Advertisement

Sādhanā

, 43:34 | Cite as

Spectral element method for wave propagation on irregular domains

  • Yan Hui Geng
  • Guo Liang Qin
  • Jia Zhong Zhang
Article
  • 51 Downloads

Abstract

A spectral element approximation of acoustic propagation problems combined with a new mapping method on irregular domains is proposed. Following this method, the Gauss–Lobatto–Chebyshev nodes in the standard space are applied to the spectral element method (SEM). The nodes in the physical space are mapped according to the length scale of the beeline segment or the curve segment. Using the Bubnov–Galerkin method, some acoustic problems with two kinds of irregular domains are simulated in detail. First, the basic problem with analytical solution is analysed numerically. Numerical results show that the SEM integrated with the length-scale method has the same precision as the isoparametric SEM. Also, it can save nearly half of the time cost. Additionally, the acoustic propagations with inlet flow are simulated numerically. All the results indicate that the SEM integrated with the length-scale method has the ability to simulate the acoustic problems with irregular domains. It is shown that the mapping method maintains the curve edges and provides a useful alternative for isoparametric element, which represents a curved edge with a straight edge.

Keywords

Spectral element method curved quadrilateral element isoparametric element Chebyshev polynomial mapping method 

Notes

Acknowledgements

The authors would like to acknowledge the support by the National Fundamental Research Program of China (No. 2012CB026004).

References

  1. 1.
    Auteri F et al 2001 Incompressible Navier–Stokes solutions by a triangular spectral/p element projection method. Comput. Methods Appl. Mech. Eng. 190: 6927–6945CrossRefzbMATHGoogle Scholar
  2. 2.
    Bécache E et al 2010 High-order absorbing boundary conditions for anisotropic and convective wave equations. J. Comput. Phys. 229(4): 1099–1129MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chuang J J and Yang D C H 2004 A boundary-blending method for the parametrization of 2D surfaces with highly irregular boundaries. J. Mech. Des. 126(2): 327–335CrossRefGoogle Scholar
  4. 4.
    Du J and Fogelson A L 2011 A Cartesian grid method for two-phase gel dynamics on an irregular domain. Int. J. Numer. Methods Fluids 67(12): 1799–1817MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Giraldo F X and Taylor M A 2006 A diagonal mass matrix triangular spectral element method based on cubature points. J. Eng. Math. 56(3): 307–322MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Giraldo F X and Warburton T 2005 A nodal triangle-based spectral element method for the shallow water equations on the sphere. J. Comput. Phys. 207(1): 129–150MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Goldstein M E 1974 Unified approach to aerodynamic sound generation in the presence of sound. J. Acoust. Soc. Am. 56(5): 499–509Google Scholar
  8. 8.
    Kim J W and Lee D J 2000 Fourth computational aeroacoustics (CAA) workshop on benchmark problems. In: Proceedings of the NASA Conference, ClevelandGoogle Scholar
  9. 9.
    Kosec G 2016 A local numerical solution of a fluid-flow problem on an irregular domain. Adv. Eng. Softw. 1–9Google Scholar
  10. 10.
    Kumar P and Ik K K 2015 Hydrodynamic modeling of moored ship motion in an irregular domain. Procedia Eng. 127: 598–604CrossRefGoogle Scholar
  11. 11.
    Mengaldo G et al 2015 Dealiasing techniques for high-order spectral element methods on regular and irregular grids. J. Comput. Phys. 299: 56–81MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mercerat E D et al 2006 Triangular spectral element simulation of two-dimensional elastic wave propagation using unstructured triangular grids. Geophys. J. Int. 166(2): 679–698CrossRefGoogle Scholar
  13. 13.
    Moxey D et al 2015 An isoparametric approach to high-order curvilinear boundary-layer meshing. Comput. Methods Appl. Mech. Eng. 283: 636–650MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pasquetti R 2016 Comparison of some isoparametric mappings for curved triangular spectral elements. J. Comput. Phys. 316: 573–577MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pasquetti R and Rapetti F 2004 Spectral element methods on triangles and quadrilaterals: comparisons and applications. J. Comput. Phys. 198(1): 349–362CrossRefzbMATHGoogle Scholar
  16. 16.
    Patera A T 1984 A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54: 468–488CrossRefzbMATHGoogle Scholar
  17. 17.
    Pontaza J P 2007 A spectral element least-squares formulation for incompressible Navier–Stokes flows using triangular nodal elements. J. Comput. Phys. 221(2): 649–665MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pozrikidis C 2014 The finite element method in two dimensions. In: Introduction to Finite and Spectral Element Methods Using MATLAB. Hoboken: CRC PressGoogle Scholar
  19. 19.
    Shao W et al 2012 Chebyshev tau meshless method based on the integration–differentiation for Biharmonic-type equations on irregular domain. Eng. Anal. Boundary Elem. 36(12): 1787–1798MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tam C K W and Hardin J C 1996 Second computational aeroacoustic (CAA) workshop on benchmark problems. In: Proceedings of the NASA Conference, TallahasseeGoogle Scholar
  21. 21.
    Theillard M et al 2013 A second-order sharp numerical method for solving the linear elasticity equations on irregular domains and adaptive grids—application to shape optimization. J. Comput. Phys. 233: 430–448MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Vosse F N V D and Minev P D 1996 Spectral element methods: theory and applications. Eindhoven: Eindhoven University of TechnologyGoogle Scholar
  23. 23.
    Wan M et al 2005 Numerical prediction of static form errors in peripheral milling of thin-walled workpieces with irregular meshes. J. Manuf. Sci. Eng. 127(1): 13–22CrossRefGoogle Scholar
  24. 24.
    Wu X and Han G 2011 Direct expansion method of boundary condition for solving 3D elliptic equations with small parameters in the irregular domain. Comput. Math. Appl. 61(10): 2971–2980MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang R X et al 2009 Spectral elements method for acoustic propagation problems based on linearized euler equations. J. Comput. Acoust. 17(4): 383–402CrossRefzbMATHGoogle Scholar
  26. 26.
    Zhu C Y et al 2011 Implicit Chebyshev spectral element method for acoustics wave equations. Finite Elem. Anal. Des. 47(2): 184–194MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  • Yan Hui Geng
    • 1
  • Guo Liang Qin
    • 1
  • Jia Zhong Zhang
    • 1
  1. 1.School of Energy and Power EngineeringXi’an Jiaotong UniversityXi’anChina

Personalised recommendations