Computational Mechanics

, Volume 56, Issue 1, pp 97–112 | Cite as

Surface effects on shape and topology optimization of nanostructures

  • S. S. Nanthakumar
  • Navid Valizadeh
  • Harold S. ParkEmail author
  • Timon RabczukEmail author
Original Paper


We present a computational method for the optimization of nanostructures, where our specific interest is in capturing and elucidating surface stress and surface elastic effects on the optimal nanodesign. XFEM is used to solve the nanomechanical boundary value problem, which involves a discontinuity in the strain field and the presence of surface effects along the interface. The boundary of the nano-structure is implicitly represented by a level set function, which is considered as the design variable in the optimization process. Two objective functions, minimizing the total potential energy of a nanostructure subjected to a material volume constraint and minimizing the least square error compared to a target displacement, are chosen for the numerical examples. We present results of optimal topologies of a nanobeam subject to cantilever and fixed boundary conditions. The numerical examples demonstrate the importance of size and aspect ratio in determining how surface effects impact the optimized topology of nanobeams.


Nanomaterials Surface effects Shape optimization Extended finite element method (XFEM) Level set method 



Timon Rabczuk and Navid Valizadeh gratefully acknowledge the financial support of the Framework Programme 7 Initial Training Network Funding under grant number 289361 “Integrating Numerical Simulation and Geometric Design Technology”. Harold Park acknowledges the support of the Mechanical Engineering department at Boston University. S. S. Nanthakumar gratefully acknowledges the financial support of DAAD.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute of Structural MechanicsBauhaus-Universität WeimarWeimarGermany
  2. 2.Department of Mechanical EngineeringBoston UniversityBostonUSA
  3. 3.School of Civil, Environmental and Architectural EngineeringKorea universitySeoulRepublic of Korea

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