Statistical Adjustments to Engineering Models - Semantic Scholar
Statistical Adjustments to Engineering Models V. Roshan Joseph and Shreyes N. Melkote JQT, October, 2009 Supported by NSF CMMI-0654369
Model-based Quality Improvement • Models are used for – Process control – Process optimization
• Two types of models – Statistical models – Engineering models
Statistical Models • Statistical models – Developed based on data – Linear/nonlinear regression models
Engineering Models • Engineering models – Developed based on engineering/physical laws – Analytical and finite element models
Engineering Models Vs Statistical Models • Statistical models – Predictions are good closer to the data, but can be poor when made away from data
• Engineering models – Physically meaningful predictions, but often are not accurate because of the assumptions
• Can we integrate them to produce better models?
Engineering - Statistical Models • Improve engineering models using data – More realistic predictions than engineering models – Less expensive than pure statistical models (fewer data)
Surface Roughness Prediction in Micro-Turning Ykinemat
Workpiece
Primary cutting edge
ic
Tool Feed x
Secondary cutting edge Nose radius
Engineering model:
Statistical model: Y = β 0 + β1 x + β 2 x 2
Existing methods • Mechanistic model calibration – Estimate unknown parameters (calibration parameters) from data – Box, Hunter, Hunter (1978), Kapoor et al. (1998) – Not a general method
• Bayesian calibration – Kennedy and O’Hagan (2001) – Reese et al. (2004), Higdon et al. (2004), Bayarri et al. (2007), Qian and Wu (2008).
Bayesian Methodology • • • •
Take engineering model as the prior mean Get data from the physical experiment Obtain posterior distribution Engineering-Statistical model is the posterior mean
Prior
Posterior
Data xxxxxxxxx
Engineering model
Prior distribution
Posterior distribution
Eng.-Stat. Model
Methodology-continued • Output: Y • Factors: • Random error:
• Objective: Find • Engineering model: • Calibration parameters: • Data:
Sequential Model Building Engineering Model
Positive relation?
No
Check & Correct
No
Engineering Model
Yes Is MI large? Yes Constant adjustment
Is MI large? Yes Functional Adjustment Model
No
Constant Adjustment Model
Methodology-continued • Check the usefulness of engineering model using graphical analysis • If it is useful
• If MI is small, then stop. Engineering model is good.
Constant adjustment model
• If MI is small, then stop. CAM is good.
Functional adjustment model
• Add terms until MI is small enough.
Constant adjustment model
Posterior distribution • posterior distribution is
• constant adjustment predictor is
• Prediction interval
Simplification • least squares estimate
Empirical Bayes estimation • Estimate
hyperparameters by maximizing
Approximate frequentist procedure • Fit the simple linear regression
Surface roughness example • Engineering model:
• There is a positive relation
Example-continued • From replicates
• Engineering model is not good for prediction
Constant adjustment model
Functional adjustment model
Two-stage estimation • Use the estimate of from the constant adjustment model •
Approximate frequentist procedure • Fit a multiple linear regression • Do a variable selection
Surface roughness example
Calibration parameters • Liu and Melkote (2006)
New engineering model
• R(x) is calculated using a combination of analytical formulas and finite element simulations
Statistical adjustments • First use least squares estimate
2 x f ( x;η~ ) = − 24.83 + 4.49 log R ( x) 8r
• MI=.209 (new engineering model is good)
Constant adjustment model
Approximate frequentist procedure • Fit a nonlinear regression
A Spot Welding Example • Higdon et al. (2004) and Bayarri et al. (2007) – Three factors: Load, Current, and Gage – One calibration parameter
Eng. Model (Black-dashed) : 0.69 Joseph&Melkote (Red-solid): 0.23 Bayarri et al. (Blue-dotted) : 0.20
Example: LAMM Laser assisted mechanical micromachining (LAMM) integrates thermal softening with mechanical micro cutting
+ Laser heating
= LAMM Mechanical micromachining
Objective Find optimum processing conditions that minimize cutting/thrust forces and thermal damage.
