×

A novel Galerkin-like weakform and a superconvergent alpha finite element method (S\(\alpha \)FEM) for mechanics problems using triangular meshes. (English) Zbl 1273.74542

Summary: A carefully designed procedure is presented to modify the piecewise constant strain field of linear triangular FEM models, and to reconstruct a strain field with an adjustable parameter \(\alpha \). A novel Galerkin-like weakform derived from the Hellinger-Reissner variational principle is proposed for establishing the discretized system equations. The new weak form is very simple, possesses the same good properties of the standard Galerkin weakform, and works particularly well for strain construction methods. A superconvergent alpha finite element method (S\(\alpha \)FEM) is then formulated by using the constructed strain field and the Galerkin-like weakform for solid mechanics problems. The implementation of the S\(\alpha \)FEM is straightforward and no additional parameters are used. We prove theoretically and show numerically that the S\(\alpha \)FEM always achieves more accurate and higher convergence rate than the standard FEM of triangular elements (T3) and even more accurate than the four-node quadrilateral elements (Q4) when the same sets of nodes are used. The S\(\alpha \)FEM can always produce both lower and upper bounds to the exact solution in the energy norm for all elasticity problems by properly choosing an \(\alpha \). In addition, a preferable-\(\alpha \) approach has also been devised to produce very accurate solutions for both displacement and energy norms and a superconvergent rate in the energy error norm. Furthermore, a model-based selective scheme is proposed to formulate a combined S\(\alpha \)FEM/NS-FEM model that handily overcomes the volumetric locking problems. Intensive numerical studies including singularity problems have been conducted to confirm the theory and properties of the S\(\alpha \)FEM.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74G15 Numerical approximation of solutions of equilibrium problems in solid mechanics

Software:

XFEM
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987), Prentice-Hall · Zbl 0634.73056
[2] Zienkiewicz, O. C.; Taylor, R. L., The Finite Element Method (2000), Butterworth Heinemann: Butterworth Heinemann Oxford · Zbl 0991.74002
[3] Liu, G. R.; Quek, S. S., The Finite Element Method: A Practical Course (2003), Butterworth Heinemann: Butterworth Heinemann Oxford · Zbl 1027.74001
[4] G.R. Liu, A weakened weak (W2) form for a unified formulation of compatible and incompatible methods, part I-Theory and part II-Application to solid mechanics problems, Int. J. Numer. Methods Eng., in press, 2009.; G.R. Liu, A weakened weak (W2) form for a unified formulation of compatible and incompatible methods, part I-Theory and part II-Application to solid mechanics problems, Int. J. Numer. Methods Eng., in press, 2009.
[5] Liu, G. R., A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods, Int. J. Comput. Methods, 5, 2, 199-236 (2008) · Zbl 1222.74044
[6] Allman, D. J., A compatible triangular element including vertex rotations for plane elasticity analysis, Comput. Struct., 19, 1-8 (1984) · Zbl 0548.73049
[7] Bergan, P. G.; Felippa, C. A., A triangular membrane element with rotational degrees of freedom, Comput. Methods Appl. Mech. Eng., 50, 25-69 (1985) · Zbl 0593.73073
[8] Allman, D. J., Evaluation of the constant strain triangle with drilling rotations, Int. J. Numer. Methods Eng., 26, 2645-2655 (1988) · Zbl 0674.73057
