×

zbMATH — the first resource for mathematics

Meshless local B-spline-FD method and its application for 2D heat conduction problems with spatially varying thermal conductivity. (English) Zbl 1334.80012
Summary: In this paper, a new class of meshless methods based on local collocation and B-spline basis functions is presented for solving elliptic problems. The proposed approach is called as meshless local B-spline basis functions based finite difference (local B-FD) method. The method was straightforward to develop and program as it was truly meshless. Only scattered nodal distribution was required hence avoiding at all mesh connectivity for field variable approximation and integration. In the method, any governing equations were discretized by B-spline approximation in the spirit of FD technique using local B-spline collocation i.e. any derivative at a point or node was stated as neighboring nodal values based on the B-spline interpolants. In addition, as B-spline basis functions pose favorable properties such as (i) easy to construct to any arbitrary order/degree, (ii) have partition of unity property, and (iii) can be easily designed to pose the Kronecker delta property, the shape function construction as well as the imposition of boundary conditions can be incorporated efficiently in the present method. The applicability and capability of the present local B-FD method were demonstrated through several heat conduction problems with heat generation and spatially varying conductivity.

MSC:
80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
80A20 Heat and mass transfer, heat flow (MSC2010)
Software:
LAF-SPEM
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Griebel, M.; Schweitzer, M. A., A particle-partition of unity method for the solution of elliptic, parabolic, and hyperbolic pdes, SIAM J. Sci. Comput., 22, 853-890, (2000) · Zbl 0974.65090
[2] Lucy, L. B., A numerical approach to the testing of the fission hypothesis, Astron. J., 82, 1013-1024, (1977)
[3] Gingold, R. A.; Monaghan, J. J., Smoothed particle hydrodynamics: theory and application to non-spherical stars, Mon. Not. R. Astron. Soc., 181, 375-389, (1977) · Zbl 0421.76032
[4] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method: diffuse approximation and diffuse elements, Comput. Mech., 10, 307-318, (1992) · Zbl 0764.65068
[5] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin method, Int. J. Numer. Methods Eng., 37, 229-256, (1994) · Zbl 0796.73077
[6] Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, Int. J. Numer. Methods Fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[7] 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
[8] Atluri, S. N.; Shen, S. P., The meshless local Petrov-Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods, CMES, 3, 1, 11-51, (2002) · Zbl 0996.65116
[9] Atluri, S. N.; Shen, S. P., The meshless local Petrov-Galerkin (MLPG) method, (2002), Tech Science Press USA · Zbl 1012.65116
[10] Kansa, E. J., Multiquadric - a scattered data approximation scheme with applications to computational fluid dynamics II, Comput. Math. Appl., 19, 8/9, 147-161, (1990) · Zbl 0850.76048
[11] Shu, C.; Ding, H.; Yeo, K. S., Local radial basis function-based differential quadrature method and its application to solve two dimensional incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 192, 941-954, (2003) · Zbl 1025.76036
[12] Tolstykh, A. I.; Shirobokov, D. A., On using radial basis functions in a ‘‘finite difference mode” with applications to elasticity problems, Comput. Mech., 33, 68-79, (2003) · Zbl 1063.74104
[13] Shan, Y. Y.; Shu, C.; Qin, N., Multiquadric finite difference (MQ-FD) method and its application, Adv. Appl. Math. Mech., 1, 615-638, (2009)
[14] Roque, C. M.C.; Cunha, D.; Shu, C.; Ferreira, A. J.M., A local radial basis functions - finite differences technique for the analysis of composite plates, Eng. Anal. Boundary Elem., 35, 363-374, (2011) · Zbl 1259.74078
[15] Le, P. B.H.; Mai-Duy, N.; Tran-Cong, T.; Baker, G., A Cartesian-grid collocation technique with integrated radial basis functions for mixed boundary value problems, Int. J. Numer. Methods Eng., 82, 435-463, (2010) · Zbl 1188.74083
[16] Liu, G. R.; Yu, G. T.; Dai, K. Y., Assessment and applications of point interpolation methods for computational mechanics, Int. J. Numer. Methods Eng., 59, 10, 1373-1397, (2004) · Zbl 1041.74562
[17] Liu, G. R., Meshfree methods: moving beyond the finite element method, (2009), CRC Press USA
[18] Yu, Y.; Chen, Z., A 3-D radial point interpolation method for meshless time-domain modelling, IEEE Trans. Microw. Theory Tech., 57, 8, 2015-2020, (2009)
[19] Yu, Y.; Chen, Z., Towards the development of an unconditionally stable time-domain meshless method, IEEE Trans. Microw. Theory Tech., 58, 3, 578-586, (2010)
[20] Yu, Y.; Chen, Z., The CPML absorbing boundary conditions for the unconditionally stable meshless modelling, IEEE Antennas Wirel. Propag. Lett., 11, 468-472, (2012)
[21] T. Kaufmann, C. Fumeaux, R. Vahldieck, The meshless radial point interpolation method for time-domain electromagnetics, in: Proceedings of IEEE MTT-S International Microwave Symposium Digest, IEEE, Atlanta, GA, USA, 2008, pp. 61-64.
