (b) λ=0.7 (c) λ=1.2
Figure 3. The obtained topologies for different load ratios.
(a) λ=0.4 (b) λ=0.7 (c) λ=1.2
Figure 4. The evolution of the value of the objective function for different load ratios.