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Isogeometric meshless finite volume method in nonlinear elasticity

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Abstract

An isogeometric meshless finite volume method has been presented to solve some nonlinear problems in elasticity. A non-uniform rational B-spline isogeometric basis function is used to construct the shape function. High computational efficiency and precision are other benefits of the method. Solving some sample problems of thin-walled structures shows the good performance of this method.

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References

  1. Kohnke P.C.: Large deflection analysis of frame structures by fictitious forces. Int. J. Numer. Methods Eng. 12, 1279–1294 (1978)

    Article  MATH  Google Scholar 

  2. Haisler W.E., Stricklin J.A., Stebbins F.J.: Development and evaluation of solution procedures for geometrically nonlinear structural analysis. AIAA J. 10, 264–272 (1972)

    Article  MATH  Google Scholar 

  3. De Freitas J.A.T., Smith D.L.: Existence, uniqueness and stability of elastoplastic solutions in the presence of large displacements. SM Arch. 9, 433–450 (1984)

    MathSciNet  MATH  Google Scholar 

  4. De Freitas J.A.T., Smith D.L.: Elastoplastic analysis of planar structures for large displacements. J. Struct. Mech. 12, 419–445 (1984)

    Article  Google Scholar 

  5. Boisse, P., Daniel, J.L., Ge’lin, J.C.: A combined membrane and bending model for the analysis of large elastic–plastic deformations of thin shells. In: Teodosiu, C., Raphanel, J.L., Rotterdam, S.F., Balkema, A.A. (eds.) Proceedings of the International Seminar MECMAT’ 91 on Large Plastic Deformations: Fundamental Aspects and Applications to Metal Forming, Fontainebleau, France, pp. 369–376 (1993)

  6. Lee J.D.: A large-strain elastic–plastic finite element analysis of rolling process. Comput. Methods Appl. Mech. Eng. 161, 315–347 (1998)

    Article  MATH  Google Scholar 

  7. Basar Y., Itskov M.: Constitutive model and finite element formulation for large strain elasto-plastic analysis of shells. Computat. Mech. 23, 466–481 (1999)

    Article  MATH  Google Scholar 

  8. Brünig M.: Formulation and numerical treatment of incompressibility constraints in large strain elastic–plastic analysis. Int. J. Numer. Methods Eng. 45, 1047–1068 (1999)

    Article  MATH  Google Scholar 

  9. Gingold R.A., Moraghan J.J.: Smooth particle hydrodynamics: theory and applications to non spherical stars. Mon. Notices R. Astron. Soc. 181, 375–389 (1977)

    Article  MATH  Google Scholar 

  10. Nayroles B., Touzot G., Villon P.: Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10, 307–318 (1992)

    Article  MATH  Google Scholar 

  11. Belytschko T., Lu Y.Y., Gu L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu W.K., Jun S., Zhang Y.: Reproducing kernel particle methods. Int. J. Numer. Methods Eng. 20, 1081–1106 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu G.R., Gu Y.T.: A point interpolation method for two-dimensional solids. Int. J. Numer. Methods Eng. 50, 937–951 (2001)

    Article  MATH  Google Scholar 

  14. Liu G.R.: Mesh Free Methods: Moving Beyond the Finite Element Method. CRC Press, Boca Raton (2002)

    Book  Google Scholar 

  15. Liu G.R., Gu Y.T.: An Introduction to Meshfree Methods and Their Programming. Springer, Berlin (2005)

    Google Scholar 

  16. Liu G.R., Gu Y.T.: Ameshfree method: meshfree weak–strong (MWS) form method, for 2-D solids. Comput. Mech. 33(1), 2–14 (2003)

    Article  MATH  Google Scholar 

  17. Atluri S.N., Zhu T.: A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22, 117–127 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Atluri S.N., Kim H.G., Cho J.Y.: A critical assessment of the truly meshless local Petrov–Galerkin (MLPG), and local boundary integral equation (LBIE) methods. Comput. Mech. 24, 348–372 (1999)

    Article  MATH  Google Scholar 

  19. Atluri S.N., Shen S.P.: The Meshless Local Petrov–Galerkin (MLPG) Method. Tech Science Press, Encino (2002)

    MATH  Google Scholar 

  20. Gu Y.T., Liu G.R.: A meshless Local Petrov–Galerkin (MLPG) formulation for static and free vibration analyses of thin plates. CMES Comput. Model. Eng. Sci. 2(4), 463–476 (2001)

