×

Meshless methods: An overview and recent developments. (English) Zbl 0891.73075

Summary: Meshless approximations based on moving least-squares, kernels, and partitions of unity are examined. It is shown that the three methods are in most cases identical except for the important fact that partitions of unity enable \(p\)-adaptivity to be achieved. Methods for constructing discontinuous approximations and approximations with discontinuous derivatives are also described. Next, several issues in implementation are reviewed: discretization (collocation and Galerkin), quadrature in Galerkin and fast ways of constructing consistent moving least-squares approximations. The paper concludes with some sample calculations.

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
76M25 Other numerical methods (fluid mechanics) (MSC2010)
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics

Software:

LS-DYNA; DYNA3D
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Lucy, L. B., A numerical approach to the testing of the fission hypothesis, The Astron. J., 8, 12, 1013-1024 (1977)
[2] Monaghan, J. J., Why particle methods work, SIAM J. Sci. Stat. Comput., 3, 4, 422 (1982) · Zbl 0498.76010
[3] Monaghan, J. J., An introducion to SPH, Comput. Phys. Comm., 48, 89-96 (1988) · Zbl 0673.76089
[4] Swegle, J. W.; Hicks, D. L.; Attaway, S. W., Smoothed particle hydrodynamics stability analysis, J. Comput. Phys., 116, 123-134 (1995) · Zbl 0818.76071
[5] Dyka, C. T., Addressing tension instability in SPH methods, (Technical Report NRL/MR/6384 (1994), NRL) · Zbl 0900.73945
[6] Johnson, G. R.; Beissel, S. R., Normalized smoothing funtions for SPH impact computations, Int. J. Numer. Methods Engrg. (1996) · Zbl 0880.73076
[7] Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, Int. J. Numer. Methods Engrg., 20, 1081-1106 (1995) · Zbl 0881.76072
[8] 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
[9] Belytschko, T.; Gu, L.; Lu, Y. Y., Fracture and crack growth by element-free Galerkin methods, Model. Simul. Mater. Sci. Engrg., 2, 519-534 (1994)
[10] Duarte, C. A.; Oden, J. T., Hp clouds—a meshless method to solve boundary-value problems, (Technical Report 95-05 (1995), Texas Institute for Computational and Applied Mathematics, University of Texas: Texas Institute for Computational and Applied Mathematics, University of Texas Austin) · Zbl 0976.74071
[12] Liu, W. K.; Li, S.; Belytschko, T., Reproducing least-square kernel Galerkin method. (i) Methodology and convergence, Comput. Methods Appl. Mech. Engrg. (1996), in preparation
[13] Perrone, N.; Kao, R., A general finite difference method for arbitrary meshes, Comput. Struct., 5, 45-58 (1975)
[14] Liszka, T.; Orkisz, J., The finite difference method at arbitrary irregular grids and its application in applied mechanics, Comput. Struct., 11, 83-95 (1980) · Zbl 0427.73077
[15] Monaghan, J. J., Smooth particle hydrodynamics, Annu. Rev. Astron. Astrophys., 30, 543-574 (1992)
[16] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least-squares methods, Math. Comput., 37, 141-158 (1981) · Zbl 0469.41005
[17] Cleveland, W. S., Visualizing Data (1993), AT&T Bell Laboratories: AT&T Bell Laboratories Murray Hill, NJ
[18] Shepard, D., A two-dimensional interpolation function for irregularly spaced points, (ACM National Conference (1968)), 517-524
[22] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int. J. Numer. Methods Engrg., 37, 229-256 (1994) · Zbl 0796.73077
[23] Liu, W. K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. Numer. Methods Engrg., 38, 1655-1679 (1995) · Zbl 0840.73078
[24] Liu, W. K.; Chen, Y., Wavelet and multiple scale reproducing kernel methods, Int. J. Numer. Methods Fluids, 21, 901-931 (1995) · Zbl 0885.76078
[25] Liu, W. K.; Chen, Y.; Chang, C. T.; Belytschko, T., Advances in multiple scale kernel particle methods, Comput. Mech., An International Journal (1996) · Zbl 0868.73091
[32] Yagawa, G., Some remarks on free mesh method: A kind of meshless finite element method, (International Conference on Computational Engrg. Science. International Conference on Computational Engrg. Science, Hawaii, USA (1995)) · Zbl 0894.73182
[33] Lu, Y. Y.; Belytschko, T.; Gu, L., A new implementation of the element free Galerkin method, Comput. Methods Appl. Mech. Engrg., 113, 397-414 (1994) · Zbl 0847.73064
