Two Dimensional Complex Wavenumber Dispersion Analysis using B-Spline Finite Elements Method
Authors
Abstract
Grid dispersion is one of the criteria of validating the finite element method (FEM) in simulating acoustic or elastic wave propagation. The difficulty usually arisen when using this method for simulation of wave propagation problems, roots in the discontinuous field which causes the magnitude and the direction of the wave speed vector, to vary from one element to the adjacent one. To solve this problem and improve the response accuracy, two approaches are usually suggested: changing the integration method and changing shape functions. The Finite Element iso-geometric analysis (IGA) is used in this research. In the IGA, the B-spline or non-uniform rational B-spline (NURBS) functions are used which improve the response accuracy, especially in one-dimensional structural dynamics problems. At the boundary of two adjacent elements, the degree of continuity of the shape functions used in IGA can be higher than zero. In this research, for the first time, a two dimensional grid dispersion analysis has been used for wave propagation in plane strain problems using B-spline FEM is presented. Results indicate that, for the same degree of freedom, the grid dispersion of B-spline FEM is about half of the grid dispersion of the classic FEM.
Keywords
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17. Cottrell, J. A., Hughes, T. J., and Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons, 2009.
18. Kolman, R., Plešek, J., Okrouhlík, M., and Gabriel, D., “Dispersion Errors of B-spline Based Finite Element Method in One-Dimensional Elastic Wave Propagation”, The 3rd International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering. Papadrakakis M. et al, pp. 1-12, 2011.
19. Aki, K., and Richards, P. G., Quantitative Seismology, University Science Books, Sausalito, USA, 2002.
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2. Duczek, S., Joulaian, M., Düster, A., and Gabbert, U., “Numerical Analysis of Lamb Waves Using the Finite and Spectral Cell Methods”, International Journal for Numerical Methods in Engineering, No. 1, pp. 26-53, 2014.
3. Chakrabarti, P., and Chopra, A. K., “Earthquake Analysis of Gravity Dams Including Hydrodynamic Interaction”, Earthquake Engineering & Structural Dynamics, Vol. 2, pp. 143-160, 1973.
4. Lysmer, J., and Drake, L. A., “A Finite Element Method for Seismology”, Methods in Computational Physics, Vol. 11, pp. 181-216, 1972.
5. Smith, W. D., “The Application of Finite Element Analysis to Body Wave Propagation Problems”, Geophysical Journal International, Vol. 42, pp. 747-768, 1975.
6. Mullen, R., and Belytschko, T. “Dispersion Analysisof Finite Element Semidiscretizations of the Two Dimensional Wave Equation”, International Journal for Numerical Methods in Engineering, Vol. 18, pp. 11-29, 1982.
7. Marfurt, K. J. “Accuracy of Finite-Difference and Finite-Element Modeling of the Scalar and Elasticwave Equations”, Geophysics, Vol. 49, pp. 533-549, 1984.
8. Seriani, G., and Priolo, E., “Spectral Element Method for Acoustic Wave Simulation in Heterogeneous Media”, Finite Elements in Analysis and Design, Vol. 16, pp. 337-348, 1994.
9. DeBasabe Delgado, J. d. D., “High-order finite element methods for seismic wave propagation”, Ph.D. Dissertation, The University of Texas at Austin, 2009.
10. Cohen, G., Joly, P., and Tordjman, N., “Higher-Order Finite Elements with Mass-Lumping for the 1D Wave Equation”, Finite Elements in Analysis and Design, Vol. 16. pp. 329-336, 1994.
11. Komatitsch, D., Ritsema, J., and Tromp, J., “The Spectral-Element Method, Beowulf Computing, and Global Seismology” Science, Vol. 298, pp. 1737-1742, 2002.
12. Chaljub, E., Komatitsch, D., Vilotte, J. P., Capdeville, Y., Valette, B., and Festa, G., “Spectral-Element Analysis in Seismology”, Advances in Geophysics, Vol. 48, pp. 365-419, 2007.
13. Düster, A., Demkowicz, L., and Rank, E., “High Order Finite Elements Applied to the Discrete Boltzmann Equation”, International Journal for Numerical Methods in Engineering, Vol. 67, pp. 1094-1121, 2006.
14. Hughes, T. J., Cottrell, J. A., and Bazilevs, Y “Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement”, Computer Methods in Applied Mechanics and Engineering, Vol. 194, pp. 4135-4195, 2005.
15. Cottrell, J., Reali, A., Bazilevs, Y., and Hughes, T. “Isogeometric Analysis of Structural Vibrations”, Computer Methods in Applied Mechanics and Engineering, Vol. 195, pp. 5257-5296, 2006.
16. Hughes, T. J., Reali, G. A., and Sangalli, G, “Duality and Unified Analysis of Discrete Approximations in Structural Dynamics and Wave Propagation: Comparison of p-Method Finite Elements with k-Method NURBS”, Computer methods in Applied Mechanics and Engineering, Vol. 197, pp. 4104-4124, 2008.
17. Cottrell, J. A., Hughes, T. J., and Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons, 2009.
18. Kolman, R., Plešek, J., Okrouhlík, M., and Gabriel, D., “Dispersion Errors of B-spline Based Finite Element Method in One-Dimensional Elastic Wave Propagation”, The 3rd International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering. Papadrakakis M. et al, pp. 1-12, 2011.
19. Aki, K., and Richards, P. G., Quantitative Seismology, University Science Books, Sausalito, USA, 2002.
20. Daryabor, P., Farzin, M. and Honarvar, F. “Calculating the Lamb Wave Modes in an Aluminum Sheet Bonded to a Composite Layer with FEM and Experiment”, Modares Mechanical Engineering, Vol. 13, No. 11, pp. 95-106, 2013.
21. Achenbach, J., Wave Propagation in Elastic Solids., Elsevier Science Ltd, 1984.
22. Willberg, C., Duczek, S., Perez, J. V., Schmicker, D., and Gabbert, U. “Comparison of Different Higher Order Finite Element Schemes for the Simulation of Lamb Waves”, Computer Methods in Applied Mechanics and Engineering, Vol. 241, pp. 261-246, 2012.
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