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Journal of Computational
Science and
Technology
Vol.4, No.1, 2010
Shape and Reinforcement Optimization of
Underground Tunnels∗
Kazem GHABRAIE∗∗,YiMinXIE
∗∗∗, Xiaodong HUANG∗∗∗
and Gang REN∗∗∗
∗∗ Faculty of Engineering and Surveying, University of Southern Queensland
West Street, Toowoomba, QLD 4350, Australia
E-mail: kazem.ghabraie@usq.edu.au
∗∗∗ School of Civil, Environmental and Chemical Engineering, RMIT University
GPO Box 2476V, Melbourne, VIC 3001, Australia
Abstract
Design of support system and selecting an optimum shape for the opening are two im-
portant steps in designing excavations in rock masses. Currently selecting the shape
and support design are mainly based on designer’s judgment and experience. Both
of these problems can be viewed as material distribution problems where one needs
to find the optimum distribution of a material in a domain. Topology optimization
techniques have proved to be useful in solving these kinds of problems in structural
design. Recently the application of topology optimization techniques in reinforcement
design around underground excavations has been studied by some researchers. In this
paper a three-phase material model will be introduced changing between normal rock,
reinforced rock, and void. Using such a material model both problems of shape and
reinforcement design can be solved together. A well-known topology optimization
technique used in structural design is bi-directional evolutionary structural optimiza-
tion (BESO). In this paper the BESO technique has been extended to simultaneously
optimize the shape of the opening and the distribution of reinforcements. Validity and
capability of the proposed approach have been investigated through some examples.
Key words : Underground Excavation, Tunnel Reinforcement, Topology Optimization,
Shape Optimization, Finite Element Analysis
1. Introduction
During the last two decades topology optimization has attracted considerable attention
and the techniques in this field have been improved significantly. 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 potential 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 underground openings(1), (2) while others tried to optimize
the topology of reinforcement around a tunnel with a predefined shape(3) – (6). In this paper,
attempt is made to optimize the shape of the opening and the topology of the surrounding
reinforcement simultaneously. It will be shown that the sensitivities of these two optimization
problems only differ in constant values. Hence the two optimization problems can be solved
together with almost no extra computational effort.
The topology optimization techniques used in aforementioned works are the Solid
Isotropic Material with Penalization (SIMP) method, the Evolutionary Structural Optimiza-
tion (ESO) technique and its Bi-directional version, the BESO method. These methods are
the most popular topology optimization techniques due to their ease of use and reasonable
results. All these methods control the topology using a material distribution approach, that is,
they change the topology by adding and removing material to and from the design domain.
∗Received 20 Nov., 2009 (No. 09-0708)
[DOI: 10.1299/jcst.4.51]
Copyright c
2010 by JSME
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Journal of Computational
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Originally proposed by Bendsøe in 1989(7), the SIMP method is based on the homog-
enization technique which was introduced earlier by Bendsøe and Kikuchi(8). The SIMP
material model assumes that the Young’s modulus of the designing material is a function
of the material’s relative density (E=E(ρ)). The relative density function varies between a
very small positive value ρmin (representing voids) and 1 (representing solids). These relative
densities are treated as design variables and the SIMP method adjusts them iteratively using a
gradient-based update scheme to achieve the optimum of the chosen objective function(7) .The
relation between the relative density and the modulus of elasticity in the SIMP method is a
power-law relationship defined as E=ρpE0where pis a penalty factor and E0is the modulus
of elasticity of the base material(7), (9). The penalty factor penalizes the intermediate values
of the relative densities and pushes the final topology towards a binary solid/empty topology.
Bigger penalties result in more clear topologies but they will cause some convergency prob-
lems. On the other hand small penalties will result in blurred topologies with lots of elements
having intermediate (virtual) density values. Even with large penalty factors the final result
will still have some elements with intermediate densities.
The BESO method was proposed in late 90s(10), (11) as an improved version of the ESO
method which was originally introduced in early 90s by Xie and Steven(12), (13).TheESO
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(14).
Themaindifference between the SIMP and the BESO methods is in the nature of the
design variables they use. While the SIMP material model uses continuous design variables,
the BESO method uses discrete values. This property of the BESO technique assures that the
results are always clear because no intermediate density is in use. The clearness of the results
is important in shape optimization since the boundaries in a blurred image are not easily
definable. For this reason the BESO technique is easier to implement for shape optimization
of the opening. In this paper the BESO method is used for solving both problems of shape
optimization of the opening and topology optimization of reinforcements. 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, unlike the SIMP material model, elements can be removed com-
pletely. Topology optimization methods which remove elements completely are sometimes
referred to as hard kill methods as oppose to soft kill methods like the SIMP method. 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 hard kill methods the sensitivity of void ele-
ments cannot be calculated directly from the analysis results and should be extrapolated 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 material. In this manner the sensitivities of voids
are directly calculable.
