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On the completeness of meshfree particle methods. (English) Zbl 0939.74076
Summary: We investigate the completeness of smooth particle hydrodynamics (SPH) and its modifications. Completeness, or the reproducing conditions, in Galerkin approximations play the same role as consistency in finite-difference approximations. We examine several techniques which restore various levels of completeness by satisfying reproducing conditions on the approximation or the derivatives of the approximation. A Petrov-Galerkin formulation for a particle method is developed using approximations with corrected derivatives. It is compared to a normalized SPH formulation based on kernel approximations and to a Galerkin method based on moving least-square approximations. It is shown that the major difference is that in the SPH discretization, the function which plays the role of the test function is not integrable. Numerical results show that approximations which do not satisfy the completeness and integrability conditions fail to converge for linear elastostatics, so convergence is not expected in nonlinear continuum mechanics.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K20 Plates
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References:
[1] Gingold, Mon. Not. Roy. Astron. Soc. 181 pp 375– (1977) · Zbl 0421.76032
[2] Liu, Int. J. Numer. Meth. Engng. 38 pp 1655– (1995) · Zbl 0840.73078
[3] Johnson, Int. J. Numer. Meth. Engng. 39 pp 2725– (1996) · Zbl 0880.73076
[4] Randles, Comput. Meth. Appl. Mech. Engng. 139 pp 375– (1996) · Zbl 0896.73075
[5] ’Moving-least-squares-particle hydrodynamics i: Consistency and stability’, Int. J. Numer. Meth. Engng., 1997, Submitted for publication.
[6] Belytschko, Comput. Meth. Appl. Mech. Engng. 139 pp 3– (1996) · Zbl 0891.73075
[7] Belytschko, Comput. Meth. Appl. Mech. Engng. 119 pp 1– · Zbl 0849.73064
[8] Krongauz, Comput Meth. Appl. Mech. Engng. 146 pp 371– (1997a) · Zbl 0894.73156
[9] Monaghan, Annu. Rev. Astron. Astrophys. 30 pp 543– (1992)
[10] The Finite Element Method, Prentice-Hall, Englewood Cliffs, N.J., 1987.
[11] Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brooks, Pacific Grove, CA, 1989. · Zbl 0681.65064
[12] and , An Analysis of Finite Element Method, Prentice-Hall, Englewood Cliffs, N.J., 1973.
[13] ’A two dimensional function for irregularly spaced data’, ACM National Conf., 1968.
[14] Belytschko, Int. J. Numer. Meth. Engng. 37 pp 229– · Zbl 0796.73077
[15] Monaghan, Comput. Phys. Commun. 48 pp 89– (1988) · Zbl 0673.76089
[16] Krongauz, Comput. Meth. Appl. Mech. Engng. 131 pp 133– (1996) · Zbl 0881.65098
[17] Morris, Publ. Astron. Soc. Aust. 13 (1996)
[18] Krongauz, Comput. Mech. 19 pp 371– (1997b) · Zbl 1031.74527
[19] Nayroles, Comput. Mech. 10 pp 307– (1992) · Zbl 0764.65068
[20] private communication, 1995.
[21] Mas-Gallic, Numer. Math. 51 pp 323– (1987) · Zbl 0625.65084
[22] Laguna, Astrophys. J. 439 pp 814– (1995)
[23] and , Mechanics of Materials, 3rd edn, PWS-KENT Pub. Co., Boston, 1990.
[24] and , Theory of Elasticity, 3rd edn, McGraw-Hill, New York, 1987.
[25] Belytschko, Comput. Meth. Appl. Mech. Engng. 88 pp 311– (1991) · Zbl 0742.73019
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