Generalized mixed variational principles and solutions of ill-conditioned problems in computational mechanics: I: Volumetric locking. (English) Zbl 1054.74737
Summary: Although the finite element method (FEM) has been extensively applied to various areas of engineering, the ill-conditioned problems occurring in many situations are still thorny to deal with. This study attempts to provide a high-performing and simple approach to the solutions of ill-conditioned problems. The theoretical foundation of it is the parametrized variational principles, called the generalized mixed variational principles (GMVPs) initiated by Rong in 1981. GMVPs can solve many kinds of ill-conditioned problems in computational mechanics. Among them, four cases are investigated in detail: the volumetric locking, the shear locking, the inhomogeneousness and the membrane locking problems, composing four parts of the study, Part I–IV, respectively. This paper is Part I, wherein a GMVP specially suited to the nearly incompressible and incompressible materials is constructed. A derived condition to overcome the rank deficiency of the global matrix of FEM for the perfect incompressibility is put forward. Based on these, a new approach, named the order-one shooting method, is proposed to avoid the spurious checkerboard pressure modes produced by certain FEM formulations.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
Citations:
Zbl 1055.74563References:
| [1] | Rong, T. Y., A variational principle with split modulus in elasticity, J. Southwest Jiaotong Univ., 1, 48-56 (1981), (in Chinese) |
| [2] | T.Y. Rong, A variational principle with split stiffness in the mechanics of thin elastic shells, plates and bar systems, in: Collected Papers from 1981 Symposium on Scientific Research, Edited by the Department of Scientific Affairs, Southwest Jiaotong Univ., Chengdu, China, 1981, pp. 213-215. Also presented at the First National Conf. on Computing in Civil Engrg., Kunming, China, Dec. 1981 (in Chinese); T.Y. Rong, A variational principle with split stiffness in the mechanics of thin elastic shells, plates and bar systems, in: Collected Papers from 1981 Symposium on Scientific Research, Edited by the Department of Scientific Affairs, Southwest Jiaotong Univ., Chengdu, China, 1981, pp. 213-215. Also presented at the First National Conf. on Computing in Civil Engrg., Kunming, China, Dec. 1981 (in Chinese) |
| [3] | T.Y. Rong, Generalized mixed variational principles and finite element methods in elasticity, in: Proceedings of the First Conference on Computational Mechanics in Sichuan Province, vol. 1, Chongqing, China, 1983, pp. 1-13 (in chinese); T.Y. Rong, Generalized mixed variational principles and finite element methods in elasticity, in: Proceedings of the First Conference on Computational Mechanics in Sichuan Province, vol. 1, Chongqing, China, 1983, pp. 1-13 (in chinese) |
| [4] | Rong, T. Y., Generalized mixed variational principles and new FEM models in solid mechanics, Int. J. Solids Struct., 24, 1131-1140 (1988) · Zbl 0686.73050 |
| [5] | Rong, T. Y.; Lu, A. Q., Parametrized Lagrange multiplier method and proof of full independence of the three-field variables in Hu-Washizu variational principle and others, J. Southwest Jiaotong Univ., 1, 84-92 (1997), (in Chinese) |
| [6] | Rong, T. Y.; Lu, A. Q., Parameterized Lagrange multiplier method and construction of generalized mixed variational principles for computational mechanics, Comput. Methods Appl. Mech. Engrg., 164, 287-296 (1988) · Zbl 0963.74080 |
| [7] | Fraeijs De Veubeke, B., Displacement and equilibrium models in the finite element mehod, (Zienkiewicz, O. C.; Holister, G. S., Stress Analysis (1965), Wiley: Wiley New York) · Zbl 0359.76021 |
| [8] | J. Todd, The condition of certain matrices, I. Quart. J. Mech. Appl. Math. 2 (4) (1949) 469-472; J. Todd, The condition of certain matrices, I. Quart. J. Mech. Appl. Math. 2 (4) (1949) 469-472 · Zbl 0034.37602 |