2. Material Model
Although the geomechanical materials are naturally inhomogeneous, non-linear,
anisotropic, and inelastic(15), in excavation design in rocks, modeling them as an isotropic,
homogeneous material and assuming linear elastic behavior can be instructive and sometimes
can predict the real behavior with acceptable accuracy(16). In fact the simplified linear elastic
material model is still the most common material model used in geomechanics(15). Moreover
the results of a linear analysis can be used as a first-order approximation of non-linear cases.
Most of previous works(1), (3) – (6) which applied the topology optimization techniques in exca-
vation design, have adopted the linear elastic material model. As the application of topology
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Journal of Computational
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Vol.4, No.1, 2010
optimization techniques in excavation design has not been investigated thoroughly and in or-
der to produce comparable results with previous works and verifying the current approach,
the linear elastic material model has been used here. The authors are currently working on
implementing elasto-plastic material models.
The homogeneity assumption is valid in case of intact rock and highly weathered rock
mass. In case of massive rocks with few major discontinuities, the overall behavior of the
rock mass is predominantly influenced by the discontinuities. Hence, in these types of rocks,
the homogeneity assumption is far from reality. Reinforcement optimization of underground
excavations in such types of rocks is addressed in Ref. (17).
In underground excavation it rarely happens that the ground material can adequately
resist the consequences of stress relief. The use of supports is thus usually unavoidable. Active
support design is based on the idea that the soil or rock mass is actively contributing in carrying
the load caused by excavation(4). One way of such supporting design is using rock bolts. Using
rock bolt systems, the rock mass can effectively be reinforced only where it is not strong
enough. Homogenized properties can be used to model the behavior of the reinforced parts of
rock mass(18). In this paper, in line with previous publications(3) – (6) , a linear elastic behavior
is assumed for reinforced rock. Further discussion on validity of this type of analysis can be
found in Ref. (4). The moduli of elasticity of host rock and reinforced rock are represented
by EOand ERrespectively. As mentioned before, the void areas are considered to be made of
a very weak material. The modulus of elasticity of this weak material is represented by EV
and it is assumed that EV=0.001EO. It is also assumed that all these materials have same
Poisson’s ratio equal to 0.3.
3. Objective Function and Problem Statement
Consider a simple design case depicted in Fig. 1. In this figure, Γrepresents 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 figure. The dark shaded area Ωwith the outer boundary of ∂Ωand
inner boundary of Γis 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. In the shape
optimization of the opening one deals with finding Γwhile in reinforcement optimization the
shape and topology of ∂Ωand Ωare of interest. The simultaneous shape and reinforcement
optimization can be viewed as finding the optimal Ωwhen both its inner and outer boundaries
Γand ∂Ωare changing.
Fig. 1 A simple design case.
Ren et al.(1) and Ghabraie et al.(2) have used a full stress design strategy to optimize the
shape of the underground opening. In full stress design strategy, it is (intuitively) assumed
that in the optimal design all parts of the structure are fully stressed. However, in spite of
reasonable results, this approach is based on intuition lacking a mathematically proven basis.
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On the other hand calculating sensitivities for stress based objective functions is not an easy
task. In this paper the mean compliance has been used as objective function which is the most
common objective function used in topology optimization problems. Considering a volume
constraint on reinforcement material and restricting the size of the opening, the problem of
concern can be expressed as
minx1,x2,...,xnc=fTu
such that VR≤¯
VR,
VV=¯
VV
(1)
where x1,x2,...,xnare design variables, cis the mean compliance, fis the nodal force vector,
ustands for nodal displacement vector, and VR,VV,¯
VRand ¯
VVare reinforcement and void
volume and their corresponding limits respectively. It can be seen from Eq. (1) that the mean
compliance is equivalent to twice of the strain energy. Minimizing compliance will be equiv-
alent to maximizing the stiffness of structure. The problem thus will be finding the stiffest
design with prescribed opening size and predefined upper limit of the volume of reinforce-
ment material.