[22] Kaufmann, T.; Yu, Y.; Engstrom, C.; Chen, Z.; Fumeaux, C., Recent developments of the meshless radial point interpolation method for time-domain electromagnetics, Int. J. Numer. Model. Electron. Networks Devices Fields, 25, 468-489, (2012)
[23] Kaufmann, T.; Engstrom, C., High-order absorbing boundary conditions for the meshless radial point interpolation method in the frequency domain, Int. J. Numer. Model. Electron. Networks Devices Fields, 26, 478-492, (2013)
[24] Ala, G.; Francomano, E., An improved smoothed particle electromagnetics method in 3D time domain simulations, Int. J. Numer. Model. Electron. Networks Devices Fields, 25, 4, 325-337, (2012)
[25] Ala, G.; Francomano, E., A marching-on in time meshless kernel based solver for full-wave electromagnetic simulation, Numer. Algorithms, 62, 4, 541-558, (2013) · Zbl 1269.78016
[26] Ala, G.; Francomano, E., A multisphere particle numerical model for non-invasive investigations of neuronal human brain activity, Progr. Electromagn. Res. Lett., 36, 143-153, (2013)
[27] Ala, G.; Di Blasi, G.; Francomano, E., A numerical meshless particle method in solving the magnetoencephalography forward problem, Int. J. Numer. Model. Electron. Networks Devices Fields, 25, 5-6, 428-440, (2012)
[28] Wang, H.; Qin, Q. H.; Kang, Y. L., A meshless model for transient heat conduction in functionally graded materials, Comput. Mech., 38, 51-60, (2006) · Zbl 1097.80001
[29] Gao, X. W., A meshless BEM for isotropic heat conduction problems with heat generation and spatially varying conductivity, Int. J. Numer. Methods Eng., 66, 1411-1431, (2006) · Zbl 1116.80021
[30] Wu, X. H.; Shen, S. P.; Tao, W. Q., Meshless local Petrov-Galerkin collocation method for two-dimensional heat conduction problems, CMES, 22, 65-76, (2007) · Zbl 1152.80329
[31] Singh, A.; Singh, I. V.; Prakash, R., Meshless element free Galerkin method for unsteady nonlinear heat transfer problems, Int. J. Heat Mass Transfer, 50, 1212-1219, (2007) · Zbl 1124.80366
[32] Singh, I. V.; Tanaka, M., Heat transfer analysis of composite slabs using meshless element free Galerkin method, Comput. Mech., 38, 521-532, (2006) · Zbl 1168.80309
[33] Sladek, J.; Sladek, V.; Tan, C. L.; Atluri, S. N., Analysis of transient heat conduction in 3D anisotropic functionally graded solids, by the MLPG method, CMES, 32, 161-174, (2008) · Zbl 1232.80006
[34] Li, Q. H.; Chen, S. S.; Zeng, J. H., A meshless model for transient heat conduction analyses of 3D axisymmetric functionally graded solids, Chin. Phys. B, 22, 12, (2013), 120204-1-5
[35] Li, Q. H.; Chen, S. S.; Kou, G. X., Transient heat conduction analysis using the MLPG method and modified precise time step integration method, J. Comput. Phys., 30, 2736-2750, (2011) · Zbl 1218.65108
[36] Chen, L.; Liew, K. M., A local Petrov-Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems, Comput. Mech., 47, 455-467, (2011) · Zbl 1241.80005
[37] Soleimani, S.; Jalaal, M.; Bararnia, H.; Ghasemi, E.; Ganji, D. D.; Mohammadi, F., Local RBF-DQ method for two-dimensional transient heat conduction problem, Int. Commun. Heat Mass Transfer, 37, 1411-1418, (2010)
[38] Dai, B.; Zheng, B.; Liang, Q.; Wang, L., Numerical solution of transient heat conduction problems using improved meshless local Petrov-Galerkin method, Appl. Math. Comput., 219, 10044-10052, (2013) · Zbl 1307.80008
