On the Numerical Modeling of Deformations of Pressurized Martensitic Thin Films

Pavel Bělík; Timothy Brule; Mitchell Luskin

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 3, page 525-548
  • ISSN: 0764-583X

Abstract

top
We propose, analyze, and compare several numerical methods for the computation of the deformation of a pressurized martensitic thin film. Numerical results have been obtained for the hysteresis of the deformation as the film transforms reversibly from austenite to martensite.

How to cite

top

Bělík, Pavel, Brule, Timothy, and Luskin, Mitchell. "On the Numerical Modeling of Deformations of Pressurized Martensitic Thin Films." ESAIM: Mathematical Modelling and Numerical Analysis 35.3 (2010): 525-548. <http://eudml.org/doc/197493>.

@article{Bělík2010,
abstract = { We propose, analyze, and compare several numerical methods for the computation of the deformation of a pressurized martensitic thin film. Numerical results have been obtained for the hysteresis of the deformation as the film transforms reversibly from austenite to martensite. },
author = {Bělík, Pavel, Brule, Timothy, Luskin, Mitchell},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Thin film; finite element; martensitic transformation; active materials.; hysteresis; convergence; finite element approximation},
language = {eng},
month = {3},
number = {3},
pages = {525-548},
publisher = {EDP Sciences},
title = {On the Numerical Modeling of Deformations of Pressurized Martensitic Thin Films},
url = {http://eudml.org/doc/197493},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Bělík, Pavel
AU - Brule, Timothy
AU - Luskin, Mitchell
TI - On the Numerical Modeling of Deformations of Pressurized Martensitic Thin Films
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 3
SP - 525
EP - 548
AB - We propose, analyze, and compare several numerical methods for the computation of the deformation of a pressurized martensitic thin film. Numerical results have been obtained for the hysteresis of the deformation as the film transforms reversibly from austenite to martensite.
LA - eng
KW - Thin film; finite element; martensitic transformation; active materials.; hysteresis; convergence; finite element approximation
UR - http://eudml.org/doc/197493
ER -

