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Coupled FEM and BEM code for simulating acoustically excited bubbles near deformable structures

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Abstract

An understanding of biotissue–bubble interactions and the stresses induced in the tissue is needed to identify potential mechanisms of tissue damage, such as vessel rupture, by acoustically excited bubbles. Interactions between acoustically excited bubbles and nearby rigid structures have been studied effectively using the boundary element method. However, if the nearby structure is a biotissue, structure deformations will affect the bubble response. In this paper a coupled finite element and boundary element code, developed to investigate the interactions between an acoustically excited bubble and a deformable structure, is presented. In particular, this model was developed to investigate the response of bubbles within deformable tubes. This code is validated by comparison to other simulation and experimental results and employed to obtain the response of an acoustically excited bubble centered within a tube. General characteristics of bubble–tube interactions and stresses induced in the tube wall are described by considering typical simulation results.

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References

  1. Abramowitz M, Stegun IA (1974) Handbook of mathematical functions. Dover, New York

    Google Scholar 

  2. Amini S, Harris PJ, Wilton DT (1992) Coupled boundary and finite element methods for the solution of the dynamic fluid–structure interaction problem. Springer, Heidelberg

    Google Scholar 

  3. Anderson DG (1965) Gaussian quadrature formulae for \({\int_{0}^{1} - {\rm ln} (x )f(x)dx}\). Math Comput 19(91): 477–481

    Article  MATH  Google Scholar 

  4. Bathe KJ (1982) Finite element procedures in engineering analysis. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  5. Blake Jr, Taib BB, Doherty G (1986) Transient cavities near boundaries. Part 1. Rigid boundary. J Fluid Mech 170: 479–497

    Article  MATH  Google Scholar 

  6. Blake Jr, Taib BB, Doherty G (1987) Transient cavities near boundaries. Part I1. Free surface. J Fluid Mech 181: 197–212

    Article  Google Scholar 

  7. Brebbia CA (ed.) (1984) Boundary element techniques in computer-aided engineering. Martinus Nijhoff Publishers, Dordrecht

    Google Scholar 

  8. Brujan EA, Nahen K, Schmidt P, Vogel A (2001) Dynamics of laser-induced cavitation bubbles near an elastic boundary. J Fluid Mech 433: 251–281

    MATH  Google Scholar 

  9. Brujan EA, Nahen K, Schmidt P, Vogel A (2001) Dynamics of laser-induced cavitation bubbles near elastic boundaries: influence of the elastic modulus. J Fluid Mech 433: 283–314

    MATH  Google Scholar 

  10. Brujan EA (2004) The role of cavitation microjets in the therapeutic applications of ultrasound. Ultrasound Med Biol 30: 381–387

    Article  Google Scholar 

  11. Chahine GL, Kalumuck KM (1998) BEM software for free surface flow simulation including fluid–structure interaction effects. Int J Comp Appl Tech 11: 177–198

    Google Scholar 

  12. Chen WS, Brayman AA, Matula TJ, Crum LA, Miller MW (2003) The pulse length-dependence of inertial cavitation dose and hemolysis. Ultrasound Med Biol 29: 739–748

    Article  Google Scholar 

  13. Dalecki D, Raeman CH, Child SZ, Cox C, Francis CW, Meltzer RS, Carstensen EL (1997) Hemolysis in vivo from exposure to pulsed ultrasound. Ultrasound Med Biol 23: 307–313

    Article  Google Scholar 

  14. Duck FA (1990) Physical properties of tissue. Academic, London

    Google Scholar 

  15. Duncan JH, Zhang S (1991) On the interaction of a collapsing cavity and a compliant wall. J Fluid Mech 226: 401–423

    Article  MATH  Google Scholar 

  16. Duncan JH, Milligan CD, Zhang S (1996) On the interaction between a bubble and a submerged compliant structure. J Sound Vibration 197(1): 17–44

