Two-dimensional rate-independent plasticity using the element-based finite volume method

  • Paulo Vicente de Cassia Lima Pimenta
  • Francisco MarcondesEmail author


The element-based finite volume method (EbFVM) is well established in computational fluid dynamics; in the last decade, it has been extended to several areas of engineering and physics interest, such as electromagnetism, acoustics, and structural mechanics analysis with complex geometrical shapes. This paper describes the treatment of the conservative EbFVM approach for two-dimensional isotropic elastic–plastic rate-independent problems. In particular, we use plane strain and plane stress approaches upon incremental thermal and mechanical loads. In order to verify the performance of the EbFVM, numerical results are compared with a commercial simulator. Finally, from the present implementation and the comparisons performed, we can ensure that EbFVM makes accurate prediction as the traditional numerical approach commonly employed for the solution of mechanics problems.


EbFVM Mechanics expansion Thermal expansion Nonlinear material 



The first and second authors would like to thank CAPES (Coordination for the Improvement of Higher Education Personnel) and CNPq (the National Council for Scientific and Technological Development of Brazil), respectively, for the financial support of this work.


  1. 1.
    Voller VR (2000) Basic control volume finite element methods for fluids and solids, vol 1. World Scientific, SingaporezbMATHGoogle Scholar
  2. 2.
    Zienkiewicz O, Taylor R (2000) The finite element method: solid mechanics, vol 2, 5th edn. Butterworth-Heinemann, OxfordzbMATHGoogle Scholar
  3. 3.
    Demirdžić I, Martinović D (1993) Finite volume method for thermo-elastoplastic stress analysis. Comput Methods Appl Mech Eng 109(3–4):331–334. CrossRefzbMATHGoogle Scholar
  4. 4.
    Maliska C (2004) Heat transfer and computational fluid mechanics. LTC, Florianópolis (in Portuguese) Google Scholar
  5. 5.
    Taylor G, Bailey C, Cross M (2003) A vertex-based finite volume method applied to non-linear material problems in computational solid mechanics. Int J Numer Methods Eng 56(4):507–529. CrossRefzbMATHGoogle Scholar
  6. 6.
    Taylor G, Bailey C, Cross M (1995) Solution of the elastic/visco-plastic constitutive equations: a finite volume approach. Appl Math Model 19(12):746–760. CrossRefzbMATHGoogle Scholar
  7. 7.
    Zienkiewick O, Taylor R (1989) The finite element method: basic formulation and linear problems, vol 1. McGraw-Hill Book Company, MaidenheadGoogle Scholar
  8. 8.
    Bilbao S, Hamilton B (2017) Wave-based room acoustics simulation: explicit/implicit finite volume modeling of viscothermal losses and frequency-dependent boundaries. J Audio Eng Soc 65(1/2):78–89. CrossRefGoogle Scholar
  9. 9.
    Slone A, Bailey C, Cross M (2003) Dynamic solid mechanics using finite volume methods. Appl Math Model 27(2):69–87. CrossRefzbMATHGoogle Scholar
  10. 10.
    Zhang B, Xu C-L, Wang S-M (2017) Generalized source finite volume method for radiative transfer equation in participating media. J Quant Spectrosc Radiat Transf 189:189–197. CrossRefGoogle Scholar
  11. 11.
    Fernandes BRB, Marcondes F, Sepehrnoori K (2013) Investigation of several interpolation functions for unstructured meshes in conjunction with compositional reservoir simulation. Numer Heat Transf Part A Appl 64(12):974–993. CrossRefGoogle Scholar
  12. 12.
    Patankar SV (1980) Numerical heat transfer and fluid flow: computational methods in mechanics and thermal science, vol 1. CRC Press, Boca RatonGoogle Scholar
  13. 13.
    Baliga B, Patankar S (1980) A new finite-element formulation for convection diffusion problems. Numer Heat Transf 3(4):393–409. CrossRefGoogle Scholar
  14. 14.
    Filippini G, Maliska C, Vaz M (2014) A physical perspective of the element-based finite volume method and FEM-Galerkin methods within the framework of the space of finite elements. Int J Numer Meth Eng 98(1):24–43. MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wheel M (1996) A geometrically versatile finite volume formulation for plane elastostatic stress analysis. J Strain Anal Eng Design 31(2):111–116CrossRefGoogle Scholar
  16. 16.