[34] Chu, Y. A.; Moran, B., A computational model for nucleation of solid-solid phase transformations, Model. Simul. Mater. Sci. Engrg., 3, 455-471 (1995)
[36] Johnson, G. R., Linking Lagrangian particle methods to standard finite element methods for high velocity impact computations, Nucl. Engrg. Des., 150, 265-274 (1994)
[37] Attaway, S. W.; Heinstein, M. W.; Swegle, J. W., Coupling of smooth particle hydrodynamics with the finite element method, Nucl. Engrg Des., 150, 199-205 (1994)
[38] Hegen, D., Numerical techniques for the simulation of crack growth, Technical report, Eindhoven University of Technology. Final report of the postgraduate programme Mathematics for Industry (1994)
[39] Belytschko, T.; Organ, D.; Krongauz, Y., A coupled finite element-element-free Galerkin method, Comput. Mech., 17, 186-195 (1995) · Zbl 0840.73058
[40] Liu, W. K.; Chen, Y.; Uras, R. A., Enrichment of the finite element method with the reproducing kernel particle method, (Cory, J. J.F.; Gordon, J. L., Current Topics in Computuational Mechanics. Current Topics in Computuational Mechanics, ASME-PVP, Vol. 305 (1995)), 253-258
[41] Liu, W. K.; Chen, Y.; Jun, S.; Chen, J. S.; Belytschko, T.; Pan, C.; Uras, R. A.; Chang, C. T., Overview and applications of the reproducing kernel particle methods, Archives of Computational Methods in Engineering: State of the art reviews, 3, 3-80 (1996)
[43] Hein, P., Diffuse element method applied to Kirchhoff plates, (Technical report (1993), Dept. Civil Engrg, Northwestern University: Dept. Civil Engrg, Northwestern University Evanston, IL)
[44] Lu, Y. Y.; Gu, L.; Belytschko, T., Internal report (1996)
[45] Krysl, P.; Belytschko, T., Analysis of thin plates by the element-free Galerkin method, Comput. Mech., 17, 26-35 (1996) · Zbl 0841.73064
[47] Belytschko, T.; Stolarski, H.; Liu, W. K.; Carpenter, N.; Ong, J. S.-J., Stress projection for membrane and shear locking in shell finite elements, Comput. Methods Appl. Mech. Engrg., 51, 221-258 (1985) · Zbl 0581.73091
[48] Stummel, F., The generalized patch test, SIAM J. Numer. Anal., 3, 449-471 (1979) · Zbl 0418.65058
[49] Strang, G.; Fix, G., An Analysis of the Finite Element Method (1973), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0278.65116
[50] Ciarlet, P. G., Basic error estimates for elliptic problems, (Lions, C. P.G.; Lions, J. L., Finite Element Methods, Part 1. Finite Element Methods, Part 1, Handbook of Numerical Analysis, Vol. II (1991), Elsevier, North-Holland: Elsevier, North-Holland Amsterdam) · Zbl 0875.65086
[51] Timoshenko, S. P.; Goodier, J. N., Theory of Elasticity (1970), McGraw-Hill: McGraw-Hill New York · Zbl 0266.73008
[52] Sumi, Y.; Nemat-Nasser, S.; Keer, L. M., On crack path stability in a finite body, Engrg. Fracture Mech., 22, 759-771 (1985)
[53] Sumi, Y., Computational crack path prediction, Theoret. Appl. Fracture Mech., 4, 149-156 (1985)
[55] Kalthoff, J. F.; Winkler, S., Failure mode transition at high rates of shear loading, (Chiem, C. Y.; Kunze, H. D.; Meyer, L. W., International Conference on Impact Loading and Dynamic Behavior of Materials, Vol. 1 (1987)), 185-195
[56] Organ, D., Numerical solutions to dynamic fracture problems using the element-free Galerkin method, (Ph.D. Thesis (1996), Northwestern University)
[57] John, R.; Shah, S. P., Mixed mode fracture of concrete subjected to impact loading, ASCE J. Struct. Engrg., 116, 585-602 (1990), accepted for publication
[58] John, R., Mixed mode fracture of concrete subjected to impact loading, (Ph.D. Thesis (1988), Northwestern University)
[59] Li, F. Z.; Shih, C. F.; Needleman, A., A comparison of methods for calculating energy release rates, Engrg. Fracture Mech., 21, 2, 405-421 (1985)
[60] Moran, B.; Shill, C. F., Crack tip and associated domain integral from momentum and energy balance, Engrg. Fracture Mech., 22, 6, 615-641 (1987)
[61] Hallquist, J. O., (DYNA2D and DYNA3D user’s manuals (1980), Lawrence Livermore National Laboratories: Lawrence Livermore National Laboratories Livermore, CA)
[62] Liu, W. K.; Chang, H.; Chen, J. S.; Belytschko, T., Arbitrary Lagrangian-Eulerian Petrov-Galerkin finite elements for nonlinear continua, Comput. Methods Appl. Mech. Engrg., 68, 259-310 (1988) · Zbl 0626.73076
[63] Belytschko, T.; Tabbara, M., Dynamic fracture using element-free Galerkin methods, Int. J. Numer. Methods Engrg., 39 (1996) · Zbl 0953.74077
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.