Least-squares finite element approximations to the Timoshenko beam problem. (English) Zbl 1112.74519
Summary: A least-squares finite element method for the Timoshenko beam problem is proposed and analyzed. The method is shown to be convergent and stable without requiring extra smoothness of the exact solutions. For sufficiently regular exact solutions, the method achieves optimal order of convergence in the \(H^1\)-norm for all the unknowns (displacement, rotation, shear, moment), uniformly in the small parameter which is generally proportional to the ratio of thickness to length. Thus the locking phenomenon disappears as the parameter tends to zero. A sharp a posteriori error estimator which is exact in the energy norm and equivalent in the \(H^1\)-norm is also briefly discussed.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
Keywords:
Timoshenko beam problem; Least-squares; Finite element method; Locking phenomenon; A posteriori error estimatorReferences:
| [1] | Ainsworth, M.; Oden, J. T., A unified approach to a posteriori error estimation using element residual methods, Numer. Math., 65, 23-50 (1993) · Zbl 0797.65080 |
| [2] | Arnold, D. N., Discretization by finite elements of a model parameter dependent problem, Numer. Math., 37, 405-421 (1981) · Zbl 0446.73066 |
| [3] | Babuška, I.; Rheinboldt, W. C., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15, 736-754 (1978) · Zbl 0398.65069 |
| [4] | Babuška, I.; Rheinboldt, W. C., A posteriori error analysis of finite element solutions for one-dimensional problems, SIAM J. Numer. Anal., 18, 565-589 (1981) · Zbl 0487.65060 |
| [5] | Babuška, I.; Suri, M., On locking and robustness in the finite element method, SIAM J. Numer. Anal., 29, 1261-1293 (1992) · Zbl 0763.65085 |
| [6] | Bank, R. E.; Smith, R. K., A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal., 30, 921-935 (1993) · Zbl 0787.65078 |
| [7] | Bank, R. E.; Weiser, A., Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44, 283-301 (1985) · Zbl 0569.65079 |
| [8] | Bedivan, D. M.; Fix, G. J., Least squares methods for optimal shape design problems, Comput. Math. Appl., 30, 17-25 (1995) · Zbl 0838.65125 |
| [9] | Bochev, P. B.; Gunzburger, M. D., Analysis of least squares finite element methods for the Stokes equations, Math. Comp., 63, 479-506 (1994) · Zbl 0816.65082 |
| [10] | Bornemann, F. A.; Erdmann, B.; Kornhuber, R., A posteriori error estimates for elliptic problems in two and three space dimensions, SIAM J. Numer. Anal., 33, 1188-1204 (1996) · Zbl 0863.65069 |
| [11] | S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 1994; S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 1994 · Zbl 0804.65101 |
| [12] | Cai, Z.; Manteuffel, T. A.; McCormick, S. F., First-order system least squares for velocity-vorticity-pressure form of the Stokes equations, with application to linear elasticity, Electronic Trans. Numer. Anal., 3, 150-159 (1995) · Zbl 0856.76010 |
| [13] | Cai, Z.; Manteuffel, T. A.; McCormick, S. F., First-order system least squares for the Stokes equations, with application to linear elasticity, SIAM J. Numer. Anal., 34, 1727-1741 (1997) · Zbl 0901.76052 |
| [14] | Cai, Z.; Manteuffel, T. A.; McCormick, S. F.; Parter, S., First-order system least squares (FOSLS) for planar linear elasticity: pure traction, SIAM J. Numer. Anal., 35, 320-335 (1998) · Zbl 0968.74061 |
| [15] | Chang, C. L., Finite element method for the solution of Maxwell’s equations in multiple media, Appl. Math. Comput., 25, 89-99 (1988) · Zbl 0635.65129 |
| [16] | Chang, C. L.; Jiang, B.-N., An error analysis of least squares finite element method of velocity-pressure-vorticity formulation for Stokes problem, Comput. Meth. Appl. Mech. Eng., 84, 247-255 (1990) · Zbl 0733.76042 |
| [17] | Chang, C. L.; Yang, S.-Y.; Hsu, C.-H., A least-squares finite element method for incompressible flow in stress-velocity-pressure version, Comput. Meth. Appl. Mech. Eng., 128, 1-9 (1995) · Zbl 0866.76043 |
| [18] | Chang, C. L.; Nelson, J. J., Least-squares finite element method for the Stokes problem with zero residual of mass conservation, SIAM J. Numer. Anal., 34, 480-489 (1997) · Zbl 0890.76036 |
| [19] | Cheng, X.-L.; Han, W.; Huang, H.-C., Finite element methods for Timoshenko beam, circular arch and Reissner-Mindlin plate problems, J. Comput. Appl. Math., 79, 215-234 (1997) · Zbl 0878.73068 |
| [20] | P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978; P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978 · Zbl 0383.65058 |
| [21] | J.M. Fiard, T.A. Manteuffel, S.F. McCormick, First-order system least squares (FOSLS) for convection-diffusion problems: numerical results, preprint, October 1996; J.M. Fiard, T.A. Manteuffel, S.F. McCormick, First-order system least squares (FOSLS) for convection-diffusion problems: numerical results, preprint, October 1996 · Zbl 0911.65108 |
| [22] | Fix, G. J.; Rose, M. E., A comparative study of finite element and finite difference methods for Cauchy-Riemann type equations, SIAM J. Numer. Anal., 22, 250-261 (1985) · Zbl 0569.65077 |
| [23] | Jiang, B.-N.; Carey, G. F., Adaptive refinement for least-squares finite elements with element-by-element conjugate gradient solution, Int. J. Numer. Meth. Eng., 24, 569-580 (1987) · Zbl 0624.65113 |
| [24] | Jiang, B.-N.; Wu, J.; Povinelli, L. A., The origin of spurious solutions in computational electromagnetics, J. Comput. Phys., 125, 104-123 (1996) · Zbl 0848.65086 |
| [25] | Li, L., Discretization of the Timoshenko beam problem by the \(p\) and the \(h\)−\(p\) versions of the finite element method, Numer. Math., 57, 413-420 (1990) · Zbl 0683.73041 |
| [26] | Loula, A. F.D.; Hughes, T. J.R.; Franca, L. P., Petrov-Galerkin formulations of the Timoshenko beam problem, Comput. Meth. Appl. Mech. Eng., 63, 115-132 (1987) · Zbl 0645.73030 |
| [27] | Loula, A. F.D.; Hughes, T. J.R.; Franca, L. P.; Miranda, I., Mixed Petrov-Galerkin methods for the Timoshenko beam problem, Comput. Meth. Appl. Mech. Eng., 63, 133-154 (1987) · Zbl 0607.73076 |
| [28] | J.-L. Liu, Exact a posteriori error analysis of the least squares finite element method, Appl. Math. Comput., to appear; J.-L. Liu, Exact a posteriori error analysis of the least squares finite element method, Appl. Math. Comput., to appear · Zbl 1023.65117 |
| [29] | Yang, S.-Y.; Liu, J.-L., Least squares finite element methods for the elasticity problem, J. Comput. Appl. Math., 87, 39-60 (1997) · Zbl 0902.73076 |
| [30] | Yang, S.-Y.; Chang, C. L., Analysis of a two-stage least squares finite element method for the planar elasticity problem, Math. Meth. Appl. Sci., 22, 713-732 (1999) · Zbl 0930.74068 |
| [31] | Zienkiewicz, O. C.; Zhu, J. Z., A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Meth. Eng., 24, 337-357 (1987) · Zbl 0602.73063 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
