A gradient smoothing method (GSM) with directional correction for solid mechanics problems. (English) Zbl 1162.74502
Summary: A novel gradient smoothing method (GSM) is proposed in this paper, in which a gradient smoothing together with a directional derivative technique is adopted to develop the first- and second-order derivative approximations for a node of interest by systematically computing weights for a set of field nodes surrounding. A simple collocation procedure is then applied to the governing strong-from of system equations at each node scattered in the problem domain using the approximated derivatives. In contrast with the conventional finite difference and generalized finite difference methods with topological restrictions, the GSM can be easily applied to arbitrarily irregular meshes for complex geometry. Several numerical examples are presented to demonstrate the computational accuracy and stability of the GSM for solid mechanics problems with regular and irregular nodes. The GSM is examined in detail by comparison with other established numerical approaches such as the finite element method, producing convincing results.
MSC:
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |
Keywords:
numerical methods; gradient smoothing method (GSM); meshfree method; solid mechanics; numerical analysisReferences:
| [1] | Richtmyer RD (1957). Difference methods for initial-value problems. Wiley, New York · Zbl 0079.33702 |
| [2] | Forsythe GE and Wasow WR (1960). Finite difference methods for partial differential equations. Wiley, New York · Zbl 0099.11103 |
| [3] | Richtmyer RD and Morton KW (1967). Difference methods for boundary-value problems. Wiley, New York · Zbl 0155.47502 |
| [4] | Samarskij AA (1977) Theory of finite difference schemes (in Russian). Nauka, Moscow |
| [5] | Gawain TH and Ball RE (1978). Improved finite difference formulas for boundary-value problems. Int J Num Meth Eng 12: 1151–1160 · Zbl 0377.73062 · doi:10.1002/nme.1620120706 |
| [6] | Anderson DA, Tannenhill JC and Fletcher RH (1984). Computational fluid mechanics and heat transfer. McGraw-Hill, Washington |
| [7] | Frey WH (1977). Flexible finite-difference stencils from isoparametric finite elements. Int J Numer Methods Eng 11: 1653–1665 · Zbl 0368.65051 · doi:10.1002/nme.1620111103 |
| [8] | Lau PC (1979). Curvilinear finite difference method for biharmonic equation. Int J Numer Methods Eng 14: 791–812 · Zbl 0401.65059 · doi:10.1002/nme.1620140604 |
| [9] | Lau PC (1979). Curvilinear finite difference methods for three-dimensional potential problems. J Comput Phys 32: 325–344 · Zbl 0434.65075 · doi:10.1016/0021-9991(79)90149-9 |
| [10] | Kwok SK (1984). An improved curvilinear finite difference (CFD) method for arbitrary mesh systems. Comput Struct 18: 719–731 · Zbl 0532.65067 · doi:10.1016/0045-7949(84)90017-8 |
| [11] | Nay RA and Utku S (1973). An alternative for the finite element method. Variat Mech Eng 3: 62–74 · Zbl 0312.73087 |
| [12] | Wyatt MJ, Davies G and Snell C (1975). A new difference based finite element method. Instn Eng 59: 395–409 |
| [13] | Mullord P (1979). A general mesh finite difference method using combined nodal and element interpolation. Appl Math Modell 3: 433–440 · Zbl 0431.73068 · doi:10.1016/S0307-904X(79)80026-8 |
| [14] | Snell C, Vesey DG and Mullord P (1981). The application of a general finite difference method to some boundary value problems. Comput Struct 13: 547–552 · Zbl 0465.73113 · doi:10.1016/0045-7949(81)90050-X |
| [15] | Tworzydlo W (1987). Analysis of large deformation of membrane shells by the generalized finite difference method. Comput Struct 27: 39–59 · Zbl 0624.73101 · doi:10.1016/0045-7949(87)90180-5 |
| [16] | Liu GR (2003). Meshfree method: moving beyond the finite element method. CRC Press, Boca Raton |
| [17] | Liu GR and Gu YT (2005). An introduction to meshfree methods and their programming. Springer, Dordrecht |
| [18] | Belytschko T, Krongauz Y, Organ D, Fleming M and Krysl P (1996). Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139: 3–47 · Zbl 0891.73075 · doi:10.1016/S0045-7825(96)01078-X |
