SPIE Smart Structures + NDE 2023

Gaussian process regression surrogate modelfor dynamic analysis to account for uncertainties

in seismic loading

Taisei Saida (University of Tsukuba)
Mayuko Nishio (University of Tsukuba)

2/21

Uncertainties in infrastructures• Infrastructures such as bridges are designed for load and strength.
• However, during the service life, structures may deteriorate and sufferdamage.
• This is due to the difference between design and reality. In reality, there aremany uncertainties.
• A reliability analysis is required that considers uncertainties related to loadsand structural strength.

Deteriorate

Piers has no damage

Large
displacement

All bearings ruptured（MLIT, Measures to prevent roads from aging, Aging Status）（MLIT, Anti-aging Intiatives）（JSCE, Steel Structure Committee）

Reliability Analysis Flow
TargetProbability

Considering
Uncertaintiesex)Mass of Piers

3/26

4/26

Reliability Analysis Flow
General Reliability Analysis
StiffnessMonte Carlo
Sampling

Mass

Probability

Considering
Uncertaintiesex)Mass of Piers

Analyticalmodel

Outputdistribution
Probability

TargetdangerMaximum
StressDisaster Risk
Evaluation

5/26

Reliability Analysis Flow
TargetGeneral Reliability Analysis
Outputdistribution
Probability

Analyticalmodel

StiffnessMonte Carlo
Sampling

Reliability Analysis using Surrogate Model
AnalyticalmodelAnalysis
DataMaximum
StressMass

Surrogate
ModelMaximum
Stress

Outputdistribution
Probability

DoE
Sampling
Stiffness

Probabilityex)Mass of Piers

Disaster Risk
Evaluation

Maximum
Stress

Mass

Considering
Uncertainties

dangerdangerMaximum
StressSurrogate models can reduce
computational cost of reliability analysis

6/21【Previous Studies】Surrogate model for seismic response analysis

Abbiati et al. 2021
• Using parameters of artificial ground motions
and structure as inputs• Constructed surrogate model for seismic risk
analysis of piping(Journal of Loss Prevention in the Process Industries, Vol.72)

Cannot input actual
ground motion

7/21【Previous Studies】Surrogate model for seismic response analysis

Abbiati et al. 2021
• Using parameters of artificial ground motions
and structure as inputs• Constructed surrogate model for seismic risk
analysis of piping

Cannot input actual
ground motion(Journal of Loss Prevention in the Process Industries, Vol.72)

Zhang et al. 2020
• Seismic waveforms are input
using convolutional neural networks (CNN)• Constructed surrogate models
for seismic response analysis of buildings••

Not consider
structural uncertainty
Unclear why theresult is obtained(Engineering Structures, Vol.206)•Issue•Considers both actual ground motion andstructural parameters
Be able to explain why the predicted results
are obtained (Explainability)

【Objective】Deep kernel learning surrogate model
Feature extraction of seismic loads

Stiffness(K)

●
●

●
●

●

●

●

●

●

●

・・・・・・・・・・●

●

●

●

・・●

●

●

●

CNN extracts the feature of loadsExplainability
Seismic Loads: Grad-CAM
Structural Parameter: ARD

Maximum Displacement

Mass(M)●

Gaussian Process Regression (GPR)

HztResponse Spectrum

Seismic Wave

Structural質量Parameter●Constructing explainable deep kernel learning surrogate model withCNN and GPR to reduce computational costs on seismic risk analysis

9/26

Gaussian Process Regression (GPR) with ARD Kernel
GPR• Nonparametric• Non-linear regression

x : input vectory : output vector
k：kernel function
K : kernel matrix

Kernel Matrix
K nm = k ( x n , x m )Knm : elements of kernel matrix

10/26Gaussian Process Regression (GPR) with ARD Kernel
GPR

ARD Kernel Function• Nonparametric• Non-linear regression

ARD：Automatic Relevance Determinationx : input vectory : output vector
k：kernel function
K : kernel matrix

Kernel Matrix
K nm = k ( x n , x m )Knm : elements of kernel matrix

Matern5/2 kernelk (r ) = σ  1 +Dri  5  +  2  exp  − 5  
3 i =1 li i =1 lii =1 li Dri

5 D ri211/26Gaussian Process Regression (GPR) with ARD Kernel
GPRARD Kernel Function• Nonparametric• Non-linear regression

ARD：Automatic Relevance Determinationx : input vectory : output vector
k：kernel function
K : kernel matrix

Kernel Matrix
K nm = k ( x n , x m )Matern5/2 kernelk (r ) = σ  1 +Dri  
5  +  2  exp  − 5  
3 i =1 li i =1 li
i =1 li Dri

5 D ri2Length Scale（li）
Represents the contributionof each input variable to the output

ARD KernelEx)
Poisson’s ratio

Estimate the contributionof input parameters

Young’s modulus

Thickness

Knm : elements of kernel matrix
Contribution (%)