| [9] | Rong, T. Y., Generalized mixed variational principle and FEM in elasticity, Adv. Mech., 2, 186-199 (1987), (in Chinese) |
| [10] | Rong, T. Y., Generalized mixed variational principles in elasticity and the finite element method, Acta Mech. Sinica, 2, 153-164 (1988), (in Chinese) |
| [11] | Felippa, C. A., Parameterized multifield variational principles in elasticity: I. Mixed functionals, Comm. Appl. Numer. Methods, 5, 79-88 (1989) · Zbl 0659.73015 |
| [12] | Felippa, C. A., A survey of parametrized variational principles and applications to computational mechanics, Comput. Methods Appl. Mech. Engrg., 113, 109-139 (1994) · Zbl 0848.73063 |
| [13] | Rong, T. Y., The finite element method with split elastic modulus in elasticity, J. Southwest Jiaotong Univ., 1, 56-63 (1981), (in Chinese) |
| [14] | T.Y. Rong, The matrix method with split stiffness for structural analysis, in: Collected Papers from 1981 Symposium on Scientific Research, Edited by the Department of Scientific Affairs, Southwest Jiaotong Univ., Chengdu, China, 1981, pp. 216-225. Also presented at the First National Conf. on Computing in Civil Engrg., Kunming, China, Dec. 1981 (in Chinese); T.Y. Rong, The matrix method with split stiffness for structural analysis, in: Collected Papers from 1981 Symposium on Scientific Research, Edited by the Department of Scientific Affairs, Southwest Jiaotong Univ., Chengdu, China, 1981, pp. 216-225. Also presented at the First National Conf. on Computing in Civil Engrg., Kunming, China, Dec. 1981 (in Chinese) |
| [15] | T.Y. Rong, A new generalized mixed FEM model and its application to fracture mechanics, in: Proceedings of the Second International Conference on Computing in Civil Engineering, June, 1985, Hangzhou, China, pp. 939-946; T.Y. Rong, A new generalized mixed FEM model and its application to fracture mechanics, in: Proceedings of the Second International Conference on Computing in Civil Engineering, June, 1985, Hangzhou, China, pp. 939-946 |
| [16] | Forsythe, G. E., Computer Methods for Mathematical Computations (1977), Printice-Hall: Printice-Hall Englewood Cliffs, NJ · Zbl 0361.65002 |
| [17] | Herrmann, L. R., Elasticity equations for incompressible and nearly incompressible materials by a variational theorem, AIAA J., 3, 1896-1900 (1965) |
| [18] | Key, S. W., A variational principle for incompressible and nearly incompressible anisotropic elasticity, Int. J. Solid. Struct., 15, 951-964 (1969) · Zbl 0175.22101 |
| [19] | Taylor, R. L.; Pister, K. S.; Herrmann, L. R., On a variational theorem for incompressible and nearly incompressible elasticity, Int. J. Solid. Struct., 4, 875-883 (1968) · Zbl 0167.52804 |
| [20] | Babuska, I., The finite element method with Lagrange multipliers, Numer. Math., 20, 179-192 (1973) · Zbl 0258.65108 |
| [21] | Argyris, J. H.; Dunne, P. C.; Angelopoulos, T.; Bichat, B., Large natural strains and some special difficulties due to non-linearity and incompressibility in finite elements, Comput. Methods Appl. Mech. Engrg., 4, 219-278 (1974) · Zbl 0284.73049 |
| [22] | Thompson, E. G., Average and complete incompressibility in finite element method, Int. J. Numer. Methods Engrg., 9, 925-932 (1975) |
| [23] | Babuska, I.; Oden, J. T.; Lee, J. K., Mixed-hybrid finite element approximation of second order elliptic boundary-value problems, Comput. Methods Appl. Mech. Engrg., 11, 175-206 (1977) · Zbl 0382.65056 |
| [24] | Scharhost, T.; Pian, T. H.H., Finite element analysis of rubber-like materials by a mixed model, Int. J. Numer. Methods Engrg., 12, 665-676 (1978) · Zbl 0372.73076 |
| [25] | Lee, R. L.; Gresho, P. M.; Sani, R. L., Smoothing techniques for certain primitive variable solutions of the Navier-Stokes equations, Int. J. Numer. Methods Engrg., 14, 1785-1804 (1979) · Zbl 0426.76035 |
