A Cartesian-grid collocation technique with integrated radial basis functions for mixed boundary value problems.

*(English)*Zbl 1188.74083Summary: In this paper, high-order systems are reformulated as first-order systems, which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretization with 1D-integrated radial basis function networks (1D-IRBFN) (Numer. Meth. Partial Differential Equations 2007; 23:1192-1210). The present method is enhanced by a new boundary interpolation technique based on 1D-IRBFN, which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well-known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates.

##### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74B05 | Classical linear elasticity |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

##### Keywords:

RBF; collocation method; elasticity; Cartesian grid; mixed formulation; first-order system; volumetric locking; incompressibility
PDF
BibTeX
Cite

\textit{P. B. H. Le} et al., Int. J. Numer. Methods Eng. 82, No. 4, 435--463 (2010; Zbl 1188.74083)

Full Text:
DOI

##### References:

[1] | Bordas, A simple error estimator for extended finite elements, Communications in Numerical Methods in Engineering 24 pp 961– (2008) · Zbl 1156.65093 |

[2] | Liu, Meshfree Methods: Moving Beyond the Finite Element Method (2003) |

[3] | Belytschko, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) · Zbl 0796.73077 |

[4] | Le P, Mai-Duy N, Tran-Cong T, Baker G. A meshless IRBF-based numerical simulation of adiabatic shear band formation in one dimension. In Conference on Nonlinear Analysis and Engineering Mechanics Today, Quoc Son N, Dung N (eds), HoChiMinh City, Vietnam, December 2006; 10, CD paper No. 28. |

[5] | Le P, Mai-Duy N, Tran-Cong T, Baker G. Meshless IRBF-based numerical simulation of dynamic strain localization in quasi-britle materials. Ninth U.S. National Congress on Computation Mechanics, San Francisco, U.S.A., July 2007; CD page 63. |

[6] | Le P, Mai-Duy N, Tran-Cong T, Baker G. An IRBFN Cartesian grid method based on displacement???stress formulation for 2D Elasticity problems. Eighth World Congress on Computational Mechanics (WCCM8), Venice, Italy, June???July 2008. |

[7] | Liu, An Introduction to Meshfree Methods and their Programming (2005) |

[8] | Zhang, Meshless method based on collocation with radial basis functions, Computational Mechanics 26 pp 333– (2000) · Zbl 0986.74079 |

[9] | Li, Hermite-cloud: a novel true meshless method, Computational Mechanics 33 pp 30– (2003) · Zbl 1061.65128 |

[10] | Zhang, Least-squares collocation meshless method, International Journal for Numerical Methods in Engineering 51 pp 1089– (2001) · Zbl 1056.74064 |

[11] | Onate, A finite point method for elasticity problem, Computers and Structures 79 pp 2151– (2001) |

[12] | Liu, A meshfree method: Weak???Strong (MWS) form method, for 2D solids, Computational Mechanics 33 pp 2– (2003) · Zbl 1063.74105 |

[13] | Gu, A meshfree Weak???Strong (MWS) form method for time dependent problems, Computational Mechanics 35 pp 134– (2005) · Zbl 1109.74371 |

[14] | Pan, Meshless Galerkin least-squares method, Computational Mechanics 35 pp 182– (2005) · Zbl 1143.74390 |

[15] | Hu, Weighted radial basis collocation method for boundary value problems, International Journal for Numerical Methods in Engineering 69 pp 2736– (2006) |

[16] | Chen, Reproducing kernel enhanced radial basis collocation method, International Journal for Numerical Methods in Engineering 75 pp 600– (2008) · Zbl 1195.74278 |

[17] | Atluri, A new implementation of the meshless finite volume method, through the MLPG ???mixed??? approach, CMES: Computer Modeling in Engineering and Sciences 6 (6) pp 491– (2004) · Zbl 1151.74424 |

[18] | Atluri, Meshless local Petrov???Galerkin (MLPG) mixed collocation method for elasticity problems, CMES: Computer Modeling in Engineering and Sciences 14 (3) pp 141– (2006) · Zbl 1357.74079 |

[19] | Libre, A stabilized RBF collocation scheme for Neumann type boundary value problems, CMES: Computer Modeling in Engineering and Sciences 24 pp 63– (2008) · Zbl 1232.65156 |