Grad-CAM for contribution of seismic loads
Deep kernel learning k=1 2M●●●・・Conv・・・
●
●

●

GAP

●

●

●

●

●

・・MLP・・・・

●

●

●

●

GPR・

●

Predict Val (P)●●A1 , A 2 , … , A N

Seismic Wavet

Response Spectrum●

StructuralParameter●

Grad-CAMMk 
LGrad-CAM = ReLU    k A  kMapping

Response Spectrum1 N Pk =  k
N i Ai

Visualize
contribution partto the output

Analytical model of an isolated RC pier

13/26

（Japan Road Association, 1997）

AnalysisParameters

Upper Structure

Integration：
Isolated Bearing

BilinearModel•

Newmark - β•Newton Raphson

Time Increment：0.001sTakeda
Model
RC Pier
2DOF model of
Isolated RC Pier

Rayleigh damping
Damping Ratio：•

2% for pier•

0% for bearing14/26

Uncertainty parameter setting
Parameters

Mu

SeismicIsolation
Bearing

Kb1,
Kb2,Qb
Krc1,Krc2,Qrc

Superstructure

Mrc

NominalWeight (Mu)604,000 kg

Primary stiffness (Kb1)

40,023.2 kN/m

Secondary stiffness (Kb2)

6,154.4 kN/mYield load (Qb)

1,117.2 kN

Weight (Mrc)

346,300 kg

Primary stiffness (Krc1)

110,000 kN/m

Secondary stiffness (Krc2)

8,250 kN/mYield load (Qrc)

3,399 kN

Uncertainty

Uniform
Distribution± 10 %RC Pier（Reference: Japan Road Association, 1997）

Reliability Analysis Overview and Input/Output
Uncertain
Parameters

Seismic Response
AnalysisMaximum Displacement(Pier and Bearing)

8 structural parameters
20 waves uncertainty••••••••••

15/26

PierJMA KOBE (1996)
Tsurui-nishi (2003)
Tsurui-higashi (2003)
Taiki (2003)
Tokamachi (2004)
Ojiya (2004)
Nagaoka (2004)
Ichinoseki-nishi(2008)
Otsu (2016)
Oguni (2016)

Max Displacement (m)Bearing

Output maximum
displacement

Max Displacement (m)Surrogate model inputs and outputs
Inputs
Structural Parameter
Response Spectrum

Surrogate Model

Outputs
Maximum Displacementsof Pier and Bearing

【Result】Predict Maximum Displacement
Predicts by surrogate model
Pier
R2: 0.976

Predicted Disp(m)Test Disp(m)Test Disp(m)Train data：300 Test data(from analysis)：10000

Bearing
R2: 0.977
Highly accurate predictions.
Predicted Disp(m)16/21

【Result】Predict Maximum Displacement
Predicts by surrogate model

Test Disp(m)Pier
R2: 0.976

Bearing
R2: 0.977
Highly accurate predictions.

Predicted Disp(m)Predicted Disp(m)Pier■：Predict■：TestBearing■：Predict■：TestMax Displacement (m)Frequency

Frequency

Test Disp(m)Train data：300 Test data(from analysis)：10000

Max displacement
distribution ispredictable.

Max Displacement (m)Surrogate models can predict with high accuracies

17/21

【Result】Estimated Contribution
Estimated Contribution to Pier’s Max Disp
Train num：300 Test (from analysis) num：10000
EigenPeriod

HighAcceptability

18/21

19/21【Result】Estimated Contribution
Estimated Contribution to Pier’s Max Disp
Train num：300 Test (from analysis) num：10000
1st Mode(about 1.03s)

High
Acceptability

Low
Acceptability

1st Mode(about 1.03s)

20/21【Result】Estimated Contribution
Estimated Contribution to Pier’s Max Disp
Train num：300 Test (from analysis) num：10000

Estimated
Contribution byARD（%）

1st Mode(about 1.03s)

High
Acceptability

Low
Acceptability

1st Mode(about 1.03s)

Contribution■：Seismic Load■：Struct Param
Large contributionfrom seismic loads

Contributions of seismic loads and structural parameters can be estimatedConclusion and Future Works
Conclusion• A surrogate model for seismic response analysis using deep kernel learning wasconstructed.
• Seismic load features were extracted by CNN.
• The contributions of seismic loads were estimated by Grad-CAM and structuralparameters by ARD.

• The constructed surrogate model exceeded 0.97 in the R2 index, and the predicteddistribution was qualitatively consistent with the test data.
• In some cases, the Grad-CAM showed a larger contribution close to the naturalperiod, while in other cases it did not.

• The ARD-estimated contributions were in agreement with the engineering findings
as well, with the external forces having a larger contribution.

Future Work• Combined with adaptive sampling, which preferentially samples points that have asignificant impact on the performance of the surrogate model, the computational
cost could be further reduced

Thank you for listening.

Acknowledgement
This study was supported by the JST FOREST Program, Japan