8 mm
215
mm
Nested Multiscale Simulations
10
๐ฃ๐ฃ๐ญ๐ญ๐ญ๐ญ IP๐ข๐ข๐ญ๐ญ๐ญ๐ญ IP
๐ฃ๐ฃ๐ญ๐ญ๐ญ๐ญ IP
* Resin not shown
640,000 IPs
15,000 IPs
900,000 IPs
๐ข๐ข๐ญ๐ญ๐ญ๐ญ IP
๐ฃ๐ฃ๐ญ๐ญ๐ญ๐ญ IP
๐ข๐ข๐ญ๐ญ๐ญ๐ญ IP
10 ฮผm
80,000 IPs
โข Random process based UQ in a single length-scaleโข Neglecting spatial & part-to-part variations
Prior Works:
โข Integration points (IPs) arestructures at finer scales.
๐ผ๐ผ = 90ยฐvf = 62%
๐ผ๐ผ = 80ยฐvf = 68%
vf = 65%
vf = 65%vf = 60%
vf = 70%vf = 64%
๐ผ๐ผ: Yarn angle
vf: Avg. fiber volume fraction
(3) Cured UD-RVE at Microscale
(2) Cured Woven-RVE at Mesoscale
(1) Woven Composite at Macroscale
Our Approach For UQ & UP in Multiscale Simulations
11
Meso StructureMeso SRPMeso Simulator
Macro StructureMacro SRPMacro Simulator
โข Modeling uncertainties sources and their spatial variations via spatial random processes (SRPs)
โข Coupling SRPs across length-scales via top-down sampling
โข Creating metamodels of homogenized constitutive relations w microstructure variations
Bostanabad, R., et. al. (2018) CMAME, 338, 506-532.
Managing Complexity โOn-the-fly Machine Learning
Metamodeling
Homogenization
Homogenization Speed-up
Sensitivity Analysis
Input Variation
Out
put V
aria
tion
Micro Structure
Micro SRPMicro Simulator
Homogenization Top-down Sampling
Top-down Sampling
Sampling Dimension Reduction
๐ฅ๐ฅ1
๐ฅ๐ฅ2
Multi-Response Gaussian Processes (MRGPs) for Quantifying Correlated Uncertain Sources
12
Multi-response Gaussian process (MRGP):
GP
MRGP
Only need mean and covariance functions.
Single response Gaussian processes:
๐๐~๐บ๐บ๐บ๐บ ๐๐ ๏ฟฝ ,๐๐2๐๐ ๏ฟฝ,๏ฟฝ
๐๐~๐๐๐๐๐บ๐บ๐บ๐บ ๐๐ ๏ฟฝ ,๐ฎ๐ฎโจ๐๐ ๏ฟฝ,๏ฟฝ
Mean Function Covariance Function
๐ฅ๐ฅ1
๐ฃ๐ฃ๐๐
๐๐
๐ฅ๐ฅ1
๐ฃ๐ฃ๐๐
โข Captures the between-response correlations &variations around themeans.
๐ฎ๐ฎ = 4 โ10โ10 50
๐๐ = 4,๐๐ = 1.5,๐๐ = 55GP:
MRGP:
๐๐ = 1.5,๐๐ = 550
๐๐ ๐๐,๐๐โฒ = ๐๐โ โ๐๐=1๐๐ 10๐๐๐๐ ๐ฅ๐ฅ๐๐โ๐ฅ๐ฅ๐๐
โฒ 2
Top-down Sampling
13
Hyperparameters of the MRGP at the higher scale are considered as the responses of the MRGP at thelower scale.
โข Hierarchy of random fields.โข Non-stationary random field
at the fine scale.โข No limit on the number of
levels/scales.
โข Superscripts โ Level โข Subscripts โ Counter (e.g., for realization)
Level 1 (Coase Scale) Level 2 (Fine Scale)
Needs ๐ท๐ท,๐ฎ๐ฎ,๐๐for its MRGP
Needs ๐ท๐ท,๐ฎ๐ฎ,๐๐for its MRGP
Macroscale MRGP๐๐๐๐~๐๐๐๐๐บ๐บ๐บ๐บ ๐ท๐ท๐๐,๐ฎ๐ฎ1โจ๐๐๐๐ ๏ฟฝ,๏ฟฝ
Realization 1
Realization 2
๐๐๐๐๐๐๐ท๐ท๐๐,๐ฎ๐ฎ2, ๐๐๐๐
๐๐๐๐๐๐๐ท๐ท๐๐,๐ฎ๐ฎ2, ๐๐๐๐
๐๐๐๐ = ln๐๐๐๐
1 + ๐๐๐๐2๐๐๐๐2
๐๐๐๐2 = ln 1 +๐๐๐๐2
๐๐๐๐2Transformation: required if the underlying distribution is not normal.
