An Automated Machine Learning Engine with Inverse Analysis for Seismic Design of Dams
Abstract
:1. Introduction
2. Automated Machine Learning (AutoML)
2.1. MLBased Response Evaluation of Dams
2.2. Underpinning Theory
 Linear models: minimizing a regularized empirical loss with stochastic gradient descent (SGD), and Bayesian Automatic Relevance Determination (ARD) regression;
 Ensemble models: Adaboost, random forest (and decision trees), extra trees, gradient boosting;
 Probabilistic model: Gaussian process (GP) regression;
 Knearest neighbor (KNN) and support vector regression (SVR);
 Neural networks: Multilayer perceptron (MLP).
3. Design Variables
3.1. Dam Shape
 ${L}_{1}\in $ (50, 150) m.
 ${L}_{2}$ = ${L}_{1}\times {\alpha}_{1}$; ${\alpha}_{1}\in (0.00,0.05)$⟶${L}_{2}\in $ (0, 7) m.
 ${L}_{3}$ = ${L}_{4}\times {\alpha}_{3}$; ${\alpha}_{3}\in (1.00,1.20)$⟶${L}_{3}\in $ (7, 40) m.
 ${L}_{4}$ = ${L}_{1}\times {\alpha}_{2}$; ${\alpha}_{2}\in (0.12,0.24)$⟶${L}_{4}\in $ (6, 35) m.
 ${L}_{5}$ = ${L}_{6}\times {\alpha}_{5}$; ${\alpha}_{5}\in (0.30,0.70)$⟶${L}_{6}\in $ (20, 140) m.
 ${L}_{6}$ = ${L}_{1}\times {\alpha}_{4}$; ${\alpha}_{4}\in (1.10,1.60)$⟶${L}_{5}\in $ (55, 235) m.
 ${L}_{7}$ = ${L}_{5}\times {\alpha}_{6}$; ${\alpha}_{6}\in (0.75,0.90)$⟶${L}_{7}\in $ (40, 200) m.
 $WL$ = ${L}_{5}\times {\alpha}_{7}$; ${\alpha}_{7}\in (0.50,1.00)$
3.2. Material Properties
3.3. Loads
4. Data Structure
4.1. Software
4.2. InputOutput Coverage
 Scalar quantities cover the maximum (or minimum) response of the dam at a particular location and the entire duration of the applied ground motion. For example, maximum crest displacement shows the “global” behavior of the dam under the applied motion. Similarly, the maximum first principal stress at the dam heel is a “local” metric that presents the onset of cracking (if exceeds the tensile strength). Other peaks (i.e., maximum or minimum) response quantities can be extracted from displacement, stress, and strain results.
 Vector quantities cover the responses over time, or they present the spatial distribution of the response parameters. Cumulative inelastic duration (CID) shows the time intervals in which the stress at a particular location exceeds the tensile strength. The overstressed area (OA) illustrates the spatial distribution of regions within the dam body where the tensile strength exceeds the tensile strength (or a multiplayer of it).
 Out1: maximum horizontal crest displacement, ${\Delta}_{max}$
 Out2: maximum first principal stress at heel, ${\sigma}_{p1,max}^{heel}$
 Out3: maximum first principal stress at upstream face 5% from the heel, ${\sigma}_{p1,max}^{UP5\%}$
 Out4: maximum first principal stress at toe, ${\sigma}_{p1,max}^{toe}$
 Out5: minimum third principal stress at the heel, ${\sigma}_{p2,min}^{heel}$
 Out6: minimum third principal stress at the toe, ${\sigma}_{p2,min}^{toe}$
 Out7: CID for demand capacity ratio exceeds one at the heel, $CI{D}_{DCR=1}^{heel}$
 Out8: CID for demand capacity ratio exceeds two at the heel, $CI{D}_{DCR=2}^{heel}$
 Out9: Overstressed area for demand capacity ratio exceeds one at the heel, $O{A}_{DCR=1}^{heel}$
 Out10: Overstressed area for demand capacity ratio exceeds two at the heel, $O{A}_{DCR=2}^{heel}$
5. Results: Surrogate Model
5.1. Scenario 1: Single Output
5.2. Scenario 2: MultiOutput, Out1 through Out6
5.3. Scenario 3: MultiOutput, Out1 through Out10
6. Results: Dam Design Engine
 ICOLD recommendations There are two basic seismic loads for the design of new dams [70]: Operating Basis Earthquake (OBE) which represents the seismic intensity level at the dam site for which only minor (easily repairable) damage is acceptable and the dam should remain functional. The OBE corresponds to the return period of 145 years (50% probability of exceedance in 100 years). Safety Evaluation Earthquake (SEE) represents the seismic intensity level at the dam site for which a dam must be able to resist without the uncontrolled release of the reservoir water. The SEE ground motion can be obtained from a probabilistic and/or deterministic seismic hazard analysis. For large and high consequence dams, SEE is defined as (a) Maximum Credible Earthquake (MCE) from DSHA where the parameters should be estimated at the 84th percentile level, (b) Maximum Design Earthquake (MDE) from PSHA corresponding to return period of 10,000 years (1% probability of exceedance in 100 years) [71,72].
 FEMA recommendations Timebased performance assessment evaluates a dam’s performance over a period considering all earthquakes that may occur in that period, and the probability that each will occur [73]. This procedure follows the following main steps: (a) generate a seismic hazard curve, i.e., $\lambda $ vs. ${S}_{a}\left({T}_{1}\right)$, (b) compute seismic intensity range and split it into ${N}_{i}$ equal intervals, (c) develop a target response spectrum, ${S}_{a}^{trg}\left(T\right)$, for each intensity range, and (d) select and scale suites of ${N}_{gm}$ ground motions for each spectrum.
Algorithm 1: Estimation of the fundamental period of the coupled system [74,75]. 
Inputs:${L}_{w}$ [m], ${L}_{6}$ [m], ${E}_{c}$ [MPa], ${E}_{r}$ [MPa], ${\alpha}_{w}$ Output:${T}_{dwf}$ [sec].

