Engineering with Computers

, Volume 35, Issue 2, pp 423–449 | Cite as

A framework of shape optimisation based on the isogeometric boundary element method toward designing thin-silicon photovoltaic devices

  • Toru TakahashiEmail author
  • Tatsuro Yamamoto
  • Yuta Shimba
  • Hiroshi Isakari
  • Toshiro Matsumoto
Original Article


We propose a gradient-based optimisation framework to design the shape of the interfaces in periodic layered structures, which can model photovoltaic devices, optical gratings, etc., according to a desired electromagnetic property. To this end, we first develop an isogeometric boundary element method (IGBEM) to analyse singly periodic boundary value problems for 2D Helmholtz equation in layered media. By nature, the IGBEM is not only numerically accurate but also suitable for the shape optimisation in comparison with the conventional boundary- and domain-type solvers. Next, we derive the shape derivative (or shape sensitivity) of the objective function in terms of the magnetic field strength or energy absorption rate on the basis of the adjoint variable method. The shape derivative can be computed by solving the primal and adjoint problems with the IGBEM. Subsequently, we map our shape optimisation problems to nonlinear programming problems in order to exploit a general-purpose software for the latter problems. After verifying our optimisation framework rigorously by comparing with the exact solutions, we demonstrate a shape optimisation of the silicon-metal interface in a model of thin-silicon photovoltaic devices: it is crucial to determine the interface so that the energy of the incident light can be confined in the silicon layer as much as possible when the thickness of the layer is unconventionally small; one micrometre in our model. The optimal shape achieved 8.6 times higher absorption rate than the reference (flat) shape. This simulation shows that the proposed framework is capable of making a fundamental contribution to analysing and designing photovoltaic devices as well as photonic and plasmonic devices that consist of singly periodic layers.


Boundary element method Isogeometric analysis Periodic structure Shape optimisation Adjoint variable method Nonlinear programming problem Quasi-Newton method Photovoltaic device Surface plasmon Plasmonics 



This research was supported by JSPS KAKENHI (No. 15K06683). We would like to thank Mr. Tetsuro Hirai at Nagoya University for his helpful discussions on the proposed IGBEM, especially for the singular integrals. We also would like to thank Mr. Shinichi Hayashi of former graduate student for his contributions to the present research project at the early stage.


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  • Toru Takahashi
    • 1
    Email author
  • Tatsuro Yamamoto
    • 2
  • Yuta Shimba
    • 1
  • Hiroshi Isakari
    • 1
  • Toshiro Matsumoto
    • 1
  1. 1.Department of Mechanical Systems EngineeringGraduate School of Engineering, Nagoya UniversityNagoyaJapan
  2. 2.Department of Micro-Nano Systems EngineeringGraduate School of Engineering, Nagoya UniversityNagoyaJapan

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