# zbMATH — the first resource for mathematics

Reproducing kernel element method. I: Theoretical formulation. (English) Zbl 1060.74670
Summary: In this paper and its sequels [for part II see Li, Shaofan et al., Comput. Methods Appl. Mech. Eng. 193, No. 12–14, 953–987 (2004; Zbl 1093.74062); for part III see the following entry] we introduce and analyze a new class of methods, collectively called the reproducing kernel element method (RKEM). The central idea in the development of the new method is to combine the strengths of both finite element methods (FEM) and meshfree methods. Two distinguished features of RKEM are: the arbitrarily high order smoothness and the interpolation property of shape functions. These properties are desirable, especially in solving Galerkin weak forms of higher-order partial differential equations and in treating Dirichlet boundary conditions. So, unlike the FEM, there is no need for special treatment, with the RKEM, of solution of high-order equations. Compared to meshfree methods, Dirichlet boundary conditions do not present any difficulty in using the RKEM. A rigorous error analysis and convergence study of the method are presented. The performance of the method is illustrated and assessed through some numerical examples.

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74S05 Finite element methods applied to problems in solid mechanics
Full Text:
##### References:
 [1] Babuška, I.; Oden, J.T.; Lee, J.K., Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems, Comput. methods appl. mech. engrg., 11, 175-206, (1977) · Zbl 0382.65056 [2] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077 [3] Belytschko, T.; Liu, W.K.; Moran, B., Nonlinear finite elements for continua and structures, (2000), Wiley England · Zbl 0959.74001 [4] Brenner, S.C.; Scott, L.R., The mathematical theory of finite element methods, (1994), Springer-Verlag New York · Zbl 0804.65101 [5] Chen, J.S.; Han, W.; You, Y.; Meng, X., A reproducing kernel method with nodal interpolation property, Int. J. numer. methods engrg., 56, 935-960, (2003) · Zbl 1106.74424 [6] Chen, J.S.; Wang, H.P., New boundary condition treatments in meshfree computation of contact problems, Comput. methods appl. mech. engrg., 187, 441-468, (2000) · Zbl 0980.74077 [7] Chen, J.S.; Pan, C.; Wu, C.T.; Liu, W.K., Reproducing kernel particle methods for large deformation analysis of nonlinear structures, Comput. methods appl. mech. engrg., 139, 195-229, (1996) [8] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043 [9] Gosz, J.; Liu, W.K., Admissible approximations for essential boundary conditions in the reproducing kernel particle method, Comput. mech., 19, 120-135, (1996) · Zbl 0889.73078 [10] Günther, F.C.; Liu, W.K., Implementation of boundary conditions for meshless methods, Comput. methods appl. mech. engrg., 163, 205-230, (1998) · Zbl 0963.76068 [11] Han, W.; Meng, X., Error analysis of the reproducing kernel particle method, Comput. methods appl. mech. engrg., 190, 6157-6181, (2001) · Zbl 0992.65119 [12] Han, W.; Meng, X., (), 193-210 [13] Han, W.; Wagner, G.J.; Liu, W.K., Convergence analysis of a hierarchical enrichment of Dirichlet boundary condition in a meshfree method, Int. J. numer. methods engrg., 53, 1323-1336, (2002) · Zbl 0995.65108 [14] S. Hao, W.K. Liu, Revisit of moving particle finite element method, Fifth World Congress on Computational Mechanics, Vienna, Austria, July 7-12, 2002s [15] S. Hao, W.K. Liu, Moving particle finite element method with global superconvergence, Int. J. Numer. Methods Engrg. submitted for publication [16] Hao, S.; Park, H.S.; Liu, W.K., Moving particle finite element method, Int. J. numer. methods engrg., 53, 1937-1958, (2002) · Zbl 1169.74606 [17] Kaljevic, I.; Saigal, S., An improved element free Galerkin formulation, Int. J. numer. methods engrg., 40, 2953-2974, (1997) · Zbl 0895.73079 [18] Li, S.; Liu, W.K., Moving least square reproducing kernel method (II) Fourier analysis, Comput. methods appl. mech. engrg., 139, 159-193, (1996) · Zbl 0883.65089 [19] Li, S.; Liu, W.K., Meshfree and particle methods and their applications, Appl. mech. rev., 55, 1-34, (2002) [20] S. Li, H. Lu, W. Han, W.K. Liu, Reproducing kernel element, Part II. Globally conforming $$I\^{}\{m\}/C\^{}\{n\}$$ hierarchies, Comput. Methods Appl. Mech. Engrg., accepted for publication · Zbl 1060.74670 [21] Liu, W.K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. numer. methods engrg., 38, 1655-1679, (1995) · Zbl 0840.73078 [22] Liu, W.K.; Jun, S.; Zhang, Y.F., Reproducing kernel particle methods, Int. J. numer. methods fluids, 20, 1081-1106, (1995) · Zbl 0881.76072 [23] Liu, W.K.; Li, S.; Belytschko, T., Moving least square reproducing kernel method. part I: methodology and convergence, Comput. methods appl. mech. engrg., 143, 422-453, (1997) [24] Liu, W.K.; Uras, R.A.; Chen, Y., Enrichment of the finite element method with the reproducing kernel particle method, J. appl. mech., ASME, 64, 861-870, (1997) · Zbl 0920.73366 [25] Liu, W.K.; Chen, Y.; Uras, R.A.; Chang, C.T., Generalized multiple scale reproducing kernel particle methods, Comput. methods appl. mech. engrg., 139, 91-158, (1996) · Zbl 0896.76069 [26] Lu, H.; Chen, J.S., (), 251-267 [27] H. Lu, W.K. Liu, J.S. Chen, J. Cao, Consistent smoothing technique and treatment of material discontinuity in the reproducing kernel element, Int. J. Numer. Methods Engrg., submitted for publication [28] Rachford, H.H.; Wheeler, M.F., An $$H\^{}\{−1\}$$-Galerkin procedure for the two-point boundary value problem, (), 353-382 [29] Wagner, G.J.; Liu, W.K., Hierarchical enrichment for bridging scales and meshfree boundary conditions, Int. J. numer. methods engrg., 50, 507-524, (2000) · Zbl 1006.76073 [30] Wagner, G.J.; Liu, W.K., Application of essential boundary conditions in mesh-free methods: a corrected collocation method, Int. J. numer. methods engrg., 47, 1367-1379, (2000) · Zbl 0965.76069 [31] Zhang, L.T.; Wagner, G.J.; Liu, W.K., A parallel meshfree method with boundary enrichment for large-scale CFD, J. comput. phys., 176, 483-506, (2002) · Zbl 1060.76096
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.