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


Abstract and Figures

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.
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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
In excavation design, optimizin
g the reinforcement and finding the optimal shape of the ope
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.
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
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
and it is assumed that
EE 001.0=. It is also assumed that all these ma-
terials have same Poisson's ratio equal to 0.3.
Consider a simple design case depicted in Figure 1.
In this figure,
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
with the outer bound-
ary of
and inner boundary of
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
when both its inner and
outer boundaries
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
xxx ,,,
are design variables,
is the
mean compliance, f is the nodal force vector, u
stands for nodal displacement vector, and
VVV ,,
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
stands for stiffness matrix and
x is the i-
th design variable. The stiffness matrix in element
level can be related to design variables by
xKK )(
)( = (3)
is the elasticity modulus of the element
which is a function of the element's design variable.
K is the stiffness matrix of element
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.
For a general two-phase material case, the interpo-
lated modulus of elasticity can be defined as
)( EExExE +=
are Young's moduli of the two
materials. The value of
x results in
EE =
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
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
= (6)
If the material of an element changes, one can calcu-
late the approximate change in objective function by
substituting corresponding
value in Equation 6.
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
)( (7)
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
xxx ) one can write
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
KK =.
Similarly for an element changing from rein-
forced rock to void
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
VRR ii
uKu uKu
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=
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
into Equation 11:
ORR ii
uKu uKu
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.
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
ˆ is the filtered sensitivity number of the i-th
element, n is the number of elements and
drH = ,0max (14)
r is the filtering radius and
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.
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-
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
. The upper limit for the vol-
ume of the reinforcement material is chosen equal to
. 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 (
) has been considered. Figure
3 shows the final topologies for
, 7.0
and 2.1
. 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%
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.
Bendsøe, M.P. & Sigmund, O. 2004, Topology Optimization -
Theory, Methods and Applications, Springer, Berlin.
Bernaud, D., Debuhan, P. & Maghous, S. 1995, Numerical
simulation of the convergence of a bolt-supported tunnel
through a homogenization method. International Journal
for Numerical and Analytical Methods in Geomechanics
Brady, B.H.G. & Brown, E.T. 2004, Rock Mechanics for Un-
derground Mining, Kluwer Academic Publishers,
Ghabraie, K., Xie, Y.M. & Huang, X. 2007, Shape optimiza-
tion of underground excavation using ESO method, In: Xie,
Y.M. & Patnaikuni, I. (eds.) Innovations in Structural En-
gineering and Construction, Vol. 2: 877-882, Taylor and
Francis, London.
Huang, X. & Xie, Y.M. 2007, Convergent and mesh-
independent solutions for the bi-directional evolutionary
structural optimization method. Finite Elements in Analysis
and Design 43(14):1039-1049.
Jing, L. 2003, A review of techniques, advances and out-
standing issues in numerical modelling for rock mechanics
and rock engineering. International Journal of Rock Me-
chanics and Mining Sciences 40(3):283-353.
Li, Q., Steven, G.P. & Xie, Y.M. 1999, On equivalence be-
tween stress criterion and stiffness criterion in evolutionary
structural optimization. Structural Optimization 18: 67-73.
Li, Q., Steven, G.P. & Xie, Y.M. 2001, A simple checkerboard
suppression algorithm for evolutionary structural optimiza-
tion. Structural and Multidisciplinary Optimization
Liu, Y., Jin, F., Li, Q. & Zhou, S. 2008, A fixed-grid bidirec-
tional evolutionary structural optimization method and its
applications in tunnelling engineering. International Jour-
nal for Numerical Methods in Engineering 73(12):1788-
Querin, O.M, Steven, G.P & Xie, Y.M 1998, Evolutionary
structural optimization (ESO) using a bi-directional algo-
rithm. Engineering Computations 15(8): 1031-1048.
Ren, G., Smith, J.V, Tang, J.W & Xie, Y.M 2005, Under-
ground excavation shape optimization using an evolution-
ary procedure. Computers & Geotechnics 32:122-132.
Sigmund, O. & Petersson, J. 1998, Numerical instabilities in
topology optimization: A survey on procedures dealing
with checkerboards, mesh-dependencies and local minima.
Structural Optimization 16(1):68-75.
