^{1}

^{*}

^{1}

^{2}

This paper reviews several recently-developed techniques for the minimum-cost optimal design of water-retaining structures (WRSs), which integrate the effects of seepage. These include the incorporation of uncertainty in heterogeneous soil parameter estimates and quantification of reliability. This review is limited to methods based on coupled simulation-optimization (S-O) models. In this context, the design of WRSs is mainly affected by hydraulic design variables such as seepage quantities, which are difficult to determine from closed-form solutions or approximation theories. An S-O model is built by integrating numerical seepage modeling responses to an optimization algorithm based on efficient surrogate models. The surrogate models (meta-models) are trained on simulated data obtained from finite element numerical code solutions. The proposed methodology is applied using several machine learning techniques and optimization solvers to optimize the design of WRS by incorporating different design variables and boundary conditions. Additionally, the effects of several scenarios of flow domain hydraulic conductivity are integrated into the S-O model. Also, reliability based optimum design concepts are incorporated in the S-O model to quantify uncertainty in seepage quantities due to uncertainty in hydraulic conductivity estimates. We can conclude that the S-O model can efficiently optimize WRS designs. The ANN, SVM, and GPR machine learning technique-based surrogate models are efficiently and expeditiously incorporated into the S-O models to imitate the numerical responses of simulations of various problems.

Construction of water-retaining structures (WRSs) [

Hydraulic structures that impound a considerable amount of water (head) and are constructed on permeable soil foundations are associated with water seepage. Seepage forces threaten the hydraulic efficiency and structural stability of such structures. Another consequence of seeping water is pore-water pressure, which applies uplift (upward) pressure on the structure’s floor (apron) and may cause it to collapse.

The mathematical relationship between seepage design variables with flow domain characteristics is complex and nonlinear. An analytical solution may be obtained only for simple and symmetrical cases. Many approximation and empirical theories have been proposed for estimating seepage quantities (particularly uplift pressure and exit gradient). These theories include Bligh’s creep theory, Lane’s weighted creep theory, the flow-net method, the fragment method and Khosla’s theory [

Recently, as a result of the development of numerical methods and computerized processes, many seepage problems related to WRS have been accurately simulated and solved by numerical methods such as the finite element method (FEM). Furthermore, hydraulic conductivity variation and other soil parameters can be integrated into a numerical model to study the consequences of varying soil parameters on WRS design. However, numerical techniques only provide solutions when certain parameters of the hydraulic structure are known, including the boundary conditions and geometry of the flow domain. This means that numerical models do not provide generalized performance equations similar to those provided by analytical solutions.

Considering the above-mentioned arguments, the contradicting goals of efficiency, accuracy, safety and cost must be simultaneously integrated into the design process to attain optimum, safe and economical WRSs. Hence, the optimization approach can be used to identify optimal WRS designs. As a result, the minimum cost and safest WRSs can be achieved. Directly coupling a numerical model to an optimization model to attain an optimum WRS design is computationally inefficient and time-consuming. Alternatively, the numerical model could be replaced by an approximate machine learning model (surrogate model) that accurately and expeditiously imitates the numerical model’s responses. Such surrogate models (meta-models) can be trained based on numerically simulated data (input and output) sets.

Linked simulation optimization (S-O) models based on surrogated models have been applied to solving groundwater and water resource problems and other applications related to groundwater management and identification of contaminant sources [

The current paper presents a few innovative applications of a newly developed S-O technique that can determine optimal WRS designs (safe, reliable and economic) based on adequately trained meta-models. Induced seepage forces, and many safety factors and design requirements related to WRSs such as overturning, sliding safety factors, and prevention of eccentric load conditions, are considered in the S-O approaches. Also, uncertainty in seepage characteristic estimates due to uncertainty in hydraulic conductivity is integrated into the reliability based optimum design using S-O model. For each S-O model, the type of machine learning technique and optimization solver are selected based on prediction accuracy and efficiency.

In the following sections, different scenarios and applications of linked S-O approaches were used to find the optimum WRS design. Section two encompasses the linked S-O models for simple WRS including two end cut-offs and apron. The coupled S-O models in section three is used for comprehensive (complex) WRS incorporating 10 cut-offs and stratified flow domain based on different hydraulic conductivity values. The fourth section incorporates the reliability concept in the S-O models using RBOD framework to address uncertainty in estimate the seepage characteristics due to uncertainty of hydraulic conductivity. The fifths section includes more advance application of RBOD using multi-objective multi-realization optimization (MOMRO) models.

A simple conceptual seepage model was proposed to illustrate the WRS design problem [_{1}) and downstream side (d_{2}); 1 - 60 for upstream water head (H) and half the WRS apron width (b).

Training of meta-models is based on datasets simulated by a numerical seepage modeling code (Geo-Studio/SEEP/W) [_{1}, d_{2}, 2b, H) and are randomly generated using the Latin hypercube sampling (LHS) method [

from numerical seepage modeling for each input set. The most important output data are uplift pressure on the floor at the upstream cut-off (PC), uplift pressure on the floor at the downstream cut-off (PE) and the exit gradient (ie) at the toe of the WRS.

