Structural and Multidisciplinary Optimization

, Volume 49, Issue 3, pp 523–535 | Cite as

Design multiple-layer gradient coils using least-squares finite element method

  • Feng Jia
  • Zhenyu Liu
  • Maxim Zaitsev
  • Jürgen Hennig
  • Jan G. Korvink


The design of gradient coils for magnetic resonance imaging is an optimization task in which a specified distribution of the magnetic field inside a region of interest is generated by choosing an optimal distribution of a current density geometrically restricted to specified non-intersecting design surfaces, thereby defining the preferred coil conductor shapes. Instead of boundary integral type methods, which are widely used to design coils, this paper proposes an optimization method for designing multiple layer gradient coils based on a finite element discretization. The topology of the gradient coil is expressed by a scalar stream function. The distribution of the magnetic field inside the computational domain is calculated using the least-squares finite element method. The first-order sensitivity of the objective function is calculated using an adjoint equation method. The numerical operations needed, in order to obtain an effective optimization procedure, are discussed in detail. In order to illustrate the benefit of the proposed optimization method, example gradient coils located on multiple surfaces are computed and characterised.


Magnetic resonance imaging Gradient coil design Stream function Least-squares finite element method Adjoint method 



This research is supported by German Federal Ministry of Education and Research (BMBF) INUMAC grant 13N9208, an operating grant of the University of Freiburg, and the National Nature Science Foundation of China (No. 51275504).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Feng Jia
    • 1
  • Zhenyu Liu
    • 2
  • Maxim Zaitsev
    • 1
  • Jürgen Hennig
    • 1
  • Jan G. Korvink
    • 3
    • 4
  1. 1.Department of Radiology, Medical PhysicsUniversity Hospital FreiburgFreiburgGermany
  2. 2.State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics (CIOMP)Chinese Academy of SciencesChangchunChina
  3. 3.Department of Microsystems Engineering (IMTEK)University of FreiburgFreiburgGermany
  4. 4.Freiburg Institute for Advanced Studies (FRIAS)University of FreiburgFreiburgGermany

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