COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Multi-Objective Optimization for Shape Design of Arch Dams
Linsong Sun 1*, Weihua Zhang 1, Nenggang Xie 2
1
2
College of Hydraulic Science & Engineering, Yangzhou University, Yangzhou, 225009 China
College of Mechanical Engineering, Anhui University of Technology, Maanshan, 243002 China
Email: [email protected]
Abstract A multi-objective optimization model is established for the shape design of arch dams. In this
model, the geometric parameters describing dam shape is taken as design variables; four objectives are
considered of dam volume, maximal tensile stress, maximal pressure stress and relative depth of the zone
with tensile stress that is larger than 1.0MPa; and the constraints include geometric constraints, dam
volume constraint and mechanical property constrains, i.e., stress constraints, stability constraints, etc.
Traditionally, multi-objective optimization schemes transform multiple objective functions into a single
objective function in some manner, weighted sum method and utopia point method for examples, and the
resulting problem is solved as a single objective optimization problem. In this paper the cooperative game
model analogy to the multi-objective optimization is proposed with each player correspond to one of the
objective functions. After defining the utility of each player, the Nash arbitrary scheme is used to solve the
cooperative game. The optimization of BAIHETAN arch dam, which is located in Sichuan province of
China, is calculated as an engineering example. The results are compared to those obtained by weighted
sum method and utopia point method and indicate that the cooperative game method is superior to
traditional methods. The optimal design saves 9.77×104m3 of dam volume, compared to initial design,
along with the decrease of maximal tensile stress with 38.22%, the decrease of maximal pressure stress
with 28.83%, and the decrease of relative depth of large tensile stress zone with 10.15%.
Key words: multi-objective optimization; shape design; cooperative game; arch dam
INTRODUCTION
Shape optimization of arch dams starts from 1970s. The early studies focus on economic aspect that is to
minimize dam volume [1]. In recent decades the safety aspect has been taken into account more and more
in shape optimization of arch dams. Sun et al [2] discussed the optimal design of arch dam with maximum
tensile stress as objective to be minimized. Li [3] discussed the equivalence of safe model and economic
model in arch dam optimization, and a bin-objective optimization model is established to minimize both
dam volume and maximum stress. Xie et al [4] proposed a multi-objective optimization model for arch
dam design with dam volume, maximum tensile stress and maximum strain energy to be minimized. Wang
et al [5] introduced a flexible modeling method for multi-objective optimization of arch dams. The dam
volume is taken as economic index. Stress level, probability of failure and area of high stress zone are
taken as safety indices. The different types of optimal model can be formed according to decision-maker’s
favor.
Multi- objective optimization methods appeared for the first time in economics, and a growing interest
arose in engineering. Many multi-objective structural optimization examples, solved with different
methods, may be found in a survey paper of Marler et al [6] Sun et al [7] solved the bin-objective
optimization problem for shape design of arch dams with fuzzy theory. The non-inferior solutions are
obtained by weighted sum method first, and the optimal solution is selected according to grade of fuzzy
closeness. In the work of Wang et al [5], the optimal solution was obtained with utopia point method. Xie
⎯ 1009 ⎯
et al [4] proposed a fuzzy evaluation function, which is the combination of grade of membership of
objectives, for the multi-objective optimization of arch dams under the actions of static and dynamic load.
In this paper, the multi-objective optimization of arch dams is discussed with four objectives, i.e., dam
volume, maximum tensile stress, maximum pressure stress and relative depth of large stress zone. The
optimization of BAHETAN arch dam located in Sichuan province of China is presented as an example.
The cooperative game method is used to treat the presence of multiple objectives. The results obtained are
presented and compared with those obtained by other multi-objective optimization techniques as weighted
sum method and utopia point method.
GEOMETRIC MODEL OF PARABOLIC DOUBLE-CURVATURE ARCH DAM’S SHAPE
The shape of arch dams is usually described by center vertical cantilever and some horizontal arch rings in
reference elevations.
ϕL
ϕR
ϕm
Figure 1: The cantilever of arch dam
Figure 2: Horizontal arch ring
1. The geometric description of center cantilever The center cantilever, which is shown in Figure1, can
be described by its upstream curve ycu and width Tc. Suppose they are cubic polynomial function of z, i.e.
y cu ( z ) = a0 + a1 z + a 2 z 2 + a3 z 3
(1)
Tc ( z ) = b0 + b1 z + b2 z 2 + b3 z 3
(2)
Where a0 ~ a3 and b0 ~ b3 can be expressed as follows by ycu ( z ) and Tc (z ) at the four reference
elevations with z coordinates of z1, z2, z3 and z4.