The ANN technique is used to develop the meta-models in this section. ANN can explore complex, discontinuous and nonlinear relationships between datasets [

The optimization model was formulated as follows:

Find X = { x 1 , x 2 , x 3 , x 4 } = { d 1 , d 2 , b , b * }

Minimize

f ( X ) = c f b ( t 1 + t 2 ) 2 + t c ∑ s = 1 2 c s d s (1)

Subject to:

F S i e = ε ( H , d 1 , d 2 , b , i e c r t )

F S i e ≥ F S e x i t (2)

F s f l − u s = ϵ ( H , d 1 , d 2 , b )

F s f l − u s ≥ F S u p l i f t (3)

F s f l − d s = γ ( H , d 1 , d 2 , b )

F s f l − d s ≥ F S u p l i f t (4)

E c c ≥ b 3 (5)

E c c ≤ 2 b 3 (6)

M p a s = f ( H , b , b * , t 1 , t 2 , G c , G w , P C , P E )

M a c t = f ( H , b , b * , t 1 , t 2 , G c , G w , P C , P E )

V l = f ( H , b , b * , t 1 , t 2 , G c , G w , P C , P E )

H l = f ( H , G w )

E c c = M p a s − M a c t V l o a d

F s o v e r ≥ 1.5 (7)

F s o v e r = M p a s M a c t

F s s l i d ≥ 1.5 (8)

F s s l i d = C × b + f × V l H l

where t 1 , t 2 represent the floor thicknesses at the upstream and downstream sides, respectively. c f is the construction cost of the floor per cubic meter ($400/m^{3}), c c is the construction cost of the cut-offs per cubic meter, which is a function of the cut-off depth (d_{1}, d_{2}) as shown in Equation (9) and Equation (10), and t c is the cut-off thickness, which equals 1.0 m.

C 1 = x 1 3 + 20 x 1 2 + 200 x 1 + 400 (9)

C 2 = x 2 3 + 20 x 2 2 + 200 x 2 + 400 (10)

F s i e is exit gradient safety factor determined by the meta-models {ε( )}. F s f l − u s , F s f l − d s are safety factors imposing weight on the upstream and downstream floors of the WRS to safely counterbalance the uplift pressures (PC, PE) [

The S-O technique was implemented for different H values ranging from 2 m to 40 m. The design requirement of the WRS and all the constraints were satisfied for each optimum solution. The optimum solutions for the different H values show that d_{1} and d_{2} make considerable contributions to the hydraulic safety of a WRS (see _{2} is more important than that of d_{1}, because d_{2} has a substantial impact on the ie value, which is the critical factor in the hydraulic design of WRSs [

The optimum width (2b) of a WRS effectively influences its optimum hydraulic design, because the total width is directly integrated into the specified safety factors. Reduction of the optimum WRS width (2b) could be attributed to increasing floor thicknesses at the upstream and downstream sides (t_{1}, t_{2}) with increases in head, which results in an expensive design. Therefore, the S-O model reduces the total width and simultaneously augments the depths of d_{1} and d_{2}, which is an efficient and cost-effective solution to reducing the tremendous uplift pressure and exit gradient effects experienced at large H values. Moreover, b^{*} considerably contributes to the optimum hydraulic design of the WRS. The weight of water head above b^{*} counterbalances the uplift pressure and enhances the stability of the WRS. This means that the value of b^{*} substantially contributes to WRS safety and provides a cheaper solution.

Often, the seepage prevention components of WRS projects are the end cut-offs

(upstream and downstream) and the apron between them. Limitations in the orientation, length, and number of cut-offs, and the width of the apron reduce the opportunity of finding a feasible optimum solution using the linked S-O technique. Including the effects of different scenarios of hydraulic conductivity and anisotropic ratios on WRSs is an important concept that must be considered in their optimum design. Moreover, studying soil stratification based on different values of hydraulic conductivity and quantifying its effects on optimum WRS design is another concept that needs to be considered.

Hence, a comprehensive conceptual model is proposed, as shown in

The geometry of the assumed numerical model comprised ten cut-offs (sheet piles) with varied positions, lengths, and orientations. Additionally, three subsoil layers were assumed and the principle (horizontal) hydraulic conductivity (kx) and anisotropic ratio (ky/kx) were varied for each layer and each case. As a result, the contribution of each variable of the comprehensive model to the optimal design can be explored for various boundary conditions.

The prescribed range of each design variable and parameter is shown in

Symbol | H | b_{1}, b_{2}, …, b_{11} | d_{1}, d_{2}, …, d_{10} | β_{1}, β_{2}, …, β_{10} | LD_{1}, LD_{2}, LD_{3} | k_{x}_{1}, k_{x}_{2}, k_{x}_{3} | (k_{y}/k_{x})_{1}, (k_{y}/k_{x})_{2}, (k_{y}/k_{x})_{3} |
---|---|---|---|---|---|---|---|

Units | m | m | m | degrees | m | m^{3}/day | - |

Range | 2 - 100 | 1 - 120 | 0 - 60 | 30 - 150 | 5 - 100 | 0.01 - 20 | 0.1 - 1.5 |

The ranges of hydraulic conductivity and anisotropic ratios used cover a wide range of typical real-life values [_{1}, LD_{2}, LD_{3}), their hydraulic conductivities in the horizontal direction ( k x 1 , k x 2 , k x 3 ), and their anisotropic ratios (k_{y}/k_{x})_{1}, (k_{y}/k_{x})_{2}, (k_{y}/k_{x})_{3}, respectively. The most important seepage design characteristics for each numerical seepage model were the uplift pressures in front of (PEi) and behind (PCi) each single cut-off ( S 1 , S 2 , ⋯ , S 10 ) in addition to the exit gradient (ie) at the toe of the hydraulic structure. Hence, 21 meta-models were required, one for each seepage characteristic.