Thermal Model Natural B.C. on front face
Y X Z Symmetry B.C. on bottom face
• Mapped dense mesh (25 μm x 12.5 μm x 20μm) • An 8 noded 3-D thermal element (Solid70) is used • Gaussian distribution of heat flux applied to a 5x5 element matrix which sweeps the mesh on the front face
Geometric Model γ chip
cos( α avg + θ PD ) 2 sin θ PD = + sin( π / 4 + θ PD ) cos( α avg − φ ) sin( φ + θ PD )
γ work =
2 sin θ PD + sin(π / 4 + θ PD )
sin(θ PD + θ / 2) sin θ / 2 + sin(θ PB + θ / 2) sin(θ PB + θ PD ) sinψ sin(ψ + θ / 2)
(Manjunathiah et. al, 2000) •
γ chip = 2V •
γ work = 2V
γ chip 2 sin( π / 4 + θ PD )PD γ work 2 sin(π / 4 + θ PD ) PD +
γ eff =
v chip γ chip + v work γ work •
•
γ eff = sin(ψ + θ / 2) PC sinψ
v chip + v work
•
vchip γ chip + vwork γ work vchip + vwork
For plane strain conditions,
ε = γ eff / 3 •
•
ε = γ eff / 3
Shear Flow Strength (
)
⎛ ⎛ ε& σ ( ε ,ε& ,T , HRC ) = A + Bε + C ln( ε + ε 0 ) + D ⎜1 + E ln⎜⎜ ⎜ ⎝ ε&0 ⎝
32 28 24 680
67 0
66 0
65 0
20
64 0
63 0
62 0
0 61
Depth below the surface μ( m)
68 0
67 0
65 0
64 0
63 0
62 0
36
66 0
40
( )
⎞ ⎞⎛ m ⎟ ⎟⎜1 − T * ⎞⎟ ⎟ ⎟⎝ ⎠ ⎠ Yan et⎠ al., 2007 69 0
S =σ / 3
n
16 12 8
125
680
100
670
75
660
50
650
25
640
0
63 0
0
62 0
61 0
4
150
Distance from the center of the tool face along tool edge at 100 μm from the center of the laser beam (μm)
10W laser power, 10 mm/min speed 100 μm laser-tool distance and 110 μm spot size
Forces • Cutting and thrust forces, n
∑S( i )w( i )
Fc = {( h − p )cotφ + h + rn sinθ − ( k −1)δ }
i=1
n
∑S( i )w( i )
Ft = {( h − p )cotφ − h + rn sinθ + ( k −1)δ cotψ }
i=1
Equilibrium Forces/Deflection
Force model
Force prediction
• Positive relation, but predictions are smaller than actual
Force prediction-continued
• Better than cutting force, but slightly smaller than actual
Engineering-Statistical Force Models
Plot of measured vs. predicted cutting and thrust forces
Optimization Problem • For a given depth of cut (t), find the optimum levels of set depth of cut, laser power, laser speed, and distance from tool to minimize cutting/thrust forces while making sure there is no heat affected zone.
min x1 , x2 , x3 , x4 yˆ + yˆ 2 c
2 t
subject to doc = t T2 ≤ Ac1
Nonlinear programming {
} { 2
}
min 1.54 x10.89 exp(0.0014 x 2 − 0.009 x3 e −0.0034 x4 ) + 1.03 x10.8 exp(0.0014 x 2 − 0.043 x3 e −0.0034 x4 )
x1 − 0.57 x10.8 exp(0.0014 x 2 − 0.196 x3 e −0.0034 x4 ) = t 25 + 196.4 x3 exp(−0.0021x1 x3 − 0.00045 x 2 x3 ) ≤ 800
10 ≤ x1 ≤ 25, 10 ≤ x2 ≤ 50, 0 ≤ x3 ≤ 10, 100 ≤ x4 ≤ 200
2
Optimization Results • For example, for depth of cut = 10 μm • Set depth of cut (x1) = 12.30 μm • Cutting speed (x2) = 10 mm/min • Laser power (x3) = 4.5 W • Laser location from the tool edge (x4) = 100 μm
Validation 50
Before machining
30
After machining
20
10 μm
10
0 0
0.1
0.2 0.3 0.4 Distance (mm)
0.5
0.6
45 40
Hardness (HRC)
Height (μm)
40
35 30 10 μm groove depth
25
25 μm groove depth
20 0
50
100
150
200
250
Distance from the edge of groove (μm)
300
Conclusions • Engineering models can be improved by using data • Engineering-Statistical models perform better than engineering models and statistical models • Need relatively less amount of data • They use the physics of the process
Process Optimization
Engineering knowledge
Factors & Levels
Experiment
Engineering model
Engineering-Statistical model
Statistical model
Optimize
Conclusions-continued • Simple procedure – Fit two linear/nonlinear regressions – Do variable selection
• Easy-to-implement – No additional programming is required
Model-based Quality Improvement • Models are used for – Process control – Process optimization
• Two types of models – Statistical models – Engineering models
Statistical Models • Statistical models – Developed based on data – Linear/nonlinear regression models
Engineering Models • Engineering models – Developed based on engineering/physical laws – Analytical and finite element models
Engineering Models Vs Statistical Models • Statistical models – Predictions are good closer to the data, but can be poor when made away from data
• Engineering models – Physically meaningful predictions, but often are not accurate because of the assumptions
• Can we integrate them to produce better models?
