New improved hourglass control for bilinear and trilinear elements in anisotropic linear elasticity. (English) Zbl 0621.73104
One-point reduced integration method is studied for 4-node quadrilateral and 8-node brick elements together with correction terms of the numerical integration rule for selective and directional reduced integration schemes for anisotropic linear elasticity. These correction terms were previously called hourglass control to the reduced integration method by Belytschko and others [T. Belytschko, J. S-J. Ong, W. K. Liu and J. Kennedy, ibid. 43, 251-276 (1984; Zbl 0522.73063; Zbl 0532.73074)]. In the present work the idea of existing hourglass control is carefully examined for its convergence and accuracy, and is extended to include both selective and directional reduced integration methods.
MSC:
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 74E10 | Anisotropy in solid mechanics |
| 65K10 | Numerical optimization and variational techniques |
Keywords:
instabilities; hourglassing; selective reduced integration; two dimensions; three dimensions; One-point reduced integration method; 4- node quadrilateral; 8-node brick elements; hourglass control; convergence; accuracy; directional reduced integration methodsReferences:
| [1] | Maechnen, G.; Sack, S., The tensor code, (Alder, B.; etal., Methods of Computational Physics (1964), Academic Press: Academic Press New York), 181-260 |
| [2] | Key, S. W., HONDO—A finite element computer program for the large deformation dynamic response of axisymmetric solids, R.74-0039 (1974), Sandia National Laboratories: Sandia National Laboratories Albuquerque, NM |
| [3] | Flanagan, D. P.; Belytschko, T., A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, Internat. J. Numer. Meths. Engrg., 17, 696-706 (1981) · Zbl 0478.73049 |
| [4] | Goudreau, G. L.; Hallquist, J. O., Recent developments in large-scale finite element Lagrangian hydrocode technology, Comput. Meths. Appl. Mech. Engrg., 33, 725-757 (1982) · Zbl 0493.73072 |
| [5] | Hallquist, J. O., (Theoretical manual for DYNA3D, UCID-19401 Lawrence Livermore National Lab (1983), University of California), 19-20 |
| [6] | Belytschko, T.; Ong, J. S.; Liu, W. K.; Kennedy, J. D., Hourglass control in linear and nonlinear problems, Comput. Meths. Appl. Mech. Engrg., 43, 251-276 (1984) · Zbl 0522.73063 |
| [7] | Jacquotte, O. P.; Oden, J. T., Analysis of hourglass instabilities and control in underintegrated finite element methods, Comput. Meths. Appl. Mech. Engrg., 44, 339-363 (1984) · Zbl 0543.73104 |
| [8] | Jacquotte, O. P., Stability, accuracy, and efficiency of some underintegrated methods in finite element computations, Comput. Meths. Appl. Mech. Engrg., 50, 275-293 (1985) · Zbl 0579.65110 |
| [9] | Jacquotte, O. P.; Oden, J. T.; Becker, E. B., Numerical control of the hourglass instability, Internat. J. Numer. Meths. Engrg., 22, 219-228 (1986) · Zbl 0593.73084 |
| [10] | Schulz, J. C., Finite element hourglassing control, Internat. J. Numer. Meths. Engrg., 21, 1039-1048 (1985) · Zbl 0566.73066 |
| [11] | Liu, W. K.; Ong, J. S.; Uras, R. A., Finite element stabilization matrices — a unification approach, Comput. Meths. Appl. Mech. Engrg., 53, 13-46 (1985) · Zbl 0553.73065 |
| [12] | Belytschko, T.; Bachrach, W. E., Simple quadrilaterals with high coarse mesh accuracy, (AMD-73 (1985), ASME: ASME New York), 39-56 · Zbl 0623.73074 |
| [13] | Bachrach, W. E.; Liu, W. K.; Uras, R. A., Consolidation of various approaches in developing naturally based quadrilateral, Comput. Meths. Appl. Mech. Engrg., 55, 43-62 (1986) · Zbl 0571.73077 |
| [14] | Cook, R. D.; Feng, Z. H., Control of spurious mode in the nine node quadrilateral element, Internat. J. Numer. Meths. Engrg., 18, 1576-1580 (1982) · Zbl 0489.73075 |
| [15] | Kikuchi, N., Finite Element Methods in Mechanics (1986), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0587.73102 |
| [16] | Malkus, D. M.; Hughes, T. J.R., Mixed finite element methods—reduced and selective integration techniques: a unification concepts, Comput. Meths. Appl. Mech. Engrg., 15, 63-82 (1978) · Zbl 0381.73075 |
| [17] | Hughes, T. J.R., Equivalence of finite elements for nearly incompressible material, J. Appl. Mech., 44, 181-183 (1977) |
| [18] | Hughes, T. J.R., Generalization of selective reduced integration procedures to anisotropic nonlinear media, Internal. J. Numer. Meths. Engrg, 15, 1413-1418 (1980) · Zbl 0437.73053 |
| [19] | Oden, J. T.; Kikuchi, N., Finite element methods for constrained problems in elasticity, Internat. J. Numer. Meths. Engrg., 18, 701-725 (1982) · Zbl 0486.73068 |
| [20] | Taylor, R. L.; Beresford, P. J.; Wilson, E. L., A nonconforming element for stress analysis, Internat. J. Numer. Meths. Engrg., 18, 1211-1219 (1976) · Zbl 0338.73041 |
| [21] | Zienkiewicz, O. C.; Taylor, R. L.; Too, J. M., Reduced integration technique in general analysis of plates and shells, Internat. J. Numer. Meths. Engrg., 18, 275-290 (1971) · Zbl 0253.73048 |
| [22] | Pawsey, S. T.; Clough, R. W., Improved numerical integration of thick shell finite elements, Internat. J. Numer. Meths. Engrg., 3, 565-586 (1971) · Zbl 0248.73035 |
| [23] | Takemoto, H.; Cook, R. D., Some modifications of an isoparametric element, Internat. J. Numer. Meths. Engrg., 3, 401-405 (1973) · Zbl 0264.73102 |
| [24] | Hughes, T. J.R.; Taylor, R. L.; Kanoknukulchai, W., A simple and efficient finite element for plate bending, Internat. J. Numer. Meths. Engrg., 11, 1529-1543 (1977) · Zbl 0363.73067 |
| [25] | Liu, W. K.; Belytschko, T.; Ong, J. S.; Law, S. E., Use of stabilization matrices in non-linear analysis, Engrg. Comp., 2, 47-55 (1985) |
| [26] | Ashwell, D. G.; Sabir, A. B., A new cylindrical shell finite element based on simple independent strain functions, Internat. J. Mech. Sci., 44, 171-183 (1972) · Zbl 0231.73033 |
| [27] | Timoshenko, S. P.; Goodier, J. N., The Theory of Elasticity (1970), Mcgraw-Hil: Mcgraw-Hil New York · Zbl 0266.73008 |
| [28] | Timoshenko, S. P.; Krieger, S. Woinowsky, Theory of Plates and Shells (1959), Mcgraw-Hill: Mcgraw-Hill New York · Zbl 0114.40801 |
| [29] | Zienkiewicz, O. C., The Finite Element Method (1977), Mcgraw-Hill: Mcgraw-Hill New York · Zbl 0435.73072 |
| [30] | Jones, R. M., Mechanics of Composite Materials (1975), Mcgraw-Hill: Mcgraw-Hill New York |
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