The support vector machine (SVM) is one of the most popular machine learning techniques and has recently been used for various nonlinear and complex engineering problems [

For each uplift pressure dependent variable, two different SVM models were built and trained on different training/testing datasets randomly selected from source data. A basic version of the ensemble surrogate model based on an average of the two models was developed. This procedure provides a more robust and accurate prediction. Also, any uncertainty arising from source data or meta-model prediction can be reduced by using the ensemble surrogate model. For exit gradient, three different models were developed. The coefficient of determination (RSQ) and mean square error (MSE) for the training and testing phases were used as measures of the predictive accuracy of the meta-models based on detached data. The parameters of each SVR model were carefully selected after performing trial-and-error iterations until the best RSQ and MSE values were obtained.

The optimization model described in this section includes 31 decision variables ( b 1 , b 2 , ⋯ , b 11 , d 1 , d 2 , ⋯ , β 1 , β 2 , ⋯ , β 10 ) and several constraints. Also, the optimization solver evaluates the objective function and constraint values based on 21 ensemble surrogate model responses. This makes the optimization problem a complex task. Safety factors and other hydraulic design requirements represent the imposed constraints of the optimization model within the S-O model. The best value of each design/decision variable was selected by the optimization algorithm to provide a safe and economic design. For this complex optimization task, the hybrid genetic algorithm (HGA) was used. The HGA is a combination of two optimization algorithms: GA and the interior point algorithm (IPA) [

Formulation of the optimization model in this section is similar to that in Section 2. However, because many cutoffs were used, flotation constraints (safety factors) were applied at all key uplift pressure points associated with different cut-off locations. Moreover, the construction cost incorporates the effect of inclination angle. Hence, Equation (11) is used in determining the construction cost of the cut-offs.

C c = 0.05 d i 2 + 200 d i 2 + 0.0698 β i 2 − 12.558 β i + 565.93 ∀ i (11)

The linked S-O was implemented for different head values ranging from 20 m to 100 m. Other parameters were kept constant, such as hydraulic conductivity for all layers (k_{x} = 5 m/day), anisotropic ratio ((k_{y}/k_{x})_{1} = 1) and the depth of soil layers (50 m). The obtained optimum solutions can determine the vital values of all the provided design (decision) variables. This means that the optimization solver selects the design variables that provide the safest and most cost-efficient WRS design for the optimum solution.

In general, the optimum solutions obtained demonstrate that the contributions of variables b_{1} to b_{8} and d_{2} to d_{8} to WRS safety are insignificant. The optimum values for these variables approached zero. In contrast, values b_{9}, b_{10}, b_{11}, d_{9} and d_{10} had vital roles in optimum WRS design, as shown in _{9} is to reduce PC9 and PE10 uplift pressure and exit gradient value. More importantly, the function of d_{10} is to directly reduce the exit gradient value, which is the most critical seepage characteristic. The functions of b_{9} and b_{10} are to provide sufficient weight to stabilize the WRS, reduce uplift pressure, and provide sufficient width to counterbalance overturning and sliding forces.

Other important variables were β_{9} and β_{10}, which are related to d_{9} and d_{10}, respectively. The optimum value of β_{10} is 150˚. This is logical, as making the inclination angle of the last cut-off toward downstream (greater than 90˚) substantially decreases the exit gradient value. This can be attributed to the augmentation of

the streamline length of seeping water, particularly when β_{10} reaches 150˚. Thus, the duration and distance of seeping water would increase, which can reduce the exit gradient value. The optimum value of β_{9} was 30˚ in all implemented cases. Such an inclination angle can reduce uplift pressure under b_{10}. This helps decrease the construction costs of WRSs.

In general, it seems that effective and optimum WRS designs must include two upstream and downstream cut-offs, the width (b_{10}) between them, and the width (b_{9}) on the upstream side. These widths are necessary to provide sufficient weight for the WRS to resist external hydrostatic loads and uplift pressure and play a vital role in satisfying WRS design requirements (constraints) such as sliding, overturning and eccentric load conditions.

All the optimum solutions satisfied the safety factors and requirements of WRS design. For all implemented cases, the optimum solution attained the minimum allowable value of the exit gradient safety factor, which is five. This reflects the significance of the exit gradient safety factor in WRS design and how it affects the construction cost because it is mainly controlled by the depth and inclination angle of the last cut-off (d_{10}, β_{10}), which are indispensable and expensive components used to reduce the exit gradient impact.

The same procedure which was applied to study the effects of upstream water head was implemented to quantify the effects of the hydraulic conductivity (k_{x}_{1}) and anisotropic ratio (k_{y}/k_{x})_{1} of the first layer. The first layer is the nearest layer to the foundation of the WRS (_{y}/k_{x})_{1} was studied by assuming eight different values ranging from 0.1 to 1.5. Ten values of k_{x}_{1} ranging from 0.01 m/day to 20 m/day were specified and processed using the S-O technique. The values of the other design variables and parameters were left constant.