    Woodward CS, Dawson CN (2000) Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media. SIAM J Numer Anal 37(3):701–724MathSciNetCrossRefGoogle Scholar
  17. 17.
    Forsyth PA, Kropinski M (1997) Monotonicity considerations for saturated unsaturated subsurface flow. SIAM J Sci Comput 18(5):1328–1354MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bause M, Knabner P (2004) Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods. Adv Water Resour 27(6):565–581CrossRefGoogle Scholar
  19. 19.
    Srivastava R, Yeh T-CJ (1992) A three-dimensional numerical model for water flow and transport of chemically reactive solute through porous media under variably saturated conditions. Adv Water Resour 15(5):275–287CrossRefGoogle Scholar
  20. 20.
    Jasak H, Weller H (2000) Application of the finite volume method and unstructured meshes to linear elasticity. Int J Numer Meth Eng 48(2):267–287. CrossRefzbMATHGoogle Scholar
  21. 21.
    Fallah N, Bailey C, Cross M, Taylor G (2000) Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis. Appl Math Model 24(7):439–455. CrossRefzbMATHGoogle Scholar
  22. 22.
    Moczo P, Robertsson JO, Eisner L (2007) The finite-difference time-domain method for modeling of seismic wave propagation. Adv Geophys 48:421–516CrossRefGoogle Scholar
  23. 23.
    Verzicco R, Orlandi P (1996) A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. J Comput Phys 123(2):402–414MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ryskin G, Leal L (1984) Numerical solution of free-boundary problems in fluid mechanics. Part 1. The finite-difference technique. J Fluid Mech 148:1–17CrossRefGoogle Scholar
  25. 25.
    Suliman R, Oxtoby OF, Malan A, Kok S (2014) An enhanced finite volume method to model 2d linear elastic structures. Appl Math Model 38(7):2265–2279MathSciNetCrossRefGoogle Scholar
  26. 26.
    Marcondes F, Sepehrnoori K (2010) An element-based finite-volume method approach for heterogeneous and anisotropic compositional reservoir simulation. J Petrol Sci Eng 73(1):99–106. CrossRefGoogle Scholar
  27. 27.
    Winslow AM (1966) Numerical solution of the quasilinear poisson equation in a nonuniform triangle mesh. J Comput Phys 1(2):149–172. MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Schneider G, Zedan M (1983) Control-volume-based finite element formulation of the heat conduction equation. Spacecr Therm Control Design Oper Prog Astronaut Aeronaut 86:305–327Google Scholar
  29. 29.
    Fryer YD, Bailey C, Cross M, Lai C-H (1991) A control volume procedure for solving the elastic stress-strain equations on an unstructured mesh. Appl Math Model 15(11–12):639–645. CrossRefzbMATHGoogle Scholar
  30. 30.
    Bailey C, Cross M (1995) A finite volume procedure to solve elastic solid mechanics problems in three dimensions on an unstructured mesh. Int J Numer Methods Eng 38(10):1757–1776. CrossRefzbMATHGoogle Scholar
  31. 31.
    Idelsohn SR, Oñate E (1994) Finite volumes and finite elements: two good friends. Int J Numer Methods Eng 37(19):3323–3341. CrossRefzbMATHGoogle Scholar
  32. 32.
    Vaz M, Muñoz-Rojas PA, Filippini G (2009) On the accuracy of nodal stress computation in plane elasticity using finite volumes and finite elements. Comput Struct 87(17):1044–1057. CrossRefGoogle Scholar
  33. 33.
    Demirdžić I, Muzaferija S (1994) Finite volume method for stress analysis in complex domains. Int J Numer Methods Eng 37(21):3751–3766. CrossRefzbMATHGoogle Scholar
  34. 34.
    de Souza Neto EA, Peric D, Owen DR (2011) Computational methods for plasticity: theory and applications. Wiley, New YorkGoogle Scholar
  35. 35.
    Simo JC, Hughes TJ (2006) Computational inelasticity, vol 7. Springer, BerlinzbMATHGoogle Scholar
  36. 36.
    Crisfield MA, Remmers JJ, Verhoosel CV et al (2012) Nonlinear finite element analysis of solids and structures. Wiley, New YorkzbMATHGoogle Scholar
  37. 37.
    Kim N-H (2014) Introduction to nonlinear finite element analysis. Springer, BerlinGoogle Scholar
  38. 38.
    Logan D (2002) First course in finite element analysis. Brooks/Cole, BostonGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • Paulo Vicente de Cassia Lima Pimenta
    • 1
  • Francisco Marcondes
    • 1
    Email author
  1. 1.Department of Metallurgical Engineering and Material ScienceFederal University of CearáFortalezaBrazil

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