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Published in: Acta Mechanica Sinica 3/2018

02-02-2018 | Research Paper

The dimension split element-free Galerkin method for three-dimensional potential problems

Authors: Z. J. Meng, H. Cheng, L. D. Ma, Y. M. Cheng

Published in: Acta Mechanica Sinica | Issue 3/2018

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Abstract

This paper presents the dimension split element-free Galerkin (DSEFG) method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares (IMLS) approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method.

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Literature
1.
go back to reference Belytschko, T., Krongauz, Y., Organ, D., et al.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139, 3–47 (1996)CrossRefMATH Belytschko, T., Krongauz, Y., Organ, D., et al.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139, 3–47 (1996)CrossRefMATH
2.
go back to reference Dolbow, J., Belytschko, T.: An introduction to programming the meshless element free Galerkin method. Arch. Comput. Methods Eng. 5, 207–241 (1998)MathSciNetCrossRef Dolbow, J., Belytschko, T.: An introduction to programming the meshless element free Galerkin method. Arch. Comput. Methods Eng. 5, 207–241 (1998)MathSciNetCrossRef
3.
go back to reference Canelas, A., Laurain, A., Novotny, A.A.: A new reconstruction method for the inverse potential problem. J. Comput. Phys. 268, 417–431 (2014)MathSciNetCrossRefMATH Canelas, A., Laurain, A., Novotny, A.A.: A new reconstruction method for the inverse potential problem. J. Comput. Phys. 268, 417–431 (2014)MathSciNetCrossRefMATH
4.
go back to reference Sun, L.L., Chen, W., Zhang, C.Z.: A new formulation of regularized meshless method applied to interior and exterior anisotropic potential problems. Appl. Math. Model. 37, 7452–7464 (2013)MathSciNetCrossRef Sun, L.L., Chen, W., Zhang, C.Z.: A new formulation of regularized meshless method applied to interior and exterior anisotropic potential problems. Appl. Math. Model. 37, 7452–7464 (2013)MathSciNetCrossRef
5.
go back to reference Khaji, N., Javaran, S.H.: New complex Fourier shape functions for the analysis of two-dimensional potential problems using boundary element method. Eng. Anal. Bound. Elem. 37, 260–272 (2013)MathSciNetCrossRefMATH Khaji, N., Javaran, S.H.: New complex Fourier shape functions for the analysis of two-dimensional potential problems using boundary element method. Eng. Anal. Bound. Elem. 37, 260–272 (2013)MathSciNetCrossRefMATH
6.
go back to reference Gong, Y.P., Dong, C.Y., Qin, X.C.: An isogeometric boundary element method for three dimensional potential problems. J. Comput. Appl. Math. 313, 454–468 (2017)MathSciNetCrossRefMATH Gong, Y.P., Dong, C.Y., Qin, X.C.: An isogeometric boundary element method for three dimensional potential problems. J. Comput. Appl. Math. 313, 454–468 (2017)MathSciNetCrossRefMATH
7.
go back to reference Sun, F.L., Zhang, Y.M., Young, D.L., et al.: A new boundary meshfree method for potential problems. Adv. Eng. Softw. 100, 32–42 (2016)CrossRef Sun, F.L., Zhang, Y.M., Young, D.L., et al.: A new boundary meshfree method for potential problems. Adv. Eng. Softw. 100, 32–42 (2016)CrossRef
8.
go back to reference Mukherjee, Y.X., Mukherjee, S.: The boundary node method for potential problems. Int. J. Numer. Methods Eng. 40, 797–815 (1997)CrossRefMATH Mukherjee, Y.X., Mukherjee, S.: The boundary node method for potential problems. Int. J. Numer. Methods Eng. 40, 797–815 (1997)CrossRefMATH
9.
