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Research Papers: Design Automation

Hierarchical Design of Negative Stiffness Metamaterials Using a Bayesian Network Classifier1

[+] Author and Article Information
Jordan Matthews, Clinton Morris

Mechanical Engineering Department,
The University of Texas at Austin,
Austin, TX 78712

Timothy Klatt, Michael Haberman

Mechanical Engineering Department and
Applied Research Laboratories,
The University of Texas at Austin,
Austin, TX 78712

Carolyn C. Seepersad

Mechanical Engineering Department,
The University of Texas at Austin,
Austin, TX 78712
e-mail: ccseepersad@mail.utexas.edu

David Shahan

HRL Laboratories,
Malibu, CA 90265

2Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received August 17, 2015; final manuscript received February 8, 2016; published online March 4, 2016. Assoc. Editor: Nam H. Kim.

J. Mech. Des 138(4), 041404 (Mar 04, 2016) (12 pages) Paper No: MD-15-1576; doi: 10.1115/1.4032774 History: Received August 17, 2015; Revised February 08, 2016

A set-based approach is presented for exploring multilevel design problems. The approach is applied to design negative stiffness metamaterials with mechanical stiffness and loss properties that surpass those of conventional composites. Negative stiffness metamaterials derive their properties from their internal structure, specifically by embedding small volume fractions of negative stiffness inclusions in a continuous host material. Achieving high stiffness and loss from these materials by design involves managing complex interdependencies among design variables across a range of length scales. Hierarchical material models are created for length scales ranging from the structure of the microscale negative stiffness inclusions to the effective properties of mesoscale metamaterials to the performance of an illustrative macroscale component. Bayesian network classifiers (BNCs) are used to map promising regions of the design space at each hierarchical modeling level, and the maps are intersected to identify sets of multilevel solutions that are likely to provide desirable system performance. The approach is particularly appropriate for highly efficient, top-down, performance-driven, multilevel design, as opposed to bottom-up, trial-and-error multilevel modeling.

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Figures

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Fig. 1

An illustration of the hierarchical levels of the metamaterial, with micro-, meso-, and macroscales indicated by the subscripts μ, m, and M, respectively. The negative stiffness inclusion is illustrated in the upper left, and the beam coating application is illustrated on the right.

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Fig. 2

A simplified 2D representation of a multilevel metamaterials modeling and design problem

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Fig. 3

KDEs scaled by the prior class probability (left) are transformed into a posterior class discriminant (right) through Bayes decision theory. Differently shaded points represent different performance classes, arrows are high-performance (low-performance) points, and the dotted line represents the decision boundary.

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Fig. 4

A candidate inclusion design showing FE modeling

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Fig. 5

Cross section view of the parameterized geometry of the inclusion used in this study

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Fig. 6

Meso- to microscale design space mapping

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Fig. 7

Meso- to microscale performance space mapping

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Fig. 8

Meso- to microscale high-performance space mapping

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Fig. 9

Conceptual schematic of the homogenization approach of the self-consistent micromechanical model

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Fig. 10

Macro- to mesoscale design space mapping

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Fig. 11

Macroscale performance space mapping

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Fig. 12

A trend line of the negative sloping macro- to mesoscale design space

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Fig. 16

Macro- to mesoscale performance space (right) and meso- to microscale design space (left), corresponding to various performance thresholds for the effective loss factor

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Fig. 15

A multilayer cantilever beam with a metamaterial coating (left) subjected to an impulsive load on its free end (right)

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Fig. 14

A preliminary application of BNC mappings to the NS metamaterial design problem. The meso- to microscale (inclusion) design space axes are normalized inclusion dimensions. The meso- to microscale (composite) design space axes are effective stiffness in the first principal direction and in shear. The macro- to mesoscale (composite) performance space axes are the effective loss and stiffness, normalized by the matrix properties. Macro- to mesoscale (composite) performance requirements are backpropagated to the macro- to mesoscale (composite) design space, which is then backpropagated to the meso- to microscale (inclusion) design space, to indicate inclusion design parameters that provide the desired higher-level performance.

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Fig. 13

The effective stiffness and loss ratio of the metamaterial as a function of C12m,eff, with C11m,eff=−20 MPa

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Fig. 17

The shock response of the cantilever beam with high- and low-performance coatings

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