⎧a0 ⎫ ⎡1
⎪a ⎪ ⎢
⎪ 1 ⎪ ⎢1
⎨ ⎬=⎢
⎪a 2 ⎪ ⎢1
⎪⎩a3 ⎪⎭ ⎢⎣1
z1
z12
z2
z3
z
z
z4
z
2
2
2
3
2
4
⎧b0 ⎫ ⎡1
⎪b ⎪ ⎢
⎪ 1 ⎪ ⎢1
⎨ ⎬=⎢
⎪b2 ⎪ ⎢1
⎪⎩b3 ⎪⎭ ⎢⎣1
z1
z12
z2
z3
z 22
z 32
z4
z 42
z13 ⎤
⎥
z 23 ⎥
z 33 ⎥
⎥
z 43 ⎥⎦
−1
z13 ⎤
⎥
z 23 ⎥
z 33 ⎥
⎥
z 43 ⎥⎦
−1
⎧ ycu ( z1 ) ⎫
⎪ y ( z )⎪
⎪ cu 2 ⎪
⎬
⎨
⎪ ycu ( z 3 ) ⎪
⎪⎩ ycu ( z 4 )⎪⎭
(3)
⎧Tc ( z1 ) ⎫
⎪T ( z )⎪
⎪ c 2 ⎪
⎬
⎨
⎪Tc ( z 3 ) ⎪
⎪⎩Tc ( z 4 )⎪⎭
(4)
From Eqs. (1) and (2), The downstream curve of cantilever is expressed as following:
⎯ 1010 ⎯
y cd ( z ) = y cu ( z ) + Tc ( z )
(5)
So, the center cantilever can be defined by character parameters of y cu ( z i ) and Tc ( z i ) at the four
reference elevations.
It should be note that, for the convenient of construction, the overhangs on upstream and downstream, i.e.
KU and K D respectively, must be constrained, and they can be expressed as
′ ( H ) = a1 + 2a 2 H + 3a 3 H 2
K U = y cu
(6)
′ (0) = y cu
′ (0) + Tc′(0) = a1 + b1
K D = y cd
(7)
2. The geometric description of horizontal arch ring The arch ring shown in Figure 2 can be described
by axial curve and ring thickness. Take left half for example, the axis of parabolic arch ring can be
expressed by parameter variable ϕm as
⎧ xm = RCL tan ϕ m
⎪
x m2
⎨
y
y
=
+
c
⎪ m
2 RCL
⎩
(8)
1
Where yc = ycu + Tc is the y coordinate of ring axis at arch center, RCL is the radius of axis at arch
2
center.
The thickness of arch ring is supposed to change by ϕm as
Tm = Tc + [TL − Tc ] ⋅
1 − cos ϕ m
1 − cos ϕ L
(9)
where Tc and TL are the thickness at arch center and arch end respectively. ϕ L is called semi-center
angle and can be expressed as
ϕ L = tan −1[
XL
]
RCL
(10)
in which, X L is the length of left chord of arch axis and can be treated as a constant at certain elevation.
After the determining of arch axis and thickness, the upstream curve and downstream curve can be
calculated by the following formulas
xmu = xm + 0.5Tm sin ϕ m ⎫
⎬
y mu = y m − 0.5Tm cos ϕ m ⎭
(11)
xmd = xm − 0.5Tm sin ϕ m ⎫
⎬
y md = y m + 0.5Tm cos ϕ m ⎭
(12)
So, if the center cantilever has been defined, the shape of left half of certain arch ring can be determined by
RCL and TL , and similarly, the shape of right half can be determined by RCR and TR corresponding to
the radius of right half axis at arch center and the thickness at right arch end respectively.
As to the whole dam, it is supposed that RCL , TL , RCR and TR are cubic polynomial functions of z, i.e.
RCL ( z ) = c0 + c1 z + c2 z 2 + c3 z 3
(13)
TL ( z ) = d 0 + d1 z + d 2 z 2 + d 3 z 3
(14)
RCR ( z ) = e0 + e1 z + e2 z 2 + e3 z 3
(15)
⎯ 1011 ⎯
TR ( z ) = f 0 + f1 z + f 2 z 2 + f 3 z 3
(16)
in which, the coefficients c0 ～ c3 , d 0 ～ d 3 , e0 ～ e3 and f 0 ～ f 3 can, similarly to center cantilever, be
expressed by corresponding parameters RCL ( z i ) , TL ( z i ) , RCR ( z i ) and TR ( z i ) at the four reference
elevations with z coordinates of z1, z2, z3 and z4.