go back to reference Chati, M.K., Mukherjee, S.: The boundary node method for three-dimensional problems in potential theory. Int. J. Numer. Methods Eng. 47, 1523–1547 (2000)MathSciNetCrossRefMATH Chati, M.K., Mukherjee, S.: The boundary node method for three-dimensional problems in potential theory. Int. J. Numer. Methods Eng. 47, 1523–1547 (2000)MathSciNetCrossRefMATH
10.
go back to reference Zhang, Y.M., Sun, F.L., Young, D.L., et al.: Average source boundary node method for potential problems. Eng. Anal. Bound. Elem. 70, 114–125 (2016)MathSciNetCrossRef Zhang, Y.M., Sun, F.L., Young, D.L., et al.: Average source boundary node method for potential problems. Eng. Anal. Bound. Elem. 70, 114–125 (2016)MathSciNetCrossRef
11.
go back to reference Li, X.L.: Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces. Appl. Numer. Math. 99, 77–97 (2016)MathSciNetCrossRefMATH Li, X.L.: Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces. Appl. Numer. Math. 99, 77–97 (2016)MathSciNetCrossRefMATH
12.
go back to reference Lu, Y.Y., Belytschko, T., Gu, L.: A new implementation of the element free Galerkin method. Comput. Methods Appl. Mech. Eng. 113, 397–414 (1994)MathSciNetCrossRefMATH Lu, Y.Y., Belytschko, T., Gu, L.: A new implementation of the element free Galerkin method. Comput. Methods Appl. Mech. Eng. 113, 397–414 (1994)MathSciNetCrossRefMATH
13.
go back to reference Li, X.L., Zhang, S.G., Wang, Y., et al.: Analysis and application of the element-free Galerkin method for nonlinear Sine-Gordon and generalized Sinh-Gordon equations. Comput. Math. Appl. 71, 1655–1678 (2016)MathSciNetCrossRef Li, X.L., Zhang, S.G., Wang, Y., et al.: Analysis and application of the element-free Galerkin method for nonlinear Sine-Gordon and generalized Sinh-Gordon equations. Comput. Math. Appl. 71, 1655–1678 (2016)MathSciNetCrossRef
14.
go back to reference Vahid, S.: Topology optimization using bi-directional evolutionary structural optimization based on the element-free Galerkin method. Eng. Optim. 48, 380–396 (2016)MathSciNetCrossRef Vahid, S.: Topology optimization using bi-directional evolutionary structural optimization based on the element-free Galerkin method. Eng. Optim. 48, 380–396 (2016)MathSciNetCrossRef
15.
go back to reference Dehghan, M., Abbaszadeh, M., Mohebbi, A.: The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate. J. Comput. Appl. Math. 286, 211–231 (2015)MathSciNetCrossRefMATH Dehghan, M., Abbaszadeh, M., Mohebbi, A.: The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate. J. Comput. Appl. Math. 286, 211–231 (2015)MathSciNetCrossRefMATH
16.
go back to reference Joldes, G.R., Wittek, A., Miller, K.: Adaptive numerical integration in element-free Galerkin methods for elliptic boundary value problems. Eng. Anal. Bound. Elem. 51, 52–63 (2015)MathSciNetCrossRef Joldes, G.R., Wittek, A., Miller, K.: Adaptive numerical integration in element-free Galerkin methods for elliptic boundary value problems. Eng. Anal. Bound. Elem. 51, 52–63 (2015)MathSciNetCrossRef
17.
go back to reference Chen, L., Liu, C., Ma, H.P., et al.: An interpolating local Petrov–Galerkin method for potential problems. Int. J. Appl. Mech. 6, 1450009 (2014)CrossRef Chen, L., Liu, C., Ma, H.P., et al.: An interpolating local Petrov–Galerkin method for potential problems. Int. J. Appl. Mech. 6, 1450009 (2014)CrossRef
18.
go back to reference Deng, Y.J., Liu, C., Peng, M.J., et al.: The interpolating complex variable element-free Galerkin method for temperature field problems. Int. J. Appl. Mech. 7, 1550017 (2015)CrossRef Deng, Y.J., Liu, C., Peng, M.J., et al.: The interpolating complex variable element-free Galerkin method for temperature field problems. Int. J. Appl. Mech. 7, 1550017 (2015)CrossRef
