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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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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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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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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.

References

( 1 ) Ren, G., Smith, J.V., Tang, J.W. and Xie, Y.M., Underground excavation shape optimiza-

tion using an evolutionary procedure. Computers and Geotechnics, Vol.32,No.2 (2005),

pp.122–132.

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