Generally, the optimum solutions obtained revealed that increases in (k_{y})_{1} and the (k_{y}/k_{y})_{1} ratio significantly decrease the total cost of the WRS, as shown in _{x}_{1} increases with a constant anisotropic ratio ((k_{y}/k_{x})_{1} = 1), seeping water can easily move from the high-pressure zone (upstream) to the low-pressure zone (downstream). Consequently, the pore-water pressure underneath the WRS and the exit gradient values decrease. Thus, great cut-off depths and significant widths between cut-offs are unnecessary. Similarly, when the anisotropic ratio (k_{y}/k_{x})_{1} is large with a specified hydraulic conductivity (k_{x}_{1} = 5), the seeping water motion in the vertical direction becomes faster and the exit gradient becomes smaller compared to that

obtained for small (k_{y}/k_{x})_{1} ratios. Hence, for high values of (k_{x})_{1} and (k_{y}/k_{x})_{1}, the optimum values of d_{9}, d_{10}, b_{9}, and b_{10}, which are the most effective variables, decrease and, consequently, the optimal cost declines.

For small values of (k_{y}/k_{x})_{1} and k_{x}_{1}, the exit gradient safety factor and safe eccentric distance play crucial roles in the optimum solution, compared to the other safety factors. This is evident as these safety factors reached the maximum or minimum allowable limit to satisfy the design requirements, while the optimum design attained the minimum construction cost. _{y}/k_{x})_{1} and k_{x}_{1}, respectively.

The S-O results demonstrate that the contributions of b_{1} to b_{8} and d_{1} to d_{8} to WRS safety are insignificant because their optimum values approached zero. Optimum solutions for WRS were based on increasing the values of b_{9} and b_{10} to counterbalance the uplift pressures, and on augmenting d_{9}, d_{10} and β_{10} to decrease the exit gradient, as shown in Figures 9-12. Also, there is a significant contribution of d_{9} associated with the minimum value of β_{9} required to decrease uplift pressure beneath b_{10}, which represents a large portion of the WRS floor.

k_{x}_{1} | Exit gradient safety factor | Overturning safety factor | Sliding safety factor |
---|---|---|---|

0.10 | 5.00 | 1.83 | 2.38 |

0.50 | 5.00 | 1.82 | 2.33 |

0.90 | 5.00 | 1.81 | 2.30 |

1.50 | 5.00 | 1.79 | 2.25 |

4.00 | 5.00 | 1.72 | 2.04 |

7.00 | 5.00 | 1.61 | 1.50 |

10.00 | 5.00 | 1.59 | 1.50 |

13.00 | 5.00 | 1.59 | 1.50 |

17.00 | 5.00 | 1.60 | 1.50 |

20.00 | 5.00 | 1.61 | 1.50 |

(k_{y}/k_{x})_{1} | Exit gradient safety factor | Overturning safety factor | Sliding safety factor |
---|---|---|---|

0.1 | 5.0 | 2.0 | 6.5 |

0.3 | 5.0 | 1.9 | 5.7 |

0.5 | 5.0 | 1.9 | 3.8 |

0.7 | 5.0 | 1.8 | 2.2 |

0.9 | 5.0 | 1.7 | 2.1 |

1.1 | 5.0 | 1.7 | 2.0 |

1.3 | 5.0 | 1.7 | 2.1 |

1.5 | 5.0 | 1.8 | 2.2 |

In particular, the optimization solver increased the d_{10} and β_{10} values to achieve a safe exit gradient, even though it is a more expensive option (Equation (11)). These variables were more effective at reducing the exit gradient, which is the most critical safety factor. Hence, the optimum value of β_{10} was 150˚, which is the maximum specified limit for this variable. For the same reason, the inclination angle of the upstream cut-offs (β_{9}) approached the minimum allowable limit (30˚) for all implemented cases.

Simultaneously, to corroborate WRS stability and satisfy related safety factors, the required optimum WRS width was provided by b_{9} and b_{10}. Furthermore, the uplift pressure on the downstream side decreased with increases in total width, which contributed to reductions in exit gradient. Therefore, the values of b_{9} and b_{10} mainly provide an efficient cross-section and weight to resist external loads and uplift pressures, and partially reduce the uplift pressure and exit gradient.

Soil properties, and especially hydraulic conductivity, have large covariance values of 200% - 300%, which means the level of uncertainty in hydraulic conductivity is high [

This section concentrates on developing a reliability based optimum design (RBOD) framework to find optimum WRS designs at minimum cost and with a particular level of reliability while considering uncertainty in hydraulic conductivity and seepage. This can be achieved by formulating a constrained multi-realization optimization model linked with an S-O technique utilizing a GA optimization solver and incorporating many stochastic ensemble GPR meta-models. The minimum-cost objective function and stochastic constraints within the S-O model are based on the responses of ensemble meta-models. Reliability constraints are simultaneously integrated into the S-O model and are based on the ensemble surrogate responses to quantify the reliability of the design. Each meta-model in the ensemble model is trained and tested based on large datasets simulated by a numerical seepage modeling code (SEEP/W) [

Reliability level was formulated as an additional constraint. All stochastic constraints were continually controlled until the desired reliability level was achieved for each single iteration of the optimization model. Reliability constraints, stochastic constraints, and deterministic constraints were simultaneously evaluated with the objective function to attain the optimum solution. The majority of the constraints and the objective function were based on the ensemble surrogate model responses within the S-O model.

The optimization model is considered a complex one. Hence, the optimization solver and machine learning technique had to be efficient and accurate enough to provide reliable and accurate solutions. Therefore, GA was utilized as an optimization solver for this task. Based on the conceptual model shown in _{1}), downstream cut-off (d_{2}), total width of WRS (b), upstream water head (H), and hydraulic conductivity.