    Article  Google Scholar 

  17. Fong SW, Klaseboer E, Turangan CK, Khoo BC, Hung KC (2006) Numerical analysis of a gas bubble near bio-materials in an ultrasound field. Ultrasound Med Biol 32(6): 925–942

    Article  Google Scholar 

  18. Gilmore FR (1952) Hydrodynamics Laboratory report 26-4, California Institute of Technology

  19. Harris PJ (1993) A numerical method for determining the motion of a bubble close to a fixed rigid structure in a fluid. Int J Numer Methods Eng 33: 1813–1822

    Article  Google Scholar 

  20. Hartland S (2004) Surface and interfacial tension: measurement, theory, and applications. Marcel Dekker, New York

    Google Scholar 

  21. Jaswon MA, Symm GT (1977) Integral equation methods in potential theory and elastotatics. Academic Press, London

    Google Scholar 

  22. Jeffrey A (1995) Handbook of mathematical formulas and integrals. Academic Press, London

    Google Scholar 

  23. Kalumuck KM, Duraiswami R, Chahine GL (1995) Bubble dynamics fluid–structure interaction simulation by coupling fluid BEM and structural FEM codes. J Fluids Struct 9: 861–883

    Article  Google Scholar 

  24. Klaseboer E, Hung KC, Wang C, Wang CW, Khoo BC, Boyce P, Debono S, Charlier H (2005) Experimental and numerical investigation of the dynamics of an underwater explosion bubble near a resilient/rigid structure. J Fluid Mech 537: 387–413

    Article  MATH  Google Scholar 

  25. Lennon GP, Liu PLF, Liggett JA (1979) Boundary integral equation solution to axisymmetric potential flows I. Basic formulation. Water Resour Res 15(5): 1102–1106

    Article  Google Scholar 

  26. Li P, Cao T, Cou C, Armstrong WF, Miller D (2003) Impact of myocardial contrast echocardiography on vascular permeability: an in vivo dose response study of delivery mode, pressure amplitude, and contrast dose. Ultrasound Med Biol 29(9): 1341–1349

    Article  Google Scholar 

  27. Li P, Armstrong WF, Miller DL (2004) Impact of myocardial contrast echocardiography on vascular permeability: comparison of three different contrast agents. Ultrasound Med Biol 30(1): 83–91

    Article  Google Scholar 

  28. Longuet-Higgins MS, Cokelet ED (1976) The deformation of steep surface waves on water. I A numerical method of computation. Proc Roy Soc Lond A 350: 1–26

    Article  MATH  MathSciNet  Google Scholar 

  29. Melbin J, Noordergraaf A (1971) Elastic deformation in orthotropic vessels Theoretical and experimental results. Circ Res XXVIII: 680–692

    Google Scholar 

  30. Miao H (2006) Numerical study of ultrasound bioeffects by solving gas–liquid–solid interaction problems with coupled FEM and BEM. PhD dissertation, University of Rochester, Rochester

  31. Miller DL, Gies RA (1998) Gas–body-based contrast agent enhances vascular bioeffects of 1.09 MHz ultrasound on mouse intestine. Ultrasound Med Biol 24: 1201–1208

    Article  Google Scholar 

  32. Miller DL, Gies RA (2000) The influence of ultrasound frequency and gas–body composition on the contrast agent-mediated enhancement of vascular bioeffect in mouse intestine. Ultrasound Med Biol 26: 307–313

    Article  Google Scholar 

  33. Miller DL, Quddus J (2000) Diagnostic ultrasound activation of contrast agent gas bodies induces capillary rupture in mice. Proc Nat Acad Sci 97: 10179–10184

    Article  Google Scholar 

  34. Miller DL, Thomas RM (1993) Contrast agent gas bodies enhance hemolysis induced by lithotripter shockwaves and high-intensity focused ultrasound in whole blood. Ultrasound Med Biol 22: 1089–1095

