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A mesh-free algorithm for dynamic impact analysis of hyperelasticity

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Abstract

A mesh-free method based on local Petrov-Galerkin formulation is presented to solve dynamic impact problems of hyperelastic material. In the present method, a simple Heaviside test function is chosen for simplifying domain integrals. Trial function is constructed by using a radial basis function (RBF) coupled with a polynomial basis function, in which the shape function possesses the kronecker delta function property. So, additional treatment is not required for imposing essential boundary conditions. Governing equations of impact problems are established and solved node by node by using an explicit time integration algorithm in a local domain, which is very similar to that of the collocation method except that numerical integration can be implemented over local domain in the present method. Numerical results for several examples show that the present method performs well in dealing with the dynamic impact problem of hyperelastic material.

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References

  1. Green, A.E. and Zerna, W., Theoretical Elasticity. London: Oxford University Press, 1966.

    MATH  Google Scholar 

  2. Papadrakakis, M. and Topping, B.H.V., Advances in Non-linear Finite Element Methods. Edinburgh: Civil-Comp Press, 1994.

    Google Scholar 

  3. Belytschko, T., Lu, Y.Y. and Gu, L., Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37: 229–256.

    Article  MathSciNet  Google Scholar 

  4. Liu, W.K., Chen, Y., Uras, R.A. and Chang, C.T., Generalized multiple-scale kernel particle reproducing methods. Computer Methods in Applied Mechanics and Engineering, 1996, 139: 91–157.

    Article  MathSciNet  Google Scholar 

  5. Gingold, R.A. and Monaghan, J.J., Smoothed particle hydrodynamics: Theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society, 1977, 181: 375–389.

    Article  Google Scholar 

  6. Liu, G.R., Meshfree Methods: Moving beyond the Finite Element Method. Singapore, CRC Press, 2003.

    Google Scholar 

  7. Atluri, S.N. and Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics, 1998, 22: 117–127.

    Article  MathSciNet  Google Scholar 

  8. Lin, H. and Atluri, S.N., Meshless local Petrov-Galerkin (MLPG) method for convection-diffusion problems. Computer Modeling in Engineering & Sciences, 2000, 1: 45–60.

    MathSciNet  Google Scholar 

  9. Atluri, S.N. and Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation. Computational Modeling and Simulation in Engineering, 1998, 3: 187–196.

    Google Scholar 

  10. Kim, H.G. and Atluri, S.N., Arbitrary placement of secondary nodes, and error control, in the meshless local Petrov-Galerkin (MLPG) method. Computer Modeling in Engineering & Sciences, 2000, 1: 11–32.

    MathSciNet  MATH  Google Scholar 

  11. Lin, H. and Atluri, S.N., The meshless local Petrov-Galerkin (MLPG) method for solving incompressible Navier-stokes equations. Computer Modeling in Engineering & Sciences, 2001, 2: 117–142.

    MathSciNet  Google Scholar 

  12. Long, S.Y., A local Petrov-Galerkin method for the elasticity problem. Acta Mechanica Sinica, 2001, 33: 508–518.

    MathSciNet  Google Scholar 

  13. Long, S.Y. and Atluri, S.N., A meshless local Petrov-Galerkin method for solving the bending problems of a thin plate. Computer Modeling in Engineering & Sciences, 2002, 3: 53–63.

    MATH  Google Scholar 

  14. Sladek, J., Sladek, V. and Zhang, Ch., Stress analysis in anisotropic functionally graded materials by the MLPG method. Engineering Analysis with Boundary Elements, 2005, 29(6): 597–609.

    Article  Google Scholar 

  15. Sladek, J., Sladek, V., Krivacek, J., Wen, P.H. and Zhang, Ch., Meshless local Petrov-Galerkin (MLPG) method for Reissner–Mindlin plates under dynamic load. Computer Methods in Applied Mechanics and Engineering, 2007, 196(25–28): 2681–2691.

    Article  MathSciNet  Google Scholar 

  16. Vaghefi, R., Baradaran, G.H. and Koohkan, H., Three-dimensional static analysis of thick functionally graded plates by using meshless local Petrov-Galerkin (MLPG) method. Engineering Analysis with Boundary Elements, 2010, 34(6): 564–573.

    Article  MathSciNet  Google Scholar 

  17. Atluri, S.N. and Shen, S.P., The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple & Less Costly Alternative to the Finite Element and Boundary Element Methods. Computer Modeling in Engineering & Science, 2002, 3: 11–51.

    MathSciNet  MATH  Google Scholar 

  18. Crisfield, M.A., Non-linear Finite Element Analysis of Solids and Structures. London: Thomson Press, 1997.

    MATH  Google Scholar 

  19. Belytschko, T., A survey of numerical methods and computer programs for dynamic structural analysis. Nuclear Engineering and Design, 1976, 37: 23–34.

    Article  Google Scholar 

  20. Chen, J.S. and Pan, C.H., A pressure projection method for nearly incompressible rubber hyperelasticity, Part I: Theory. Journal of Applied Mechanics, 1996, 63: 862–868.

    Article  Google Scholar 

  21. Chen, J.S., Wang, H.P., Yoon, S. and You, Y., Some recent improvements in meshfreee methods for incompressible finite elasticity boundary value problems with contact. Computational Mechanics, 2000, 25: 137–156.

    Article  Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, 410082, Changsha, China

    De’an Hu, Zhanhua Sun, Chao Liang & Xu Han

Authors
  1. De’an Hu
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  2. Zhanhua Sun
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  3. Chao Liang
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  4. Xu Han
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Correspondence to De’an Hu.

Additional information

Project supported by the National Natural Science Foundation of China (No. 10902038).

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Hu, D., Sun, Z., Liang, C. et al. A mesh-free algorithm for dynamic impact analysis of hyperelasticity. Acta Mech. Solida Sin. 26, 362–372 (2013). https://doi.org/10.1016/S0894-9166(13)60033-6

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  • DOI: https://doi.org/10.1016/S0894-9166(13)60033-6

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