The input data comprised 150 sets of seepage design variables (d_{1}, d_{2}, b, H), which were randomly generated utilizing the Halton sequences (HS) method

[

Based on each standard deviation value, a random field of hydraulic conductivity was generated and incorporated in the numerical seepage model. As unlimited realizations could be generated from a log-normal distribution for a certain standard deviation value, each input dataset (d_{1}, d_{2}, b, H) was simulated with four different random realizations (random fields) of the same standard deviation value. Some 600 simulated datasets were used for training each surrogate model for a particular seepage characteristic. This procedure ensures that the numerical responses for different hydraulic conductivity realizations are recorded and incorporated into the surrogate model’s training data.

Accordingly, the varied seepage quantities, such as uplift pressure on the upstream side (Pc_{1}), downstream uplift pressure (Pe_{2}) and exit gradient (ie), were determined by the numerical seepage modeling code four times for each input dataset (case). Furthermore, because exit gradient value is more critical than other quantities and hydraulic conductivity varies randomly, four points, shown in _{1}, d_{2}, b, H) and four stochastically varied sets of output data (Pc_{1}, Pe_{2}, ie_{1}, ie_{2}, ie_{3}, ie_{4}). Therefore, the responses of meta-models reflect variation of seepage characteristics obtained from the four scenarios of random hydraulic conductivity. For each seepage design variable, five meta-models were trained to imitate different responses, reflecting the influence of five different hydraulic conductivity random fields drawn from the five log-normal distributions. As a result, thirty meta-models were built and grouped to develop six ensemble stochastic meta-models linked to the optimization model within the RBOD framework. Deterministic meta-models were developed separately to compare stochastic optimum solutions with deterministic solutions. Deterministic responses were used to train three meta-models (Pc_{1}, Pe_{2}, ie) based on expected hydraulic conductivity (σ = 0, μ = 2). Deterministic meta-models were incorporated in the deterministic S-O model to find the optimum solution of WRS for different head values.

The meta-models were built based on Gaussian process regression (GPR) [

Formulation of the optimization model was based on the multi-realization

Error measure | ie_{1} | ie_{2} | ie_{2} | ie_{2} | pc_{1} | pe_{2} | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | |

MSE | 0.17 | 0.28 | 0.107 | 0.121 | 0.09 | 0.25 | 0.14 | 0.12 | 3.09 | 5.69 | 9.76 | 8.78 |

NSE | 0.54 | 0.37 | 0.696 | 0.428 | 0.74 | 0.73 | 0.64 | 0.54 | 1.00 | 0.99 | 0.99 | 0.98 |

R | 0.73 | 0.64 | 0.830 | 0.710 | 0.86 | 0.88 | 0.80 | 0.77 | 0.99 | 0.99 | 0.99 | 0.99 |

(“stacking”) optimization technique [

It is also important to note that some stochastic design variables, such as the thicknesses of upstream and downstream floors (t_{1}, t_{2}), involved in computation of the objective function are based on stochastic ensemble meta-models. Therefore, to provide a safe design, the maximum values of each thickness are considered in determining the objective function. The multiple-realization optimization-based RBOD using the stochastic S-O model is formulated as:

Find X = { x 1 , x 2 , x 3 , x 4 } = { d 1 , d 2 , b , b * }

Minimize

f ( X ) = c f x 3 max ( t 1 m ) + max ( t 2 m ) 2 + t c ∑ s = 1 2 c s c x s (12)

The term { max ( t m ) } is a function returning the maximum of many stochastic values related to upstream and downstream floor thickness (t_{1}, t_{2}) of WRS. Similar constraints to those shown in Section 2 are utilized to formulate the optimization model described in this section. However, because we used stochastic responses due to uncertainty in seepage characteristics, there were 5 stochastic constraints for each safety factor. Based on a satisfaction percentage of these constraints, the desired reliability level can be achieved as shown below.

Z q l o g i c a l m = F s q m ≥ / ≤ F s q m a l l o w a b l e ∀ q , m (13)

g ( x ) q = ∑ m Z l o g i c a l m ≤ D R ∀ q (14)

where t 1 m , t 2 m represents the stochastic floor thicknesses at the upstream and downstream sides (^{3}); c c is the construction cost of the cut-offs per cubic meter, which is a function of cut-off depth, as shown in Equation (15), while t c is the cut-off thickness and equals 1.0 m.

c s c = x s 3 + 20 x s 2 + 200 x s + 400 ∀ s (15)

A RBOD framework based on stochastic S-O methodology was applied to hypothetical cases to study the effect of varying reliability on optimum WRS design. In these cases, the upstream head values (H) considered were 10 m, 20 m, 40 m, 60 m, 80 m and 100 m. The S-O models were implemented with different reliability levels (20%, 40%, 60%, 80% and 100%). The percentage of reliability only reflects the uncertainty of seepage quantities under WRS due to uncertainty associated with heterogeneous hydraulic conductivity.

