The solution of elastostatic and elastodynamic problems with Chebyshev spectral finite elements. (English) Zbl 0963.74059
Summary: Chebyshev spectral finite elements are applied to the solution of elastostatic and elastodynamic problems in two dimensions. The accuracy of the spectral approach is judged by examining the dispersion of solutions. It is shown that the spectral approach can achieve nearly zero dispersion for a wide range of spatial and temporal discretizations. Even coarse mesh solutions from the explicit lumped capacitance approach demonstrate exceptional accuracy. We also include a brief description of the development of Chebyshev spectral elements.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74S25 | Spectral and related methods applied to problems in solid mechanics |
| 74H15 | Numerical approximation of solutions of dynamical problems in solid mechanics |
| 74G15 | Numerical approximation of solutions of equilibrium problems in solid mechanics |
Keywords:
solution dispersion; dynamic stress intensity factor; Chebyshev spectral finite elements; coarse mesh solutionsReferences:
| [1] | W. Dauksher A.F. Emery, The use of spectral methods in predicting the reflection and transmission of ultrasonic signals through flaws, in: D.O. Thompson, D.E. Chimenti, (Eds.), Review of Progress in Quantitative Nondestructive Evaluation 15, Center for NDE, University of Iowa, 1996; W. Dauksher A.F. Emery, The use of spectral methods in predicting the reflection and transmission of ultrasonic signals through flaws, in: D.O. Thompson, D.E. Chimenti, (Eds.), Review of Progress in Quantitative Nondestructive Evaluation 15, Center for NDE, University of Iowa, 1996 |
| [2] | Dauksher, W.; Emery, A. F., Accuracy in modeling the acoustic wave equation with Chebyshev spectral finite elements, Fin. El. Aanl. Des., 26, 115-128 (1997) · Zbl 0896.65067 |
| [3] | J.N. Reddy, An Introduction to the Finite Element Method, 2nd ed., McGraw-Hill, New York, 1993; J.N. Reddy, An Introduction to the Finite Element Method, 2nd ed., McGraw-Hill, New York, 1993 |
| [4] | D. Gottlieb, S.A. Orszag, Numerical analysis of spectral methods: Theory and Applications, CBMS Regional Conference Series in Applied Mathematics 26, SIAM, Philadelphia, 1977; D. Gottlieb, S.A. Orszag, Numerical analysis of spectral methods: Theory and Applications, CBMS Regional Conference Series in Applied Mathematics 26, SIAM, Philadelphia, 1977 · Zbl 0412.65058 |
| [5] | Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods in Fluid Dynamics (1988), Springer: Springer Berlin · Zbl 0658.76001 |
| [6] | Orszag, S. A., Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech., 50, 689-703 (1971) · Zbl 0237.76027 |
| [7] | Frankel, J. I., A Galerkin solution to a regularized Cauchy singular integro-differential equation, Quart. App. Math, 53, 245-258 (1995) · Zbl 0823.65145 |
| [8] | Kaya, A. C.; Erdogan, F., On the solution of integral equations with strongly singular kernals, Quart. App. Math, 45, 105-122 (1987) · Zbl 0631.65139 |
| [9] | Boyd, J. P., Chebyshev and Fourier Spectral Methods (1989), Springer: Springer Berlin · Zbl 0681.65079 |
| [10] | Dauksher, W.; Emery, A. F., An evaluation of the cost effectiveness of Chebyshev spectral and p-finite element solutions to the scalar wave equation, Int. J. Num. Meth., 45, 1099-1113 (1999) · Zbl 0942.74070 |
| [11] | Patera, A. T., A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. Comp. Phys., 54, 468-488 (1984) · Zbl 0535.76035 |
| [12] | Bathe, K. J., Finite Element Procedures in Engineering Analysis (1982), Prentice-Hall: Prentice-Hall Englewood Cliffs · Zbl 0528.65053 |
| [13] | Hallquist, J. O., LS-DYNA3D Theoretical Manual (1993), Livermore Software Technology Corporation: Livermore Software Technology Corporation Livermore |
| [14] | Marfurt, K. J., Accuracy of finite difference and finite element modeling of the scalar and elastic wave equations, Geophys., 49, 533-549 (1984) |
| [15] | K. Aki, P.G. Richards, Quantitative Seismology, vol. 2, W.H. Freeman, San Francisco, 1980; K. Aki, P.G. Richards, Quantitative Seismology, vol. 2, W.H. Freeman, San Francisco, 1980 |
| [16] | Alford, R. M.; Kelly, K. R.; Boore, D. M., Accuracy of finite difference modeling of the acoustic wave equation, Geophys, 6, 834-842 (1974) |
| [17] | Krieg, R. D.; Key, S. W., Transient shell response by numerical time integration, Int. J. Num. Meth. Eng., 7, 273-286 (1973) |
| [18] | Hilber, H. M.; Hughes, T. J.R.; Taylor, R. M., Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Eng. Struc. Dyn., 5, 283-292 (1977) |
