1. Towards Nonlinear Multimaterial Topology Optimization using Machine Learning and Metamodel-based Optimization
Kai Liu Purdue University
Andrés Tovar Indiana Univ. - Purdue Univ. Indianapolis
Emily NutWell Honda R&D Americas
Duane Detwiler Honda R&D Americas

2. Three Stages Design Optimization
Systematic Design Optimization Approach
Conceptual Design
Generating conceptual design using structural optimization
Continuous design distribution
Design Parameterization
Parameterizing conceptual design using machine learning
Simple and efficient
Parametric Optimization
Utilizing parametric optimization with reduced number of design variables
Further improves performances and manufacturability

3. Conceptual Design Generalized Structural optimization problem
The resulting is a distribution of up to n materials within the structure.
This is preliminary result. Can we manufacture this result? Can we further improve the performance?

4. Design Parameterization
Generalized Structural optimization problem
K-means clustering is a simple widely used unsupervised machine learning technique.
K-means
This is preliminary result. Can we manufacture this result? Can we further improve the performance?

5. Design Parameterization
K-means clustering
Objective
Given a set of objects, K-means clustering aims to partition the n objects into K sets so as to minimize the within-cluster sum of squares:

6. Design Parameterization
K-means clustering
Algorithm
Step 1. Given n objects, initialize k cluster centers
Step 2. Assign each object to its closest cluster center
Step 3. Update the center for each cluster
Step 4. Repeat 2 and 3 until no change in cluster centers

7. Parametric Optimization
Problem statement
The parametric optimization problem can be posted with reduced number of variables.
To find the material parameters that extremize a set of objective functions, subject to a set of functional constraints.

8. Parametric Optimization
Metamodel-based multi-objective optimization
The set of objectives may involve nonlinear and computationally expensive functions.
To solve such problems, we proposed a metamodel-based multi-objective optimization algorithm.

9. Examples Minimum compliance Compliant mechanism Crashworthiness design
MBB-Beam
Compliant mechanism
Force Inverter
Crashworthiness design
S-rail tubular component

10. Examples Conceptual Design Minimum Compliance elements: 60x20 Q4
objective: min compliance
mass fraction: 0.5
material: E = 1.0, nu = 0.3
optimization problem:

11. Examples Design Parameterization Cluster Number K Minimum Compliance
Using Kmeans to cluster conceptual design into 2 groups
Cluster Number K
The bigger value of K, the closer clustered solution to the conceptual design.

12. Examples Parametric Optimization Minimum Compliance
Minimum compliance with reduced number of design variables
Optimization problem:

13. Examples Minimum Compliance Conceptual Design Design Parameterization
Objective:
xmin: 0.001
xmax: 1.000
# values: 890
Design Parameterization
Objective:
xmin: 0.300
xmax: 0.971
# values: 2
Parametric Optimization
Objective:
xmin: 0.290
xmax: 1.000
# values: 2

14. Examples Clustered Multimaterial[1]
Minimum Compliance - comparison of multimaterial topology optimization solutions
Clustered
Multimaterial[1]
16 Topology + 1 Kmeans + 7 SQP
236 Outer Inner Topology
Iterations
167.35
169.35
Objective
1.00
0.25
0.45
1.00
0.25
0.45
0.29
0.53
0.18
0.29
0.53
0.18 3
986
# distinct values

15. Examples Conceptual Design Compliant Mechanism elements: 150x75 Q4
objective: max output displacement
mass fraction: 0.35
material: E = 1.0, nu = 0.3
optimization problem:

16. Examples Design Parameterization Parametric Optimization
Compliant Mechanism
Design Parameterization
Using Kmeans to cluster conceptual design into 2 groups
Parametric Optimization
Maximize output displacement with reduced number of design variables

17. Examples Minimum Compliance Conceptual Design Design Parameterization
Parametric Optimization
Objective: -1.02
xmin: 0.001
xmax: 1.00
# values: 3027
Objective: 0.133
xmin: 0.042
xmax: 0.940
# values: 2
Objective:
xmin: 0.011
xmax: 1.000
# values: 2

18. Examples Clustered Multimaterial[1]
Compliant Mechanism - comparison of multimaterial topology optimization solutions.
Clustered
Multimaterial[1]
200 Outer Inner Topology
Iterations
153 Topology + 1 Kmeans + 2 SQP
-0.713
-0.674
Objective
1.000
0.001
0.590
1.000
0.001
0.590
0.298
0.615
0.087
0.298
0.615
0.087 3
3070
# distinct values

19. Examples Conceptual Design Crashworthiness Design Geometry
Initial Design

20. Examples
Crashworthiness Design
Conceptual Design
Optimization Problem

21. Examples Design Parameterization Crashworthiness Design
Using Kmeans to cluster conceptual design into 11 groups[2]

22. Examples Parametric Optimization Crashworthiness Design
Maximize crashworthiness
Optimization problem
specific energy absorption, peak crushing force

23. Pareto Fronts

24. Summary 1 Conceptual 1 2 Param. 2 3 3 Optimization
Three Stages Design Optimization
1 Conceptual
structural optimization
thousands variables
good performance 1
2 Param.
Kmeans clustering
reduced variables
worst performance
Design Cycle 2 3
3 Optimization
multiobjective optimization
sequential metamodel update
improved performance
improved manufacturability

25. References Acknowledgement
Tavakoli, R., and Mohseni, S. M., “Alternating active-phase algorithm for multimaterial topology optimization problems: A 115-line MATLAB implementation”, Struct Multidisc Optim, 49(4), pp. 621–642.
Liu, K., Tovar, A., Nutwell, E., and Detwiler, D., “Thin-walled compliant mechanism component design assisted by machine learning and multiple surrogates”, SAE Technical Paper , 2015.
Acknowledgement
Honda R&D Americas supported this research effort. Any opinions, findings, conclusions, and recommendations expressed in this investigation are those of the writers and do not necessarily reflect the views of the sponsors.