[9] Felippa, C. A., A study of optimal membrane triangular with drilling freedoms, Comput. Methods Appl. Mech. Eng., 192, 2125-2168 (2003) · Zbl 1140.74551
[10] Piltner, R.; Taylor, R. L., Triangular finite elements with rotational degrees of freedom and enhanced strain modes, Comput. Struct., 75, 361-368 (2000)
[11] Rong, T.; Genki, Y., Allman’s triangle, rotational DOF and partition of unity, Int. J. Numer. Methods Eng., 69, 837-858 (2007) · Zbl 1194.74477
[12] Simo, J. C.; Rifai, M. S., A class of mixed assumed strain methods and the method of incompatible modes, Int. J. Numer. Methods Eng., 29, 1595-1638 (1990) · Zbl 0724.73222
[13] Andelfinger, U.; Ramm, E., EAS-elements for two-dimensional threedimensional plate and shells and their equivalence to HR-elements, Int. J. Numer. Methods Eng., 36, 1413-1449 (1993) · Zbl 0772.73071
[14] 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 Eng., 39, 3083-3099 (1996) · Zbl 0884.73075
[15] Bischoff, M.; Ramm, E.; Braess, D., A class equivalent enhanced assumed strain and hybrid stress finite elements, Comput. Mech., 22, 444-449 (1999) · Zbl 0958.74058
[16] César de Sá, J. M.A.; Natal Jorge, R. M., New enhanced strain elements for incompatible problems, Int. J. Numer. Methods Eng., 44, 229-248 (1999) · Zbl 0937.74062
[17] César de Sá, J. M.A.; Natal Jorge, RM.; Fontes Valente, R. A.; A Areias, P. M., Development of shear locking-free shell elements using an enhanced assumed strain formulation, Int. J. Numer. Methods Eng., 53, 1721-1750 (2002) · Zbl 1114.74484
[18] Cardoso, R. P.R.; Yoon, J. W.; Mahardika; Choudhry, S.; Alves de Sousa, R. J.; Fontes Valente, R. A., Enhanced assumed strain (EAS) and assumed natural strain (ANS) methods for one-point quadrature solid-shell elements, Int. J. Numer. Methods Eng., 75, 156-187 (2008) · Zbl 1195.74165
[19] Pian, T. H.H.; Wu, C. C., Hybrid and Incompatible Finite Element Methods (2006), CRC Press: CRC Press Boca Raton, FL · Zbl 1110.65003
[20] Liu, G. R.; Zhang, G. Y.; Dai, K. Y.; Wang, Y. Y.; Zhong, Z. H.; Li, G. Y.; Han, X., A linearly a u conforming point interpolation method (LC-PIM) for 2D solid mechanics problems, Int. J. Comput. Methods, 2, 645-665 (2005) · Zbl 1137.74303
[21] Liu, G. R.; Gu, Y. T., A point interpolation method for two-dimensional solids, Int. J. Numer. Methods Eng., 50, 937-951 (2001) · Zbl 1050.74057
[22] Wang, J. G.; Liu, G. R., A point interpolation meshless method based on radial basis functions, Int. J. Numer. Methods Eng., 54, 1623-1648 (2002) · Zbl 1098.74741
[23] Chen, J. S.; Wu, C. T.; Yoon, S.; You, Y., A stabilized conforming nodal integration for Galerkin mesh-free methods, Int. J. Numer. Methods Eng., 50, 435-466 (2001) · Zbl 1011.74081
[24] Liu, G. R.; Dai, K. Y.; Nguyen, T. T., A smoothed element method for mechanics problems, Comput. Mech., 39, 859-877 (2007) · Zbl 1169.74047
[25] Liu, G. R.; Nguyen, T. T.; Dai, K. Y.; Lam, K. Y., Theoretical aspects of the smoothed finite element method (SFEM), Int. J. Numer. Methods Eng., 71, 902-930 (2007) · Zbl 1194.74432
[26] Nguyen-Xuan, Hung; Bordas, Stéphane; Nguyen-Dang, Hung, Smooth finite element methods: convergence, accuracy and properties, Int. J. Numer. Methods Eng., 74, 175-208 (2008) · Zbl 1159.74435
[27] KY, Dai; GR, Liu; TT, Nguyen, An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics, Finite Elem. Anal. Des., 43, 847-860 (2007)
[28] Dai, K. Y.; Liu, G. R., Free and forced vibration analysis using the smoothed finite element method (SFEM), J. Sound Vib., 301, 803-820 (2007)
[29] Nguyen-Xuan, H.; Rabczuk, T.; Bordas, S.; Debongnie, J. F., A smoothed finite element method for plate analysis, Comput. Methods Appl. Mech. Eng., 197, 1184-1203 (2008) · Zbl 1159.74434