The effect of reliability on optimum WRS design could be clearly seen by comparing the minimum construction costs obtained for different reliability levels, as shown in

in

The main role of d_{1} is to directly reduce the uplift pressure under the floor of the WRS and, indirectly, to reduce the exit gradient value. This is because the exit gradient is proportional to the uplift pressure located before the downstream cut-offs. In general, the optimum length of d_{1} decreases with reductions in head value, as shown in _{1} is augmented by increasing the degree of reliability. However, for some values, especially with 100% reliability at H = 80, 40 m, the optimum length was less than for other reliability levels. This can be explained by considering that the objective function minimizes construction cost. Therefore, the optimization algorithm identifies the minimum construction cost solution for each case separately, as long as the design (decision) vector satisfies the constraints. On the other hand, because the meta-model responses are stochastic, it is difficult to expect the optimum value with different reliability ranks. Furthermore, if the optimization solver could provide an optimum solution that satisfies, for example, three out of five (60% reliability) of the stochastic constraints, this does not promise that the optimum solution with 80% reliability is in vicinity of the 60% solution. The justification is that additional stochastic constraints may require larger values of the variable, e.g. d_{1}, which significantly increases the objective function value. Consequently, the optimization solver (GA) changes the direction of the search and continues in a more promising direction that provides a lower cost. Moreover, while the objective function is for minimum construction cost, the optimum solution for a certain reliability level does not necessarily follow the general trend of the other reliability levels. For instance, the optimum value of d_{1} at H = 80 m and 100% reliability was unexpectedly less than for other values. That may be logical if the values of d_{2}, b and b^{*} are considered simultaneously for this case. The value of d_{2}, shown in _{2} is more important for reducing the crucial exit gradient value to the safe limit.

Head (m) | 20% | 40% | 60% | 80% | 100% | Deterministic |
---|---|---|---|---|---|---|

100 | 75.4 | 52.2 | 60.1 | 66.5 | 83.7 | 47.0 |

80 | 45.2 | 59.4 | 55.9 | 68.1 | 44.7 | 40.7 |

60 | 23.3 | 36.2 | 40.7 | 42.2 | 40.5 | 26.0 |

40 | 18.4 | 21.0 | 27.7 | 24.9 | 13.6 | 13.4 |

20 | 10.1 | 6.2 | 7.6 | 9.5 | 9.8 | 9.2 |

10 | 2.3 | 3.9 | 6.7 | 7.0 | 8.1 | 6.6 |

Head (m) | 20% | 40% | 60% | 80% | 100% | Deterministic |
---|---|---|---|---|---|---|

100 | 57.3 | 91.2 | 90.1 | 89.6 | 89.2 | 90.0 |

80 | 42.3 | 43.2 | 62.0 | 62.2 | 81.6 | 74.8 |

60 | 37.4 | 37.6 | 41.1 | 40.5 | 56.6 | 45.4 |

40 | 21.2 | 20.3 | 22.3 | 25.1 | 35.5 | 22.0 |

20 | 7.6 | 7.7 | 15.9 | 15.8 | 16.2 | 12.1 |

10 | 3.2 | 3.0 | 5.2 | 5.5 | 4.8 | 7.9 |

On the other hand, the optimum value of d_{2}, shown in _{2} with 100% reliability presents the highest value for different values of H to satisfy all stochastic seepage responses and relevant constraints.

^{*}, which provided additional weight from upstream water (

To improve the efficiency and accuracy of the RBOD model and direct search optimization solver, a new approach was utilized. This approach is based on MOMRO model. The advantage of this approach is that some stochastic optimization constraints based on many ensemble meta-models are formulated as a second objective function to be minimized in the MOMRO model. The stochastic constraints used to achieve a safe exit gradient are formulated as a second stochastic objective function. The multi-objective optimization solver minimizes two stochastic objectives: exit gradient and construction cost. Desired reliability levels are implicitly incorporated into the objective functions and explicitly for the constraints. This significantly improves the solver’s search efficiency and aids in exploring more feasible candidate solutions in the search space. In this case, the multi-objective non-dominated sorting genetic algorithm-II (NSGA-II) was used.

The objective of this section is to find a safe, reliable and minimum-cost optimum WRS design that incorporates uncertainty in HHC estimates. The RBOD framework was implemented based on a more efficient and productive approach using the MOMRO technique. MOMRO integrates many stochastic responses

Head (m) | 20% | 40% | 60% | 80% | 100% | Deterministic |
---|---|---|---|---|---|---|

100 | 149.1 | 147.9 | 89.5 | 90.8 | 91.3 | 99.1 |

80 | 109.3 | 97.6 | 78.5 | 73.1 | 72.2 | 71.4 |

60 | 104.3 | 91.1 | 54.4 | 54.5 | 53.5 | 53.2 |

40 | 58.8 | 93.2 | 58.8 | 94.8 | 35.0 | 59.5 |

20 | 29.7 | 70.3 | 17.4 | 17.5 | 30.1 | 32.5 |

10 | 8.5 | 8.5 | 36.3 | 37.7 | 65.4 | 18.6 |

Head (m) | 20% | 40% | 60% | 80% | 100% | Deterministic |
---|---|---|---|---|---|---|

100 | 39.6 | 50.2 | 86.0 | 89.3 | 87.5 | 64.1 |

80 | 39.7 | 65.7 | 67.5 | 71.2 | 71.4 | 66.3 |

60 | 5.8 | 31.9 | 52.3 | 52.7 | 49.9 | 49.8 |

40 | 0.0 | 1.7 | 7.8 | 10.8 | 31.2 | 30.7 |

20 | 3.9 | 0.3 | 15.0 | 15.5 | 1.0 | 17.4 |

10 | 5.4 | 6.6 | 0.4 | 2.4 | 6.1 | 4.7 |

from well-trained meta-models based on GPR machine learning techniques. These stochastic responses represent the uncertainties in estimates of particular seepage design variables, which are embedded in stochastic constraints and objective functions of MOMRO.