References

top
  1. R. Adams, Sobolev spaces. Academic Press, New York (1975).  Zbl0314.46030
  2. J.H. Argyris, I. Fried and D.W. Scharpf, The TUBA family of plate elements for the matrix displacement method. Aero. J. Roy. Aero. Soc.72 (1968) 701-709.  
  3. N.W. Ashcroft and N.D. Mermin, Solid State Physics. Saunders College Publishing, Orlando (1976).  Zbl1107.82300
  4. J. Ball and R. James, Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal.100 (1987) 13-52.  Zbl0629.49020
  5. J. Ball and R. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. Lond. A338 (1992) 389-450.  Zbl0758.73009
  6. M. Bernadou and K. Hassan, Basis functions for general Hsieh-Clough-Tocher triangles, complete or reduced. Internat. J. Numer. Methods Engrg.17 (1981) 784-789.  Zbl0478.73050
  7. K. Bhattacharya and R.D. James, A theory of thin films of martensitic materials with applications to microactuators. J. Mech. Phys. Solids47 (1999) 531-576.  Zbl0960.74046
  8. K. Bhattacharya, B. Li and M. Luskin, The simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation. Arch. Rat. Mech. Anal.149 (1999) 123-154.  Zbl0942.74056
  9. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer-Verlag, New York (1994).  Zbl0804.65101
  10. P. Belík, T. Brule and M. Luskin, Numerical modelling of a temperature-operated martensitic microvalve. http://www.math.umn.edu/ luskin/research/valve/.  Zbl1062.74047
  11. P. Belík and M. Luskin, Stability of microstructure for tetragonal to monoclinic martensitic transformations. ESAIM: M2AN34 (2000) 663-685.  Zbl0981.74042
  12. C. Carstensen and P. Plechác, Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp.66 (1997) 997-1026.  Zbl0870.65055
  13. C. Carstensen and P. Plechác, Adaptive algorithms for scalar non-convex variational problems. Appl. Numer. Math.26 (1998) 203-216.  Zbl0894.65029
  14. M. Chipot, C. Collins and D. Kinderlehrer, Numerical analysis of oscillations in multiple well problems. Numer. Math.70 (1995) 259-282.  Zbl0824.65045
  15. M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rat. Mech. Anal.103 (1988) 237-277.  Zbl0673.73012
  16. M. Chipot and S. Müller, Sharp energy estimates for finite element approximations of nonconvex problems. Preprint (1997).  
  17. P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978).  Zbl0383.65058
  18. R.W. Clough and J.L. Tocher, Finite element stiffness matrices for analysis of plates in bending. In Proceedings of the conference on matrix methods in structural mechanics. Wright Patterson A.F.B., Ohio (1965) 515-545.  
  19. C. Collins, Computation of twinning. In Microstructure and phase transitions. J. Ericksen, R. James, D. Kinderlehrer and M. Luskin Eds. IMA Vol. Math. Applic.54, Springer-Verlag, New York (1993) 39-50.  Zbl0797.73049
  20. C. Collins and M. Luskin, The computation of the austenitic-martensitic phase transition. In Partial differential equations and continuum models of phase transitions. M. Rascle, D. Serre and M. Slemrod Eds. Lect. Notes Phys.344, Springer-Verlag, Berlin (1989) 34-50.  Zbl0991.80502
  21. C. Collins, M. Luskin and J. Riordan, Computational results for a two-dimensional model of crystalline microstructure. In Microstructure and phase transitions. J. Ericksen, R. James, D. Kinderlehrer and M. Luskin Eds. IMA Vol. Math. Applic.54, Springer-Verlag, New York (1993) 51-56.  Zbl0797.73048
  22. G. Dolzmann, Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal.36 (1999) 1621-1635.  Zbl0941.65062
  23. J.W. Dong, L.C. Chen, C.J. Palmstrøm, R.D. James and S. McKernann, Molecular beam epitaxy growth of ferromagnetic single crystal (001) Ni2MnGa on (001) GaAs. Appl. Phys. Lett.75 (1999) 1443-45.  
  24. D.A. Dunavant, k High degree efficient symmetrical Gaussian quadrature rules for the triangle. Internat. J. Numer. Methods Engrg.21 (1985) 1129-1148.  Zbl0589.65021
  25. G.B. Folland, Real analysis. Modern techniques and their applications. John Wiley & Sons, Inc., New York (1984).  Zbl0549.28001
  26. M. Giaquinta. Calculus of variations. Springer-Verlag, Berlin (1996).  Zbl0853.49001
  27. D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin (1998).  Zbl1042.35002
  28. R. Glowinski, Numerical methods for nonlinear variational problems. Springer-Verlag, New York (1984).  Zbl0536.65054
  29. P.-A. Gremaud, Numerical analysis of a nonconvex variational problem related to solid-solid phase transitions. SIAM J. Numer. Anal.31 (1994) 111-127.  Zbl0797.65052
  30. M. Gurtin, Topics in finite elasticity. SIAM, Philadelphia (1981).  Zbl0486.73030
  31. R.D. James and R. Rizzoni, Pressurized shape memory thin films. J. Elasticity59, special issue in honor of Roger Fosdick, D. Carlson Ed. (2000) 399-436.  Zbl0990.74038
  32. P. Krulevitch, A.P. Lee, P.B. Ramsey, J.C. Trevino, J. Hamilton and M.A. Northrup, Thin film shape memory alloy microactuators. Journal of Microelectromechanical Systems5 (1996) 270.  
  33. M. Kruzík, Numerical approach to double well problems. SIAM J. Numer. Anal.35 (1998) 1833-1849.  Zbl0929.49016
  34. P. Lascaux and P. Lesaint, Some nonconforming finite elements for the plate bending problem. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. R-1 (1975) 9-53.  Zbl0319.73042
  35. B. Li and M. Luskin, Finite element analysis of microstructure for the cubic to tetragonal transformation. SIAM J. Numer. Anal.35 (1998) 376-392.  Zbl0919.49020
  36. B. Li and M. Luskin, Nonconforming finite element approximation of crystalline microstructure. Math. Comp.67 (1998) 917-946.  Zbl0901.73076
  37. B. Li and M. Luskin, Approximation of a martensitic laminate with varying volume fractions. ESAIM: M2AN33 (1999) 67-87.  Zbl0928.74012
  38. Z. Li, Simultaneous numerical approximation of microstructures and relaxed minimizers. Numer. Math.78 (1997) 21-38.  Zbl0890.65067
  39. D.G. Luenberger, Introduction to linear and nonlinear programming. Addison-Wesley, Reading, Mass. (1973).  Zbl0297.90044
  40. M. Luskin, Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math.75 (1996) 205-221.  Zbl0874.73060
  41. M. Luskin, On the computation of crystalline microstructure. Acta Numer.5 (1996) 191-257.  Zbl0867.65033
  42. M. Luskin and L. Ma, Analysis of the finite element approximation of microstructure in micromagnetics. SIAM J. Numer. Anal.29 (1992) 320-331.  Zbl0760.65113
  43. L.S.D. Morley, The triangular equilibrium element in the solution of plate bending problems. Aero. Quart.19 (1968) 149-169.  
  44. P. Pedregal, On the numerical analysis of non-convex variational problems. Numer. Math.74 (1996) 325-336.  Zbl0858.65059
  45. E. Polak, Computational methods in optimization. Academic Press, New York (1971).  
  46. T. Roubícek, Numerical approximation of relaxed variational problems. J. Convex Anal.3 (1996) 329-347.  Zbl0881.65058
  47. H.L. Royden, Real analysis. 3rd edn, Macmillan Publishing Company, New York (1988).  Zbl0704.26006
  48. W. Rudin, Functional analysis. McGraw-Hill, New York (1973).  Zbl0253.46001
  49. Z. Shi, Error estimates of Morley element. Chinese J. Num. Math. Appl.12 (1990) 102-108.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.