# zbMATH — the first resource for mathematics

A meshless local boundary integral equation method for two-dimensional steady elliptic problems. (English) Zbl 1206.74023
Summary: A novel meshless local boundary integral equation (LBIE) method is proposed for the numerical solution of two-dimensional steady elliptic problems, such as heat conduction, electrostatics or linear elasticity. The domain is discretized by a distribution of boundary and internal nodes. From this nodal points’ cloud a “background” mesh is created by a triangulation algorithm. A local form of the singular boundary integral equation of the conventional boundary element method is adopted. Its local form is derived by considering a local domain of each node, comprising by the union of neighboring “background” triangles. Therefore, the boundary shape of this local domain is a polygonal closed line. A combination of interpolation schemes is taken into account. Interpolation of boundary unknown field variables is accomplished through boundary element shape functions. On the other hand, the radial basis point interpolation function method is employed for interpolating the unknown interior fields. Essential boundary conditions are imposed directly due to the Kronecker delta-function property of the boundary element interpolation functions. After the numerical evaluation of all boundary integrals, a banded stiffness matrix is constructed, as in the finite element method. Several potential and elastostatic benchmark problems in two dimensions are solved numerically. The proposed meshless LBIE method is also compared with other numerical methods in order to demonstrate its efficiency, accuracy and convergence.

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74S15 Boundary element methods applied to problems in solid mechanics
Full Text:
##### References:
 [1] Gingold RA, Moraghan JJ (1977) Smoothed particle hydrodynamics: theory and applications to non-spherical stars. Mon Notice R Astron Soc 18: 375–389 · Zbl 0421.76032 [2] Randles PW, Libersky LD (1996) Smoothed particle hydrodynamics: some recent improvements and applications. Comput Methods Appl Mech Eng 139: 375–408 · Zbl 0896.73075 [3] Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37: 229–256 · Zbl 0796.73077 [4] Lu YY, Belytschko T, Tabbara M (1995) Element-free Galerkin method for wave propagation and dynamic fracture. Comput Methods Appl Mech Eng 26: 31–153 · Zbl 1067.74599 [5] Gu YT, Liu GR (2001) A coupled element free Galerkin/boundary element method for stress analysis of two-dimensional solids. Comput Methods Appl Mech Eng 190: 4405–4419 [6] Chen JS, Pan C, Wu CT, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of nonlinear structures. Comput Methods Appl Mech Eng 139: 195–227 · Zbl 0918.73330 [7] Melenk JM, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139: 289–314 · Zbl 0881.65099 [8] Babuška I, Melenk JM (1997) The partition of unity method. Int J Numer Methods Eng 40: 727–758 · Zbl 0949.65117 [9] Oñate E, Idelsohn S, Zienkiewicz OC, Taylor RL (1996) A finite point method in computational mechanics: applications to convective transport and fluid flow. Int J Numer Methods Eng 39: 3839–3866 · Zbl 0884.76068 [10] Gu YT, Liu GR (2001) A local point interpolation method for static and dynamic analysis of thin beams. Comput Methods Appl Mech Eng 190: 5515–5528 · Zbl 1059.74060 [11] Atluri SN, Zhu T (1998) A new meshless local Petrov-Galerkin (MLPG) approach in computation mechanics. Comput Mech 22: 117–127 · Zbl 0932.76067 [12] Atluri SN, Kim H-G, Cho JY (1999) A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG) and local boundary integral equation (LBIE) methods. Comput Mech 24: 348–372 · Zbl 0977.74593 [13] Wendland H (1999) Meshless Galerkin method using radial basis functions. Math Comput 68: 1521–1531 · Zbl 1020.65084 [14] Atluri SN, Shen S (2002) The meshless local Petrov-Galerkin (MLPG) method: a simple and less-costly alternative to the finite element and boundary element methods. Comput Model Eng Sci 3: 11–51 · Zbl 0996.65116 [15] Zhu T, Zhang J-D, Atluri SN (1998) A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Comput Mech 21: 223–235 · Zbl 0920.76054 [16] Zhu T, Zhang J, Atluri SN (1999) A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems. Eng Anal Boundary Elem 23: 375–389 · Zbl 0957.74077 [17] Atluri SN, Sladek J, Sladek V, Zhu T (2000) The local boundary integral equation (LBIE) and it’s meshless implementation for linear elasticity. Comput Mech 25: 180–198 · Zbl 0976.74078 [18] Mukherjee YX, Mukherjee S (1997) The boundary node method for potential problems. Int J