19.
go back to reference Peng, M.J., Cheng, Y.M.: A boundary element-free method (BEFM) for two-dimensional potential problems. Eng. Anal. Bound. Elem. 33, 77–82 (2009)MathSciNetCrossRefMATH Peng, M.J., Cheng, Y.M.: A boundary element-free method (BEFM) for two-dimensional potential problems. Eng. Anal. Bound. Elem. 33, 77–82 (2009)MathSciNetCrossRefMATH
20.
go back to reference Lian, H., Kerfriden, P., Bordas, S.: Implementation of regularized isogeometric boundary element methods for gradient-based shape optimization in two-dimensional linear elasticity. Int. J. Numer. Methods Eng. 106, 972–1017 (2016)MathSciNetCrossRefMATH Lian, H., Kerfriden, P., Bordas, S.: Implementation of regularized isogeometric boundary element methods for gradient-based shape optimization in two-dimensional linear elasticity. Int. J. Numer. Methods Eng. 106, 972–1017 (2016)MathSciNetCrossRefMATH
21.
go back to reference Chen, T., Raju, I.S.: A coupled finite element and meshless local Petrov–Galerkin method for two-dimensional potential problems. Comput. Methods Appl. Mech. Eng. 192, 4533–4550 (2003)CrossRefMATH Chen, T., Raju, I.S.: A coupled finite element and meshless local Petrov–Galerkin method for two-dimensional potential problems. Comput. Methods Appl. Mech. Eng. 192, 4533–4550 (2003)CrossRefMATH
22.
go back to reference Cheng, Y.M., Chen, M.J.: A boundary element-free method for linear elasticity. Acta. Mech. Sin. 19, 181–186 (2003)CrossRef Cheng, Y.M., Chen, M.J.: A boundary element-free method for linear elasticity. Acta. Mech. Sin. 19, 181–186 (2003)CrossRef
23.
go back to reference Zhang, Z., Liew, K.M., Cheng, Y.M.: Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems. Eng. Anal. Bound. Elem. 32, 100–107 (2008)CrossRefMATH Zhang, Z., Liew, K.M., Cheng, Y.M.: Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems. Eng. Anal. Bound. Elem. 32, 100–107 (2008)CrossRefMATH
24.
go back to reference Zhang, Z., Liew, K.M., Cheng, Y.M., et al.: Analyzing 2D fracture problems with the improved element-free Galerkin method. Eng. Anal. Bound. Elem. 32, 241–250 (2008)CrossRefMATH Zhang, Z., Liew, K.M., Cheng, Y.M., et al.: Analyzing 2D fracture problems with the improved element-free Galerkin method. Eng. Anal. Bound. Elem. 32, 241–250 (2008)CrossRefMATH
25.
go back to reference Zhang, Z., Li, D.M., Cheng, Y.M., et al.: The improved element-free Galerkin method for three-dimensional wave equation. Acta Mech. Sin. 28, 808–818 (2012)MathSciNetCrossRefMATH Zhang, Z., Li, D.M., Cheng, Y.M., et al.: The improved element-free Galerkin method for three-dimensional wave equation. Acta Mech. Sin. 28, 808–818 (2012)MathSciNetCrossRefMATH
26.
go back to reference Zhang, Z., Hao, S.Y., Liew, K.M., et al.: The improved element-free Galerkin method for two-dimensional elastodynamics problems. Eng. Anal. Bound. Elem. 37, 1576–1584 (2013)MathSciNetCrossRefMATH Zhang, Z., Hao, S.Y., Liew, K.M., et al.: The improved element-free Galerkin method for two-dimensional elastodynamics problems. Eng. Anal. Bound. Elem. 37, 1576–1584 (2013)MathSciNetCrossRefMATH
27.
go back to reference Li, K.T., Huang, A.X.: Mathematical aspect of the stream-function equations of compressible turbomachinery flows and their finite element approximation using optimal control. Comput. Methods Appl. Mech. Eng. 41, 175–194 (1983)MathSciNetCrossRefMATH Li, K.T., Huang, A.X.: Mathematical aspect of the stream-function equations of compressible turbomachinery flows and their finite element approximation using optimal control. Comput. Methods Appl. Mech. Eng. 41, 175–194 (1983)MathSciNetCrossRefMATH