| [26] | Sani, R. L.; Gresho, P. M.; Lee, R. L.; Griffiths, D. F., The cause and cure(?) of the spurious pressures generated by certain FEM solutions of the incompressible Navier-Stokes equations: Part 1, Int. J. Numer. Methods Fluids, 1, 17-43 (1981) · Zbl 0461.76021 |
| [27] | Sani, R. L.; Gresho, P. M.; Lee, R. L.; Griffiths, D. F.; Engelman, M., The cause and cure(!) of the spurious pressures generated by certain FEM solutions of the incompressible Navier-Stokes equations: Part 2, Int. J. Numer. Methods Fluids, 1, 171-204 (1981) · Zbl 0461.76022 |
| [28] | Häggbland, B.; Sundberg, J. A., Large strain solutions of rubber components, Comput. Struct., 17, 835-843 (1983) |
| [29] | Simo, J. C.; Taylor, R. L.; Pister, K. S., Variational and projection methods for the volume constraint in finite deformation elasto-plasticity, Comput. Methods Appl. Mech. Engrg., 51, 177-208 (1985) · Zbl 0554.73036 |
| [30] | Belytschko, T.; Bachrach, W. E., Efficient implementation of quadrilaterals with high coarse-mesh accuracy, Comput. Methods Appl. Mech. Engrg., 54, 279-301 (1986) · Zbl 0579.73075 |
| [31] | Rice, J. G.; Schnipke, R. J., An equal-order velocity-pressure formulation that does not exhibit spurious pressure modes, Comput. Methods Appl. Mech. Engrg., 58, 135-149 (1986) · Zbl 0639.76030 |
| [32] | Sussman, T. S.; Bathe, K. J., A finite element formulation for nonlinear impressible elastic and inelastic analysis, Comput. Struct., 26, 357-409 (1987) · Zbl 0609.73073 |
| [33] | Chang, T. Y.P.; Saleeb, A. F.; Li, G., Large strain analysis of rubber-like materials based on a perturbed Lagrangian variational principle, Comput. Mech., 8, 221-233 (1991) · Zbl 0850.73281 |
| [34] | Chen, J. S.; Pan, C.; Chang, T. Y.P., On the control of pressure oscillation in bilinear-displacement constant-pressure element, Comput. Methods Appl. Mech. Engrg., 128, 137-152 (1995) · Zbl 0860.73065 |
| [35] | Chen, J. S.; Han, W.; Wu, C. T.; Duan, W., On the perturbed Lagrange formulation for nearly incompressible and incompressible hyperelasticity, Comput. Methods Appl. Mech. Engrg., 142, 335-351 (1997) · Zbl 0892.73057 |
| [36] | Braess, D., Enhanced assumed strain elements and locking in membrane problems, Comput. Methods Appl. Mech. Engrg., 165, 155-174 (1998) · Zbl 0949.74065 |
| [37] | Canga, M. E.; Becker, E. B., An iterative technique for finite element analysis of near-incompressible materials, Comput. Methods Appl. Mech. Engrg., 170, 79-101 (1999) · Zbl 0946.74058 |
| [38] | Hughes, T. J.R.; Liu, W. K.; Brooks, A., Finite element analysis of incompressible viscous flows by the penalty function formulation, J. Comput. Phys., 30, 1-60 (1979) · Zbl 0412.76023 |
| [39] | Jankovich, E.; Leblenc, F.; Durand, M.; Bercovier, M., A finite element method for the analysis of rubber parts, experimental and analytical assessment, Comput. Struct., 14, 385-391 (1981) · Zbl 0467.73100 |
| [40] | Oden, J. T.; Kikuchi, N., Finite elements for constrained problems in elasticity, Int. J. Numer. Methods Engrg., 18, 701-725 (1982) · Zbl 0486.73068 |
| [41] | Naylor, J., Stresses in nearly incompressible materials by finite elements with application to the calculation of excess pore pressures, Int. J. Numer. Methods Engrg., 8, 443-460 (1974) · Zbl 0282.73048 |
| [42] | Hughes, T. J.R., Equivalence of finite elements for nearly incompressible elasticity, J. Appl. Mech., 44, 181-183 (1977) |
| [43] | Malkus, D. S.; Hughes, T. J.R., Mixed finite element methods-reduced and selective integration techniques: a unification of concepts, Comput. Methods Appl. Mech. Engrg., 15, 63-81 (1978) · Zbl 0381.73075 |
| [44] | Hughes, T. J.R., Generalization of selective integration procedures to anisotropic and nonlinear media, Int. J. Numer. Methods Engrg., 15, 1413-1418 (1980) · Zbl 0437.73053 |
| [45] | Flangan, D. P.; Belytschko, T., A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, Int. J. Numer. Methods Engrg., 17, 679-706 (1981) · Zbl 0478.73049 |