Numer Methods Eng 40: 797–815 · Zbl 0885.65124 [19] Kothnur VS, Mukherjee S, Mukherjee YX (1999) Two-dimensional linear elasticity by the boundary node method. Int J Solids Struct 36: 1129–1147 · Zbl 0937.74074 [20] Sukumar N, Moran B, Belytschko T (1998) The natural element method in solid mechanics. Int J Numer Methods Eng 43: 839–887 · Zbl 0940.74078 [21] Sukumar N, Moran B, Yu Semenov A, Belikov VV (2001) Natural neighbour Galerkin methods. Int J Numer Methods Eng 50: 1–27 · Zbl 1082.74554 [22] Sladek J, Sladek V, Atluri SN (2000) Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties. Comput Mech 24: 456–462 · Zbl 0961.74073 [23] Sellountos EJ, Vavourakis V, Polyzos D (2005) A new singular/hypersingular MLPG (LBIE) method for 2D elastostatics. Comput Model Eng Sci 7: 35–48 · Zbl 1179.74179 [24] Bodin A, Ma J, Xin XJ, Krishnaswami P (2006) A meshless integral method based on regularized boundary integral equation. Comput Methods Appl Mech Eng 195: 6258–6286 · Zbl 1157.74047 [25] Vavourakis V, Polyzos D (2007) A MLPG4(LBIE) formulation in elastostatics. Comput Mater Continua 5: 185–196 · Zbl 1153.74385 [26] Sladek J, Sladek V, Atluri SN (2001) A pure contour formulation for the meshless local boundary integral equation method in thermoelasticity. Comput Model Eng Sci 2: 423–434 · Zbl 1060.74069 [27] Sladek J, Sladek V (2003) Application of local boundary integral equation method into micropolar elasticity. Eng Anal Boundary Elem 27: 81–90 · Zbl 1021.74047 [28] Sladek J, Sladek V, Van Keer R (2003) Meshless local boundary integral equation method for 2D elastodynamic problems. Int J Numer Methods Eng 57: 235–249 · Zbl 1062.74643 [29] Sladek J, Sladek V, Mang HA (2003) Meshless LBIE formulations for simply supported and clamped plates under dynamic load. Comput Struct 81: 1643–1651 [30] Sellountos EJ, Polyzos D (2003) A MLPG (LBIE) method for solving frequency domain elastic problems. Comput Model Eng Sci 4: 619–636 · Zbl 1064.74170 [31] Sellountos EJ, Polyzos D (2005) A meshless local boundary integral equation method for solving transient elastodynamic problems. Comput Mech 35: 265–276 · Zbl 1109.74369 [32] Sellountos EJ, Sequeira A (2008) A hybrid multi-region BEM/ LBIE-RBF velocity-vorticity scheme for the two-dimensional Navier-Stokes equations. Comput Model Eng Sci 23: 127–147 · Zbl 1232.76041 [33] Sladek J, Sladek V, Krivacek J, Zhang Ch (2003) Local BIEM for transient heat conduction analysis in 3-D axisymmetric functionally graded solids. Comput Mech 32: 169–176 · Zbl 1038.80510 [34] Vavourakis V, Polyzos D (2006) A MLPG4 (LBIE) formulation for solving axisymmetric problems. In: Sladek J, Sladek V (eds) Advances in meshless methods. Tech Science Press, Forsyth, USA · Zbl 1180.74068 [35] Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37: 141–158 · Zbl 0469.41005 [36] Vavourakis V, Sellountos EJ, Polyzos D (2006) A comparison study on different MLPG(LBIE) formulations. Comput Model Eng Sci 13: 171–184 · Zbl 1232.65163 [37] Atluri SN, Han ZD, Rajendran AM (2004) A new implementation of the meshless finite volume method, through the MLPG ”mixed” approach. Comput Model Eng Sci 6: 491–513 · Zbl 1151.74424 [38] Augarde CE, Deeks AJ (2005) On the effects of nodal distributions for imposition of essential boundary conditions in the MLPG meshfree method. Int J Numer Methods Eng 21: 389–395 · Zbl 1073.65140 [39] Wang JG, Liu GR (2002) On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Comput Methods Appl Mech Eng 191: 2611–2630 · Zbl 1065.74074 [40] Vavourakis V (2008) A local hypersingular boundary integral equation method using a triangular background mesh. Comput Model Eng Sci 36: 119–146 · Zbl 1232.74120 [41] Banerjee PK (1994) The boundary element methods in engineering. McGraw-Hill, New York [42] Brebbia CA, Dominguez J (1998) Boundary elements: an introductory course. Computational Mechanics Publications, Southampton · Zbl 0691.73033 [43] Guiggiani M, Krishnasamy G, Rudolphi TJ, Rizzo FJ (1992) A general algorithm for the numerical solution of hypersingular boundary integral equations. ASME J Appl Mech 59: 604–614 · Zbl 0765.73072 [44] Guiggiani M (1998) Formulation and numerical treatment of boundary integral equations with hypersingular kernels. In: Sladek V, Sladek J (eds) Singular integrals in boundary element methods. Computational Mechanics Publications, Southampton [45] Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York [46] Sellountos EJ, Polyzos D (2005) A MLPG(LBIE) approach in combination with BEM. Comput Methods Appl Mech Eng 194: 859–875 · Zbl 1112.74555
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.