Because the mean compliance is a convex and self-adjoint function its sensitivity analysis
is relatively easy and computationally efficient(19). 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(19) or direct differentiation(20) as
∂c
∂xi
=−uT∂K
∂xi
u(2)
where Kstands for stiffness matrix and xiis the i-th design variable. The stiffness matrix in
element level can be related to design variables by
Ki(xi)=E(xi)
EOKO
i(3)
where Eis the elasticity modulus of the element iwhich is a function of the element’s design
variable. KO
iis the stiffness matrix of element ias if it was made of original rock, that is
KO
i=Vi
BT
iDOBidV(4)
where DOrepresents the constitutive matrix for original rock, Birepresents the strain-
displacement matrix and Viis the volume of element i.
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 assumed that there is a shotcrete lining around the opening with material
properties similar to that of reinforced rock. In this manner, in the shape optimization of the
opening, the material can be changed from void to reinforced rock and vice versa. In the
reinforcement optimization, on the other hand the two material phases are original rock and
reinforced rock.
4. Material Interpolation Scheme
For a general two-phase material case, the interpolated modulus of elasticity can be de-
fined as
E(x)=E1+x(E2−E1)(5)
where E1and E2are Young’s moduli of the two materials. Using Eq. (5) in Eq. (2), the latter
can be rewritten as
∂c
∂xi
=−E2−E1
EOuT
iKO
iui(6)
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where uiindicates local nodal displacements at the element level for the i-th element. The
change in objective function due to a change in an element can be approximated as
Δc=Δxi
∂c
∂xi
(7)
If the material of an element changes, one can calculate the approximate change in objective
function by substituting corresponding Δxivalue in Eq. (7).
4.1. Shape optimization of the tunnel
In shape optimization of the opening the two phases of the material are void and rein-
forced rock so Eqs. (5) and (7) can be rewritten as
E(x)=EV+x(ER−EV)(8)
and
Δc=−Δxi·ER−EV
EOuT
iKO
iui(9)
respectively. Note that in Eq. (8) the void and the reinforced rock phases are represented by
values of 0 and 1 for xrespectively. Now for an element changing from void to reinforced
rock (Δxi=xR−xV=+1) one can write
Δc=−ER−EV
EVuT
iKiui,i∈V (10)
with Vstanding for the set of the numbers of currently void elements. Note that in Eq. (10)
the i-th element is void so Ki=KV
i.
In the same way when an element switches from reinforced rock to void, the change in
objective function can be approximated as
Δc=ER−EV
ERuT
iKiui,i∈R (11)
Here Ris the set of the numbers of currently reinforced elements. The positive sign in Eq. (11)
shows that changing a reinforced element to a void will increase compliance (decrease stiff-
ness) while the negative sign in Eq. (10) states that replacing a void element with a reinforced
one will make the structure stiffer.
Based on Eqs. (10) and (11) one can define the following sensitivity number for shape
optimization of the opening
αS=⎧
⎪
⎨
⎪
⎩
EV(ER−EV)uT
iKiui,i∈V
ER(ER−EV)uT
iKiui,i∈R (12)
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
EV=0. Furthermore in this way, the stiffer materials get higher sensitivity numbers than the
softer materials.
Considering the sensitivity numbers defined in Eq. (12) the reinforced elements with the
lowest sensitivity numbers are the least efficient elements and should be change to voids while
the void elements with the highest sensitivity numbers are the most efficient ones and should
be switched to reinforced rock.
4.2. Reinforcement optimization
In topology optimization of reinforcements, original rock elements can be turned into
reinforced elements and vice versa. The interpolation equation Eq. (5) can hence be written
as
E(x)=EO+x(ER−EV) (13)
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This case is completely similar to that of shape optimization. The only difference is that EVis
replaced by EOhere. Similar to the shape optimization case, a change in the objective function
due to reinforcing an element can be calculated as
Δc=−ER−EO
EOuT
iKiui,i∈O (14)
where Ois the set of the original rock elements’ numbers. On the other hand if an element
changes from reinforced rock to original rock the change in objective function will be
Δc=ER−EO
ERuT
iKiui,i∈R (15)
Based on Eqs. (14) and (15) the following sensitivity number can be defined for the reinforce-
ments optimization
αR=⎧
⎪
⎨
⎪
⎩
EO(ER−EO)uT
iKiui,i∈O
ER(ER−EO)uT
iKiui,i∈R (16)
The reinforced elements 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 efficient ones and should be changed to reinforced rock.
Sensitivity numbers defined in Eqs. (12) and (16) only differ in constant coefficients and
both can be calculated by multiplying the strain energy of the elements by the calculated
coefficients. That means the computational time to solve these two problems is nearly same
as that of a single problem.
