×

zbMATH — the first resource for mathematics

Interpolating meshless local Petrov-Galerkin method for steady state heat conduction problem. (English) Zbl 1464.80031
Summary: In many meshfree methods, moving least squares scheme (MLS) has been used to generate meshfree shape functions. Imposition of Dirichlet boundary conditions is difficult task in these methods as the MLS approximation is devoid of Kronecker delta property. A new variant of the MLS approximation scheme, namely interpolating moving least squares scheme, possesses Kronecker delta property. In the current work, a novel interpolating meshless local Petrov-Galerkin (IMLPG) method has been developed based on the interpolating MLS approximation for two and three dimensional steady state heat conduction in regular and complex domain. The interpolating MLPG method shows two advantages over standard meshless local Petrov-Galerkin (MLPG) method i.e. higher computational efficiency and ease to impose EBCs at similar accuracy level. Performance of three different test functions in-conjunction with interpolating MLPG method has been shown.

MSC:
80M15 Boundary element methods applied to problems in thermodynamics and heat transfer
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
80A19 Diffusive and convective heat and mass transfer, heat flow
80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Garg, R.; Thakur, H. C.; Tripathi, B., A Review of applications of meshfree methods in the area of heat transfer and fluid flow : MLPG method in particular, IRJET, 2, 329-338, (2015)
[2] Sladek, J.; Stanak, P.; Han, Z. D., Applications of the MLPG method in engineering & sciences: a review, Comput Model Eng Sci, 92, 423-475, (2013)
[3] Fries, T.; Matthias, H., Classification and overview of meshfree methods, (2004)
[4] Liu, G. R., Meshfree methods: moving beyond the finite element method, second ed, (2003), CRC Press
[5] Thakur, H.; Singh, K. M.; Sahoo, P. K., Meshless local Petrov-Galerkin method for nonlinear heat conduction problems, Numer Heat Transf Part B Fundam, 56, 393-410, (2009)
[6] Thakur, H.; Singh, K. M.; Sahoo, P. K., MLPG analysis of nonlinear heat conduction in irregular domains, Comput Model Eng Sci, 68, 117-149, (2010)
[7] Thakur, H.; Singh, K. M.; Sahoo, P. K., Phase change problems using the MLPG method, Numer Heat Transf Part A Appl, 59, 438-458, (2011)
[8] Atluri, S. N.; Kim, H. G.; Cho, J. Y., Critical assessment of the truly meshless local Petrov-Galerkin (MLPG) and local boundary integral equation (LBIE) methods, Comput Mech, 24, 348-372, (1999) · Zbl 0977.74593
[9] Sartoretto, F.; Torino, V., The DMLPG meshless technique for poisson problems, Appl Math Sci, 8, 8233-8250, (2014) · Zbl 1383.62262
[10] Zhang, Z.; Wang, J.; Cheng, Y.; Liew, K. M., The improved element-free Galerkin method for three-dimensional transient heat conduction problems, Sci China Physics Mech Astron, 56, 1568-1580, (2013)
[11] Zhang, Z.; Liew, K. M.; Cheng, Y.; Lee, Y. Y., Analyzing 2D fracture problems with the improved element-free Galerkin method, Eng Anal Bound Elem, 32, 241-250, (2008) · Zbl 1244.74240
[12] Ren, H. P.; Cheng, Y. M.; Zhang, W., An improved boundary element-free method (IBEFM) for two-dimensional potential problems, Chin Phys B, 18, 4065-4073, (2009)
[13] Ren, H. P.; Cheng, Y. M.; Zhang, W., An interpolating boundary element-free method (IBEFM) for elasticity problems, Sci China Physics Mech Astron, 53, 758-766, (2010)
[14] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math Comput, 37, (1981), 141-141 doi:10.1090/S0025-5718-1981-0616367-1 · Zbl 0469.41005
[15] Zhang, Y. J.; Wang, E. Z., IMLS method of square sphere, Eng Mech, 23, 60-64, (2006)
[16] Netuzhylov, H., Meshfree collocation solution of boundary value problems via interpolating moving least squares, Commun Numer Methods Eng, 22, 893-899, (2006) · Zbl 1105.65356
[17] Netuzhylov, H., Enforcement of boundary conditions in meshfree methods using interpolating moving least squares, Eng Anal Bound Elem, 32, 512-516, (2008) · Zbl 1244.65190
[18] Netuzhylov, H.; Zilian, A., Space-time meshfree collocation method: methodology and application to initial-boundary value problems, Int J Numer Methods Eng, 80, 355-380, (2009) · Zbl 1176.65114
[19] Netuzhylov, H.; Zilian, A., Meshfree collocation method for implicit time integration of ODEs, Int J Comput Methods, 8, 119-137, (2011) · Zbl 1270.65043
[20] Ren, H.; Cheng, Y., The interpolating element-free Galerkin (IEFG) method for two-dimensional elasticity problems, Int J Appl Mech, 3, 735-738, (2011)
[21] Ren, H.; Cheng, J.; Huang, A., The complex variable interpolating moving least-squares method, Appl Math Comput, 219, 1724-1736, (2012) · Zbl 1291.65138
