, Volume 95, Supplement 1, pp 381–408 | Cite as

Numerical solution of coupled problems using code Agros2D

  • Pavel Karban
  • František Mach
  • Pavel Kůs
  • David Pánek
  • Ivo Doležel


New code Agros2D for 2D numerical solution of coupled problems is presented. This code is based on the fully adaptive higher-order finite element method and works with library Hermes2D containing the most advanced numerical algorithms for the numerical processing of systems of second-order partial differential equations. It is characterized by several quite unique features such as work with hanging nodes of any level, multimesh technology (every physical field can be calculated on a different mesh generally varying in time) and a possibility of combining triangular, quadrilateral and curved elements. The power of the code is illustrated by three typical coupled problems.


Higher-order finite element method Coupled problems  Monolithic solution hp-adaptivity Hanging nodes 

Mathematics Subject Classification

65M60 68N01 



This work was supported by the European Regional Development Fund and Ministry of Education, Youth and Sports of the Czech Republic (Project No. CZ.1.05/2.1.00/03.0094: Regional Innovation Centre for Electrical Engineering - RICE) and Grant project GACR P102/11/0498.


  1. 1.
  2. 2.
    CEDRAT Flux.
  3. 3.
    COMSOL Multiphysics.
  4. 4.
  5. 5.
  6. 6.
  7. 7.
  8. 8.
  9. 9.
  10. 10.
  11. 11.
  12. 12.
  13. 13.
  14. 14.
  15. 15.
  16. 16.
  17. 17.
    Cavaliere P (2002) Hot and warm forming of 2618 aluminum alloy. J Light Met 2:247–252CrossRefGoogle Scholar
  18. 18.
    Davis TA (2004) A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans Math Soft 30(2):165–195zbMATHCrossRefGoogle Scholar
  19. 19.
    Demmel JW, Eisenstat SC, Gilbert JR, Li XS, Liu JWH (1999) A supernodal approach to sparse partial pivoting. SIAM J Matrix Anal Appl 20(3):720–755MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Dolezel I, Karban P, Ulrych B, Pantelyat M, Matyukhin M, Gontarowskiy P, Shulzhenko M (2008) Limit operation regimes of actuators working on principle of thermoelasticity. IEEE Trans Magn 44:810–813CrossRefGoogle Scholar
  21. 21.
    Dubcova L, Solin P, Cerveny J, Kus P (2010) Space and time adaptive two-mesh hp-fem for transient microwave heating problems. Electromagnetics 30:23–40CrossRefGoogle Scholar
  22. 22.
    Fabbri M, Morandi A, Ribany L (2008) Dc induction heating of aluminum billets using superconducting magnets. COMPEL 27:480–490zbMATHCrossRefGoogle Scholar
  23. 23.
    Geuzaine C, Remacle JF (2009) Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Meth Eng 79:1309–1331MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Karban P, et al. Multiplatform C++ application for the solution of PDEs. (online)
  25. 25.
    Logg A, Mardal KA, Wells GN et al (2012) Automated Solution of Differential Equations by the Finite Element Method, Springer, Berlin. doi: 10.1007/978-3-642-23099-8
  26. 26.
    Long KR, Kirby RC (2010) Unified embedded parallel finite element computations via software-based frechet differentiation. SIAM J Sci Comput 32(6):3323–3351MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Magnusson N, Bersas R, Runde M (2004) Induction heating of aluminum billets using hts dc coils. Ins Phy Conf Ser 2:1104–1109Google Scholar
  28. 28.
    Moesner FM, Toshiro H (1997) Contactless manipulation of microparts by electric field traps. Proc SPIE’s Int Symp Microrobot Microsyst Fabr 3202:168–175CrossRefGoogle Scholar
  29. 29.
    Saeki M (2008) Triboelectric separation of three-component plastic mixture. Part Sci Technol 26:494–506CrossRefGoogle Scholar
  30. 30.
    Shewchuk JR (1996) Triangle: engineering a 2d quality mesh generator and delaunay triangulator. Applied Computational Geometry: towards Geometric Engineering 1148:203–222.
  31. 31.
    Skopek M, Ulrych B, Dolezel I (2001) Optimized regime of induction heating of a disk before its pressing on shaft. IEEE Trans Magn 37:3380–3383Google Scholar
  32. 32.
    Solin P, Andrs D, Cerveny J, Simko M (2010) Pde-independent adaptive hp-fem based on hierarchic extension of finite element spaces. J Comput Appl Math 233:3086–3094MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Solin P, Cerveny J, Dolezel I (2008) Arbitrary-level hanging nodes and automatic adaptivity in the hp-fem. Math Comput Simul 77:117–132MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Solin P, Korous L (2012) Adaptive higher-order finite element methods for transient pde problems based on embedded higher-order implicit runge-kutta methods. J Comput Phy 231:1635–1649MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Solin P, Segeth K, Dolezel I (2003) Higher-Order Finite Element Methods. CRC Press, Boca RatonGoogle Scholar
  36. 36.
    Solin P et al (2012) Hermes-higher-order modular finite element system (user’s guide). (online)
  37. 37.
    Yanar DK, Kwetkus BA (1995) Journal of electrostatics. IEEE Trans Magn 36:257–266Google Scholar

Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  • Pavel Karban
    • 1
  • František Mach
    • 1
  • Pavel Kůs
    • 1
  • David Pánek
    • 1
  • Ivo Doležel
    • 1
  1. 1.Department of Theory of Electrical EngineeringUniversity of West BohemiaPilsenCzech Republic

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