[20] | Lee, Meshfree point collocation method for elasticity and crack problems, International Journal for Numerical Methods in Engineering 61 pp 22– (2004) · Zbl 1079.74667 |

[21] | Cai, First order system least-square for second-order partial differential equations: part 1, SIAM Journal on Numerical Analysis 31 pp 1785– (1994) |

[22] | Cai, First order system least-squares for second-order partial differential equations: part 2, SIAM Journal on Numerical Analysis 34 pp 425– (1997) · Zbl 0912.65089 |

[23] | Cai, First order system least-squares for Stokes equations with application to linear elasticity, SIAM Journal on Numerical Analysis 34 pp 1727– (1997) · Zbl 0901.76052 |

[24] | Cai, First order system least-squares (FOSLS) for planar linear elasticity: pure traction problem, SIAM Journal on Numerical Analysis 35 pp 320– (1998) · Zbl 0968.74061 |

[25] | Cai, First order system least-squares for linear elasticity: numerical results, SIAM Journal on Scientific Computing 21 pp 1706– (2000) · Zbl 0988.74061 |

[26] | Cai, First order system least-squares for stress???displacement formulation linear elasticity, SIAM Journal on Numerical Analysis 41 pp 715– (2003) · Zbl 1063.74099 |

[27] | Lee, First order system least-squares for elliptic problems with Robin boundary conditions, SIAM Journal on Numerical Analysis 37 pp 70– (1999) · Zbl 0942.65140 |

[28] | Jiang, The least-squares finite element method in elasticity???part 1: plane stress with drilling degrees of freedom, International Journal for Numerical Methods in Engineering 53 pp 621– (2002) · Zbl 1112.74518 |

[29] | Park, The least-squares meshfree method, International Journal for Numerical Methods in Engineering 52 pp 997– (2001) · Zbl 0992.65123 |

[30] | Kwon, The least-squares meshfree method for soving linear elastic problems, Computational Mechanics 30 pp 196– (2003) |

[31] | Tran-Cong, A Cartesian-grid collocation method based on radial-basis-function networks for solving PDEs in irregular domains, Numerical Methods for Partial Differential Equations 23 pp 1192– (2007) · Zbl 1129.65089 |

[32] | Roache, Computational Fluid Dynamics (1980) |

[33] | Le, A numerical study of strain localization in elasto-therno-viscoplastic materials using radial basis function networks, CMC: Computers, Materials and Continua 5 pp 129– (2007) |

[34] | Le, A meshless modeling of dynamic strain localization in quasi-britle materials using radial basis function networks, CMES: Computer Modeling in Engineering and Sciences 25 (1) pp 43– (2008) |

[35] | Dolbow, Volumetric locking in the element-free Galerkin method, International Journal for Numerical Methods in Engineering 46 pp 925– (1999) · Zbl 0967.74079 |

[36] | Chen, An improve reproducing kernel particle methods for nearly incompressible finite elasticity, Computer Methods in Applied Mechanics and Engineering 181 pp 117– (2000) |

[37] | Madych, Miscellaneous error bounds for multiquadric and related interpolators, Computers and Mathematics with Applications 24 pp 121– (1992) · Zbl 0766.41003 |

[38] | Mai-Duy, Numerical solution of differential equations using multiquadric radial basic function networks, Neural Networks 14 pp 185– (2001) · Zbl 1047.76101 |

[39] | Mai-Duy, Approximation of function and its derivatives using radial basis function networks, Applied Mathematical Modeling 27 pp 197– (2003) · Zbl 1024.65012 |

[40] | Mai-Duy, An efficient indirect RBFN-based method for numerical solution of PDEs, Numerical Methods for Partial Differential Equations 21 pp 770– (2005) · Zbl 1077.65125 |

[41] | Wertz, The role of the multiquadric shape parameters in solving elliptic partial differential equations, Computers and Mathematics with Applications 51 pp 1335– (2006) · Zbl 1146.65078 |

[42] | Timoshenko, Theory of Elasticity (1970) |

[43] | Liu, Edge-based smooth point interpolation methods, International Journal of Computational Methods 5 (4) pp 621– (2008) |

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.