28.
go back to reference Li, K.T., Huang, A.X., Zhang, W.L.: A dimension split method for the 3-D compressible Navier–Stokes equations in turbomachine. Commun. Numer. Methods Eng. 18, 1–14 (2002)MathSciNetCrossRefMATH Li, K.T., Huang, A.X., Zhang, W.L.: A dimension split method for the 3-D compressible Navier–Stokes equations in turbomachine. Commun. Numer. Methods Eng. 18, 1–14 (2002)MathSciNetCrossRefMATH
29.
go back to reference Li, K.T., Yu, J.P., Shi, F., et al.: Dimension splitting method for the three dimensional rotating Navier–Stokes equations. Acta Math. Appl. Sinica 28, 417–442 (2012)MathSciNetCrossRefMATH Li, K.T., Yu, J.P., Shi, F., et al.: Dimension splitting method for the three dimensional rotating Navier–Stokes equations. Acta Math. Appl. Sinica 28, 417–442 (2012)MathSciNetCrossRefMATH
30.
go back to reference Chen, H., Li, K.T., Wang, S.: A dimension split method for the incompressible Navier–Stokes equations in three dimensions. Int. J. Numer. Methods Fluids 73, 409–435 (2013)MathSciNetCrossRef Chen, H., Li, K.T., Wang, S.: A dimension split method for the incompressible Navier–Stokes equations in three dimensions. Int. J. Numer. Methods Fluids 73, 409–435 (2013)MathSciNetCrossRef
31.
33.
34.
go back to reference Hou, R.Y., Wei, H.B.: Dimension splitting algorithm for a three-dimensional elliptic equation. Int. J. Comput. Math. 89, 112–127 (2012)MathSciNetCrossRefMATH Hou, R.Y., Wei, H.B.: Dimension splitting algorithm for a three-dimensional elliptic equation. Int. J. Comput. Math. 89, 112–127 (2012)MathSciNetCrossRefMATH
35.
36.
go back to reference Bragin, M.D., Rogov, B.V.: On exact dimensional splitting for a multidimensional scalar quasilinear Hyperbolic conservation law. Dokl. Math. 94, 382–386 (2016)MathSciNetCrossRefMATH Bragin, M.D., Rogov, B.V.: On exact dimensional splitting for a multidimensional scalar quasilinear Hyperbolic conservation law. Dokl. Math. 94, 382–386 (2016)MathSciNetCrossRefMATH
37.
go back to reference D’souza, R.M., Margolus, N.H., Smith, M.A.: Dimension-splitting for simplifying diffusion in lattice-gas models. J. Stat. Phys. 107, 401–422 (2002)CrossRefMATH D’souza, R.M., Margolus, N.H., Smith, M.A.: Dimension-splitting for simplifying diffusion in lattice-gas models. J. Stat. Phys. 107, 401–422 (2002)CrossRefMATH
38.
go back to reference Cheng, H., Peng, M.J., Cheng, Y.M.: The hybrid improved complex variable element-free Galerkin method for three-dimensional potential problems. Eng. Anal. Bound. Elem. 84, 52–62 (2017)MathSciNetCrossRef Cheng, H., Peng, M.J., Cheng, Y.M.: The hybrid improved complex variable element-free Galerkin method for three-dimensional potential problems. Eng. Anal. Bound. Elem. 84, 52–62 (2017)MathSciNetCrossRef
39.
go back to reference Cheng, H., Peng, M.J., Cheng, Y.M.: A fast complex variable element-free Galerkin method for three-dimensional wave propagation problems. Int. J. Appl. Mech. 9, 1750090 (2017)CrossRef Cheng, H., Peng, M.J., Cheng, Y.M.: A fast complex variable element-free Galerkin method for three-dimensional wave propagation problems. Int. J. Appl. Mech. 9, 1750090 (2017)CrossRef
Metadata
Title
The dimension split element-free Galerkin method for three-dimensional potential problems
Authors
Z. J. Meng
H. Cheng
L. D. Ma
Y. M. Cheng
Publication date
02-02-2018
Publisher
The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences
Published in
Acta Mechanica Sinica / Issue 3/2018
Print ISSN: 0567-7718
Electronic ISSN: 1614-3116
DOI
https://doi.org/10.1007/s10409-017-0747-7

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