Overview
This pipeline utilizes linear regression modeling to predict properties of groups outside the training set. A validation set is utilized to check the predictions. Results are dependent on the before used filter method in Data_Overview.qmd.
Model Creation
Model Definitions
Overview Linear Regression Models
| M01_RoM_Minimal |
Rule of Mixtures: Mat × Vol |
Material_Hard * Lattice_Vol_Sc |
| M02_RoM_CellSize |
Rule of Mixtures + Cell Size |
Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc |
| M03_RoM_Structure |
Rule of Mixtures + Structure |
Material_Hard * Lattice_Vol_Sc + Structure |
| M04_RoM_Full |
Rule of Mixtures + Structure + Cell Size |
Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc |
| M05_StrVol_Full |
Structure × Volume Interaction |
Material_Hard * Lattice_Vol_Sc + Structure * Lattice_Vol_Sc |
| M06_StrVol_Nested |
Structure–Volume (Nested) |
Material_Hard * Lattice_Vol_Sc + Structure:Lattice_Vol_Sc |
| M07_MatCell |
Material × Cell Size |
Material_Hard * Lattice_Vol_Sc + Material_Hard * Cell_Size_Sc + Structure |
| M08_PolyVol |
Polynomial Volume (Quadratic) |
Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Structure |
| M09_PolyCellSize |
Polynomial Cell Size (Quadratic) |
Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc + I(Cell_Size_Sc^2) |
| M10_GibsonAshby |
Gibson–Ashby Scaling (Vol^1.5) |
Material_Hard * I(Lattice_Vol_Sc^1.5) + Structure |
| M11_GA_StrVol |
Gibson–Ashby + Structure × Vol^1.5 |
Material_Hard * I(Lattice_Vol_Sc^1.5) + Structure * I(Lattice_Vol_Sc^1.5) + Cell_Size_Sc |
| M12_Slenderness |
Strut Slenderness (Vol / Cell Size) |
Material_Hard * Lattice_Vol_Sc + I(Lattice_Vol_Sc / Cell_Size_Sc) + Structure |
| M13_Slenderness_MS |
Strut Slenderness + Mat × Structure |
Material_Hard * Lattice_Vol_Sc + Material_Hard * Structure + I(Lattice_Vol_Sc / Cell_Size_Sc) |
| M14_PhysHybrid |
Physics-Informed Hybrid (GA + Slenderness) |
Material_Hard * I(Lattice_Vol_Sc^1.5) + Structure:I(Lattice_Vol_Sc^1.5) + I(Lattice_Vol_Sc / Cell_Size_Sc) |
| M15_MatStrVol |
Three-Way: Mat × Structure × Volume |
Material_Hard * Structure * Lattice_Vol_Sc |
| M16_MatStr_Add |
Mat × Structure + Mat × Vol (Additive) |
Material_Hard * Structure + Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc |
| M17_PolyVol_MatStr |
Polynomial Volume + Mat × Structure |
Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Material_Hard * Structure |
Preliminary Models
Table Preliminary Model Building
| Epsilon_b |
Strut Slenderness (Vol / Cell Size) |
E_Break ~ Material_Hard * Lattice_Vol_Sc + I(Lattice_Vol_Sc / Cell_Size_Sc) + Structure |
0.158 |
0.195 |
5.234 |
0.588 |
| Epsilon_b |
Rule of Mixtures + Cell Size |
E_Break ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc |
0.118 |
0.133 |
5.372 |
0.408 |
| Epsilon_b |
Polynomial Cell Size (Quadratic) |
E_Break ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc + I(Cell_Size_Sc^2) |
0.106 |
0.106 |
5.370 |
0.373 |
| Epsilon_m |
Rule of Mixtures + Cell Size |
E_Max ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc |
0.447 |
0.084 |
0.523 |
0.039 |
| Epsilon_m |
Rule of Mixtures + Structure + Cell Size |
E_Max ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc |
0.444 |
0.085 |
0.524 |
0.039 |
| Epsilon_m |
Polynomial Cell Size (Quadratic) |
E_Max ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc + I(Cell_Size_Sc^2) |
0.417 |
0.062 |
0.538 |
0.028 |
| Sigma_b |
Strut Slenderness (Vol / Cell Size) |
S_Break ~ Material_Hard * Lattice_Vol_Sc + I(Lattice_Vol_Sc / Cell_Size_Sc) + Structure |
0.689 |
0.050 |
4.587 |
0.367 |
| Sigma_b |
Polynomial Cell Size (Quadratic) |
S_Break ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc + I(Cell_Size_Sc^2) |
0.641 |
0.047 |
4.934 |
0.328 |
| Sigma_b |
Rule of Mixtures: Mat × Vol |
S_Break ~ Material_Hard * Lattice_Vol_Sc |
0.610 |
0.061 |
5.141 |
0.402 |
| Sigma_m |
Rule of Mixtures: Mat × Vol |
S_Max ~ Material_Hard * Lattice_Vol_Sc |
0.744 |
0.019 |
3.364 |
0.122 |
| Sigma_m |