[30] H. Nguyen-Xuan, T. Nguyen-Thoi, A stabilized smoothed finite element method for free vibration analysis of Mindlin-Reissner plates, Commun. Numer. Methods Eng., in press, doi:10.1002/cnm.1137; H. Nguyen-Xuan, T. Nguyen-Thoi, A stabilized smoothed finite element method for free vibration analysis of Mindlin-Reissner plates, Commun. Numer. Methods Eng., in press, doi:10.1002/cnm.1137 · Zbl 1172.74047
[31] Nguyen-Thanh, N.; Rabczuk, T.; Nguyen-Xuan, H.; Bordas, S., A smoothed finite element method for shell analysis, Comput. Methods Appl. Mech. Eng., 198, 165-177 (2008) · Zbl 1194.74453
[32] Cui, X. Y.; Liu, G. R.; Li, G. Y.; Zhao, X.; Nguyen-Thoi, T.; Sun, G. Y., A smoothed finite element method (SFEM) for linear and geometrically nonlinear analysis of plates and shells, CMES-Comput. Model. Eng. Sci., 28, 2, 109-125 (2008) · Zbl 1232.74099
[33] Moes, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, Int. J. Numer. Methods Eng., 46, 1, 131-150 (1999) · Zbl 0955.74066
[34] Belytschko, T.; Moes, N.; Usui, S.; Parimi, C., Arbitrary discontinuities in finite elements, Int. J. Numer. Methods Eng., 50, 993-1013 (2001) · Zbl 0981.74062
[35] Bordas, S.; Nguyen, P. V.; Dunant, C.; Guidoum, A.; Nguyen-Dang, H., An extended finite element library, Int. J. Numer. Methods Eng., 71, 6, 703-732 (2007) · Zbl 1194.74367
[36] Melenk, J. M.; Babuška, I., The partition of unity finite element method: basic theory and applications, Comput. Methods Appl. Mech. Eng., 139, 1-4, 289-314 (1996) · Zbl 0881.65099
[37] Babuška, I.; Melenk, J. M., The partition of unity method, Int. J. Numer. Methods Eng., 40, 4, 727-758 (1997) · Zbl 0949.65117
[38] S. Bordas, T. Rabczuk, H. Nguyen-Xuan, P. Nguyen Vinh, S. Natarajan, T. Bog, Q. Do Minh, H. Nguyen Vinh, Strain smoothing in FEM and XFEM, Comput. Struct., in press, doi:10.1016/j.compstruc.2008.07.006; S. Bordas, T. Rabczuk, H. Nguyen-Xuan, P. Nguyen Vinh, S. Natarajan, T. Bog, Q. Do Minh, H. Nguyen Vinh, Strain smoothing in FEM and XFEM, Comput. Struct., in press, doi:10.1016/j.compstruc.2008.07.006
[39] Liu, G. R.; Nguyen-Thoi, T.; Nguyen-Xuan, H.; Lam KY, K. Y., A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems, Comput. Struct., 87, 14-26 (2009)
[40] Liu, G. R.; Zhang, G. Y., Upper bound solution to elasticity problems: a unique property of the linearly conforming point interpolation method (LC-PIM), Int. J. Numer. Methods Eng., 74, 1128-1161 (2008) · Zbl 1158.74532
[41] Fraeijs de Veubeke, B. M., Displacement and equilibrium models in the finite element method, (Zienkiewicz, O. C.; Holister, G., Stress Analysis (1965), John Wiley and Sons: John Wiley and Sons Berlin), 15-197, Chapter 9, Reprinted in Int. J. Numer. Methods Eng. 52 (2001) 287-342
[42] Debongnie, J. F.; Zhong, H. G.; Beckers, P., Dual analysis with general boundary conditions, Comput. Methods Appl. Mech. Eng., 122, 183-192 (1995) · Zbl 0851.73057
[43] Puso, M. A.; Solberg, J., A stabilized nodally integrated tetrahedral, Int. J. Numer. Methods Eng., 67, 841-867 (2006) · Zbl 1113.74075
[44] Puso, M. A.; Chen, J. S.; Zywicz, E.; Elmer, W., Meshfree and finite element nodal integration methods, Int. J. Numer. Methods Eng., 74, 416-446 (2008) · Zbl 1159.74456
[45] G.R. Liu, T. Nguyen-Thoi, K.Y. Lam, An edge-based smoothed finite element method (ES-FEM) for static and dynamic problems of solid mechanics, J. Sound Vib. 320 (2009) 1100-1130.; G.R. Liu, T. Nguyen-Thoi, K.Y. Lam, An edge-based smoothed finite element method (ES-FEM) for static and dynamic problems of solid mechanics, J. Sound Vib. 320 (2009) 1100-1130.