    Article  Google Scholar 

  35. Noltingk BE, Neppiras EA (1950) Cavitation produced by ultrasonics. Proc Phys Cos Lond B 63: 674–685

    Article  Google Scholar 

  36. O‘Rourke J (2001) Computational geometry in C. Cambridge University Press, New York

    Google Scholar 

  37. Plesset MS (1949) The dynamics of cavitation bubbles. J Appl Mech ASME Trans 16: 277–282

    Google Scholar 

  38. Prosperetti A (1991) The thermal behavior of oscillating gas bubble. J Fluid Mech 22: 587–616

    Article  MathSciNet  Google Scholar 

  39. Rayleigh Lord (1917) On the pressure developed in a liquid during the collapse of a spherical cavity. Philos Mag 34: 94–98

    Google Scholar 

  40. Rowe AJ, Finlay HM, Canham PB (2003) Collagen biomechanics in cerebral arteries and bifurcations assessed by polarizing microscopy. J Vasc Res 40: 406–415

    Article  Google Scholar 

  41. Sato K, Tomita Y, Shima A (1994) Numerical analysis of a gas bubble near a rigid boundary in an oscillatory pressure field. J Acoust Soc Am 95: 2416–2424

    Article  Google Scholar 

  42. Snowhill P, Frederick S (2005) A mechanical model of porcine vascular tissues. Part II: Stress–strain and mechanical properties of juvenile porcine blood vessels. Cardiovasc Eng Int J 5(4): 157–169

    Article  Google Scholar 

  43. Tomita Y, Robinson PB, Tong RP, Blake JR (2002) Growth and collapse of cavitation bubbles near a curved rigid boundary. J Fluid Mech 466: 259–283

    Article  MATH  Google Scholar 

  44. Utku S (1968) Explicit expressions for triangular torus element stiffness matrix. J Am Inst Aeronaut Astron 6(6): 1174–1176

    Google Scholar 

  45. Wang QX, Yeo KS, Khoo BC, Lam KY (1996) Nonlinear interaction between gas bubble and free surface. Comput Fluids 25(7): 607–628

    Article  MATH  Google Scholar 

  46. Wang XC, Shao M (2002) Basic principle of finite element method and numerical method, 2nd edn in Chinese. Tsinghau University Press, Beijing

    Google Scholar 

  47. Wang C, Khoo BC, Yeo KS (2003) Elastic mesh technique for 3D BIM simulation with an application to underwater explosion bubble dynamics. Comput Fluids 32: 1195–1212

    Article  MATH  Google Scholar 

  48. Wang C, Khoo BC (2004) An indirect boundary element method for three-dimensional explosion bubbles. J Comput Phys 194(2): 451–480

    Article  MATH  Google Scholar 

  49. Wible JH, Galen KP, Wojdyla JK, Hughes MS, Klibanov AL, Brandenburger GH (2002) Microbubbles induce renal hemorrhage when exposed to diagnostic ultrasound in anesthetized rats. Ultrasound Med Biol 28: 1535–1546

    Article  Google Scholar 

  50. Wu TW (ed) (2000) Boundary element acoustics. WIT Press, Boston

    MATH  Google Scholar 

  51. Yamada H (1970) Strength of Biological Materials. Williams& Wilkins, Baltimore

    Google Scholar 

  52. Young FR (1989) Cavitation. McGraw-Hill, New York

    Google Scholar 

  53. Zhong P, Zhou YF, Zhu SL (2001) Dynamics of bubble oscillation in constrained media and mechanisms of vessel rupture in SWL. Ultrasound Med Biol 27(1): 119–134

    Article  Google Scholar 

  54. Zienkiewicz OC, Taylor RL (2000) Finite element method, 5th edn, vol 1, the basis. Butterworth-Heinemann, London

    Google Scholar 

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Correspondence to Sheryl M. Gracewski.

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Miao, H., Gracewski, S.M. Coupled FEM and BEM code for simulating acoustically excited bubbles near deformable structures. Comput Mech 42, 95–106 (2008). https://doi.org/10.1007/s00466-007-0238-y

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  • DOI: https://doi.org/10.1007/s00466-007-0238-y

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