Rule of Mixtures + Cell Size |
S_Max ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc |
0.740 |
0.027 |
3.391 |
0.173 |
| Sigma_m |
Strut Slenderness (Vol / Cell Size) |
S_Max ~ Material_Hard * Lattice_Vol_Sc + I(Lattice_Vol_Sc / Cell_Size_Sc) + Structure |
0.707 |
0.046 |
3.592 |
0.270 |
Final Models (Complete Training Data)
| BC_E_Break ~ Material_Hard * I(Lattice_Vol_Sc^1.5) + Structure:I(Lattice_Vol_Sc^1.5) + I(Lattice_Vol_Sc / Cell_Size_Sc) |
Physics-Informed Hybrid (GA + Slenderness) |
Epsilon_b |
0.837 |
3.665 |
| BC_E_Break ~ Material_Hard * Structure * Lattice_Vol_Sc |
Three-Way: Mat × Structure × Volume |
Epsilon_b |
0.736 |
2.947 |
| BC_E_Break ~ Material_Hard * Structure + Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc |
Mat × Structure + Mat × Vol (Additive) |
Epsilon_b |
0.678 |
3.253 |
| BC_E_Max ~ Material_Hard * I(Lattice_Vol_Sc^1.5) + Structure:I(Lattice_Vol_Sc^1.5) + I(Lattice_Vol_Sc / Cell_Size_Sc) |
Physics-Informed Hybrid (GA + Slenderness) |
Epsilon_m |
0.849 |
0.433 |
| BC_E_Max ~ Material_Hard * Structure * Lattice_Vol_Sc |
Three-Way: Mat × Structure × Volume |
Epsilon_m |
0.733 |
0.364 |
| BC_E_Max ~ Material_Hard * Structure + Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc |
Mat × Structure + Mat × Vol (Additive) |
Epsilon_m |
0.700 |
0.386 |
| BC_S_Break ~ Material_Hard * I(Lattice_Vol_Sc^1.5) + Structure:I(Lattice_Vol_Sc^1.5) + I(Lattice_Vol_Sc / Cell_Size_Sc) |
Physics-Informed Hybrid (GA + Slenderness) |
Sigma_b |
0.942 |
3.131 |
| BC_S_Break ~ Material_Hard * Structure * Lattice_Vol_Sc |
Three-Way: Mat × Structure × Volume |
Sigma_b |
0.885 |
2.803 |
| BC_S_Break ~ Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Material_Hard * Structure |
Polynomial Volume + Mat × Structure |
Sigma_b |
0.863 |
3.053 |
| BC_S_Max ~ Material_Hard * I(Lattice_Vol_Sc^1.5) + Structure:I(Lattice_Vol_Sc^1.5) + I(Lattice_Vol_Sc / Cell_Size_Sc) |
Physics-Informed Hybrid (GA + Slenderness) |
Sigma_m |
0.946 |
2.444 |
| BC_S_Max ~ Material_Hard * Structure * Lattice_Vol_Sc |
Three-Way: Mat × Structure × Volume |
Sigma_m |
0.896 |
2.141 |
| BC_S_Max ~ Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Material_Hard * Structure |
Polynomial Volume + Mat × Structure |
Sigma_m |
0.878 |
2.327 |
Predictions
Compare Predictions
Validation summary created
Target: \(\sigma_m\)
|
Model
|
Formula
|
R2_Training
|
R2_Validation
|
|
Structure–Volume (Nested)
|
BC_S_Max ~ Material_Hard * Lattice_Vol_Sc + Structure:Lattice_Vol_Sc
|
0.820
|
0.716
|
|
Rule of Mixtures: Mat × Vol
|
BC_S_Max ~ Material_Hard * Lattice_Vol_Sc
|
0.795
|
0.703
|
|
Rule of Mixtures + Cell Size
|
BC_S_Max ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc
|
0.797
|
0.674
|
Target: \(\epsilon_m\)
|
Model
|
Formula
|
R2_Training
|
R2_Validation
|
|
Rule of Mixtures + Structure + Cell Size
|
BC_E_Max ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc
|
0.65
|
0.636
|
|
Polynomial Cell Size (Quadratic)
|
BC_E_Max ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc + I(Cell_Size_Sc^2)
|
0.65
|
0.636
|
|
Material × Cell Size
|
BC_E_Max ~ Material_Hard * Lattice_Vol_Sc + Material_Hard * Cell_Size_Sc + Structure
|
0.65
|
0.632
|
Target: \(\sigma_b\)
|
Model
|
Formula
|
R2_Training
|
R2_Validation
|
|
Rule of Mixtures: Mat × Vol
|
BC_S_Break ~ Material_Hard * Lattice_Vol_Sc
|
0.728
|
0.687
|
|
Structure–Volume (Nested)
|
BC_S_Break ~ Material_Hard * Lattice_Vol_Sc + Structure:Lattice_Vol_Sc
|
0.769
|
0.655
|
|
Rule of Mixtures + Cell Size
|
BC_S_Break ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc
|
0.745
|
0.652
|
Target: \(\epsilon_b\)
|
Model
|
Formula
|
R2_Training
|
R2_Validation
|
|
Rule of Mixtures: Mat × Vol
|
BC_E_Break ~ Material_Hard * Lattice_Vol_Sc
|
0.300
|
0.515
|
|
Rule of Mixtures + Cell Size
|
BC_E_Break ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc
|
0.346
|
0.486
|
|
Structure–Volume (Nested)
|
BC_E_Break ~ Material_Hard * Lattice_Vol_Sc + Structure:Lattice_Vol_Sc
|
0.334
|
0.234
|
