Pseudo-spectral methods in one-dimensional magnetostriction.

*(English)*Zbl 1329.74184Summary: In this paper a pseudo-spectral method is proposed to solve a one-dimensional model of a saturated hard ferromagnetic thin-film structure within the Euler-Bernoulli kinematics. The model accounts for the non-local nature of the magneto-elastic coupling and interaction is in the form of a logarithmic potential. The proposed solution method adopts global polynomial interpolation at a main grid, given by the Gauss-Lobatto points, and it employs a secondary grid, consisting of the Gauss points, to perform the Gaussian quadrature. The two grids are non-overlapping to avoid the singularity. Interpolation relates the unknowns, evaluated at the secondary grid, to their values at the collocation grid. Furthermore, the integration interval is parted about the singularity point. The procedure is assessed through the relative equilibrium residual for different values of the approximating polynomial degree and of the quadrature order. Maximum, average and standard deviation of the error are presented. An asymptotic analysis yields the Boundary Solution to the problem and results are compared when the latter is introduced in the numerical scheme. It is shown that its contribution is important in reducing the overall error. The equilibrium residual is plotted and its behavior discussed. It is further shown that numerical precision significantly affects the results at midspan, owing to the self-equilibrium of the system, thereby a limit exists to the best accuracy which may be gained through a more accurate interpolation.

##### MSC:

74K35 | Thin films |

74S25 | Spectral and related methods applied to problems in solid mechanics |

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

74F15 | Electromagnetic effects in solid mechanics |

PDF
BibTeX
XML
Cite

\textit{A. Nobili} and \textit{A. Tarantino}, Meccanica 50, No. 1, 99--108 (2015; Zbl 1329.74184)

Full Text:
DOI

##### References:

[1] | Boyd JP (2000) Chebyshev and Fourier spectral methods, 2nd edn. Dover Publications, Inc, New York |

[2] | Brown WF Jr (1966) Magnetoelastic interaction, tracts in natural philosophy, vol 9. Springer-Verlag, New York |

[3] | Capriz G (1989) Continua with microstructure, springer tracts in natural philosophy, vol 35. Springer-Verlag, New York |

[4] | Carman, GP; Mitrovic, M, Nonlinear constitutive relations for magnetostrictive materials with applications to 1-d problems, J Intell Mater Syst Struct, 6, 673-683, (1995) |

[5] | Funaro D (1992) Polynomial approximation of differential equations,lecture notes in physics, vol 8. Springer-Verlag, New York · Zbl 0774.41010 |

[6] | Garbow BS, Hillstrom KE, More JJ (1980) MinPack project |

[7] | Muskhelishvili NI (1992) Singular integral equations. Dover Publications, Inc., New York |

[8] | Napoli, G; Nobili, A, Mechanically induced Helfrich-hurault effect in lamellar systems, Phys Rev E, 80, 710, (2009) |

[9] | Nobili A, Tarantino AM (2006) A hard ferromagnetic and elastic beam-plate sandwich structure. Zeitschrift für angewandte Mathematik und Physik (ZAMP) 57(4) · Zbl 1161.74023 |

[10] | Nobili A, Tarantino AM (2008) Magnetostriction of a hard ferromagnetic and elastic thin-film structure. Math Mech Solids 13(2):95-123. Doi10.1177/1081286506073716. http://mms.sagepub.com/content/13/2/95.abstract, http://mms.sagepub.com/content/13/2/95.full.pdf+html · Zbl 1175.74035 |

[11] | Pandolfi, A; Napoli, G, A numerical investigation on configurational distortions in nematic liquid crystals, J Nonlinear Sci, 21, 785-809, (2011) · Zbl 1226.76004 |

[12] | Prez-Aparicio JL, Sosa H (2004) A continuum three-dimensional, fully coupled, dynamic, non-linear finite element formulation for magnetostrictive materials. Smart Mater Struct 13(3):493. http://stacks.iop.org/0964-1726/13/i=3/a=007 |

[13] | Tiersten, HF, Coupled magnetomechanical equations for magnetically saturated insulators, J Math Phys, 5, 1298-1318, (1994) · Zbl 0131.42803 |

[14] | Nobili A, Lanzoni L (2010) Electromechanical instability in layered materials. Mech Mat 42(5):581-591. doi:10.1016/j.mechmat.2010.02.006 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.