โข For ๐๐ normal and ๐๐ = exp ๐๐ :
Sensitivity Analysis of Cascaded Effects to the Mesoscale: Moduli
14
โข The average (๏ฟฝฬ ๏ฟฝ๐ฃ) is important.โข Spatial variations are not important.
1. Fiber volume fraction: 2. Misalignment angle:โข The average zenith angle (๏ฟฝฬ ๏ฟฝ๐) is important.โข Spatial variations of ๐๐ are not important.โข The average and spatial variations of azimuth angle (๐๐) are not important.
๏ฟฝฬ ๏ฟฝ๐ฃ ๏ฟฝฬ ๏ฟฝ๐๏ฟฝ๐๐
MPaNOTE: Each point inthe figure is averagedover 30 simulations
Sensitivity analyses for dimension reduction: Identify the hyperparameters that affect thehomogenized response of a woven RVE:โข ๐ท๐ท important โMean values importantโข ๐ฎ๐ฎ or ๐๐ important โ Spatial variations important
Microscale & Mesoscale Metamodels
15
3๏ฟฝ๐ช๐ช =
๐ถ๐ถ1 ๐ถ๐ถ7 ๐ถ๐ถ9๐ถ๐ถ7 ๐ถ๐ถ2 ๐ถ๐ถ8๐ถ๐ถ9 ๐ถ๐ถ8 ๐ถ๐ถ3
0 0 00 0 00 0 0
0 0 00 0 00 0 0
๐ถ๐ถ4 0 00 ๐ถ๐ถ5 00 0 ๐ถ๐ถ6
Stiffness Matrix of UD-RVE
2๏ฟฝ๐ช๐ช =
๐ถ๐ถ1 ๐ถ๐ถ7 ๐ถ๐ถ11๐ถ๐ถ7 ๐ถ๐ถ2 ๐ถ๐ถ8๐ถ๐ถ11 ๐ถ๐ถ8 ๐ถ๐ถ3
๐ถ๐ถ13 0 0๐ถ๐ถ12 0 0๐ถ๐ถ9 0 0
๐ถ๐ถ13 ๐ถ๐ถ12 ๐ถ๐ถ90 0 00 0 0
๐ถ๐ถ4 0 00 ๐ถ๐ถ5 ๐ถ๐ถ100 ๐ถ๐ถ10 ๐ถ๐ถ6
Stiffness Matrix of the Woven RVE
log(๐๐)
Size of the training dataset
Prediction Error
Micro: Predicting the stiffnessmatrix of a UD- RVE for anyvalues of 3๏ฟฝฬ ๏ฟฝ๐ฃ,๐ธ๐ธ๐๐,๐ธ๐ธ๐๐.
โข MRGP training: 30 to 80 samples
โข Validation: 20 samples
Meso: Predicting the stiffness matrix of a woven-RVE for any values of ๐ผ๐ผ, 2๏ฟฝฬ ๏ฟฝ๐ฃ and 2๏ฟฝฬ ๏ฟฝ๐.
โข MRGP training: 10 to 60 samples
โข Validation: 20 samples
๐๐ = 1001
20๏ฟฝ๐๐=1
20
1 โ๏ฟฝ๐ฆ๐ฆ๐ฆ๐ฆ
2
%
Prediction Error
๐๐
Size of the training dataset
Uncertainty Impacts at the Macroscale
16
Effect on global response: Reaction force(17%) affected by the spatial variations.
Effect on local stress in the mid-sectionโข The coefficient of variation 60
600ร 100 = 10%.
Comparison of Macroscale Stress Fields
17
MeanStressField
Standard Deviation of Stress Field (30
Simulations)
Ref
๐๐๐บ๐บ๐๐
๐ถ๐ถ AllUncorrelated
AllCorrelated
โข Spatial variations of yarn angle are more important than the other parameters.
โข Correlated spatial variations result in higher stress values.
โข Ref: Reference solution without any spatial variations.
โข The only difference between the cases is spatial variations (i.e., the averages are the same).
Remarks on Our Approach
18
โข Non-intrusive: No need to edit the finite element codes
โข Uncertainty Sources: Accounting for multi-response non-stationary uncertain sources
โข Multiscale: Effect of micro and meso uncertainties on macro response
โข Physical insights: MRGP hyperparameters can be directly linked to physical parameters and physical constraints
โข Impact: Uncertain impact will be more critical for nonlinear behavior
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