Tanskanen, P. 2002, The evolutionary structural optimization
method: theoretical aspects. Computer Methods in Applied
Mechanics and Engineering 191(47-48):5485-5498.
Xie, Y.M. & Steven, G.P. 1993, A simple evolutionary proce-
dure for structural optimization. Computers & Structures
49(5): 885-896.
Xie, Y.M. & Steven, G.P. 1997, Evolutionary Structural Opti-
mization, Springer, London.
Yang, X.Y., Xie, Y.M., Steven, G.P. & Querin, O.M. 1999,
Bidirectional evolutionary method for stiffness optimiza-
tion. AIAA Journal 37(11): 1483-1488.
Yin, L. & Yang, W. 2000a, Topology optimization for tunnel
support in layered geological structures. International
Journal for Numerical Methods in Engineering
Yin, L. & Yang, W. 2000b, Topology optimization to prevent
tunnel heaves under different stress biaxialities. Interna-
tional Journal for Numerical and Analytical Methods in
Geomechanics 24(9):783-792.
Yin, L., Yang, W. & Tianfu, G. 2000, Tunnel reinforcement
via topology optimization. International Journal for Nu-
merical and Analytical Methods in Geomechanics.
(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.
... 92, 93 The BESO algorithm is also known as a discrete method that redundant material is iteratively cleaned up from an object while the efficient part is added simultaneously. The technique was utilized to optimize specific types of structures for either traditional or AM like shape optimization for underground openings, 94 shell structure optimization, 95 and topology optimization of 3D continuum structures. 13 In some cases, BESO method has been demonstrated to be less efficient than SIMP algorithm. ...
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... In multi-phase material distribution problems, the earliest application of the ESO method is published by Rispler and Steven (1995). Despite the apparent simplicity of using soft-kill BESO in two-phase material distribution problems, however, due to the numerical problems that are explained in Ssection 2.5, this method was only recently used to solve these types of problems (see for example (Liu et al. 2008;Ghabraie 2009;Ghabraie et al. 2010a;Ghabraie et al. 2010b;Nguyen et al. 2014)). Huang and Xie (2009) extended the BESO method to solve multi-phase material distribution problems. ...
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In underground excavations, the shape of the opening is of main concern due to its impact on the stress distribution around the opening. However, because of the complex and non-linear behaviour of soil only a few works have dealt with this problem. In this paper an evolutionary structural optimization (ESO) procedure is used for optimizing the shape of underground excavations in a cohesive frictional material obeying Mohr-Coulomb material model. The presented examples show the capability of ESO method in tackling such problems.
Evolutionary structural optimization (ESO) method was originally developed based on the idea that by system-atically removing the inefficient material, the residual shape of the structure evolves toward an optimum. This paper presents an extension of the method called bidirectional ESO (BESO) for topology optimization subject to stiffness and displacement constraints. BESO allows for the material to be added as well as to be removed to modify the structural topology. Basic concepts of BESO including the sensitivity number and displacement extrapolation are proposed and optimization procedures are presented. Integrated with the finite element analysis technique, BESO is applied to several two-dimensional plane stress problems. Its effectiveness and efficiency are examined in comparison with the results obtained by ESO. It is found that BESO is more reliable and computationally more efficient than ESO in most cases. Its capability and limitation are discussed. Nomenclature C = mean compliance E = Young's modulus K* = element stiffness matrix / = thickness of plate Uj = displacement at the constrained location M* = limit of the displacement constraint ii' = element displacement vector due to real load u ij -element displacement vector due to unit virtual load acting at the location of the displacement constraint W = weight of current structure W^ = weight of maximum structure V^ b j = objective weight WSbT (1) = first local minimum of the objective weight \y|) T b 1 j l(2) = second local minimum of the objective weight W 0?t -weight of optimal topology WQ = weight of structure of the full design area W* = target weight a = sensitivity number A = increment v = Poisson's ratio
Underground excavation in either soil or rock induces complex stress redistribution around the opening principally depending on excavation geometry, in situ stresses and material properties. Finding the optimal shape for an excavation based on stress distribution has practical significance in increasing stability and lowering support costs. This paper presents a new approach to the optimization of underground excavation shapes using an evolutionary structural optimization (ESO) procedure. Illustrated examples show that the ESO method is applicable to the optimization of underground excavation shapes. Both two- and three-dimensional cases are considered and compared with theoretical solutions. Optimal shapes of underground excavations under different virgin stress states are also presented.
