An Enhanced Finite Element method for Two Dimensional Linear Viscoelasticity using Complex Fourier Elements

Document Type : Research Paper


Faculty of Civil Engineering, Shahrood University of Technology, Shahrood, Iran.


In this paper, the finite element analysis of two-dimensional linear viscoelastic problems is performed using quadrilateral complex Fourier elements and, the results are compared with those obtained by quadrilateral classic Lagrange elements. Complex Fourier shape functions contain a shape parameter which is a constant unknown parameter adopted to enhance approximation’s accuracy. Since the iso-parametric formulation utilized in the finite element code, based on the experience of authors, it is proposed that a suitable shape parameter for each problem is adopted based on an acceptable approximation of the problem’s geometry by a complex Fourier element. Several numerical examples solved, and the results showed that the finite element solutions using complex Fourier elements have excellent agreement with analytical solutions, even though noticeable fewer elements than classic Lagrange elements are employed. Furthermore, the run-times of the executions of the developed finite element code to obtain accurate results, in the same personal computer, using classic Lagrange and complex Fourier elements compared. Run-times indicate that in the finite element analysis of viscoelastic problems, complex Fourier elements reduce computational cost efficiently in comparison to their classic counterpart.


[1]           Y. Z. Wang, T. J. Tsai, Static and dynamic analysis of a viscoelastic plate by the finite element method, Applied Acoustics, Vol. 25, No. 2, pp. 77-94, 1988/01/01/, 1988.
[2]           A. Y. Aköz, F. Kadıoğlu, G. Tekin, Quasi-static and dynamic analysis of viscoelastic plates, Mechanics of Time-Dependent Materials, Vol. 19, No. 4, pp. 483-503, 2015.
[3]           W. Flügge, 2013, Viscoelasticity, Springer Berlin Heidelberg,
[4]           R. Christensen, 2012, Theory of Viscoelasticity: An Introduction, Elsevier Science,
[5]           H. F. Brinson, L. C. Brinson, 2015, Polymer Engineering Science and Viscoelasticity: An Introduction, Springer US,
[6]           D. Gutierrez-Lemini, 2014, Engineering viscoelasticity, Springer,
[7]           N. W. Tschoegl, 2012, The phenomenological theory of linear viscoelastic behavior: an introduction, Springer Science & Business Media,
[8]           J. Wang, B. Birgisson, A time domain boundary element method for modeling the quasi-static viscoelastic behavior of asphalt pavements, Engineering analysis with boundary elements, Vol. 31, No. 3, pp. 226-240, 2007.
[9]           Q. Xu, M. Rahman, Finite element analyses of layered visco‐elastic system under vertical circular loading, International journal for numerical and analytical methods in geomechanics, Vol. 32, No. 8, pp. 897-913, 2008.
[10]         G. Ghazlan, S. Caperaa, C. Petit, An incremental formulation for the linear analysis of thin viscoelastic structures using generalized variables, International journal for numerical methods in engineering, Vol. 38, No. 19, pp. 3315-3333, 1995.
[11]         J. Sorvari, J. Hämäläinen, Time integration in linear viscoelasticity—a comparative study, Mechanics of Time-Dependent Materials, Vol. 14, No. 3, pp. 307-328, 2010.
[12]         I. P. King, On the finite element analysis of two-dimensional problems with time dependent properties, Ph.D. Thesis, University of California, Berkeley, USA, 1965.
[13]         M. Zocher, S. Groves, D. H. Allen, A three‐dimensional finite element formulation for thermoviscoelastic orthotropic media, International Journal for Numerical Methods in Engineering, Vol. 40, No. 12, pp. 2267-2288, 1997.
[14]         N. Khaji, S. H. Javaran, New complex Fourier shape functions for the analysis of two-dimensional potential problems using boundary element method, Engineering Analysis with Boundary Elements, Vol. 37, No. 2, pp. 260-272, 2013.
[15]         S. Hamzehei-Javaran, Approximation of the state variables of Navier’s differential equation in transient dynamic problems using finite element method based on complex Fourier shape functions, Asian Journal of Civil Engineering, pp. 1-20, 2018.
[16]         M. A. Zocher, A thermoviscoelastic finite element formulation for the analysis of composites, Ph.D. Thesis, Texas A&M University, USA, 1995.
[17]         M. J. Powell, Radial basis functions in 1990, Adv. Numer. Anal., Vol. 2, pp. 105-210, 1992.
[18]         J. Wang, G. Liu, A point interpolation meshless method based on radial basis functions, International Journal for Numerical Methods in Engineering, Vol. 54, No. 11, pp. 1623-1648, 2002.
[19]         J. Wang, G. Liu, On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Computer methods in applied mechanics and engineering, Vol. 191, No. 23-24, pp. 2611-2630, 2002.
[20]         M. Golberg, C. Chen, H. Bowman, Some recent results and proposals for the use of radial basis functions in the BEM, Engineering Analysis with Boundary Elements, Vol. 23, No. 4, pp. 285-296, 1999.
[21]         E. J. Kansa, Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates, Computers & Mathematics with applications, Vol. 19, No. 8-9, pp. 127-145, 1990.
[22]         E. J. Kansa, Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations, Computers & mathematics with applications, Vol. 19, No. 8-9, pp. 147-161, 1990.
[23]         R. Franke, Scattered data interpolation: tests of some methods, Mathematics of computation, Vol. 38, No. 157, pp. 181-200, 1982.
[24]         S. Rippa, An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Advances in Computational Mathematics, Vol. 11, No. 2-3, pp. 193-210, 1999.
[25]         R. E. Carlson, T. A. Foley, The parameter R2 in multiquadric interpolation, Computers & Mathematics with Applications, Vol. 21, No. 9, pp. 29-42, 1991.
[26]         M. Golberg, C. Chen, S. Karur, Improved multiquadric approximation for partial differential equations, Engineering Analysis with boundary elements, Vol. 18, No. 1, pp. 9-17, 1996.
[27]         D. L. Logan, 2011, A first course in the finite element method, Cengage Learning,
[28]         J. N. Reddy, 2005, An introduction to the finite element method, McGraw-Hill Education, USA, Thirded.
[29]         O. C. Zienkiewicz, 1971, The Finite Element Method in Engineering Science, McGraw-Hill Book Co Inc. London, UK
[30]         M. H. Sadd, 2009, Elasticity: theory, applications, and numerics, Academic Press,
[31]         D. S. Simulia, ABAQUS CAE Theories Manual, ABAQUS Inc., 2014.
[32]         J. Barlow, G. A. O. Davis, Selected FE benchmarks in structural and thermal analysis,  Test No. LE1, NAFEMS Report FEBSTA,  pp. 1986.
[33]         S. Timoshenko, J. N. Goodier, 1970, Theory of Elasticity, McGraw-Hill Book Co. Inc, New York, USA., Thirded.
Volume 51, Issue 1
June 2020
Pages 157-169
  • Receive Date: 30 August 2018
  • Revise Date: 20 October 2018
  • Accept Date: 21 October 2018