# zbMATH — the first resource for mathematics

A smoothed finite element method for mechanics problems. (English) Zbl 1169.74047
Summary: In the finite element method (FEM), a necessary condition for a four-node isoparametric element is that no interior angle is greater than $$180^\circ$$ and the positivity of Jacobian determinant should be ensured in numerical implementation. In this paper, we incorporate cell-wise strain smoothing operations into conventional finite elements and propose the smoothed finite element method (SFEM) for 2D elastic problems. It is found that a quadrilateral element divided into four smoothing cells can avoid spurious modes and gives stable results for integration over the element. Compared with original FEM, the SFEM achieves more accurate results and generally higher convergence rate in energy without increasing computational cost. More importantly, as no mapping or coordinate transformation is involved in the SFEM, its element is allowed to be of arbitrary shape. Hence the restriction on the shape of bilinear isoparametric elements can be removed and problem domain can be discretized in more flexible ways, as demonstrated in the example problems.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74K20 Plates
Mfree2D
Full Text:
##### References:
 [1] Bathe KJ (1996) Finite element procedures. Prentice Hall, New Jersey [2] Beissel S, Belytschko T (1996) Nodal integration of the element - free Galerkin method. Comput Meth Appl Mech Eng 139:49–74 · Zbl 0918.73329 [3] Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Method Eng 37:229–256 · Zbl 0796.73077 [4] Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless Method: An Overview and Recent Developments. Comput Meth Appl Mech Eng 139:3–47 · Zbl 0891.73075 [5] Bonet J, Kulasegaram S (1999) Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulation. Int J Numer Method Eng 47:1189–1214 · Zbl 0964.76071 [6] Chen JS, Wu CT, Belytschko T (2000) Regularization of material instabilities by meshfree approximations with intrinsic length scales. Int J Numer Method Eng 47:1303–1322 · Zbl 0987.74079 [7] Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin meshfree method. Int J Numer Method Eng 50:435–466 · Zbl 1011.74081 [8] Dai KY, Liu GR, Lim KM, Gu YT (2003) Comparison between the radial point interpolation and the Kriging based interpolation used in mesh-free methods. Comput Mech 32:60–70 · Zbl 1035.74059 [9] Krongauz Y, Belytschko T (1997) Consistent pseudo-derivatives in meshless method. Int J Numer Method Eng 146:371–386 · Zbl 0894.73156 [10] Li Y, Liu GR, Luan MT, Dai KY, Zhong ZH, Li GY, Han X (2006) Contact analysis for solids based on linearly conforming RPIM. Comput Mech (in press) [11] Liu GR (2002) Mesh-free methods: moving beyond the finite element method. CRC Press, Boca Raton [12] Liu GR, Li Y, Dai KY, Luan MT, Xue W (2006a) A linearly conforming RPIM for 2D solid mechanics. Int J Comput Methods (in press) [13] Liu GR, Quek SS (2003) The finite element method: a practical course. Butterworth Heinemann, Oxford · Zbl 1027.74001 [14] Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li G, Han X (2006b) A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems. Int J Comput Methods (in press) [15] Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Method Eng 20:1081–1106 · Zbl 0881.76072 [16] Monaghan JJ (1982) Why particle methods work. Siam J Sci Sat Comput 3(4):423–433 · Zbl 0498.76010 [17] Sukumar N (2004) Construction of polygonal interpolants: a maximum entropy approach. Int J Numer Method Eng 61:2159–2181 · Zbl 1073.65505 [18] Sukumar N, Moran B (1999) C 1 natural neighbor interpolation for partial differential equations. Numer Methods Partial Differential Equations 15:417 · Zbl 0943.74080 [19] Sukumar N, Moran B, Belytschko T (1998) The natural element method in solid mechanics. Int J Numer Method Eng 43:839–887 · Zbl 0940.74078 [20] Sukumar N, Tabarraei (2004) Conforming polygonal finite elements. Int J Numer Method Eng 61:2045–2066 · Zbl 1073.65563 [21] Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York [22] Wang JG, Liu GR (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Method Eng 54:1623–1648 · Zbl 1098.74741 [23] Yoo JW, Moran B, Chen JS (2004) Stabilized conforming nodal integration in the natural-element method. Int J Numer Method Eng 60:861–890 · Zbl 1060.74677 [24] Zienkiewicz OC, Taylor RL (2000) The finite element method, 5th edn. Butterworth Heinemann, Oxford
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.