Fourier-Chebyshev spectral method for cavitation computation in nonlinear elasticity

Abstract

A Fourier-Chebyshev spectral method is proposed in this paper for solving the cavitation problem in nonlinear elasticity. The interpolation error for the cavitation solution is analyzed, the elastic energy error estimate for the discrete cavitation solution is obtained, and the convergence of the method is proved. An algorithm combined a gradient type method with a damped quasi-Newton method is applied to solve the discretized nonlinear equilibrium equations. Numerical experiments show that the Fourier-Chebyshev spectral method is efficient and capable of producing accurate numerical cavitation solutions.

Keywords

Fourier-Chebyshev spectral method / cavitation computation / nonlinear elasticity / interpolation error analysis / energy error estimate / convergence

References

[1] Adams R A.Sobolev Spaces. New York: Academic, 1975

[2] Ball J M. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos Trans R Soc Lond Ser A, 1982, 306: 557–611

[3] Ball J M, Currie J C,Olver P J.Null Lagrangians, weak continuity, and variational problems of arbitrary order. J Funct Anal, 1981, 41: 135–174

[4] Brezis H. Analyse Fonctionnelle: Théorie et Applications. Paris: Masson, 1983

[5] Bucknall C B, Karpodinis A, Zhang X C. A model for particle cavitation in rubber toughened plastics. J Mater Sci, 1994, 29(13): 3377–3383

[6] Celada P, Perrotta S. Polyconvex energies and cavitation. NoDEA Nonlinear Differential Equations Appl, 2013, 20: 295–321

[7] Gent A N, Lindley P B. Internal rupture of bonded rubber cylinders in tension. Proc R Soc Lond Ser A, 1958, 249: 195–205

[8] Gent A N, Park B. Failure processes in elastomers at or near a rigid spherical inclusion. J Mater Sci, 1984, 19: 1947–1956

[9] Gottlieb S,Jung J H, Kim S.A review of David Gottliebs work on the resolution of the Gibbs phenomenon. Commun Comput Phys, 2011, 9: 497–519

[10] Henao D. Cavitation, invertibility, and convergence of regularized minimizers in nonlinear elasticity. J Elast, 2009, 94: 55–68

[11] Henao D, Mora-Corral C. Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity. Arch Ration Mech Anal, 2010, 197: 619–655

[12] Henao D, Mora-Corral C. Fracture surfaces and the regularity of inverses for BV deformations. Arch Ration Mech Anal, 2011, 201: 575–629

[13] Lavrentiev M. Sur quelques probl`emes du calcul des variations. Ann Mat Pura Appl, 1926, 4: 7–28

[14] Lazzeri A, Bucknall C B. Dilatational bands in rubber-toughened polymers. J Mater Sci, 1993, 28: 6799–6808

[15] Li J,Guo B. Fourier-Chebyshev pseudospectral method for three-dimensional Navier-Stokes equations. Japan J Indust Appl Math, 1997, 14: 329–356

[16] Lian Y, Li Z. A dual-parametric finite element method for cavitation in nonlinear elasticity. J Comput Appl Math, 2011, 236: 834–842

[17] Lian Y, Li Z. A numerical study on cavitations in nonlinear elasticity-defects and configurational forces. Math Models Methods Appl Sci, 2011, 21(12): 2551–2574

[18] Meyer P A. Probability and Potentials. Waltham: Blaisdell, 1966

[19] Negrón-Marrero P V, Betancourt O. The numerical computation of singular minimizers in two-dimensional elasticity. J Comput Phys, 1994, 113: 291–303

[20] Negrón-Marrero P V, Sivaloganathan J. The numerical computation of the critical boundary displacement for radial cavitation. Math Mech Solids, 2009, 14: 696–726

[21] Nocedal J, Wright S J. Numerical Optimization. New York: Springer, 1999

[22] Shen J. A new fast Chebyshev-Fourier algorithm for Poisson-type equations in polar geometries. Appl Numer Math, 2000, 33: 183–190

[23] Shen J, Tang T, Wang L. Spectral Methods: Algorithms, Analysis and Applications. Heidelberg: Springer, 2011

[24] Sivaloganathan J, Spector S J.On cavitation, configurational forces and implications for fracture in a nonlinearly elastic material. J Elast, 2002, 67(1): 25–49

[25] Sivaloganathan J, Spector S J,Tilakraj V. The convergence of regularized minimizers for cavitation problems in nonlinear elasticity. J Appl Math, 2006, 66: 736–757

[26] Su C. The Numerical Analysis of Iso-parametric and Dual-parametric Finite Element Methods Applied in Cavitation. Ph D Dissertation. Beijing: Peking University, 2015 (in Chinese)

[27] Su C, Li Z. Error analysis of a dual-parametric bi-quadratic FEM in cavitation computation in elasticity. SIAM J Numer Anal, 2015, 53(3): 1629–1649

[28] Xu X, Henao D. An efficient numerical method for cavitation in nonlinear elasticity. Math Models Methods Appl Sci, 2011, 21(8): 1733–1760

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature