Due to an increased need in hydro-electricity, water storage, and flood protection, it is assumed that a series of new dams will be build throughout the world. The focus of this paper is on the non-probabilistic-based design of new arch-type dams by applying means of robust design optimization (RDO). This type of optimization takes into account uncertainties in the loads and in the material properties of the structure. As classical procedures of probabilistic-based optimization under uncertainties, such as RDO and reliability-based design optimization (RBDO), are in general computationally expensive and rely on estimates of the system’s response variance, we will not follow a full-probabilistic approach but work with predefined confidence levels. This leads to a bi-level optimization program where the volume of the dam is optimized under the worst combination of the uncertain parameters. As a result, robust and reliable designs are obtained and the result is independent from any assumptions on stochastic properties of the random variables in the model. The optimization of an arch-type dam is realized here by a robust optimization method under load uncertainty, where hydraulic and thermal loads are considered. The load uncertainty is modeled as an ellipsoidal expression. Comparing with any traditional deterministic optimization method, which only concerns the minimum objective value and offers a solution candidate close to limit-states, the RDO method provides a robust solution against uncertainty. To reduce the computational cost, a ranking strategy and an approximation model are further involved to do a preliminary screening. By this means, the robust design can generate an improved arch dam structure that ensures both safety and serviceability during its lifetime.

Fig.1 Description of the shape of arch type dam. (a) The central vertical section of the arch dam (b) the horizontal curves of the arch type dam

Fig.2 Temperature distributions in winter time and summer time

Fig.3 Comparison of DO-based optimum design with an RO-based optimum design

Fig.4 The procedure of the optimization

Fig.5 Approximation with Kriging metamodel. (a) The approximation of tensile stress with Kriging Metamodel; (b) the approximation of dam volume with Kriging metamodel

samplings

leave-one-out error

tensile stress approximation

LHS:800

9.0534e^{−2}

volume approximation

LHS:800

4.8018 e^{−2}

Tab.1 Approximation model of tensile stress and dam volume

Tab.2 The upper and lower boundaries of shape design variables

parameters

nominal value

change of air temperature (°C )

−5.4

change of reservoir water temperature (°C )

−4.1

water level (m)

135

density (kg/m^{3})

2400

Young’s Modulus (P_{a})

2.1e^{10}

Tab.3 Loads and material parameters of the dam model

height

thickness (m)

central angle (°)

coefficient ‘a’

140

5.59

115.45

−0.311

120

7.12

108.28

−0.302

105

8.29

101.88

−0.309

90

9.47

94.62

−0.329

75

10.68

86.49

−0.360

55

12.32

74.31

−0.421

35

13.99

60.58

−0.502

20

15.27

49.27

−0.579

0

17.01

32.85

−0.698

Tab.4 Geometrical parameters of the dam model according to the DO

height

thickness (m)

central angle (°)

coefficient ‘a’

140

3.00

123.41

−0.271

120

8.04

117.99

−0.405

105

11.28

112.38

−0.530

90

14.07

105.45

−0.677

75

16.41

97.20

−0.844

55

18.81

84.15

−1.100

35

20.41

68.75

−1.395

20

21.07

55.66

−1.640

0

21.25

36.16

−2.00

Tab.5 Geometrical parameters of the dam model according to the RO

Fig.6 Comparison of the geometries between DO optimal design and RO optimal designs

Fig.7 The sum displacement of optimal arch dam based on DO

Fig.8 The sum displacement of optimal arch dam based on RO

volume (m^{3})

sum displacement (m )

tensile stress (P_{a})

von Mises stress (P_{a})

DO

3.2382e^{5}

0.01256

1.48

2.89

RO

4.4173e^{5}

0.00987

1.40

2.79

Tab.6 Comparison

Fig.9 Comparison of mechanical properties between the arch dams acquired with DO method and RO method. (a) Comparison of tensile stress; (b) comparison of von Mises stress; (c) comparison of displacement

parameters

distribution

range

change of air temperature (°C )

uniform

[−4,−7]

change of reservoir water temperature (°C )

uniform

[−2,−6]

water level (m)

uniform

[130,140]

density ((kg/m^{3})

uniform

[2200,2600]

Young’s Modulus (P_{a})

uniform

[1.9e^{10},2.3e^{10}]

Tab.7 Range of random samples

1

S Li, L Ding, L Zhao, W Zhou. Optimization design of arch dam shape with modified complex method. Advances in Engineering Software, 2009, 40(9): 804–808 https://doi.org/10.1016/j.advengsoft.2009.01.013