MULTI-OBJECTIVE OPTIMIZATION MODEL FOR ARCH DAM DESIGN
1. General description of multi-objective optimization problem A constrained multi-objective
optimization problem can be mathematically formulated as follows:
min F ( X ) = [ f 1 ( X ) f 2 ( X ) L f m ( X )]T
s.t. X iL ≤ X i ≤ X iU i = 1,2,L , n
g j ( X ) ≤ 0 j = 1,2, L p
hk ( X ) = 0 k = 1,2, L q
⎫
⎪
⎪
⎬
⎪
⎪⎭
(17)
where F is the optimal objective vector of the scalar objective functions, in number of m, X is the design
variables vector, X iL and X iU are the lower and upper bounds for the ith design variable, in number of n,
gj(X) are the inequality constraints, in number of p, hk(X) are the equality constraints, in number of q.
2. Design variables in arch dam optimization From above text, it is known that the shape of parabolic
double curvature arch dam can be described by 24 character parameters of y cu (z ) , Tc (z ) , RCL ( z ) ,
TL ( z ) , RCR ( z ) and TR ( z ) at the four reference elevations. So, in the shape optimization of arch dams, it’s
very natural to take these parameters as design variables.
3. Objective functions in arch dam optimization The aims of arch dam optimization contain two aspects
of economy and safety. The dam volume can be taken as the economical objective function, which is
simple relatively. The objective function of safety is complicated. The maximum tensile principal stress is
taken as safety objective function in [1]. But it is not enough to use maximum stress as safety index alone.
A dam with smaller region of large stress is safer than that with larger region of large stress, even if they
have the same maximum stress. So, it's necessary to take the region of high stress as another safety
objective function. To convenient for the comparison between different design schemes, the relative depth
of large tensile stress zone, which is the radio of the depth of large tensile stress zone on dam base to
corresponding dam thickness, is taken as a safety objective function. So, the scalar objective functions in
multi-objective optimization of arch dams contain the follows:
f1 ( X) = V
f 2 ( X) = σ t max
f3 ( X) = σ c max
f 4 ( X) = d max
⎫
⎪
⎪
⎬
⎪
⎪⎭
(18)
where V is the dam volume, σ tmax is the maximal tensile principal stress, σ cmax is the absolute maximal
pressure principal stress, and dmax is the relative depth of large tensile stress zone on dam base.
The scalar objective functions in Eq.(18) are in different dimension, and their value are discrepant very
much. It's necessary to make them normalization as
f1 ( X) = V
[V ]
σ
f3 ( X) = c max
[σ c ]
σ t max
⎫
[σ t ]⎪
⎬
d
⎪
f 4 ( X) = max
[d ] ⎭
f 2 ( X) =
(19)
where [·] are the upper bounds of corresponding variable, which are determinate according to design
specification or the analysis of initial design scheme.
⎯ 1012 ⎯
4. Constraints in arch dam optimization The constraints, which should be subject to in arch dam
optimization, include geometric constraints, dam volume constraint and mechanical property constrains,
i.e., stress constraints, stability constraints, etc. Mathematically, the constraints are formulated as
K U ≤ [K U ]
K D ≤ [K D ]
V ≤ [V ]
σ t max ≤ [σ t ]
σ c max ≤ [σ c ]
d max ≤ [d ]
ϕ ≤ [ϕ ]
⎫
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎭
(20)
in which, ϕ is semi-center angle of arch ring.
MULTI-OBJECTIVE OPTIMIZATION TECHNIQUES
Multi-objective optimization schemes usually transform multiple objective functions into a single objective
function in some manner and the resulting problem is solved as a single objective optimization problem.
1. Weighted sum method In this method, the objective functions are made scalars with the weighted sum,
and the new problem is posed as follows:
m
f ( X ) = ∑ wl f l ( X )
min
l =1
X iL ≤ X i ≤ X iU i = 1,2, L, n
g j ( X ) ≤ 0 j = 1,2, L p
s.t.
hk ( X ) = 0 k = 1,2,L q
⎫
⎪
⎪⎪
⎬
⎪
⎪
⎪⎭
(21)
where wl is a positive constant indicating the weight (and hence importance ) assigned to fl. By giving a
relatively large value to wl it is possible to favor fl over other objective functions. Note that the condition
m
∑w
l =1
l
= 1 should be hold in Eq. (21).