    MATH  Google Scholar 

  21. Gu Y.T., Liu G.R.: A meshless local Petrov–Galerkin (MLPG) method for free and forced vibration analyses for solids. Comput. Mech. 27(3), 188–198 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gu Y.T., Liu G.R.: A local point interpolation method for static and dynamic analysis of thin beams. Comput. Methods Appl. Mech. Eng. 190, 5515–5528 (2001)

    Article  MATH  Google Scholar 

  23. Liu G.R., Gu Y.T.: A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids. J. Sound Vib. 246(1), 29–46 (2001)

    Article  Google Scholar 

  24. Liu G.R., Gu Y.T.: Comparisons of two meshfree local point interpolation methods for structural analyses. Comput. Mech. 29(2), 107–121 (2002)

    Article  MATH  Google Scholar 

  25. Chen J.S., Pan C., Roque C.M.O.L., Wang H.P.: A Lagrangian reproducing kernel particle method for metal forming analysis. Comput. Mech. 22, 289–307 (1998)

    Article  MATH  Google Scholar 

  26. Chen J.S., Pan C., Wu C.T.: Application of reproducing kernel particle method to large deformation contact analysis of elastomers. Rubber Chem. Technol. 71, 191–213 (1998)

    Article  Google Scholar 

  27. Chen J.S., Pan C., Wu C.T., Liu W.K.: Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput. Methods Appl. Mech. Eng. 139, 195–227 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu W.K., Jun S.: Multiple-scale reproducing kernel particle methods for large deformation problems. Int. J. Numer. Methods Eng. 41, 1339–1362 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li S., Hao W., Liu W.K.: Numerical simulations of large deformation of thin shell structures using meshfree methods. Comput. Mech. 25, 102–116 (2000)

    Article  MATH  Google Scholar 

  30. Li S., Liu W.K.: Numerical simulations of strain localization in inelastic solids using mesh-free methods. Int. J. Numer. Methods Eng. 48, 1285–1309 (2000)

    Article  MATH  Google Scholar 

  31. Liu W.K., Jun S., Zhang Y.F.: Reproducing kernel particle methods. Int. J. Numer. Methods Fluids 20, 1081–1106 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Liu W.K., Jun S., Li S., Adee J., Belytschko T.: Reproducing kernel particle methods for structural dynamics. Int. J. Numer. Methods Eng. 38, 1655–1679 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Jun S., Liu W.K., Belytschko T.: Explicit reproducing kernel particle methods for large deformation problems. Int. J. Numer. Methods Eng. 41, 137–166 (1998)

    Article  MATH  Google Scholar 

  34. Liu W.K., Chen Y., Uras R.A., Chang C.T.: Generalized multiple scale reproducing kernel particle methods. Comput. Methods Appl. Mech. Eng. 139, 91–157 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hughes T.J.R., Cottrell J.A., Bazilevs Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Hallquist, J.O.: LS-DYNA Theory Manual. Livermore Software Technology Corporation, Livermore, CA (2006)

  37. Clarke M.J., Hancock G.J.: A study of incremental-iterative strategies for nonlinear analyses. Int. J. Numer. Methods Eng. 29, 1365–1391 (1990)

    Article  Google Scholar 

  38. Adotte, G.: Second order theory in orthotropic plates. Proceedings, Journal of the Structural Division, ASCE, 93 (ST5, proc. paper 5520), October 1967

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Authors and Affiliations

  1. Mechanical Engineering Department, South Dakota State University, Brookings, SD, 57007-0294, USA

    M. R. Moosavi

  2. University of Lorraine, IJL UMR CNRS 7198, 54600, Villers Les Nancy, France

    A. Khelil

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  1. M. R. Moosavi
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  2. A. Khelil
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Correspondence to M. R. Moosavi.

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Moosavi, M.R., Khelil, A. Isogeometric meshless finite volume method in nonlinear elasticity. Acta Mech 226, 123–135 (2015). https://doi.org/10.1007/s00707-014-1166-5

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  • DOI: https://doi.org/10.1007/s00707-014-1166-5

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