5. FILTERING SENSITIVITIES
It is known that some topology optimization methods including the BESO and the SIMP
methods are prone to numerical instabilities like the formation of checkerboard patterns and
mesh dependency(22). One of the simplest approaches which is known to be capable of over-
coming these two instabilities is filtering the sensitivities(22) – (24). In filtering technique a new
sensitivity number will be calculated based on the sensitivity numbers of the element itself
and its neighboring elements. The following filtering scheme has been used in this paper to
calculate the filtered sensitivity numbers
ˆαi=n
j=1αjHij
n
j=1Hij
(17)
where ˆαiis the filtered sensitivity number of the i-th element, nis the number of elements and
Hij =max{0,rf−dij}(18)
Here rfis the filtering radius and dij is the distance between the centers of the i-th and the
j-th elements. Note that the equation Eq. (17) 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 smoother results. The smoothness of the final result increases by using
larger filtering radii. However one should note that choosing a very large filtering radius can
result in convergency problems and sub-optimal solutions.
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 Eq. (1), then these volumes will be adjusted gradually to meet
the constraints. This can be achieved by controlling the number of switches between different
materials. In Ref. (24) an algorithm has been proposed for gradually adjusting the materials’
volumes. If one starts from a feasible design there is no need to change the volume. In this case
the number of elements to be added or removed should be equal to keep the volume constant.
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In the examples solved here a feasible initial design is used. At every iteration a number of
elements will switch between reinforcements and voids to optimize the shape of the opening
based on Eq. (12). Then some other switches will be applied between normal and reinforced
rocks to optimize the topology of reinforcement’s distribution based on Eq. (16). 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 re-
ferred 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 initial iterations reaching a
flat line at the end showing 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 value of reinforced
rock and shotcrete is constrained. In order to satisfy this volume constraint, in reinforcement
optimization the number of reinforcing and weakening elements should be adjusted.
The algorithm of the BESO procedure used here is briefly reviewed in Fig. 2.
Fig. 2 The flowchart of the BESO procedure for optimizing the shape and the
reinforcement of tunnels.
7. Examples
For verification purpose the proposed BESO algorithm has been used to solve some ex-
amples. In these examples the relative values of moduli of elasticity of reinforced rock, origi-
nal rock, and void elements have been considered as 10000:3000:3 respectively. In all cases it
is assumed that the tunnel is long and straight enough to validate plane strain assumption. The
semi-infinite underground domain has been modeled by a large finite element mesh. In all
examples the outer boundaries of the design 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. In all examples it is
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assumed that the tunnel should have a flat floor. To fulfill this requirement in all examples a
layer of non-designable reinforced rock has been considered at the bottom of the opening.
7.1. Example 1
In the first example, a single tunnel under biaxial stress like the one sckeched in Fig. 1
has been considered. An initial guess design together with non-designable elements has been
depicted in Fig. 3. The minimum size of the opening is W=2.4m and H=1.6m. This
area is restricted to void elements by setting a rectangular area of non-designable voids. The
size of the opening is 7.92m2. The upper limit for the volume of the reinforcement material
is chosen equal to 14.8m2. The infinite domain has been replaced by a large finite domain of
size 20m ×20m surrounding the opening. 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. 3).
Fig. 3 An initial guess design illustrating the design domain, non-designable elements,
loading, and restraints.
Because of symmetry only half of the design domain has been considered in finite ele-
ment analysis with proper symmetry constraints. A typical 2D mesh consisting of 50 ×100
equally sized quadrilateral 4-node elements has been used to discretize the half model.
Three cases with different values of horizontal to vertical stress ratio (λ) has been consid-
ered. Figure 4 show the final results together with corresponding objective function evolution
for λ=0.4, λ=0.7, and λ=1.2. As shown in these figures in all cases the objective function
changes almost monotonically and smoothly. It can be seen that the final shape of the open-
ing and the final topology of reinforcements change dramatically with load ratio. The aspect
ratio of the optimum opening shapes show a correlation with the applied load ratios which is
also reported in Ref. (1) and (2). The initial and the final values of the objective function are
reported in Table 1.
After obtaining the results, the boundaries of the opening can be smoothed using a post-
processing subroutine. A smoothing procedure based on B´
ezier curves has been applied in
this paper. After obtaining the final shapes, the boundaries between different materials are
extracted. The boundary lines are then smoothed using B ´
ezier curves. Finally the location of
smoothed nodes are used to produce a new mesh. The smoothed results of this example are
shown in Fig. 5. Knowing the reinforced area one can choose the location and length of the
rockbolts and the thickness of shotcrete.