Floor or sidewall heave imposes a severe issue in tunnel construction. We propose a two-phase mixture model consisting of the original rock and the fully anchored rock mass. The host ground is discretized by finite elements. The relative ratios of two phases in various elements are optimized by optimality criteria method so that the floor or sidewall heave is minimized. The topology of reinforcement is designed to prevent floor heave or sidewall heave of a tunnel surrounded by homogeneous rock. The stress biaxiality k dictates whether the floor heave or the sidewall heave control the tunnel distortion. The results indicate the importance to set reinforcement near the tunnel corners to prevent sidewall and floor heaves. Part of the results is confirmed in the practices.
Anchoring is a fundamental method for supporting tunnels. It reinforces the original rock mass and reduces the deformation along the tunnel surface. The topological complexity of its layouts renders a design methodology difficult. A numerical approach to reinforce the host ground becomes desirable. The present paper proposes a topology optimization method based on a two-phase cell model and finite-element discretization of the host ground. The element consists of the original rock and the reinforcing material. The design issue involves the distribution of the reinforcing materials. The relative ratios of the two phases in various elements will be optimized to reduce the compliance of the tunnel.The method enables the computer-aided design for the support of underground structures. The capabilities of the method are demonstrated by the designs to support a deep tunnel under various in situ stresses. The results indicate that oriented reinforcement is needed along the direction of the largest absolute value of the principal stress.
The evolutionary structural optimisation (ESO) method has been under continuous development since 1992. Traditionally, the method was conceived from the engineering perspective that the topology and shape of structures were naturally conservative for safety reasons and therefore contained an excess of material. To move from the conservative design to a more optimum design would therefore involve the removal of material. Thus the ESO method started from a design space much bigger than the optimum and the final topology or shape emerged by a process of removal of unwanted/inefficient/lowly stresses material. The original algorithms allowed for two forms of evolution. One was there the understressed material could be removed from anywhere in the allowable design space, and with compensation for checker-boarding this produces an optimum topology under the prescribed environments. The second form only allows removal from the surface or parts of the surface (called nibbling in the ESO lexicon); this produces a Min–Max situation where the maximum surface stress is reduced to a minimum. It has been demonstrated that the ESO process produces a surface that is an iso-stress contour thus satisfying the Min–Max optimality criterion. The present paper addresses the opposite evolutionary process whereby the structure evolves from a base which is the minimum structural form required to carry the load regardless of the magnitude of the stress levels. Material is added in the proximity of high stress to ameliorate its effect and hence the final structural form emerges. Only singly connected regions are formed in the present analysis and thus the additive ESO process is the opposite of the nibbling SO, mentioned above, that produces optimum surface shapes. The paper presents a brief background to the current state of structural optimisation research. This is followed by a discussion of the strategies for the additive ESO (AESO) algorithm and two examples are presented.
Tunnel construction commonly proceeds in an environment of layered geological formation. Design for tunnel support relies on the tunnel location and the mismatch of different layers. The present paper proposes a topology optimization method for the design of tunnel support. The design domain is discretized by finite elements. An element is composed of the original rock (hard or soft) and the reinforcing material (rock reinforced by grouting or bolting). The design issue involves the distribution of reinforcements. We model the reinforced host ground by a power-weighted mixture law. The ratios of two phases in various elements are optimized to minimize the deformation of the tunnel. The method enables a computer-aided design for the supports of underground tunnels embedded within layered geological structures. The reinforced areas for the tunnel are displayed under the passive geological stress and the active external loading. Four kinds of rock formations are examined. They are homogeneous rock, hard–soft–hard (HSH) sandwich structure, two-layer structure with the soft rock at the top and the hard rock at the bottom (SH), and the one with the hard rock at the top and the soft rock at the bottom (HS). The simulation reveals the high efficiency of tunnel support by optimizing its topologies. Copyright © 2000 John Wiley & Sons, Ltd.
A simple evolutionary procedure is proposed for shape and layout optimization of structures. During the evolution process low stressed material is progressively eliminated from the structure. Various examples are presented to illustrate the optimum structural shapes and layouts achieved by such a procedure.