The steps used to generate data were the same as those mentioned in Section 4. The input design variables and seepage characteristics used were also the same. Some 150 random cases of input design variable data were generated. However, as random field HHC was used, the number of simulations for each input design variable was 20, including 20 different realizations of HHC for each case to cover a wide range of uncertainty in HHC. Each single realization represented a unique and randomly-varied distribution of hydraulic conductivity values from finite elements in the numerical model. Five log-normal distributions with standard deviations (σ) of 0.85 m/day, 1.55 m/day, 2.25 m/day, 2.95 m/day, and 3.65 m/day and constant a mean (μ = 2 m/day) were proposed to generate different HHCs. For each seepage quantity, a stochastic ensemble surrogate model was developed containing 20 meta-models. Therefore, for a single input dataset, 20 stochastic responses were obtained by the ensemble surrogate model for processing in the MOMRO model based on the RBOD technique. Similar to Section 4, GPR machine learning was used to develop 120 meta-models and build six stochastic meta-models. For each surrogate model, there were 150 sets of training and testing data (cases).

The training/testing performance of meta-models must be accurately evaluated before using them in the RBOD model. The developed GPR meta-models were evaluated using several error measures as described in Section 4. The majority of meta-models demonstrated perfect training and testing performance (

Formulation of a multi-realization optimization model based on a single objective function with numerous stochastic constraints may lead to sub-optimum or infeasible solutions. The RBOD approach required imposition of a large number of explicit constraints, which needed to be satisfied as binding conditions for a feasible solution. Many attempts were made to formulate an RBOD model for this study with a large number (120) of stochastic meta-model-based stochastic constraints using a single objective function, but the majority of solutions were infeasible.

As multi-realization technique-based reliability requires a large number of stochastic constraints, an optimal solution search process based on evolutionary algorithms may produce an infeasible solution. Search efficiency decreases with increasing numbers of constraints and problem complexity [

The multi-realization optimization technique was based on formulating stochastic constraints based on the developed ensemble stochastic meta-models. For each safety factor or condition in the optimization model, there was one or more ensemble stochastic surrogate model/s encompassing 20 surrogate model responses for a specified seepage design variable. The desired reliability level was attained by allowing the optimum solution to satisfy any fraction (n) of the total number (m = 20) of constraints for all stochastic constraints, where n/m is equivalent to the required reliability level.

Error measure | ie_{1} (2.95) | ie_{2} (1.55) | ie_{3} (1.55) | ie_{4} (2.95) | pc_{1} (3.65) | pe_{2} (3.65) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | |

MSE | 0.00 | 0.03 | 0.02 | 0.05 | 0.05 | 0.07 | 0.07 | 0.06 | 20.52 | 12.08 | 4.16 | 24.73 |

NSE | 1.00 | 0.71 | 0.93 | 0.67 | 0.81 | 0.74 | 0.81 | 0.70 | 0.97 | 0.99 | 0.99 | 0.98 |

R | 0.99 | 0.88 | 0.96 | 0.82 | 0.90 | 0.87 | 0.91 | 0.84 | 0.98 | 0.99 | 0.99 | 0.99 |

The second objective function, which minimizes the exit gradient value, integrates reliability by incorporating exit gradient stochastic responses in determining the objective function. As exit gradient is minimized, 20 stochastic exit gradient responses based on ensemble stochastic meta-models are determined and sorted in ascending order. The maximum value of all obtained exit gradients is then selected to be minimized. This is equivalent to 99.9% reliability because the resulting exit gradient is the safest of all other stochastic values. To attain 80% reliability, for example, the optimization solver is formulated to minimize the fifth maximum value (based on 20 responses) and allow up to four stochastic responses of exit gradient to be higher than the one selected for the objective function.

As there are four locations at which to determine the exit gradient values (ie_{1}, ie_{2}, ie_{3}, ie_{4}) the maximum value for each location is determined and their average is considered as the second objective function. The same technique is applied to the first objective function of minimizing construction cost. Construction cost is based on the upstream and downstream cut-off depths (d_{1}, d_{2}), and thicknesses (t_{1}, t_{2}), and floor width (b). Floor thicknesses are based on the stochastic responses of uplift pressure ensemble meta-models (pc_{1}, pe_{2}).

The constraints for each safety factor, based on 20 developed meta-models, are formulated as shown in Section 1. However, the exit gradient safety factors (constraints) are formulated as a second objective function. Furthermore, the reliability constraints and all other variables and parameters are exactly the same as in Section 4. The modification in the formulation of the optimization model for MOMRO is as shown below

Find X = { x 1 , x 2 , x 3 , x 4 } = { d 1 , d 2 , b , b * }

Minimize, f 1 ( X ) = c f b max ( m − ω ) ( t 1 m ) + max ( m − ω ) ( t 2 m ) 2 + t c ∑ s = 1 2 c s c d s (16)

Minimize,

f 2 ( X ) = max ( m − ω ) ( i e 1 m ) + max ( m − ω ) ( i e 2 m ) + max ( m − ω ) ( i e 3 m ) + max ( m − ω ) ( i e 4 m ) 4 (17)

i e i m = ε i m ( H , d 1 , d 2 , b , k m ) ∀ i , m (18)

where max ( m − ω ) is a function sorting stochastic responses in ascending order and returns the ( m − ω ) th value of the sorted vector. m is the number of stochastic responses (20), and ω is based on the desired reliability level.

