Articles

Goal-oriented error estimation applied to direct solution of steady-state analysis with frequency-domain finite element method

Expand
  • Applied Mechanics Laboratory, School of Aerospace, Tsinghua University, Beijing 100084, P. R. China

Received date: 2011-08-11

  Revised date: 2012-02-08

  Online published: 2012-05-10

Supported by

Project supported by the National Natural Science Foundation of China (No. 10876100)

Abstract

Based on the concept of the constitutive relation error along with the residuals of both the origin and the dual problems, a goal-oriented error estimation method with extended degrees of freedom is developed. It leads to the high quality local error bounds in the problem of the direct-solution steady-state dynamic analysis with a frequency-domain finite element, which involves the enrichments with plural variable basis functions. The solution of the steady-state dynamic procedure calculates the harmonic response directly in terms of the physical degrees of freedom in the model, which uses the mass, damping, and stiffness matrices of the system. A three-dimensional finite element example is carried out to illustrate the computational procedures.

Cite this article

Zhi-jia LIN;Xiao-chuan YOU;Zhuo ZHUANG . Goal-oriented error estimation applied to direct solution of steady-state analysis with frequency-domain finite element method[J]. Applied Mathematics and Mechanics, 2012 , 33(5) : 539 -552 . DOI: 10.1007/s10483-012-1569-x

References


[1] Zhang, R., Zhang, C., and Jiang, J. A new approach to direct solution of 2D heat transfer problemwith nonlinear source-terms in frequency domain. International Journal of Nonlinear Sciences andNumerical Simulation, 7(3), 295-298 (2006)

[2] Ainsworth, M. and Tinsley-Oden, J. A Posteriori Error Estimation in Finite Element Analysis,Wiley-InterScience, New York, 1-240 (2000)

[3] Tinsley-Oden, J. and Prudhomme, S. Estimation of modeling error in computational mechanics.Journal of Computational Physics, 182, 496-515 (2002)

[4] Tinsley-Oden, J., Prudhomme, S., and Bauman, P. On the extension of goal-oriented error estimationand hierarchical modeling to discrete lattice models. Comput. Methods Appl. Mech. Engrg.,194, 3668-3688 (2005)

[5] Fuentes, D., Littlefield, D., Tinsley-Oden, J., and Prudhomme, S. Extensions of goal-orientederror estimation methods to simulations of highly-nonlinear response of shock-loaded elastomerreinforcedstructures. Comput. Methods Appl. Mech. Engrg., 195, 4659-4680 (2006)

[6] Prudhomme, S. and Tinsley-Oden, J. On goal-oriented error estimation for elliptic problems:application to the control of pointwise errors. Comput. Methods Appl. Mech. Engrg., 176, 313-331 (1999)

[7] Tinsley-Oden, J. and Prudhomme, S. Goal-oriented error estimation and adaptivity for the finiteelement method. Computers and Mathematics with Applications, 41, 735-756 (2001)

[8] Ladeveze, P., Rougeota, P., Blanchardb, P., and Moreaub, J. P. Local error estimators for finiteelement linear analysis. Comput. Methods Appl. Mech. Engrg., 176, 231-246 (1999)

[9] Chamoina, L. and Ladev`eze, P. Strict and practical bounds through a non-intrusive and goalorientederror estimation method for linear viscoelasticity problems. Finite Elements in Analysisand Design, 45, 251-262 (2009)

[10] Panetier, J., Ladeveze, P., and Chamoin, L. Strict and effective bounds in goal-oriented errorestimation applied to fracture mechanics problems solved with XFEM. Int. J. Numer. Meth.Engng., 81, 671-700 (2010)

[11] Gratsch, T. and Bathe, K. J. A posteriori error estimation techniques in practical finite elementanalysis. Computers and Structures, 83, 235-265 (2005)

[12] Schleupen, A. and Ramm, E. Local and global error estimations in linear structural dynamics.Computers and Structures, 76, 741-756 (2000)

[13] Larsson, F., Hansbo, P., and Runesson, K. Strategies for computing goal-oriented a posteriorierror measures in non-linear elasticity. Int. J. Numer. Meth. Engng., 55, 879-894 (2002)

[14] Van der Zee, K. G. and Verhoosel, C. V. Isogeometric analysis-based goal-oriented error estimationfor free-boundary problems. Finite Elements in Analysis and Design, 47, 600-609 (2011)

[15] Van der Zee, K. G., Tinsley-Oden, J., Prudhomme, S., and Hawkins-Daarud, A. Goal-oriented errorestimation for Cahn-Hilliard models of binary phase transition. Numerical Methods for PartialDifferential Equations, 27(1), 160-196 (2011)

[16] Ni, Y. Q., Zheng, G., and Ko, J. M. Nonlinear periodically forced vibration of stay cables. Journalof Vibration and Acoustics, 126(2), 245-252 (2004)

[17] Challamel, N. On the comparison of Timoshenko and shear models in beam dynamics. Journal ofEngineering Mechanics-ASCE, 132(10), 1141-1146 (2006)

[18] Han, S. M., Benaoya, H., and Wei, T. Dynamics of transversely vibration beams using fourengineering theories. Journal of Sound and Vibration, 225(5), 935-988 (1999)

Outlines

/

APS Journals | CSTAM Journals | AMS Journals | EMS Journals | ASME Journals