2. Utopia point method With the utopia point method, the following optimization problems are solved
one at a time:
min
s.t.
fl (X )
X iL ≤ X i ≤ X iU i = 1,2, L , n
g j ( X ) ≤ 0 j = 1,2, L p
hk ( X ) = 0 k = 1,2, L q
⎫
⎪
⎪
⎬
⎪
⎪⎭
(l = 1, 2, L , m)
(22)
Let f l * = f l * ( X * ) be the optimum of lth objective function and X* is corresponding vector of design
variables. In objective function space, the point of ( f1* , f 2* ,…, f m* ) is called utopia point and the distance
between the solution point and utopia point is taken as an evaluation function. Then, the multi-objective
optimization problem (17) is transformed to the following scalar optimization problem:
m
∑( f (X ) − f
min
f (X ) =
s.t.
X ≤ X i ≤ X iU
l =1
L
i
g j (X ) ≤ 0
l
*
l
)
i = 1,2,L , n
j = 1,2, L p
hk ( X ) = 0 k = 1,2,L q
⎫
⎪
⎪⎪
⎬
⎪
⎪
⎪⎭
(23)
⎯ 1013 ⎯
The solution of Eq.(23) is Pareto optimum of Eq.(17).
3. Cooperative game theoretic model and Nash arbitration scheme The multi-objective optimal design
problem is analogy to a game in which each player corresponds to one of the objective functions. Let fl,w be
the worst value of lth objective function that the corresponding player will accept, i.e. an upper limit on the
lth objective. The utility of lth player is defined as u l ( X ) = f l,w − f l ( X ) . The aim of each player in a game
is to maximize its utility. In this situation of conflicting interests, a cooperative approach, based on the
concept of Pareto-optimality, is adopted. Nash [8] suggests an arbitration scheme to solve the cooperative
game by maximizing the product of the players’ utilities, i.e. maximizing the following function:
m
m
l =1
l =1
C ( X ) = ∏ u l ( X ) = ∏ [ f l,w − f l ( X )]
(24)
In the case of multi-objective optimization of arch dams with normalized objective functions in Eq.(19),
the worst value of each objective function is 1. So, the transformed problem by Nash arbitration scheme is
as follow:
m
max C ( X ) = ∏ [1 − f l ( X )]
l =1
s.t.
X ≤ X i ≤ X iU i = 1,2, L, n
g j ( X ) ≤ 0 j = 1,2, L p
L
i
hk ( X ) = 0 k = 1,2,L q
⎫
⎪
⎪⎪
⎬
⎪
⎪
⎪⎭
(25)
ENGINEERING EXAMPLE
BAIHETAN arch dam, which located in Sichuan province of China, is a parabolic double curvature arch
dam with the height of 277.0m, the dam crest elevation of 827.0m, and the normal high water level of
820.0m. In this study, four objectives in Eq. (2) are adopted in optimization, where dmax is the relative
depth of the zone with tensile stress larger than 1.0MPa. The finite element method is taken for the
structural analysis, and the FE mesh of dam body is arranged as 8-layer elements in height and 6-layer
elements in thickness. The load case considered is “hydrostatic pressure of normal high water +
temperature-down load + self-weight of dam body before grouting”. The material constants are shown in
Table 1, in which, rock A distributes form elevation of 780.0m to 827.0m in left bank; rock B distributes
from elevation of 600.0m to780.0m in left bank and from elevation of 700.0m to 827.0m in right bank;
rock C distributes from elevation of 600.0m to 700.0m in right bank and rock D distributes under elevation
of 600.0m.
Table 1 Material constants of dam concrete and foundation rocks
Materials
Young’s module
(GPa)
Poisson’s ratio
Unit weight
(kN/m3)
Thermal coefficient
(/°C)
Concrete
Rock A
Rock B
Rock C
Rock D
21.0
12.0
14.0
16.0
20.0
0.17
0.25
0.22
0.22
0.20
24.0
/
/
/
/
1.0×10-5
/
/
/
/
The initial design of dam shape provided by design department is analyzed first, and based on the results
listed in the last row of Table 2, the main constraints in optimization process are taken as follows:
⎯ 1014 ⎯
v max ≤ 700.0 × 10 4 m 3
σ tmax ≤ 9.0MPa
σ cmax ≤ 17.5MPa
d max ≤ 0.4
′ ≤ 0.5MPa
σ tmax
K U ≤ 0.30
K D ≤ 0.25
⎫
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎭
(26)
′
where, σ tmax
is the maximal tensile stress by self-weight of dam body before grouting, KU and KD are
overhung of upstream surface and downstream surface, respectively.
