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The partition of unity quadrature in meshless methods. (English) Zbl 1028.74047
Summary: In dealing with mesh-free formulations, a major problem is connected with the computation of quadratures appearing in the variational principle related to the differential boundary value problem. These integrals require, in the standard approach, the introduction of background quadrature subcells which somehow make these methods not ‘truly meshless’. In this paper a new general method for computing definite integrals over arbitrary bounded domains is proposed, and it is applied in particular to the evaluaton of the discrete weak form of equilibrium equations in an augmented Lagrangian element-free formulation. The approach is based on splitting the integrals over the entire domain into the sum of integrals over weight function supports without modifying in any way the variational principle or requiring background quadrature cells. The accuracy and computational cost of this technique are compared to standard Gauss subcells quadrature.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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[1] Beissel, Computer Methods in Applied Mechanics and Engineering 139 pp 49– (1996) · Zbl 0918.73329 · doi:10.1016/S0045-7825(96)01079-1
[2] Chen, International Journal for Numerical Methods in Engineering 50 pp 435– (2001) · Zbl 1011.74081 · doi:10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A
[3] Atluri, Computational Mechanics 22 pp 117– (1998) · Zbl 0932.76067 · doi:10.1007/s004660050346
[4] Atluri, International Journal for Numerical Methods in Engineering 47 pp 537– (2000) · Zbl 0988.74075 · doi:10.1002/(SICI)1097-0207(20000110/30)47:1/3<537::AID-NME783>3.0.CO;2-E
[5] Belytschko, Computer Methods in Applied Mechanics and Engineering 139 pp 3– (1996) · Zbl 0891.73075 · doi:10.1016/S0045-7825(96)01078-X
[6] Ventura, International Journal for Numerical Methods in Engineering 53 pp 825– (2002) · doi:10.1002/nme.314
[7] Belytschko, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) · Zbl 0796.73077 · doi:10.1002/nme.1620370205
[8] Material and crack discontinuities: application of an element free augmented Lagrangian method. In Damage and Fracture Mechanics, (eds). Computational Mechanics Publications: Southampton, U.K., 1998; 237-246.
[9] Carpinteri, Computer Methods in Applied Mechanics and Engineering 191 pp 941– (2001) · Zbl 1033.74048 · doi:10.1016/S0045-7825(01)00288-2
[10] Cuomo, Computer Methods in Applied Mechanics and Engineering 189 pp 313– (2000) · Zbl 0961.74058 · doi:10.1016/S0045-7825(99)00298-4
[11] Practical Methods of Optimization. Wiley: New York, 1987. · Zbl 0905.65002
[12] Constrained Optimization and Lagrange Multiplier Methods. Academic Press: London, 1982. · Zbl 0572.90067
[13] Theory of elasticity (3rd edn) McGraw-Hill: New York, 1987.
[14] Melenk, Seminar for Applied Mathematics, ETH Zurich, Research Report 96 (1996)
[15] Melenk, International Journal for Numerical Methods in Engineering 40 pp 727– (1997) · Zbl 0949.65117 · doi:10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N
[16] Dolbow, Computational Mechanics 23 pp 219– (1999) · Zbl 0963.74076 · doi:10.1007/s004660050403
[17] Numerical Quadrature and Cubature. Academic Press: London, 1980.
[18] Approximate Calculation of Multiple Integrals. Prentice-Hall: Englewood Cliffs, NJ, 1971. · Zbl 0379.65013
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