Example of AutoML Seismic Design
 Candidate 1: ${L}_{1}=72$ m, ${L}_{2}=0$, ${L}_{3}=15.4$ m, ${L}_{4}=14$ m, ${L}_{5}=40$ m, ${L}_{7}=75$ m.
 Candidate 2: ${L}_{1}=72$ m, ${L}_{2}=0$, ${L}_{3}=15.4$ m, ${L}_{4}=14$ m, ${L}_{5}=40$ m, ${L}_{7}=85$ m.
 Candidate 3: ${L}_{1}=70$ m, ${L}_{2}=0$, ${L}_{3}=15.4$ m, ${L}_{4}=14$ m, ${L}_{5}=40$ m, ${L}_{7}=80$ m.
7. Summary
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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No.  Description of IM  Symbol  Mathematical Model 

1  Peak ground acceleration  $PGA$  $max\left(\left\ddot{u}\left(t\right)\right\right)$ 
2  Peak ground velocity  $PGV$  $max\left(\left\dot{u}\left(t\right)\right\right)$ 
3  Peak ground displacement  $PGD$  $max\left(\leftu\left(t\right)\right\right)$ 
4  Acceleration spectrum intensity  $ASI$  ${\int}_{0.1}^{0.5}{S}_{a}\left(T,\xi \right)dT$ 
5  Velocity spectrum intensity  $VSI$  ${\int}_{0.1}^{2.5}{S}_{v}\left(T,\xi \right)dT$ 
6  Firstmode spectral acceleration  ${S}_{a}\left({T}_{1}\right)$  ${S}_{a}({T}_{1},\xi )$ 
7  Firstmode spectral velocity  ${S}_{v}\left({T}_{1}\right)$  ${S}_{v}({T}_{1},\xi )$ 
8  Firstmode spectral displacement  ${S}_{d}\left({T}_{1}\right)$  ${S}_{d}({T}_{1},\xi )$ 
9  Firstmode spectral acceleration  ${S}_{a}\left({T}_{2}\right)$  ${S}_{a}({T}_{2},\xi )$ 
10  Rootmeansquare of acceleration  ${a}_{RMS}$  ${\left(\frac{1}{{t}_{tot}}{\int}_{0}^{{t}_{tot}}{\left(\ddot{u}\left(t\right)\right)}^{2}dt\right)}^{1/2}$ 
11  Rootmeansquare of velocity  ${v}_{RMS}$  ${\left(\frac{1}{{t}_{tot}}{\int}_{0}^{{t}_{tot}}{\left(\dot{u}\left(t\right)\right)}^{2}dt\right)}^{1/2}$ 
12  Arias intensity  ${I}_{A}$  $\frac{\pi}{2g}{\int}_{0}^{{t}_{tot}}{\left(\ddot{u}\left(t\right)\right)}^{2}dt$ 
13  Specific energy density  $SED$  ${\int}_{0}^{{t}_{tot}}{\left(\dot{u}\left(t\right)\right)}^{2}dt$ 
14  Cumulative absolute velocity  $CAV$  ${\int}_{0}^{{t}_{tot}}\left\ddot{u}\left(t\right)\rightdt$ 
15  Significant duration  ${t}_{sig}$  ${t}_{0.95{I}_{A}}{t}_{0.05{I}_{A}}$ 
Rank  Ensemble Weight  Type  Duration 