The MOMRO technique was applied to hypothetical design scenarios/cases to evaluate the performance of the RBOD-based MOMRO technique. These cases included five different upstream head values (100 m, 80 m, 60 m, 40 m, 20 m) each subject to four reliability levels. The obtained Pareto-optimum fronts for each head value, including different scenarios of reliability level, are presented in

deterministic safe exit gradient [

The effects of reliability on optimum WRS design were significant. Increasing reliability increased construction cost. For instance, the minimum construction costs for H = 100 m, reliability levels of 40%, 60%, 80% and 100%, and exit gradient safety factor = 5 were $112,191,378, $129,171,757, $162,166,799 and $268,206,048, respectively. Similarly, for the same reliability levels and an exit gradient = 3, the construction costs were $59,951,442, $79,158,696, $106,049,766 and $160,838,745. This means that considering reliability in WRS design significantly affects the optimum design attributes. Moreover, for high reliability levels, only a few applicable (feasible) scenarios can be obtained from the Pareto optimal front. For example, for H = 100 m, 99.9% reliability, and exit gradient safety factor = 5, only a few points were found at higher construction cost ($268,206,048.88).

The deterministic optimum Pareto front related to the expected hydraulic conductivity (2 m/day) was also considered in this study. In general, the deterministic Pareto optimal was located close to 60% reliability trade-offs. However, some deterministic optimum solutions approached 40% reliability solutions. The 60% or 40% reliability of the deterministic solutions mean that there is a high probability of the exit gradient approaching the critical value, which might lead to piping failure. Based on this, we can deduce that the deterministic safety factors of 3 and 5 are insufficient to provide adequate safety for such important projects (WRSs), and are inappropriate to measure the safety of seepage designs that incorporate a degree of uncertainty. This is true if we assume that the prescribed safety factor is used to quantify uncertainty in the HHC only.

One important benefit of using multi-objective optimization in RBOM is the diversity of provided optimum solutions. The multi-objective optimization solver provides many optimum solutions for the same objective function values (approximately). These solutions could not be obtained by a single objective optimization model. These solutions provide more flexible options because some optimum solutions are more applicable in terms of design requirements, such as field limitation and construction procedures, etc.

This study presents several applications for a coupled simulation-optimization methodology that can optimize the design of WRSs. The method uses multiple meta-models of the seepage characteristics associated with such structures. These meta-models were trained on multiple datasets of simulated seepage scenarios to provide accurate seepage solutions. The methodology was extended to

H | Reliability | Construction cost ($) | Exit gradient | d_{1} | d_{2} | b | b^{*} |
---|---|---|---|---|---|---|---|

100 | 40% | 93,811,995.8 | 0.280 | 65.02 | 85.18 | 158.46 | 71.76 |

93,403,373.0 | 0.282 | 56.51 | 88.54 | 92.46 | 91.59 | ||

80 | 40% | 28,275,868.8 | 0.374 | 58.34 | 52.27 | 85.58 | 77.91 |

27,327,404.2 | 0.374 | 30.73 | 65.07 | 75.37 | 74.53 | ||

60 | 99.9% | 57,740,766.3 | 0.342 | 37.93 | 80.34 | 61.05 | 47.25 |

56,752,425.9 | 0.343 | 71.71 | 62.54 | 86.34 | 46.50 | ||

40 | 60.0% | 72,446,076.0 | 0.072 | 61.19 | 79.87 | 113.16 | 68.76 |

66,394,331.9 | 0.072 | 47.71 | 82.12 | 68.76 | 59.16 | ||

20 | 80.0% | 40,547,213.5 | 0.073 | 34.53 | 73.33 | 23.75 | 12.63 |

40,367,765.1 | 0.074 | 72.09 | 41.34 | 39.87 | 29.61 |

optimize the design of WRSs based on complex seepage models that consider non-homogenous and anisotropic hydraulic conductivity. Furthermore, the effect of uncertainty in seepage characteristics due to uncertain hydraulic conductivity was considered to determine reliable, minimum-cost designs.

The various meta-models are based on machine learning techniques such as ANN, SVM, and GRP. Many error measures were used to evaluate the predictive accuracy of the models to ascertain that their predictions were within allowable error ranges. The parameters of the optimization solver and machine learning-based meta-models were carefully selected according to multiple systematic experiments and trial-and-error iterations.

The results demonstrate that the S-O technique is a useful and efficient method of optimizing WRSs design with simple and complex seepage flow domains by integrating seepage modelling. The most important advantage of the S-O methodology is the incorporation the effect of uncertainty in seepage due to uncertainty in hydraulic conductivity on optimum design. This includes the incorporation of many levels of uncertainty or reliability based optimum design. Furthermore, several formulations based on multi-objective multi-realization optimization models were implemented to provide less restrictive formulation-based optimization models. This approach presents diverse optimum solutions, providing several design alternatives for the same objective functions (e.g. cost and exit gradient).

It is recommended for futures studies apply this S-O methodology to WRS design while incorporating different sources of uncertainty such as those of the upstream water table, critical exit gradient, and meta-model predictions. Furthermore, incorporating the requirement of structural design in WRS as additional constraints is a more advanced step to optimizing their designs.

The authors declare no conflicts of interest regarding the publication of this paper.

Al-Juboori, M. and Datta, B. (2018) An Overview of Recently Developed Coupled Simulation Optimization Approaches for Reliability Based Minimum Cost Design of Water Retaining Structures. Open Journal of Optimization, 7, 79-112. https://doi.org/10.4236/ojop.2018.74005