[22] Ren, H.; Cheng, Y., The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems, Eng Anal Bound Elem, 36, 873-880, (2012) · Zbl 1352.65539
[23] Ren, H.; Wang, L.; Zhao, N. A., An interpolating element-free Galerkin method for steady-state heat conduction problems, Int J Appl Mech, 6, (2014), ID-1450024 doi:10.1142/S1758825114500240.
[24] Zhao, N.; Ren, H., The interpolating element-free Galerkin method for 2D transient heat conduction problems, Math Probl Eng, 2014, (2014), ID-712834 doi:10.1155/2014/712834.
[25] Zhang, G.; Zhang, F.; Ren, H., Accuracy analysis of interpolating element-free Galerkin (IEFG) method in solving transient heat conduction problems, Int J Appl Mech, 8, (2016), ID-1650078 doi:10.1142/S1758825116500782.
[26] Ren, H.; Pei, K.; Wang, L., Error analysis for moving least squares approximation in 2D space, Appl Math Comput, 238, 527-546, (2014) · Zbl 1337.65157
[27] Cheng, Y.; Wang, W.; Peng, M.; Zhang, Z., Mathematical aspects of meshless methods, Math Probl Eng, 2014, (2014), ID-756297 doi:10.1155/2014/756297
[28] Sun, F. X.; Wang, J. F.; Cheng, Y. M.; Huang, A. X., Error estimates for the interpolating moving least-squares method in n-dimensional space, Appl Numer Math, 98, 79-105, (2015) · Zbl 1329.65280
[29] Wang, J. F.; Hao, S. Y.; Cheng, Y. M., The error estimates of the interpolating element-free Galerkin method for two-point boundary value problems, Math Probl Eng, 2014, (2014), ID-641592 doi:10.1155/2014/641592
[30] Wang, J. F.; Sun, F. X.; Cheng, Y. M.; Huang, A. X., Error estimates for the interpolating moving least-squares method, Appl Math Comput, 245, 321-324, (2014) · Zbl 1335.65018
[31] Cheng, Y. M.; Bai, F. N.; Peng, M. J., A novel interpolating element-free Galerkin (IEFG) method for two-dimensional elastoplasticity, Appl Math Model, 38, 5187-5197, (2014) · Zbl 1449.74196
[32] Cheng, Y.; Bai, F.; Liu, C.; Peng, M., Analyzing nonlinear large deformation with an improved element-free Galerkin method via the interpolating moving least-squares method, Int J Comput Mater Sci Eng, 5, (2016)
[33] Deng, Y.; Liu, C.; Peng, M.; Cheng, Y., The interpolating complex variable element-free Galerkin method for temperature field problems, Int J Appl Mech, 7, (2015), ID-1550017 doi:10.1142/S1758825115500179
[34] 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) · Zbl 1315.65086
[35] Dehghan, M.; Abbaszadeh, M.; Mohebbi, A., Analysis of two methods based on Galerkin weak form for fractional diffusion-wave : meshless interpolating element free Galerkin (IEFG) and finite element methods, Eng Anal Bound Elem, 64, 205-221, (2016) · Zbl 1403.65068
[36] Sun, F.; Wang, J., Interpolating element-free Galerkin method for the regularized long wave equation and its error analysis, Appl Math Comput, 315, 54-69, (2017)
[37] Itoh, T.; Ikuno, S., Interpolating moving least-squares-based meshless time-domain method for stable simulation of electromagnetic wave propagation in complex-shaped domain, IEEE Trans Magn, 52, 1-4, (2016)
[38] Li, X.; Wang, Q., Analysis of the inherent instability of the interpolating moving least squares method when using improper polynomial bases, Eng Anal Bound Elem, 73, 21-34, (2016) · Zbl 1403.65206
[39] Zhang, T.; Li, X., A variational multiscale interpolating element-free Galerkin method for convection-diffusion and Stokes problems, Eng Anal Bound Elem, 82, 185-193, (2017) · Zbl 1403.76053
[40] Chen, L.; Liu, C.; Ma, H. P.; Cheng, Y. M., An interpolating local Petrov-Galerkin method for potential problems, Int J Appl Mech, 6, (2014), ID-1450009 doi:10.1142/S1758825114500094
[41] Wang, J.-F.; Sun, F.-X.; Cheng, Y.-M., An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems, Chinese Phys B, 21, (2012), ID-090204 doi:10.1088/1674-1056/21/9/090204
[42] Sun, F. X.; Wang, J. F.; Cheng, Y. M., An improved interpolating element-free Galerkin method for elasticity, Chinese Phys B, 22, (2013), ID-120203 doi:10.1088/1674-1056/22/12/120203
[43] Sun, F.; Wang, J.; Cheng, Y., An improved interpolating element-free Galerkin method for elastoplasticity via nonsingular weight functions, Int J Appl Mech, 8, (2016), ID-1650096 doi:10.1142/S1758825116500964
[44] 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
[45] Sterk, M.; Trobec, R., Meshless solution of a diffusion equation with parameter optimization and error analysis, Eng Anal Bound Elem, 32, (2008) · Zbl 1244.76088
[46] Carslaw, H. S.; Jaeger, J. C., Conduction of heat in solids, second ed, (1959), Oxford University Press · Zbl 0095.30201
[47] Gu, X. Y.; Dong, C. Y.; Cheng, T., The transient heat conduction MPM and GIMP applied to isotropic materials, Eng Anal Bound Elem, 66, 155-167, (2016) · Zbl 1402.80008
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.