The solutions obtained with different multi-objective optimization techniques are reported in Table 2, in
which, the weights for four objectives are the same of 0.25 in weighted sum method, and the utopia point is
*
(V * , σ t*max , σ c*max , d max
) = (611.27 × 10 4 m 3 , 4.29MPa , 11.93MPa , 0.3225) which is obtained by four single
objective optimization.
Table 2 Multi-objective optimal solutions by different methods
Weighted sum
Utopia point
Game theory
Initial design
V/×104m3
σtmax/MPa
σcmax/MPa
dmax
699.40
643.72
670.17
679.94
4.41
6.13
5.30
8.58
12.05
12.31
12.12
17.03
0.3242
0.3767
0.3523
0.3921
The results show that, although a normalization of four objective functions is adopted in the form of
Eq.(19), to deal with numerical homogeneous quantities, weighted sum method gives no optimization to
dam volume, while utopia point method and game theory method both optimize all of the four objectives.
If the global design quality defined by Spallino et al [9] is used as following,
Q=
V σ t max σ c max d max
+
+
+ *
V * σ t*max σ c*max d max
(27)
Figure 3: Compare of center sections
The value of Q for utopia point method and game theory method are 4.683 and 4.440, respectively. That is
to say the optimal solution by game theory method is better than that by utopia point method. In additional,
the game theory method adopted here doesn’t need to solve single objective optimization problems, in
order to obtain the utopia point, first. So, the computational cost of game theory method is less than that of
⎯ 1015 ⎯
utopia point method. The center cantilevers of optimal designs and initial design are compared in Figure 3.
It is shown that the position of bulgy point on upstream bound of center cantilever is elevated in optimal
designs. This change is in favor of improving the dam’s state of stress. Compared to initial design, the
optimal design by game theory decreases 38.22% of maximum tensile stress, 28.83% of maximum
pressure stress and 10.15% of relative depth of zone with tensile stress larger than 1.0MPa. Table 3 shows
the dam parameters of optimal design by game theory method and Figure 4~6 show the contours of
stresses.
Table 3 Dam parameters of optimal design by game theory
Elevation
(m)
827.00
780.00
740.00
690.00
640.00
600.00
570.00
550.00
Parameters of center
cantilever
Thickness
yc
(m)
0.000
-22.018
-34.276
-41.911
-41.789
-36.673
-30.192
-24.713
15.638
35.136
44.415
50.622
55.453
61.653
69.327
76.476
Thickness of
arch end (m)
Semi-center
angle (°)
Radius of arch axis
at center (m)
Left
Right
Left
Right
Left
Right
17.068
39.416
48.933
51.844
66.092
76.996
76.089
84.103
17.450
35.241
51.247
58.035
65.551
72.399
77.491
84.282
48.538
47.652
50.377
48.742
44.545
41.589
34.468
17.530
48.170
44.636
41.045
42.233
44.094
38.162
31.213
13.444
322.419
298.259
245.776
226.129
213.754
185.601
178.707
163.446
234.237
250.309
273.350
247.236
205.344
207.682
203.521
216.391
3
3
5
2
3
1
8
6
7
5
1 -2.00
2 -1.00
3 0.00
4 1.00
5 2.00
6 3.00
7 4.00
8 5.00
min -2.72
max 5.29
4
Figure 4: Contours of principal stress σ1 on upstream surface of optimal design by game theory
4
7
6
2
5
4
3
1
1 -12.00
2 -10.50
3 -9.00
4 -7.50
5 -6.00
6 -4.50
7 -3.00
min -12.13
max -2.78
2
2
Figure 5: Contours of principal stress σ3 on downstream surface of optimal design by game theory
⎯ 1016 ⎯
1
4
1
1
1
2
4 3
6
7
1 -1.00
2 0.00
3 1.00
4 2.00
5 3.00
6 4.00
7 5.00
min -1.43
max 5.29
5
Figure 6: Contours of principal stress σ1on base of optimal design by game theory
CONCLUSIONS
This study proposed a multi-objective optimization model of arch dam design with four objectives of dam
volume, maximal tensile stress, maximal pressure stress and relative depth of large stress zone. Three
multi-objective optimization techniques are used to solve the problem. The optimization of BAIHETAN
arch dam is calculated as an engineering example. The results are compared and indicate that game theory
method is superior to weighted sum method and utopia point method in the field of multi-objective
optimization of arch dams.
Acknowledgements
The support of National Nature Science Foundation of PR China (NSFC) under grant No. 90410011 and
50409017 is gratefully acknowledged.
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⎯ 1017 ⎯