Structural and Multidisciplinary Optimization

, Volume 60, Issue 4, pp 1407–1422 | Cite as

Multifunctional topology optimization of strain-sensing nanocomposite beam structures

  • Ryan SeifertEmail author
  • Mayuresh Patil
  • Gary Seidel
  • Gregory Reich
Research Paper


Controlling volume fractions of nanoparticles in a matrix can have a substantial influence on composite performance. This paper presents a multi-start topology optimization algorithm that designs nanocomposite structures for objectives pertaining to stiffness and strain sensing. Local effective properties are obtained by controlling local volume fractions of carbon nanotubes (CNTs) in an epoxy matrix, which are assumed to be well dispersed and randomly oriented. Local Young’s modulus, conductivity, and piezoresistive constant drive the global objectives of strain energy and resistance change. Strain energy is obtained via a modified solution of Euler-Bernoulli equations and resistance change is obtained via solution of a bilinear quadrilateral finite element problem. The optimization uses a two-step restart method in which Pareto points from the first step are used as starting conditions in the second step. An efficient method for obtaining analytic sensitivities of the objective functions is presented. The method is used to solve a set of example problems pertaining to the design of a composite beam in bending. The results show that the strain energy may be optimized by placing high volume-fraction CNT elements away from the neutral axis. Resistance change is optimized through a combination of shifting the neutral axis, formation of conductive paths between electrodes, and asymmetric distribution of highly piezoresistive elements. Results also show that the strain energy is governed by the volume fraction constraint and the resistance change is dependent on a combination of the volume fraction constraint and the boundary electrode location.


Topology optimization Multifunctional optimization Carbon nanotubes Micromechanics Analytic sensitivities Strain sensing 



The authors would like to acknowledge the support of the Air Force Research Laboratory under agreement number FA8650-09-2-3938. This material was cleared under case number 88ABW-2018-3771.

Compliance with Ethical Standards

Conflict of interests

The authors claim no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Virginia Polytechnic Institute and State UniversityBlacksburgUSA
  2. 2.Wright-Patterson Air Force BaseUSA

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