1  $0.16$  Gradient Boosting  $24.10$ 
2  $0.04$  Gradient Boosting  $3.85$ 
3  $0.14$  Gradient Boosting  $11.88$ 
4  $0.24$  ARD Regression  $103.09$ 
5  $0.06$  Gradient Boosting  $11.311$ 
6  $0.12$  Gradient Boosting  $3.21$ 
7  $0.06$  Gradient Boosting  $11.13$ 
8  $0.10$  Gradient Boosting  $8.95$ 
9  $0.06$  Gradient Boosting  $147.51$ 
10  $0.02$  ARD Regression  $1.65$ 
Rank  Ensemble Weight  Type  Duration 

1  $0.84$  Extra Trees  $176.97$ 
2  $0.06$  Random Forest  $132.60$ 
3  $0.10$  KNearest Neighbor  $19.53$ 
Output  ${\mathit{R}}^{2}$  RMSE  MAE 

Out1  $0.912$  $0.007$  $0.002$ 
Out2  $0.958$  $0.009$  $0.004$ 
Out3  $0.956$  $0.009$  $0.004$ 
Out4  $0.948$  $0.010$  $0.005$ 
Out5  $0.958$  $0.009$  $0.004$ 
Out6  $0.947$  $0.010$  $0.005$ 
Overall  $0.947$  $0.009$  $0.004$ 
Rank  Ensemble Weight  Type  Duration 

1  $0.08$  Decision Tree  $1.88$ 
2  $0.04$  Decision Tree  $158.13$ 
3  $0.22$  Decision Tree  $161.25$ 
4  $0.04$  Decision Tree  $6.22$ 
5  $0.16$  Decision Tree  $7.37$ 
6  $0.06$  Decision Tree  $1.22$ 
7  $0.06$  Decision Tree  $160.86$ 
8  $0.20$  Decision Tree  $101.23$ 
9  $0.02$  Decision Tree  $1.7$ 
10  $0.04$  Decision Tree  $69.75$ 
11  $0.02$  KNearest Neighbor  $18.94$ 
12  $0.06$  KNearest Neighbor  $1.31$ 
Output  ${\mathit{R}}^{2}$  RMSE  MAE 

Out1  $0.884$  $0.008$  $0.003$ 
Out2  $0.946$  $0.010$  $0.005$ 
Out3  $0.947$  $0.010$  $0.005$ 
Out4  $0.933$  $0.011$  $0.005$ 
Out5  $0.950$  $0.009$  $0.005$ 
Out6  $0.927$  $0.012$  $0.006$ 
Out7  $0.912$  $0.013$  $0.004$ 
Out8  $0.909$  $0.009$  $0.001$ 
Out9  $0.926$  $0.032$  $0.010$ 
Out10  $0.854$  $0.016$  $0.003$ 
Overall  $0.919$  $0.013$  $0.005$ 
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HaririArdebili, M.A.; PourkamaliAnaraki, F. An Automated Machine Learning Engine with Inverse Analysis for Seismic Design of Dams. Water 2022, 14, 3898. https://doi.org/10.3390/w14233898
HaririArdebili MA, PourkamaliAnaraki F. An Automated Machine Learning Engine with Inverse Analysis for Seismic Design of Dams. Water. 2022; 14(23):3898. https://doi.org/10.3390/w14233898
Chicago/Turabian StyleHaririArdebili, Mohammad Amin, and Farhad PourkamaliAnaraki. 2022. "An Automated Machine Learning Engine with Inverse Analysis for Seismic Design of Dams" Water 14, no. 23: 3898. https://doi.org/10.3390/w14233898