| [19] | Liu GR and Gu YT (2001). A point interpolation method for two-dimensional solids. Int J Numer Methods Eng 50: 937–951 · Zbl 1050.74057 · doi:10.1002/1097-0207(20010210)50:4<937::AID-NME62>3.0.CO;2-X |
| [20] | Liu GR and Gu YT (2001). A local point interpolation method (LRPIM) for free vibration analyses of 2-D solids. J Sound Vib 246: 29–46 · doi:10.1006/jsvi.2000.3626 |
| [21] | Gu YT and Liu GR (2005). A Meshfree Weak-Strong (MWS) form method for time dependent problems. Comput Mech 35: 134–145 · Zbl 1109.74371 · doi:10.1007/s00466-004-0610-0 |
| [22] | Liu GR, Zhang Jian, Li Hua, Lam KY and Kee BT Bernard (2006). Radial point interpolation based finite difference method for mechanics problems. Int J Numer Methods Eng 68: 728–754 · Zbl 1129.74049 · doi:10.1002/nme.1733 |
| [23] | Jensen PS (1972). Finite difference techniques for variable grids. Comput Struct 2: 17–29 · doi:10.1016/0045-7949(72)90020-X |
| [24] | Perrone N and Kao R (1975). A general finite difference method for arbitrary meshes. Comput Struct 5: 45–58 · doi:10.1016/0045-7949(75)90018-8 |
| [25] | Krok J, Orkisz J (1990) A unified approach to the FE generalized variational FD method in nonlinear mechanics: concept and numerical approach. In: Discretization methods in structural mechanics. Springer, Berlin, pp 353–362 · Zbl 0714.73079 |
| [26] | Krok J, Orkisz J, Stanuszek M (1993) A unique FDM/FEM system of discrete analysis of boundary value problems in mechanics. In: Proceedings of the 11th Polish conference on computational methods in mechanics. Kielce-Cedzyna, Poland, pp 466–472 |
| [27] | Krok J, Orkisz J (1997) 3D elastic stress analysis in railroad rails and vehicle wheels by the adaptive FEM/FDM and Fourier series. In: Proceedings of the 13th Polish conference on computational methods in mechanics. Poznan, Poland, pp 661–668 |
| [28] | Orkisz J (1997) Adaptive approach to the finite difference method for arbitrary irregular grids. In: Interdisciplinary symposium on advances in computational mechanics. University of Texas, Austin |
| [29] | Orkisz J, Lezanski P, Przybylski P (1997) Multigrid approach to adaptive analysis of boundary-value problems by the meshless GFDM. In: IUTAM/IACM symposium on discretization methods in structural mechanics II. Vienna |
| [30] | Liszka T and Orkisz J (1980). The finite difference method at arbitrary irregular grids and its application in applied mechanics. Comput Struct 11: 83–95 · Zbl 0427.73077 · doi:10.1016/0045-7949(80)90149-2 |
| [31] | Liszka T (1984). An interpolation method for an irregular net of nodes. Int J Numer Methods Eng 20: 1599–1612 · Zbl 0544.65006 · doi:10.1002/nme.1620200905 |
| [32] | Orkisz J (1998) Meshless finite difference method I: Basic approach. In: Proceedings of the IACM-fourth world congress in computational mechanics, Argentina |
| [33] | (1998). Handbook of computational solid mechanics: survey and comparison of contemporary methods. Springer, Berlin · Zbl 0934.74002 |
| [34] | Chen JS, Wu CT, Yoon S and You Y (2001). A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50: 435–466 · Zbl 1011.74081 · doi:10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A |
| [35] | Liu GR, Li Y, Dai KY, Luan MT, Xue W (2006) A linearly conforming RPIM for 2D solid mechanics. Int J Comput Methods (in press) |
| [36] | Liu GR, Dai KY and Nguyen TT (2007). A smoothed finite element method for mechanics problems. Comput Mech 39: 859–877 · Zbl 1169.74047 · doi:10.1007/s00466-006-0075-4 |
| [37] | Shepard D (1968) A two dimensional interpolation function for irregularly spaced points. In: Proceeding of ACM national conference, pp 517–524 |
| [38] | Haselbacher A and Blazek J (2000). Accurate and efficient discretization of Navier–Stokes equations on mixed grids. AIAA J 38: 2094–2102 · doi:10.2514/2.871 |
| [39] | Weiss JM, Maruszewski JP and Smith WA (1999). Implicit solution of preconditioned Navier-Stokes equations using algebraic multigrid. AIAA J 37: 29–36 · doi:10.2514/2.689 |
| [40] | Blazek J (2001). Computational fluid dynamics: principles and applications. Elsevier, Oxford · Zbl 0995.76001 |
| [41] | Timoshenko SP and Goodier JN (1970). Theory of elasticity. McGraw-Hill, New York |
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.
