A point collocation method based on reproducing kernel approximations. (English) Zbl 0960.74078
From the summary: We present a point collocation method based on reproducing kernel approximations. We show that, in a point collocation approach, the assignment of nodal volumes and implementation of boundary conditions are not critical issues, and points can be sprinkled randomly making the point collocation method a true meshless approach. The point collocation method based on reproducing kernel approximations, however, requires the calculation of higher-order derivatives that would typically not be required in a Galerkin method. Thus we derive a correction function and reproducing conditions that enable the consistency of the point collocation method. The point collocation method is shown to be accurate for several one- and two-dimensional problems, and the convergence rate of the point collocation method is addressed.
MSC:
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 76D07 | Stokes and related (Oseen, etc.) flows |
Keywords:
meshless method; cantilever beam; elasticity; heat conduction; Stokes flow; point collocation method; reproducing kernel approximations; higher-order derivatives; correction function; reproducing conditions; convergence rateReferences:
| [1] | Aluru, Sensors and Actuators A 58 pp 1– (1997) · doi:10.1016/S0924-4247(97)80218-X |
| [2] | Senturia, IEEE Computational Science and Engineering 4 pp 30– (1997) · doi:10.1109/99.590854 |
| [3] | Belytschko, Computer Methods in Applied Mechanics and Engineering 139 pp 3– (1996) · Zbl 0891.73075 · doi:10.1016/S0045-7825(96)01078-X |
| [4] | Lucy, Astronomical Journal 82 pp 1013– (1977) · doi:10.1086/112164 |
| [5] | Monaghan, SIAM Journal on Scientific and Statistical Computing 3 pp 422– (1982) · Zbl 0498.76010 · doi:10.1137/0903027 |
| [6] | Monaghan, Computer Physics Communications 48 pp 89– (1988) · Zbl 0673.76089 · doi:10.1016/0010-4655(88)90026-4 |
| [7] | Liu, International Journal for Numerical Methods in Engineering 21 pp 901– (1995) · Zbl 0885.76078 · doi:10.1002/fld.1650211010 |
| [8] | Liu, Computational Mechanics 18 pp 73– (1996) · doi:10.1007/BF00350529 |
| [9] | Liu, Archives of Computational Methods in Engineering: State of the Art Reviews 3 pp 3– (1996) · doi:10.1007/BF02736130 |
| [10] | Liu, International Journal for Numerical Methods in Engineering 38 pp 1655– (1995) · Zbl 0840.73078 · doi:10.1002/nme.1620381005 |
| [11] | Liu, International Journal for Numerical Methods in Fluids 20 pp 1081– (1995) · Zbl 0881.76072 · doi:10.1002/fld.1650200824 |
| [12] | Chen, Computer Methods in Applied Mechanics and Engineering 139 pp 195– (1996) · Zbl 0918.73330 · doi:10.1016/S0045-7825(96)01083-3 |
| [13] | Nayroles, Computational Mechanics 10 pp 307– (1992) · Zbl 0764.65068 · doi:10.1007/BF00364252 |
| [14] | Belytschko, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) · Zbl 0796.73077 · doi:10.1002/nme.1620370205 |
| [15] | Krongauz, Computer Methods in Applied Mechanics and Engineering 131 pp 133– (1996) · Zbl 0881.65098 · doi:10.1016/0045-7825(95)00954-X |
| [16] | Lu, Computer Methods in Applied Mechanics and Engineering 113 pp 397– (1994) · Zbl 0847.73064 · doi:10.1016/0045-7825(94)90056-6 |
| [17] | Mukherjee, International Journal for Numerical Methods in Engineering 40 pp 797– (1997) · Zbl 0885.65124 · doi:10.1002/(SICI)1097-0207(19970315)40:5<797::AID-NME89>3.0.CO;2-# |
| [18] | Mukherjee, Computational Mechanics 19 pp 264– (1997) · Zbl 0884.65105 · doi:10.1007/s004660050175 |
| [19] | Yu, Computer Methods in Applied Mechanics and Engineering 154 pp 331– (1998) · Zbl 0956.76064 · doi:10.1016/S0045-7825(97)00126-6 |
| [20] | Onate, International Journal for Numerical Methods in Engineering 39 pp 3839– (1996) · Zbl 0884.76068 · doi:10.1002/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-R |
| [21] | Duarte, Computer Methods in Applied Mechanics and Engineering 139 pp 237– (1996) · Zbl 0918.73328 · doi:10.1016/S0045-7825(96)01085-7 |
| [22] | Melenk, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996) · Zbl 0881.65099 · doi:10.1016/S0045-7825(96)01087-0 |
| [23] | Zhu, Computational Mechanics 21 pp 223– (1998) · Zbl 0920.76054 · doi:10.1007/s004660050297 |
| [24] | Beissel, Computer Methods in Applied Mechanics and Engineering 139 pp 49– (1996) · Zbl 0918.73329 · doi:10.1016/S0045-7825(96)01079-1 |
| [25] | The Finite Element Method. Prentice-Hall: Englewood Cliffs, NJ, 1987. |
| [26] | Smooth particle hydrodynamics with strength of materials. In Advances in the Free-Lagrange Method, Lecture Notes in Physics. Springer: Berlin, 1990; 248-257. |
| [27] | Theory of Elasticity (3rd edn). McGraw-Hill, New York, 1970. |
| [28] | Aluru, Computational Mechanics 23 pp 324– (1999) · Zbl 0949.74077 · doi:10.1007/s004660050413 |
| [29] | Boundary Layer Theory. McGraw-Hill: New York, 1968. |
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.
