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Boundary contour analysis for surface stress recovery in 2-D elasticity and stokes flow

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Summary

A variant of the boundary element method, called the boundary contour method, offers a further reduction in dimensionality. Consequently, boundary contour analysis of 2-D problems does not require any numerical integration at all. In a boundary contour analysis, boundary stresses can be accurately computed using the approach proposed in Ref. [1]. However, due to singularity, this approach can be used only to calculate boundary stresses at points that do not lie at an end of a boundary element. Herein, it is shown that a technique based on the displacement/velocity shape functions can overcome this drawback. Further, the approach is much simpler to apply, requires less computational effort, and provides competitive accuracy. Numerical solutions and convergence study for some well-known problems in linear elasticity and Stokes flow are presented to show the effectiveness of the proposed approach.

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References

  1. Phan, A.-V.;Mukherjee, S.;Mayer, J.R.R.: The boundary contour method for two-dimensional linear elasticity with quadratic boundary elements. Comput Mech 20 (1997) 310–319

    Article  MATH  MathSciNet  Google Scholar 

  2. Phan, A.-V.;Mukherjee, S.;Mayer, J.R.R.: Stresses, stress sensitivities and shape optimization for two-dimensional linear elasticity by the boundary contour method. Int J Numer Methods Eng 42 (1998) 1391–1407

    Article  MATH  MathSciNet  Google Scholar 

  3. Phan, A.-V.;Kaplan, T.;Gray, L.J.;Adalsteinsson, D.;Sethian, J.A.;Barvosa-Carter, W.;Aziz, M.J.: Modelling a growth instability in a stressed solid. Model Simulation Mater Sci Eng 9 (2001) 309–325

    Google Scholar 

  4. Choi, J.K.;Kinnas, S.A.: Numerical water tunnel in two-and three-dimensions. J Ship Res 42 (1998) 86–98

    Google Scholar 

  5. Grilli, S.T.;Guyenne, P.;Dias, F.: A fully nonlinear model for three-dimensional overturning waves over arbitrary bottom. Int J Numer Methods Fluids 35 (2001) 829–867

    Article  MATH  Google Scholar 

  6. Calzado, F.J.;Mantic, V.;París, F.: Evaluation of in-boundary stress in 2D BEM for isotropic elasticity. A comparative study. Comput Mech 32 (2003) 445–450

    Article  MATH  Google Scholar 

  7. Gray, L.J.; Phan, A.-V.; Kaplan, T.: Boundary integral evaluation of surface derivatives. SIAM J Sci Comput (2004) (in press)

  8. Gray, L.J.;Maroudas, D.;Enmark, M.: Galerkin boundary integral method for evaluating surface derivatives. Comput Mech 22 (1998) 187–193

    Article  MATH  Google Scholar 

  9. Matsumoto, T.;Tanaka, M.;Hirata, H.: Boundary stress calculation using regularized boundary integral equation for displacement gradients. Int J Numer Methods Eng 36 (1993) 783–797

    Article  MATH  Google Scholar 

  10. Wilde, A.J.;Aliabadi, M.H.: Direct evaluation of boundary stresses in the 3D BEM elastostatics. Commun Numer Methods Eng 14 (1998) 505–517

    Article  MATH  Google Scholar 

  11. Zhao, Z.;Lan, S.: Boundary stress calculation—a comparison study. Comput Structures 71 (1999) 77–85

    Article  Google Scholar 

  12. Mukherjee, S.: Boundary element methods in creep and fracture. Elsevier Applied Science, London, 1982

    MATH  Google Scholar 

  13. Bonnet, M.: Boundary integral equation methods for solids and fluids. Wiley, Chichester, 1995

    Google Scholar 

  14. Lutz, E.D.: Numerical methods for hypersingular and near-singular boundary integrals in fracture mechanics. PhD Dissertation Cornell University, Ithaca, 1991

    Google Scholar 

  15. Nagarajan, A.;Lutz, E.D.;Mukherjee, S.: A novel boundary element method for linear elasticity with no numerical integration for two-dimensional and line integrals for three-dimensional problems. ASME J Appl Mech 61 (1994) 264–269

    Article  MATH  MathSciNet  Google Scholar 

  16. Nagarajan, A.;Lutz, E.D.;Mukherjee, S.: The boundary contour method for three-dimensional linear elasticity. ASME J Appl Mech 63 (1996) 278–286

    Article  MATH  Google Scholar 

  17. Nagarajan, A.: A novel boundary element method for linear elasticity. PhD Dissertation, Cornell University, Ithaca, 1994

    Google Scholar 

  18. Phan, A.-V.;Gray, L.J.;Kaplan, T.;Phan, T.-N.: The boundary contour method for two-dimensional Stokes flow and incompressible elastic materials. Comput Mech 28 (2002) 425–433

    Article  MATH  Google Scholar 

  19. Phan, A.-V.;Mukherjee, S.;Mayer, J.R.R.: The hypersingular boundary contour method for two-dimensional linear elasticity. Acta Mechanica 130 (1998) 209–225