Engineering - Statistical Models • Improve engineering models using data – More realistic predictions than engineering models – Less expensive than pure statistical models (fewer data)
Surface Roughness Prediction in Micro-Turning Ykinemat
Workpiece
Primary cutting edge
ic
Tool Feed x
Secondary cutting edge Nose radius
Engineering model:
Statistical model: Y = β 0 + β1 x + β 2 x 2
Existing methods • Mechanistic model calibration – Estimate unknown parameters (calibration parameters) from data – Box, Hunter, Hunter (1978), Kapoor et al. (1998) – Not a general method
• Bayesian calibration – Kennedy and O’Hagan (2001) – Reese et al. (2004), Higdon et al. (2004), Bayarri et al. (2007), Qian and Wu (2008).
Bayesian Methodology • • • •
Take engineering model as the prior mean Get data from the physical experiment Obtain posterior distribution Engineering-Statistical model is the posterior mean
Prior
Posterior
Data xxxxxxxxx
Engineering model
Prior distribution
Posterior distribution
Eng.-Stat. Model
Methodology-continued • Output: Y • Factors: • Random error:
• Objective: Find • Engineering model: • Calibration parameters: • Data:
Sequential Model Building Engineering Model
Positive relation?
No
Check & Correct
No
Engineering Model
Yes Is MI large? Yes Constant adjustment
Is MI large? Yes Functional Adjustment Model
No
Constant Adjustment Model
Methodology-continued • Check the usefulness of engineering model using graphical analysis • If it is useful
• If MI is small, then stop. Engineering model is good.
Constant adjustment model
• If MI is small, then stop. CAM is good.
Functional adjustment model
• Add terms until MI is small enough.
Constant adjustment model
Posterior distribution • posterior distribution is
• constant adjustment predictor is
• Prediction interval
Simplification • least squares estimate
Empirical Bayes estimation • Estimate
hyperparameters by maximizing
Approximate frequentist procedure • Fit the simple linear regression
Surface roughness example • Engineering model:
• There is a positive relation
Example-continued • From replicates
• Engineering model is not good for prediction
Constant adjustment model
Functional adjustment model
Two-stage estimation • Use the estimate of from the constant adjustment model •
Approximate frequentist procedure • Fit a multiple linear regression • Do a variable selection
Surface roughness example
Calibration parameters • Liu and Melkote (2006)
New engineering model
• R(x) is calculated using a combination of analytical formulas and finite element simulations
Statistical adjustments • First use least squares estimate
2 x f ( x;η~ ) = − 24.83 + 4.49 log R ( x) 8r
• MI=.209 (new engineering model is good)
Constant adjustment model
Approximate frequentist procedure • Fit a nonlinear regression
A Spot Welding Example • Higdon et al. (2004) and Bayarri et al. (2007) – Three factors: Load, Current, and Gage – One calibration parameter
Eng. Model (Black-dashed) : 0.69 Joseph&Melkote (Red-solid): 0.23 Bayarri et al. (Blue-dotted) : 0.20
Example: LAMM Laser assisted mechanical micromachining (LAMM) integrates thermal softening with mechanical micro cutting
+ Laser heating
= LAMM Mechanical micromachining
Objective Find optimum processing conditions that minimize cutting/thrust forces and thermal damage.