[46] Bergan, P. G.; Felippa, C. A., A triangular membrane element with rotational degrees of freedom, Comput. Methods Appl. Mech. Eng., 50, 25-69 (1985) · Zbl 0593.73073
[47] Felippa, C. A.; Militello, C., Variational formulation of high-performance finite elements: parametrized variational principles, Comput. Struct., 50, 1, 1-11 (1990) · Zbl 0709.73075
[48] Felippa, C. A., A survey of parametrized variational principles and applications to computational mechanics, Comput. Methods Appl. Mech. Eng., 113, 1-2, 109-139 (1994) · Zbl 0848.73063
[49] Djoko, J. K.; Reddy, B. D., An extended Hu-Washizu formulation for elasticity, Comput. Methods Appl. Mech. Eng., 195, 6330-6346 (2006) · Zbl 1122.74047
[50] G.R. Liu, T. Nguyen-Thoi, K.Y. Lam, A novel FEM by scaling the gradient of strains with scaling factor Α;Α;doi:10.1007/s00466-008-0311-1; G.R. Liu, T. Nguyen-Thoi, K.Y. Lam, A novel FEM by scaling the gradient of strains with scaling factor Α;Α;doi:10.1007/s00466-008-0311-1 · Zbl 1162.74469
[51] Liu, G. R.; Nguyen-Thoi, T.; Lam, K. Y., A novel alpha finite element method \((α\) FEM) for exact solution to mechanics problems using triangular and tetrahedral elements, Comput. Methods Appl. Mech. Eng., 197, 3883-3897 (2008) · Zbl 1194.74433
[52] G.R. Liu, H. Nguyen-Xuan, T. Nguyen-Thoi, A variationally consistent alpha FEM (VCΑ;; G.R. Liu, H. Nguyen-Xuan, T. Nguyen-Thoi, A variationally consistent alpha FEM (VCΑ; · Zbl 1217.74126
[53] Belytschko, T.; Bachrach, W. E., Efficient implementation of quadrilaterals with high coarse-mesh accuracy, Comput. Methods Appl. Mech. Eng., 54, 279-301 (1986) · Zbl 0579.73075
[54] Belytschko, T.; Bindeman, L. P., Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems, Comput. Methods Appl. Mech. Eng., 88, 311-340 (1993) · Zbl 0742.73019
[55] G.R. Liu, X. Xu, G.Y. Zhang, T. Nguyen-Thoi, A superconvergent point interpolation method (SC-PIM) with piecewise linear strain field using triangular mesh, Int. J. Numer. Methods Eng., in press, doi:10.1002/nme.2464; G.R. Liu, X. Xu, G.Y. Zhang, T. Nguyen-Thoi, A superconvergent point interpolation method (SC-PIM) with piecewise linear strain field using triangular mesh, Int. J. Numer. Methods Eng., in press, doi:10.1002/nme.2464 · Zbl 1156.74394
[56] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int. J. Numer. Methods Eng., 37, 229-256 (1994) · Zbl 0796.73077
[57] Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, Int. J. Numer. Methods Eng., 20, 1081-1106 (1995) · Zbl 0881.76072
[58] Atluri, S. N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech., 22, 117-127 (1998) · Zbl 0932.76067
[59] Onate, E.; Idelsohn, S.; Zienkiewicz, O. C.; Taylor, R. L., A finite point method in computational mechanics. Applications to convective transport and fluid flow, Int. J. Numer. Methods Eng., 39, 22, 3839-3866 (1996) · Zbl 0884.76068
