Towards Nonlinear Multimaterial Topology Optimization using Machine Learning and Metamodel-based Optimization Kai Liu Purdue University Emily NutWell Honda R&D Americas Andrs Tovar Indiana Univ. - Purdue Univ. Indianapolis Duane Detwiler Honda R&D Americas 1 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 2 Conceptual Design Generalized Structural optimization problem The resulting is a distribution of up to n materials within the structure. 3 Design Parameterization Generalized Structural optimization problem K-means K-means clustering is a simple widely used

unsupervised machine learning technique. 4 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: 5 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 6 Parametric Optimization Problem statement The parametric optimization problem can be posted with reduced number of variables.

7 Parametric Optimization Metamodel-based multi-objective optimization 8 Examples Minimum compliance MBB-Beam Compliant mechanism Force Inverter Crashworthiness design S-rail tubular component

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

optimization problem: 10 Examples Minimum Compliance Design Parameterization 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.

11 Examples Minimum Compliance Parametric Optimization Minimum compliance with reduced number of design variables Optimization problem: 12 Examples Minimum Compliance Conceptual Design

Design Parameterization Parametric Optimization Objective: 164.41 Objective: 170.831 Objective: 168.964 xmin: 0.001 xmin: 0.300 xmin: 0.290 xmax: 1.000

xmax: 0.971 xmax: 1.000 # values: 890 # values: 2 # values: 2 13 Examples Minimum Compliance - comparison of multimaterial topology optimization solutions Clustered Multimaterial Iterations 16 Topology + 1 Kmeans + 7

SQP 236 Outer + 708 Inner Topology Objective 167.35 169.35 # distinct values [1] 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 14 Examples

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

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

Design Parameterization Parametric Optimization Objective: -1.02 Objective: 0.133 Objective: -0.614 xmin: 0.001 xmin: 0.042 xmin: 0.011 xmax: 1.00 xmax: 0.940 xmax: 1.000

# values: 3027 # values: 2 # values: 2 17 Examples Compliant Mechanism - comparison of multimaterial topology optimization solutions. Clustered Multimaterial Iterations 153 Topology + 1 Kmeans + 2 SQP 200 Outer + 600 Inner

Topology Objective -0.713 -0.674 # distinct values [1] 1.00 0 0.00 1 0.59 0 1.00 0

0.00 1 0.59 0 0.29 8 0.61 5 0.08 7 0.29 8 0.61 5 0.08 7

3 3070 18 Examples Crashworthiness Design Conceptual Design Geometry Initial Design 19 Examples

Crashworthiness Design Conceptual Design Optimization Problem 20 Examples Crashworthiness Design Design Parameterization Using Kmeans to cluster conceptual design into 11 groups[2] 21 Examples Crashworthiness Design

Parametric Optimization Maximize crashworthiness Optimization problem specific energy absorption, crushing force peak 22 Pareto Fronts 23 Summary Three Stages Design Optimization

1 Conceptual structural optimization thousands variables good performance 1 2 Param. Design Cycle 3 Optimization multiobjective optimization sequential metamodel update improved performance improved 2 Kmeans clustering

reduced variables worst performance 3 24 References 1. Tavakoli, R., and Mohseni, S. M., 2014. Alternating active-phase algorithm for multimaterial topology optimization problems: A 115line MATLAB implementation, Struct Multidisc Optim, 49(4), pp. 621 642. 2. 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 2015-01-1369, 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. 25