    Article  MATH  Google Scholar 

  20. Novati, G.;Springhetti, R.: A Galerkin boundary contour method for two-dimensional linear elasticity. Comput Mech 23 (1999) 53–62

    Article  MATH  Google Scholar 

  21. Zhou, S.J.;Cao, Z.Y.;Sun, S.X.: Traction boundary contour method for linear elasticity. Int J Numer Methods Eng 46 (1999) 1883–1895

    Article  MATH  MathSciNet  Google Scholar 

  22. Szirbik, S.: Boundary contour method for plane problems in a dual formulation with linear elements. J Comput Appl Mech 1 (2000) 205–222

    MATH  Google Scholar 

  23. Mukherjee, Y.X.;Mukherjee, S.;Shi, X.;Nagarajan, A.: The boundary contour method for three-dimensional linear elasticity with a new quadratic boundary element. Eng Anal Bound Elem 20 (1997) 35–44

    Article  Google Scholar 

  24. Mukherjee, S.;Mukherjee, Y.X.: The hypersingular boundary contour method for three-dimensional linear elasticity. ASME J Appl Mech 65 (1998) 300–309

    Article  Google Scholar 

  25. Phan, A.-V.;Mukherjee, S.;Mayer, J.R.R.: A boundary contour formulation for design sensitivity analysis in two-dimensional linear elasticity. Int J Solids Struct 16 (1998) 1981–1999

    Article  Google Scholar 

  26. Phan, A.-V.;Mukherjee, S.: On design sensitivity analysis in linear elasticity by the boundary contour method. Eng Anal Bound Elem 23 (1999) 195–199

    Article  MATH  Google Scholar 

  27. Mukherjee, S.;Shi, X.;Mukherjee, Y.X.: Surface variables and their sensitivities in three-dimensional linear elasticity by the boundary contour method. Comput Methods Appl Mech Eng 173 (1999) 387–402

    Article  MATH  Google Scholar 

  28. Mukherjee, S.;Shi, X.;Mukherjee, Y.X.: Internal variables and their sensitivities in three-dimensional linear elasticity by the boundary contour method. Comput Methods Appl Mech Eng 187 (2000) 289–306

    Article  MATH  Google Scholar 

  29. Phan, A.-V.;Phan, T.-N.: Structural shape optimization system using the two-dimensional boundary contour method. Arch Appl Mech 69 (1999) 481–489

    Article  MATH  Google Scholar 

  30. Shi, X.;Mukherjee, S.: Shape optimization in three-dimensional linear elasticity by the boundary contour method. Eng Anal Bound Elem 23 (1999) 627–637

    Article  MATH  Google Scholar 

  31. Phan, A.-V.; Liu, Y.J.: Boundary contour analysis of thin films and layered coatings. In: Sixth US national congress on computational mechanics, 2001

  32. Zhou, S.J.;Sun, S.X.;Cao, Z.Y.: The dual boundary contour method for two-dimensional crack problems. Int J Fracture 92 (1998) 201–212

    Article  Google Scholar 

  33. Chen, S.Y.;Tang, W.X.;Zhou, S.J.;Leng, X.;Zheng, W.: Evaluation of J integrals and stress intensity factors in a 2D quadratic boundary contour method. Commun Numer Methods Eng 15 (1999) 91–100

    Article  MATH  Google Scholar 

  34. Rizzo, F.J.: An integral equation approach to boundary value problems of classical elastostatics. Q Appl Math 25 (1967) 83–95

    MATH  Google Scholar 

  35. Pozrikidis, C.: Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press, New York, 1992

    MATH  Google Scholar 

  36. Schlichting, H.: Boundary-layer theory. McGraw-Hill, New York, 1987

    Google Scholar 

  37. Cook, R.D.;Malkus, D.S.;Plesha, M.E.: Concepts and applications of finite element analysis. Wiley, New York, 1989

    MATH  Google Scholar 

  38. Polyzos, D.;Tsinopoulos, S.V.;Beskos, D.E.: Static and dynamic boundary element analysis in incompressible linear elasticity. Eur J Mech A Solids 17 (1998) 515–536

    Article  MATH  Google Scholar 

  39. Timoshenko, S.P.;Goodier, J.N.: Theory of elasticity. McGraw-Hill, New York, 1970

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of South Alabama, 36688, Mobile, AL, USA

    A. -V. Phan

  2. Mentor Graphics, 739 University Boulevard North, 36688, Mobile, AL, USA

    T. -N. Phan

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  1. A. -V. Phan
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  2. T. -N. Phan
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Correspondence to A. -V. Phan.

Additional information

This research was supported in part by the 2004 Ralph E. Powe Junior Faculty Enhancement Award from Oak Ridge Associated Universities and by the University of South Alabama Research Council.

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Phan, A.V., Phan, T.N. Boundary contour analysis for surface stress recovery in 2-D elasticity and stokes flow. Arch. Appl. Mech. 74, 427–438 (2005). https://doi.org/10.1007/BF02637040

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