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Element-free Galerkin method for wave propagation and dynamic fracture. (English) Zbl 1067.74599
Summary: The element-free Galerkin method (EFG) is extended to dynamic problems. The EFG method, which is based on moving least square interpolants, requires only nodal data; no element connectivity is needed. This makes the method particularly attractive for moving dynamic crack problems, since remeshing can be avoided. In contrast to the earlier formulation for static problems by the authors, the weak form of kinematic boundary conditions for dynamic problems is introduced in the implementation to enforce the kinematic boundary conditions. With this formulation, the stiffness matrix is symmetric and positive semi-definite, and hence the consistency, convergence and stability analyses of time integration remain the same as those in the finite element method. Numerical examples are presented to illustrate the performance of this method. The relationship between the element-free Galerkin method and the smooth particle hydrodynamics method is also discussed in this paper. Results are presented for some one-dimensional and two-dimensional problems with static and moving cracks.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
74R99 Fracture and damage
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[1] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077
[2] T. Belytschko, Y.Y. Lu and L. Gu, Crack propagation by element-free Galerkin methods, Engrg. Fract. Mech. in press. · Zbl 0796.73077
[3] T. Belytschko, T. Gu and Y.Y. Lu, Fracture and crack growth by element-free Galerkin, Modelling Simul. Mater. Sci. Engrg. in press.
[4] Lu, Y.Y.; Belytschko, T.; Gu, L., A new implementation of element-free Galerkin methods, Comput. methods appl. mech. engrg., 113, 3-4, 397-414, (1994) · Zbl 0847.73064
[5] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math. comput., 37, 141-158, (1981) · Zbl 0469.41005
[6] Shepard, D., A two-dimensional interpolation function for irregularly spaced points, (), 517-524
[7] 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
[8] Gingold, R.A.; Monaghan, J.J., Smoothed particle hydrodynamics: theory and application to non-spherical stars, Mon. not. roy. astron. soc., 181, 375-389, (1977) · Zbl 0421.76032
[9] Gingold, R.A.; Monaghan, J.J., Kernel estimates as a basis for general particle methods in hydrodynamics, J. comp. phys., 46, (1982) · Zbl 0487.76010
[10] Monaghan, J.J., Why particle methods work, SIAM J. sci. stat. comput., 3, 422-433, (1982) · Zbl 0498.76010
[11] Monaghan, J.J., An introduction to SPH, Comput. phys. commun., 48, 89-96, (1988) · Zbl 0673.76089
[12] Crowley, W.P., Free-Lagrange methods for compressible hydrodynamics in two space dimensions, () · Zbl 0581.76075
[13] Trease, H.E., Three-dimensional free Lagrangian hydrodynamics, () · Zbl 0633.76028
[14] Johnson, G.R.; Peterson, E.H.; Stryrk, R.A., Incorporation of an SPH option into the EPIC code for A wide range of high velocity impact computations, ()
[15] Hughes, T.J.R., The finite element method, (1987), Prentice-Hall International Editions Englewood Cliffs, NJ
[16] Gurtin, M.E., Variational principles for linear elastodynamics, Arch. rational mech. anal., 16, 36-50, (1964) · Zbl 0124.40001
[17] Washizu, K., Variational methods in elasticity and plasticity, (1975), Pergamon New York · Zbl 0164.26001
[18] Cormen, T.H.; Leiserson, C.E.; Rivest, R.L., Introduction to algorithms, (1990), McGraw-Hill New York · Zbl 1158.68538
[19] Lancaster, P.; Tismenetsky, M., The theory of matrices, (1985), Academic Press London · Zbl 0516.15018
[20] Lee, Y.J.; Freund, L.B., Fracture initiation due to asymmetric impact loading of an edge crack plate, J. appl. mech., 57, 104-111, (1990)
[21] Yau, J.F.; Wang, S.S.; Corten, H.T., A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity, J. appl. mech., 47, 335-341, (1980) · Zbl 0463.73103
[22] Li, F.Z.; Shih, C.F.; Needleman, A., A comparison of methods for calculating energy release rates, Engrg. fract. mech., 21, 405-421, (1985)
[23] Moran, B.; Shih, C.F., Crack tip and associated domain integrals from momentum and energy balance, Engrg. fract. mech., 27, 615-642, (1987)
[24] Moran, B.; Shih, C.F., A general treatment of crack tip contour integrals, Int. J. fracture, 35, 295-310, (1987)
[25] Freund, L.B., Dynamic fracture mechanics, (1990), Cambridge University Press Cambridge · Zbl 0712.73072
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