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Fig. 4 The final topology and the evolution of the value of the objective function for
load ratios of 0.4 (left), 0.7 (middle) and 1.2 (right) in example 1.
Table 1 The initial and final objective function’s values for three cases in example 1.
Case Initial value Final value Improvement(%)
λ=0.413.72 11.72 14.57
λ=0.714.73 13.27 9.90
λ=1.222.19 21.01 5.33
Fig. 5 The smoothed results of the first example.
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7.2. Example 2
In the second example a distributed traffic load is applied on the tunnel’s floor over the
width of non-designable void elements (2.4m). The magnitude of this traffic load has been
considered as σtra f f ic =0.25σ1. The smoothed final results for the three load cases are
illustrated in Fig. 6. As expected the reinforcement topology has changed in compare to the
results obtained in previous example. Especially for the case of λ=0.7 the unreinforced
area under the tunnel’s floor in previous example has been replaced by a reinforced bar. The
optimum shapes of the hole have also changed in compare to the first example. It can be seen
that the optimal tunnels have wider floors and are shorter than the shapes obtained in the first
example.
Fig. 6 The smoothed optimum results for problems with traffic load and load ratios of
0.4 (left), 0.7 (middle), and 1.2 (right) in example 2.
7.3. Example 3
Figure 7 shows a sketch of the loading and design restrictions for two identical parallel
tunnels. The minimum allowable dimensions of the opening is restricted to the width of Wand
height of H. It is also required that the tunnel has a flat floor. The area of the opening should
be fixed to A. The tunnel is under biaxial stress state with the horizontal to vertical stress ratio
of λ=σ3/σ1.Thetraffic loads of trains can be considered as a uniform distributed load over
the width W.
Fig. 7 A simple sketch of the loading and design restrictions of two parallel tunnels.
The initial guess design and the design domain have been illustrated in Fig. 8. It is
assumed that the minimum width and height of the opening are W=2.4m and H=2.8m.
The total area of the opening is restricted to A=16m2. The magnitude of the traffic load is
considered equal to σtra f f ic =0.25σ1. The semi-infinite domain is replaced by a 40m ×40m
area. An area of 20m ×20m surrounding the tunnels has been selected as the design domain.
The upper limit for the volume of the reinforcement material is considered as 58.88m2.Again
because of symmetry only half of the design domain has been considered in the FEA.
The optimum designs obtained for the three load ratios of 0.4, 0.7 and 1.2 are depicted
in Fig. 9. It can be seen that by increasing horizontal pressure the reinforcements around the
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Fig. 8 An initial guess design illustrating the design domain, non-designable elements,
loading, and restraints for two parallel tunnels.
two tunnels tend to join.
Fig. 9 The smoothed optimum designs for load ratios of λ=0.4, λ=0.7, and
λ=1.2 (respectively from left to right) for the problem of two parallel tunnels
in example 3.
8. Conclusion
The topology optimization of reinforcement around an underground opening in rock
mass and shape optimization of the opening itself have been solved simultaneously. Among
different topology/shape optimization methods available, the BESO method has been cho-
sen due to its clear topology results and its fast convergence. However contrary to common
BESO, in this paper a soft kill approach has been followed and a weak material has been used
to model void elements. Mean compliance has been considered as the objective function for
the optimization procedures together with constraints on maximum volume of reinforcements
and on the size of the opening.
The problem is then reduced to two two-phase material distribution problems. The sen-
sitivities of the objective function with respect to the design variables 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 optimization problems can be solved using nearly same
computational effort as required by a single problem.
A shotcrete lining has been assumed around the opening with mechanical properties
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Journal of Computational
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similar to that of reinforced rock. A filtering scheme has been used to prevent numerical
instabilities such as checkerboard patterns. The filtering approach also smooths inter-material
borders, resulting in a topology free of jagged edges.
To validate the method, some numerical examples have been solved. The evolution of
the objective function shows a smooth, relatively monotonic and converging curve. A post-
processor has been used to smooth the boundaries of resulted topologies based on B´
ezier
curves. The capability of the method has been tested by solving numerical examples with
different loading conditions. The optimization of two parallel tunnels has also been addressed.
It has been demonstrated via these examples that the extended BESO technique developed
in this paper is capable of solving a range of topology and shape optimization problems in
underground excavations.
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