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The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems. (English) Zbl 1352.65539
Summary: In this paper, the moving least-squares (MLS) approximation and the interpolating moving least-squares (IMLS) method proposed by Lancaster are discussed first. A new method for deriving the MLS approximation is presented, and the IMLS method is improved. Compared with the IMLS method proposed by Lancaster, the shape function of the improved IMLS method in this paper is simpler so that the new method has higher computing efficiency. Then combining the shape function of the improved IMLS method with Galerkin weak form of the potential problem, the interpolating element-free Galerkin (IEFG) method for the two- dimensional potential problem is presented, and the corresponding formulae are obtained. Compared with the conventional element-free Galerkin (EFG) method, the boundary conditions can be applied directly in the IEFG method, which makes the computing efficiency higher. For the purposes of demonstration, some selected numerical examples are solved using the IEFG method.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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##### References:
 [1] Krongauz, Y.; Belytschko, T.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput methods appl mech eng, 139, 3-47, (1996) · Zbl 0891.73075 [2] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int J numer methods eng, 37, 229-256, (1994) · Zbl 0796.73077 [3] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least square methods, Math computation, 37, 141-158, (1981) · Zbl 0469.41005 [4] Liew, K.M.; Cheng, Y.; Kitipornchai, S., Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems, Int J numer methods eng, 65, 1310-1332, (2006) · Zbl 1147.74047 [5] Cheng, Y.; Peng, M., Boundary element-free method for elastodynamics, Sci China G phys mech astron, 48, 641-657, (2005) [6] Kitipornchai, S.; Liew, K.M.; Cheng, Y., A boundary element-free method (BEFM) for three-dimensional elasticity problems, Comput mech, 36, 13-20, (2005) · Zbl 1109.74372 [7] Liew, K.M.; Cheng, Y.; Kitipornchai, S., Boundary element-free method (BEFM) for two-dimensional elastodynamic analysis using Laplace transform, Int J numer methods eng, 64, 1610-1627, (2005) · Zbl 1122.74533 [8] Sun, Y.; Zhang, Z.; Kitipornchai, S.; Liew, K.M., Analyzing the interaction between collinear interfacial cracks by an efficient boundary element-free method, Int J eng sci, 44, 37-48, (2006) · Zbl 1213.74270 [9] Liew, K.M.; Cheng, Y.; Kitipornchai, S., Analyzing the 2D fracture problems via the enriched boundary element-free method, Int J solids struct, 44, 4220-4233, (2007) · Zbl 1346.74162 [10] Liew, K.M.; Sun, Y.; Kitipornchai, S., Boundary element-free method for fracture analysis of 2-D anisotropic piezoelectric solids, Int J numer methods eng, 69, 729-749, (2007) · Zbl 1194.74531 [11] Sun, Y.; Hu, Y.G.; Liew, K.M., A mesh-free simulation of cracking and failure using the cohesive segments method, Int J eng sci, 45, 541-553, (2007) · Zbl 1213.74312 [12] Miers, L.S., Telles JCF. the boundary element-free method for elastoplastic implicit analysis, Int J numer methods eng, 76, 1090-1107, (2008) · Zbl 1195.74246 [13] Zhang, Z.; Liew, K.M.; Cheng, Y., Coupling of improved element-free Galerkin and boundary element methods for the 2D elasticity problems, Eng anal boundary elem, 32, 100-107, (2008) · Zbl 1244.74204 [14] Zhang, Z.; Liew, K.M.; Cheng, Y.; Lee, Y.Y., Analyzing 2D fracture problems with the improved element-free Galerkin method, Eng anal boundary elem, 32, 241-250, (2008) · Zbl 1244.74240 [15] Cheng, Y.; Li, J., A complex variable meshless method for fracture problems, Sci China G phys mech astron, 49, 46-59, (2006) · Zbl 1147.74410 [16] Liew, K.M.; Feng, C.; Cheng, Y.; Kitipornchai, S., Complex variable moving least-squares method: a meshless approximation technique, Int J numer methods eng, 70, 46-70, (2007) · Zbl 1194.74554 [17] Cheng, Y.; Peng, M.; Li, J., The complex variable moving least-square approximation and its application, Chin J theor appl mech, 37, 719-723, (2005) [18] Liew, K.M.; Cheng, Y., Complex variable boundary element-free method for two-dimensional elastodynamic problems, Comput methods appl mech eng, 198, 3925-3933, (2009) · Zbl 1231.74502 [19] Peng, M.; Liu, P.; Cheng, Y., The complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems, Int J appl mech, 1, 367-385, (2009) [20] Peng, M.; Li, D.; Cheng, Y., The complex variable element-free Galerkin (CVEFG) method for elasto-plasticity problems, Eng struct, 33, 127-135, (2011) [21] Zhu, T.; Atluri, S.N., A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method, Comput mech, 21, 211-222, (1998) · Zbl 0947.74080 [22] Krongauz, Y.; Belytschko, T., Enforcement of essential boundary conditions in meshless approximation using finite elements, Comput methods appl mech eng, 139, 3-47, (1996) · Zbl 0881.65098 [23] Liu, W.K.; Jun, S.; Zhang, Y.F., Reproducing kernel particle methods, Int J numer methods fluids, 20, 1081-1106, (1995) · Zbl 0881.76072 [24] Wu, Z.M.; Hon, Y.C., Convergence error estimate in solving free boundary diffusion problem by radial basis functions method, Eng anal boundary elem, 27, 73-79, (2003) · Zbl 1040.91058 [25] 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
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