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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.