| [19] | A. Bamberger, J.C. Guillot, P. Joly, Numerical Diffraction by an Uniform Grid, Report No. 472, Institut National de Recherche en Informatique et en Automatique, 1986; A. Bamberger, J.C. Guillot, P. Joly, Numerical Diffraction by an Uniform Grid, Report No. 472, Institut National de Recherche en Informatique et en Automatique, 1986 · Zbl 0659.65079 |
| [20] | A. Bamberger, G. Chavent, P. Lailly, Etude de Schemas Numeriques pour les Equations de L’Elastodynamique Lineaire, Report No. 41, Institut National de Recherche en Informatique et en Automatique, 1980; A. Bamberger, G. Chavent, P. Lailly, Etude de Schemas Numeriques pour les Equations de L’Elastodynamique Lineaire, Report No. 41, Institut National de Recherche en Informatique et en Automatique, 1980 |
| [21] | Dablain, M. A., The application of higher order differencing to the scalar wave equation, Geophys, 51, 54-66 (1986) |
| [22] | Jensen, M. S., High convergence order finite elements with lumped mass matrix, Int. J. Num. Meth. Eng., 39, 1879-1888 (1996) · Zbl 0880.76044 |
| [23] | Johnson, E., Dispersion properties of simple 3D finite elements, Comm. Num. Meth. Eng., 11, 97-103 (1995) · Zbl 0815.73057 |
| [24] | Trefethen, L. N., Group velocity and finite difference schemes, SIAM Review, 24, 2, 113-136 (1982) · Zbl 0487.65055 |
| [25] | R. Vichnevetsky, Fourier methods in computational fluid and field dynamics, in: J. Burger, Y. Jarney (Eds.), Simulation in Engineering Sciences, Elsevier, Holland, 1983; R. Vichnevetsky, Fourier methods in computational fluid and field dynamics, in: J. Burger, Y. Jarney (Eds.), Simulation in Engineering Sciences, Elsevier, Holland, 1983 |
| [26] | Warren, G. S.; Scott, W. R., Numerical dispersion in the vector finite element method using first order quadrilateral elements, IEEE Transactions on Antenna and Propagation, 42, 11, 1502-1508 (1994) · Zbl 0953.78504 |
| [27] | Belytschko, T.; Mullen, R., On dispersive properties of finite element solutions, (Miklowitz, J.; Achenbach, J. D., Modern Problems in Elastic Wave Propagation (1978), Wiley: Wiley New York) · Zbl 0434.73074 |
| [28] | Mullen, R.; Belytschko, T., Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation, Int. J. Num. Meth. Eng., 18, 11-29 (1982) · Zbl 0466.73102 |
| [29] | Ihlenburg, F.; Babuska, I., Dispersion analysis and error estimation of Galerkin finite element methods for the helmholtz equation, Int. J. Num. Meth. Eng., 38, 3745-3774 (1995) · Zbl 0851.73062 |
| [30] | Chin, R. C.Y.; Hedstrom, G.; Thigpen, L., Numerical methods in seismology, J. Comp. Phys., 54, 18-56 (1984) · Zbl 0527.73102 |
| [31] | Kittel, C., Introduction to Solid State Physics (1956), J. Wiley and Sons: J. Wiley and Sons New York |
| [32] | MacNeal, R. M., Finite elements: their design and performance (1994), Marcel Dekker: Marcel Dekker New York |
| [33] | Peterson, R. E., Stress Concentration Factors (1974), Wiley: Wiley New York |
| [34] | E.P. Chen, G.C. Sih, Transient response of cracks to impact loads in: G.C. Sih (Ed.), Mechanics of Fracture, vol. 4, Elastodynamic Crack Problems, Noordhoff, Leyden, 1977; E.P. Chen, G.C. Sih, Transient response of cracks to impact loads in: G.C. Sih (Ed.), Mechanics of Fracture, vol. 4, Elastodynamic Crack Problems, Noordhoff, Leyden, 1977 |
| [35] | H. Tada, P.C. Paris, G.R. Irwin, The Stress Analysis of Cracks Handbook, Del Research, Hellertown, 1973; H. Tada, P.C. Paris, G.R. Irwin, The Stress Analysis of Cracks Handbook, Del Research, Hellertown, 1973 |
| [36] | Chen, Y. M., Numerical computation of dynamic stress intensity factors by a lagrangian finite difference method (the hemp code), Eng. Fracture Mech., 7, 653-660 (1975) |
| [37] | Rice, J. R., A path independent integral and the approximate analysis of strain concentration by notches and cracks, Jour. Appl. Mech., 379-386 (1968) |
| [38] | A.S. Kobayashi, University of Washington, personal communication; A.S. Kobayashi, University of Washington, personal communication |
| [39] | J.A. Aberson, J.M. Anderson, W.W. King, Dynamic analysis of cracked structures using singularity finite elements in: G.C. Sih (Ed.), Mechanics of Fracture, vol. 4, Elastodynamic Crack Problems, Noordhoff, Leyden, 1977; J.A. Aberson, J.M. Anderson, W.W. King, Dynamic analysis of cracked structures using singularity finite elements in: G.C. Sih (Ed.), Mechanics of Fracture, vol. 4, Elastodynamic Crack Problems, Noordhoff, Leyden, 1977 |
| [40] | Tan, M.; Meguid, S. A., Dynamic analysis of cracks perpendicular to bimaterial interfaces using a new singular finite element, Fin. El. Anal. Des., 22, 69-83 (1996) · Zbl 0875.73180 |
| [41] | ANSYS User’s Manual for Revision 5.0, Swanson Analysis Systems, Houston, 1992; ANSYS User’s Manual for Revision 5.0, Swanson Analysis Systems, Houston, 1992 |
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