×

Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations. (English) Zbl 1160.65339

Summary: A meshfree method, namely the discrete least squares meshless method, is presented for the solution of elliptic partial differential equations. In this method the computational domain is discretized by some nodes and the set of simultaneous algebraic equations are built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional is defined as the sum of squared residuals of the differential equation and its boundary condition calculated at a set of points called sampling points, generally different from the nodal points.
A moving least squares technique is used to construct the shape functions. The proposed method automatically leads to a symmetric and positive-definite system of equations. The proposed method does not need any background mesh and, therefore, it is a truly meshless method. The solutions of several one- and two-dimensional examples of elliptic partial differential equations are presented to illustrate the performance of the proposed method. Sensitivity analysis on the parameters of the method is also carried out and the results are presented.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kim, H. G.; Atluri, S. N., Arbitrary placement of secondary nodes, and error control, in the meshless local Petrov-Galerkin (MLPG) method, CMES: Comput Modeling Eng Sci, 1, 3, 11-32 (2000) · Zbl 1147.65326
[2] Gingold, R. A.; Monaghan, J. J., Smoothed particle hydrodynamics: theory and applications to non-spherical stars, Mon Not R Astron Soc, 181, 375-389 (1977) · Zbl 0421.76032
[3] Belytschko, T.; Lu, Y. Y.; Gu, L., Element free Galerkin methods, Int J Numer Methods Eng, 37, 229-256 (1994) · Zbl 0796.73077
[4] Belytschko, T.; Krongauz, Y.; Organ, D., Meshless methods: an overview and recent developments, Comput Methods Appl Mech Eng, 139, 3-47 (1996) · Zbl 0891.73075
[5] Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, Int J Numer Methods Fluids, 20, 1081-1106 (1995) · Zbl 0881.76072
[6] \(O^∼\) nate, E.; Idelsohn, S.; Zienkiewicz, O. C.; Taylor, R. L., A finite point method in computational mechanics. Applications to convective transport and fluid flow, Int J Numer Methods Eng, 39, 22, 3839-3866 (1996) · Zbl 0884.76068
[7] Liszka, T. J.; Duarte, C. A.M.; Tworzydlo, W. W., HP-meshless cloud method, Comput Methods Appl Mech Eng, 139, 263-288 (1996) · Zbl 0893.73077
[8] Atluri, S. N.; Zhu, T. L., The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics, Comput Mech, 25, 169-179 (2000) · Zbl 0976.74078
[9] Atluri, S. N.; Zhu, T. L., New concepts in meshless methods, Int J Numer Methods Eng, 47, 1-3, 537-556 (2000) · Zbl 0988.74075
[10] Atluri, S. N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput Mech, 22, 117-127 (1998) · Zbl 0932.76067
[11] Long, S.; Atluri, S. N., A meshless local Petrov-Galerkin method for solving the bending problem of a thin plate, CMES, 1, 53-63 (2002) · Zbl 1147.74414
[12] Atluri, S. N.; Kim, H. G., A critical assessment of the truly meshless local Petrov-Galerkin (MLPG), and local boundary integral equation (LBIE) methods, Comput Mech, 24, 348-372 (1999) · Zbl 0977.74593
[13] Atluri, S. N.; Sladek, J., The local boundary integral equation (LBIE) and it’s meshless implementation for linear elasticity, Comput Mech, 25, 180-198 (2000) · Zbl 1020.74048
[14] Zhu, T.; Zhang, J. D.; Atluri, S. N., A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach, Comput Mech, 21, 223-235 (1998) · Zbl 0920.76054
[15] Aluru, N. R.; Gang, L., Finite cloud method: a true meshless technique based on a fixed reproducing kernel approximation, Int J Numer Methods Eng, 50, 2373-2410 (2001) · Zbl 0977.74078
[16] Kansa, E. J., Multiquadrics- a scattered data approximation scheme with applications to computational fluid-dynamics, I, Comput Math Appl, 19, 127-145 (1990) · Zbl 0692.76003
[17] Zhang, X.; Liu, X. H.; Song, K. Z.; Lu, M. W., Least-squares collocation meshless method, Int J Numer Methods Eng, 51, 1089-1100 (2001) · Zbl 1056.74064
[18] Wang, Q. X.; Li, H.; Lam, K. Y., development of a new meshless-point weighted least-squares (PWLS) method for computational mechanics, Comput Mech, 35, 170-181 (2005) · Zbl 1143.65397
[19] Pan, X. F.; Zhang, X.; Lu, M. W., Meshless Galerkin least-square method, Comput Mech, 35, 182-189 (2005) · Zbl 1143.74390
[20] Zhang X, Pan XF, Lu MW. Meshless weighted least-square method. In: Proceeding of fifth world congress on computational mechanics. Vienna , Austria, July 7-12, 2002.; Zhang X, Pan XF, Lu MW. Meshless weighted least-square method. In: Proceeding of fifth world congress on computational mechanics. Vienna , Austria, July 7-12, 2002.
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.