[39] Wen, P. H.; Aliabadi, M. H., An improved meshless collocation method for elasto-static and elasto-dynamic problems, Commun. Numer. Methods Eng., 24, 8, 635-651, (2008) · Zbl 1159.74461
[40] Gu, Y. T.; Zhang, L. C., Coupling of the meshfree and finite element methods for determination of the crack tip fields, Eng. Fract. Mech., 75, 986-1004, (2008)
[41] Zhang, X.; Zhang, P.; Zhang, L., An improved meshless method with almost interpolation property for isotropic heat conduction problems, Eng. Anal. Boundary Elem., 37, 850-859, (2013) · Zbl 1287.80007
[42] Ren, H.; Cheng, J.; Huang, A., The complex variable interpolating moving least-squares method, Appl. Math. Comput., 219, 1724-1736, (2012) · Zbl 1291.65138
[43] Sun, W. W.; Wu, J. M.; Zhang, X. P., Nonconforming spline collocation methods in irregular domains, Numer. Methods Partial Differ. Eqn., 23, 1509-1529, (2007) · Zbl 1140.65088
[44] Cooper, K. D., Domain-imbedding alternating direction method for linear elliptic equations on irregular regions using collocation, Numer. Methods Partial Differ. Eqn., 9, 93-106, (1993) · Zbl 0764.65069
[45] Van Blerk, J. J.; Botha, J. F., Numerical solution of partial differential equations on curved domains by collocation, Numer. Methods Partial Differ. Eqn., 9, 357-371, (1993) · Zbl 0780.65069
[46] Höllig, K.; Reif, U.; Wipper, J., Weighted extended B-spline approximation of Dirichlet problems, SIAM J. Numer. Anal., 39, 442-462, (2001) · Zbl 0996.65119
[47] Höllig, K.; Apprich, C.; Streit, A., Introduction to the web-method and its applications, Adv. Comput. Math., 23, 215-237, (2005) · Zbl 1070.65118
[48] de Boor, C., On calculating with B-splines, J. Approximation Theory, 6, l, 50-62, (1972) · Zbl 0239.41006
[49] Cox, M., The numerical evaluation of B-spline, J. Inst. Math. Appl., 10, 134-149, (1972) · Zbl 0252.65007
[50] de Boor, C., A practical guide to splines, (2001), Springer New York · Zbl 0987.65015
[51] Farin, G., Curves and surfaces for computer aided geometric design, (2002), Academic Press San Diego, CA
[52] Piegl, L.; Tiller, W., The NURBS book, (1995), Springer New York · Zbl 0828.68118
[53] Sarra, S. A., Integrated multiquadric radial basis function approximation methods, Comput. Math. Appl., 51, 1283-1296, (2006) · Zbl 1146.65327
[54] Kassab, A. J.; Divo, E., A generalized boundary integral equation for isotropic heat conduction with spatially varying thermal conductivity, Eng. Anal. Boundary Elem., 18, 273-286, (1996)
[55] Ochiai, Y., Two-dimensional steady heat conduction in functionally gradient materials by triple-reciprocity boundary element method, Eng. Anal. Boundary Elem., 28, 1445-1453, (2004) · Zbl 1151.80328
[56] Atluri, S. N.; Han, Z. D.; Rajendran, A. M., A new implementation of the meshless finite volume method, through the MLPG “mixed” approach, CMES, 6, 6, 491-513, (2004) · Zbl 1151.74424
[57] Atluri, S. N.; Liu, H. T.; Han, Z. D., Meshless local Petrov-Galerkin (MLPG) mixed collocation method for elasticity problems, CMES, 14, 3, 141-152, (2006) · Zbl 1357.74079
[58] Fonseca, A. R.; Correa, B. C.; Silva, E. J.; Mesquita, R. C., Improving the mixed formulation for meshless local Petrov-Galerkin method, IEEE Trans. Magn., 46, 8, 2907-2910, (2010)
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.