| [46] | Belytschko, T.; Ong, J. S.J.; Liu, W. K.; Kennedy, J. M., Hourglass control in linear and nonlinear problems, Comput. Methods Appl. Mech. Engrg., 43, 251-276 (1984) · Zbl 0522.73063 |
| [47] | Liu, W. K.; Ong, J. S.J.; Uras, R. A., Finite element stabilization matrix – a unification approach, Comput. Methods Appl. Mech. Engrg., 53, 13-46 (1985) · Zbl 0553.73065 |
| [48] | Liu, W. K.; Hu, Y. K.; Belytschko, T., Multiple quadrature underintegrated finite elements, Int. J. Numer. Methods Engrg., 37, 3263-3289 (1994) · Zbl 0811.73063 |
| [49] | Hueck, U.; Wriggers, P., A formulation for the 4-node quadrilateral element, Int. J. Numer. Methods Engrg., 38, 3007-3037 (1995) · Zbl 0845.73068 |
| [50] | Chen, J. S.; Pan, C., A pressure projection method for nearly incompressible rubber hyperelasticity, Part I: Theory, J. Appl. Mech., 63, 862-868 (1996) · Zbl 0896.73072 |
| [51] | Chen, J. S.; Wu, C. T.; Pan, C., A pressure projection method for nearly incompressible rubber hyperelasticity, Part II: Applications, J. Appl. Mech., 63, 869-876 (1996) · Zbl 0896.73073 |
| [52] | Simo, J.; Rifai, M. S., A class of mixed assumed strain methods and the method of incompatible modes, Int. J. Numer. Methods Engrg., 29, 1595-1638 (1990) · Zbl 0724.73222 |
| [53] | Simo, J.; Armero, F., Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes, Int. J. Numer. Methods Engrg., 33, 1423-1449 (1992) · Zbl 0768.73082 |
| [54] | Andelginger, U.; Ramm, E., EAS-element for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements, Int. J. Numer. Methods Engrg., 36, 1311-1337 (1993) · Zbl 0772.73071 |
| [55] | Yeo, S. T.; Lee, B. C., Equivalence between enhanced assumed strain method and assumed stress hybrid method based on the Hellinger-Reissner principle, Int. J. Numer. Methods Engrg., 39, 3083-3099 (1996) · Zbl 0884.73075 |
| [56] | Korelc, J.; Wriggers, P., Improved enhanced strain four-node element with Taylor expansion of the shape functions, Int. J. Numer. Methods Engrg., 40, 407-421 (1997) |
| [57] | Braess, D., Enhanced assumed strain elements and locking in membrane problems, Comput. Methods Appl. Mech. Engrg., 165, 155-174 (1998) · Zbl 0949.74065 |
| [58] | Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method: diffuse approximation and diffuse elements, Comput. Mech., 10, 307-318 (1992) · Zbl 0764.65068 |
| [59] | Belytschko, T.; Lu, Y. Y.; Gu, L., Element free Galerkin methods, Int. J. Numer. Methods Engrg., 37, 229-256 (1994) · Zbl 0796.73077 |
| [60] | Belytschko, T.; Krongauz, Y.; Organ, D.; Flemming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. Methods Appl. Mech. Engrg., 139, 3-47 (1996) · Zbl 0891.73075 |
| [61] | Duarte, C. A.; Oden, J. T., An h-p adaptive method using clouds, Comput. Methods Appl. Mech. Engrg, 139, 237-262 (1996) · Zbl 0918.73328 |
| [62] | Melenk, J. M.; Babuška, I., The partition of unit finite element method: basic theory and applications, Comput. Methods Appl. Mech. Engrg., 139, 289-314 (1996) · Zbl 0881.65099 |
| [63] | Chen, J. S.; Yoon, S.; Wang, H. P.; Liu, W. K., An improved reproducing kernel particle method for nearly incompressible finite elasticity, Comput. Methods Appl. Mech. Engrg., 181, 117-145 (2000) · Zbl 0973.74088 |
| [64] | T.Y. Rong, A.Q. Lu, Parametrized variational principles for incompressible and nearly incompressible materials, Technical Report No. SWJ-012/97, Southwest Jiaotong Univ., Chengdu, China, 1997 (in Chinese); T.Y. Rong, A.Q. Lu, Parametrized variational principles for incompressible and nearly incompressible materials, Technical Report No. SWJ-012/97, Southwest Jiaotong Univ., Chengdu, China, 1997 (in Chinese) |
| [65] | Zienkiewicz, O. C.; Taylor, R. L., The Finite Element Method in Engineering Science, vol. 1 (1989), McGraw-Hill: McGraw-Hill London · Zbl 0991.74002 |
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