[60] Rabczuk, T.; Belytschko, T.; Xiao, S. P., Stable particle methods based on Lagrangian kernels, Comput. Methods Appl. Mech. Eng., 193, 12-14, 1035-1063 (2004) · Zbl 1060.74672
[61] Rabczuk, T.; Bordas, S.; Zi, G., A three-dimensional meshfree method for continuous crack initiation, nucleation and propagation in statics and dynamics, Comput. Mech., 40, 3, 473-495 (2007) · Zbl 1161.74054
[62] Bordas, S.; Rabczuk, T.; Zi, G., Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment, Eng. Fract. Mech., 75, 5, 943-960 (2008)
[63] T. Rabczuk, S. Bordas, G. Zi, On three-dimensional modeling of crack growth using partition of unity methods, Comput. Struct., in press, doi:10.1016/j.compstruc.2008.08.010; T. Rabczuk, S. Bordas, G. Zi, On three-dimensional modeling of crack growth using partition of unity methods, Comput. Struct., in press, doi:10.1016/j.compstruc.2008.08.010 · Zbl 1161.74054
[64] Rabczuk, T.; Zi, G.; Bordas, S.; Nguyen-Xuan, H., A geometrically non-linear three-dimensional cohesive crack method for reinforced concrete structures, Eng. Fract. Mech., 75, 16, 4740-4758 (2008)
[65] Fredriksson, M.; Ottosen, N. S., Fast and accurate 4-node quadrilateral, Int. J. Numer. Methods Eng., 61, 1809-1834 (2004) · Zbl 1075.74644
[66] Nguyen-Dang, H., Finite element equilibrium analysis of creep using the mean value of the equivalent shear modulus, Meccanica, 15, 234-245 (1980) · Zbl 0454.73062
[67] Macneal, R. H.; Harder, R. L., A proposed standard set of problems to test finite element accuracy, Finite Elem. Anal. Des., 1, 3-20 (1985)
[68] Timoshenko, S. P.; Goodier, J. N., Theory of Elasticity (1970), McGraw: McGraw New York · Zbl 0266.73008
[69] Cook, R., Improved two-dimensional finite element, J. Struct. Div. (ASCE), 100, 1851-1863 (1974)
[70] Mijuca, D.; Berković, M., On the efficiency of the primal-mixed finite element scheme. On the efficiency of the primal-mixed finite element scheme, Advances in Computational Structured Mechanics (1998), Civil-Comp Press, pp. 61-69
[71] Flanagan, D. P.; Belytschko, T., A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, Int. J. Numer. Methods Eng., 17, 679-706 (1981) · Zbl 0478.73049
[72] Kosloff, D.; Frazier, G. A., Treatment of hourglass patterns in low order finite element codes, Int. J. Numer. Anal. Methods Geomech., 2, 57-72 (1978)
[73] Beckers, P.; Zhong, H. G.; Maunder, E., Numerical comparison of several a posteriori error estimators for 2D stress analysis, Revue Européenne des élémenis finis, 2, 155-178 (1993) · Zbl 0813.73059
[74] Bletzinger, K.; Bischoff, M.; Ramm, E., A unified approach for shear locking-free triangular and rectangular shell finite elements, Comput. Struct., 75, 321-334 (2000)
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.