Beissel, Stephen; Belytschko, Ted Nodal integration of the element-free Galerkin method. (English) Zbl 0918.73329 Comput. Methods Appl. Mech. Eng. 139, No. 1-4, 49-74 (1996). Summary: Spatial integration of the element-free Galerkin (EFG) method is achieved by evaluating the integrals of the weak form only at the nodes. In previous EFG formulations, a grid of cells was used to perform Gaussian quadrature over the domain. The absence of a cell structure for nodal integration results in a completely meshless method, similar in simplicity to particle methods such as smooth particle hydrodynamics (SPH). It is shown that nodal integration, like SPH, suffers from spurious singular modes. This spatial instability results from underintegration of the weak form, and it is treated by the addition to the potential energy functional of a stabilization term which contains the square of the residual of the equilibrium equation. Example problems illustrate the effect of the stabilization and provide the basis for convergence studies. Cited in 231 Documents MSC: 74S30 Other numerical methods in solid mechanics (MSC2010) Keywords:Gaussian quadrature; spurious singular modes; potential energy functional; stabilization term; convergence × Cite Format Result Cite Review PDF Full Text: DOI References: [1] 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 [2] Belytschko, T.; Lu, Y. Y.; Gu, L., Element free Galerkin methods, Int. J. Numer. Methods Engrg., 37, 229-256 (1994) · Zbl 0796.73077 [3] 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 [4] Barnhill, R. E., Representation and approximation of surfaces, (Mathematical Software, III (1977), Academic Press: Academic Press New York), 69-120 · Zbl 0407.68030 [5] Gordon, W. J.; Wixom, J. A., Shepard’s method of ‘metric interpolation’ to bivariate and multivariate data, Math. Comput., 32, 253-264 (1978) · Zbl 0383.41003 [6] McLain, D. H., Drawing contours from arbitrary data points, Comput. J., 17, 318-324 (1974) [7] Lancaster, P.; Salkaushkas, K., Surfaces generated by moving least squares methods, Math. Comput., 37, 141-158 (1981) · Zbl 0469.41005 [8] Shepard, D., A two-dimensional interpolation function for irregularly spaced points, (Proceedings 1968 A.C.M. National Conference (1968)), 517-524 [9] Batina, J. T., A gridless Euler/Navier-Stokes solution algorithm for complex aircraft applications, (31st Aerospace Sciences Meeting and Exhibit. 31st Aerospace Sciences Meeting and Exhibit, Reno, NV. 31st Aerospace Sciences Meeting and Exhibit. 31st Aerospace Sciences Meeting and Exhibit, Reno, NV, AIAA 93-0333 (1993)) [10] 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 [11] 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 [12] Lucy, L. B., A numerical approach to the testing of the fission hypothesis, Astron. J., 82, 12, 1013-1024 (1977) [13] Gingold, R. A.; Monaghan, J. J., Smooth particle hydrodynamics: Theory and application to non-spherical stars, Monthly Notices Royal Astr. Soc., 181 (1972) · Zbl 0421.76032 [14] Gingold, R. A.; Monaghan, J. J., Kernel estimates as a basis for general particle methods in hydrodynamics, J. Comput. Phys., 46 (1982) · Zbl 0487.76010 [15] Belytschko, T.; Gu, L.; Lu, Y. Y., Fracture and crack growth by element free Galerkin methods, Model. Simul. Mater. Sci. Engrg., 115, 277-286 (1994) [16] Swegle, J. W.; Attaway, S. W.; Heinstein, M. W.; Mello, F. J.; Hicks, D. L., An analysis of smoothed particle hydrodynamics, Sandia Report SAND93-2513 (1994) [17] Lasry, D.; Belytschko, T., Localization limiters in transient problems, Int. J. Solids Struct., 24, 581-597 (1988) · Zbl 0636.73021 [18] Flanagan, D. P.; Belytschko, T., A uniform strain hexahedron and quadrilateral and hourglass control, Int. J. Numer. Methods Engrg., 17, 679-706 (1981) · Zbl 0478.73049 [19] Belytschko, T.; Bindeman, L. P., Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems, Comput. Methods Appl. Mech. Engrg., 88, 311-340 (1991) · Zbl 0742.73019 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.