Thermal Model Natural B.C. on front face
Y X Z Symmetry B.C. on bottom face
• Mapped dense mesh (25 μm x 12.5 μm x 20μm) • An 8 noded 3-D thermal element (Solid70) is used • Gaussian distribution of heat flux applied to a 5x5 element matrix which sweeps the mesh on the front face
Geometric Model γ chip
cos( α avg + θ PD ) 2 sin θ PD = + sin( π / 4 + θ PD ) cos( α avg − φ ) sin( φ + θ PD )
γ work =
2 sin θ PD + sin(π / 4 + θ PD )
sin(θ PD + θ / 2) sin θ / 2 + sin(θ PB + θ / 2) sin(θ PB + θ PD ) sinψ sin(ψ + θ / 2)
(Manjunathiah et. al, 2000) •
γ chip = 2V •
γ work = 2V
γ chip 2 sin( π / 4 + θ PD )PD γ work 2 sin(π / 4 + θ PD ) PD +
γ eff =
v chip γ chip + v work γ work •
•
γ eff = sin(ψ + θ / 2) PC sinψ
v chip + v work
•
vchip γ chip + vwork γ work vchip + vwork
For plane strain conditions,
ε = γ eff / 3 •
•
ε = γ eff / 3
Shear Flow Strength (
)
⎛ ⎛ ε& σ ( ε ,ε& ,T , HRC ) = A + Bε + C ln( ε + ε 0 ) + D ⎜1 + E ln⎜⎜ ⎜ ⎝ ε&0 ⎝
32 28 24 680
67 0
66 0
65 0
20
64 0
63 0
62 0
0 61
Depth below the surface μ( m)
68 0
67 0
65 0
64 0
63 0
62 0
36
66 0
40
( )
⎞ ⎞⎛ m ⎟ ⎟⎜1 − T * ⎞⎟ ⎟ ⎟⎝ ⎠ ⎠ Yan et⎠ al., 2007 69 0
S =σ / 3
n
16 12 8
125
680
100
670
75
660
50
650
25
640
0
63 0
0
62 0
61 0
4
150
Distance from the center of the tool face along tool edge at 100 μm from the center of the laser beam (μm)
10W laser power, 10 mm/min speed 100 μm laser-tool distance and 110 μm spot size
Forces • Cutting and thrust forces, n
∑S( i )w( i )
Fc = {( h − p )cotφ + h + rn sinθ − ( k −1)δ }
i=1
n
∑S( i )w( i )
Ft = {( h − p )cotφ − h + rn sinθ + ( k −1)δ cotψ }
i=1
Equilibrium Forces/Deflection
Force model
Force prediction
• Positive relation, but predictions are smaller than actual
Force prediction-continued
• Better than cutting force, but slightly smaller than actual
Engineering-Statistical Force Models
Plot of measured vs. predicted cutting and thrust forces
Optimization Problem • For a given depth of cut (t), find the optimum levels of set depth of cut, laser power, laser speed, and distance from tool to minimize cutting/thrust forces while making sure there is no heat affected zone.
min x1 , x2 , x3 , x4 yˆ + yˆ 2 c
2 t
subject to doc = t T2 ≤ Ac1
Nonlinear programming {
} { 2
}
min 1.54 x10.89 exp(0.0014 x 2 − 0.009 x3 e −0.0034 x4 ) + 1.03 x10.8 exp(0.0014 x 2 − 0.043 x3 e −0.0034 x4 )
x1 − 0.57 x10.8 exp(0.0014 x 2 − 0.196 x3 e −0.0034 x4 ) = t 25 + 196.4 x3 exp(−0.0021x1 x3 − 0.00045 x 2 x3 ) ≤ 800
10 ≤ x1 ≤ 25, 10 ≤ x2 ≤ 50, 0 ≤ x3 ≤ 10, 100 ≤ x4 ≤ 200
2
Optimization Results • For example, for depth of cut = 10 μm • Set depth of cut (x1) = 12.30 μm • Cutting speed (x2) = 10 mm/min • Laser power (x3) = 4.5 W • Laser location from the tool edge (x4) = 100 μm
Validation 50
Before machining
30
After machining
20
10 μm
10
0 0
0.1
0.2 0.3 0.4 Distance (mm)
0.5
0.6
45 40
Hardness (HRC)
Height (μm)
40
35 30 10 μm groove depth
25
25 μm groove depth
20 0
50
100
150
200
250
Distance from the edge of groove (μm)
300
Conclusions • Engineering models can be improved by using data • Engineering-Statistical models perform better than engineering models and statistical models • Need relatively less amount of data • They use the physics of the process
Process Optimization
Engineering knowledge
Factors & Levels
Experiment
Engineering model
Engineering-Statistical model
Statistical model
Optimize
Conclusions-continued • Simple procedure – Fit two linear/nonlinear regressions – Do variable selection
• Easy-to-implement – No additional programming is required
