Research Papers: Design Automation

Optimal Design of Nonlinear Multimaterial Structures for Crashworthiness Using Cluster Analysis

[+] Author and Article Information
Kai Liu

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: liu915@purdue.edu

Duane Detwiler

Honda R&D Americas, Inc.,
Raymond, OH 43067
e-mail: ddetwiler@oh.hra.com

Andres Tovar

Associate Professor
Department of Mechanical Engineering,
Indiana University-Purdue
University Indianapolis,
Indianapolis, IN 46202
e-mail: tovara@iupui.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 12, 2016; final manuscript received August 2, 2017; published online August 30, 2017. Assoc. Editor: Nam H. Kim.

J. Mech. Des 139(10), 101401 (Aug 30, 2017) (11 pages) Paper No: MD-16-1637; doi: 10.1115/1.4037620 History: Received September 12, 2016; Revised August 02, 2017

This study presents an efficient multimaterial design optimization algorithm that is suitable for nonlinear structures. The proposed algorithm consists of three steps: conceptual design generation, clustering, and metamodel-based global optimization. The conceptual design is generated using a structural optimization algorithm for linear models or a heuristic design algorithm for nonlinear models. Then, the conceptual design is clustered into a predefined number of clusters (materials) using a machine learning algorithm. Finally, the global optimization problem aims to find the optimal material parameters of the clustered design using metamodels. The metamodels are built using sampling and cross-validation and sequentially updated using an expected improvement function until convergence. The proposed methodology is demonstrated using examples from multiple physics and compared with traditional multimaterial topology optimization (MTOP) method. The proposed approach is applied to a nonlinear, multi-objective design problems for crashworthiness.

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Fig. 2

Leave-one-out cross-validation at the second sampled point exemplified by fitting a function with a Kriging metamodel

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Fig. 1

LHSs with P = 5 and (a) high correlation and (b) low correlation achieved using the maximization of the minimum inter-site distances

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Fig. 3

Thermal compliance problem—design domain and Dirichlet boundary condition: constant surface temperature u = 0

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Fig. 4

Thermal compliance problem—conceptual design with f = 1.94 × 106

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Fig. 5

Thermal compliance problem—clustered design with f = 1.97 × 106

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Fig. 8

Mechanical compliance problem—conceptual design with 8320 distinct density values and f = 2217

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Fig. 9

Mechanical compliance problem—clustered design with three clusters and f = 2432

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Fig. 6

Thermal compliance problem—comparison of the predicted number of function evaluations

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Fig. 7

Mechanical compliance problem—design domain and boundary conditions for the three-dimensional beam in cantilever

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Fig. 12

Armor plate problem—conceptual design (left) and impact simulation (right) with maximum penetration f1 = 9.33 mm and mass fraction f2 = 0.50

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Fig. 10

Armor plate problem—finite element model

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Fig. 11

Armor plate problem—initial design (left) and impact simulation (right) with maximum penetration f1 = 12.05 mm and mass fraction f2 = 0.50

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Fig. 17

S-rail problem—locations of input and output ports for a thin-walled S-tube following the wavelength λ corresponding to the progressive buckling after an ideal axial crushing condition

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Fig. 13

Armor plate problem—clustered designs with K=1,…,18

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Fig. 14

Armor plate problem—maximum penetration and mass fraction as functions of the number of clusters

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Fig. 15

Armor plate problem—Pareto front of the design optimization problem and the optimized clustered design (K = 4) with maximum penetration f1 = 9.46 mm and mass fraction f2 = 0.50

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Fig. 16

S-rail problem—geometry of the thin-walled S-rail (side view). The cross section is squared of dimensions H × H and thickness xe.

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Fig. 21

S-rail problem—SEA and PCF values as a function of the number of clusters K

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Fig. 18

S-rail problem—initial design represented by a uniform thickness distribution in the “unfolded” thin-walled structure (left). The initial design depicts Euler-type buckling (right). The corresponding crashworthiness indicators are SEA = 3.39 kJ/kg and PCF = 267 kN.

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Fig. 19

S-rail problem—conceptual design represented the thickness distribution in the “unfolded” thin-walled structure (left). The conceptual design depicts progressive folding (right).The corresponding crashworthiness indicators are SEA = 5.05 kJ/kg and PCF = 359 kN.

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Fig. 20

S-rail problem—clustered designs with K=1,…,12

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Fig. 22

S-rail problem—Pareto fronts for K=1,…,4. Initial, conceptual, and clustered designs are dominated. Clustered designs have the following (−SEA, PCF) coordinates: for K = 1: (−3.82, 274), for K = 2: (−4.81, 349), for K = 3: (−3.98, 423), and for K = 4: (−3.90, 381).



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