2

S M Seyedpoor, J Salajegheh. Adaptive neuro-fuzzy inference system for high speed computing in optimal shape design of arch dams subjected to earthquake loading. Mechanics Based Design of Structures and Machines, 2009, 37(1): 31–59 https://doi.org/10.1080/15397730802688227

B D Youn, K K Choi. A new response surface methodology for reliability-based design optimization. Computers & Structures, 2004, 82(2–3): 241–256 https://doi.org/10.1016/j.compstruc.2003.09.002

5

H Karadeniz, V Toğan, , T Vrouwenvelder. An integrated reliability-based design optimization of offshore towers. Reliability Engineering & System Safety, 2009, 94(10): 1510–1516 https://doi.org/10.1016/j.ress.2009.02.008

6

H D Sherali, V Ganesan. An inverse reliability-based approach for designing under uncertainty with application to robust piston design. Journal of Global Optimization, 2006, 37(1): 47–62 https://doi.org/10.1007/s10898-006-9035-y

7

F Tan, T Lahmer. Shape optimization based design of arch-type dams under uncertainties. Engineering Optimization, 2017, 50(9): 1470–1482

K H Lee, I S Eom, G J Park, W I Lee. Robust Design for un- constrained optimization problems using the Taguchi method. AIAA Journal, 1996, 34(5): 1059–1063 https://doi.org/10.2514/3.13187

10

E Sandgren, T M Cameron. Robust design optimization of structures through consideration of variation. Computers & Structures, 2002, 80(20–21): 1605–1613 https://doi.org/10.1016/S0045-7949(02)00160-8

11

A Ben-Tal, L E I Ghaoui, A Nemirovski. Robust Optimization, Princeton and Oxford, MA: Princeton University Press, 2009

12

A Ben-Tal, A Nemirovski. Robust optimization methodology and applications. Mathematical Programming, 2002, 92(3): 453–480 https://doi.org/10.1007/s101070100286

13

S Sundaresan, K Ishii, D R Houser. A robust optimization procedure with variations on design variables and constraints. Engineering Optimization, 1995, 24(2): 101–117 https://doi.org/10.1080/03052159508941185

14

X Guo, W Bai, W Zhang, X Gao. Confidence structural robust design and optimization under stiffness and load uncertainties. Computer Methods in Applied Mechanics and Engineering, 2009, 198(41–44): 3378–3399 https://doi.org/10.1016/j.cma.2009.06.018

15

G Sun, G Li, Z Gong, X Cui, X Yang, Q Li. Multiobjective robust optimization method for drawbead design in sheet metal forming. Materials & Design, 2010, 31(4): 1917–1929 https://doi.org/10.1016/j.matdes.2009.10.050

16

Y Kanno, I Takewaki. Sequential semidefinite program for maximum robustness design of structures under load uncertainty. Journal of Optimization Theory and Applications, 2006, 130(2): 265–287 https://doi.org/10.1007/s10957-006-9102-z

17

L A Schmit, B Farshi. Some approximation concepts for structural synthesis. AIAA Journal, 1974, 12(5): 692–699 https://doi.org/10.2514/3.49321

18

B Zhu, J Gao, Z Chen, Y Li. Design and Research for Concrete Arch Dams. Beijing: China WaterPower Press, 2002

19

R Jin, X Du, W Chen. The use of metamodeling techniques for optimization under uncertainty. Structural and Multidisciplinary Optimization, 2003, 25(2): 99–116 https://doi.org/10.1007/s00158-002-0277-0

20

G A Fenton, D V Griffiths. Risk Assessment in Geotechnical En-gineering. Hoboken: John Wiley and Sons, 2008

21

C E Rasmussen, C K I Williams. Gaussian Processes for Machine Learning. London: MIT Press, 2006

22

S Marelli, C Lataniotis, B Sudret. UQLab User manual- Kriging(Gaussian Process model), Report UqLab Chair of Risk, Safty and Uncertainty Quantification, ETH Zurich, 2015, 9–105

23

S Marelli, B Sudret. UQLab: A framework for uncertainty quan-tification in Matlab. In: Proceedings of 2nd International Conf-erence on Vulnerability, Risk Analysis and Management (ICVRAM2014). Liverpool: 2014, 2554–2563

24

S Boyd, L Vandenberghe. Convex Optimization. Cambridge: Cambridge University Press, 2004