Abstract
As an improvement of the Meshless Local Petrov–Galerkin (MLPG), the Direct Meshless Local Petrov–Galerkin (DMLPG) method is applied here to the numerical solution of transient heat conduction problem. The new technique is based on direct recoveries of test functionals (local weak forms) from values at nodes without any detour via classical moving least squares (MLS) shape functions. This leads to an absolutely cheaper scheme where the numerical integrations will be done over low–degree polynomials rather than complicated MLS shape functions. This eliminates the main disadvantage of MLS based methods in comparison with finite element methods (FEM), namely the costs of numerical integration.
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Atluri, S.N.: The meshless method (MLPG) for domain and BIE discretizations. Tech Science Press, Encino (2005)
Babuska, I., Banerjee, U., Osborn, J., Zhang, Q.: Effect of numerical integration on meshless methods. Comput. Methods Appl. Mech Engrg 198, 27–40 (2009)
Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng., special issue 139, 3–47 (1996)
Belytschko, T., Lu, Y., Gu, L.: Element-Free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994)
Minkowycz, W., Sparrow, E., Schneider, G., Pletcher, R.: Handbook of numerical heat transfer. Wiley, New York (1988)
Mirzaei, D., Dehghan, M.: MLPG method for transient heat conduction problem with MLS as trial approximation in both time and space domains. CMES–Comput. Model. Eng. Sci. 72, 185–210 (2011)
Mirzaei, D., Dehghan, M.: New implementation of MLBIE method for heat conduction analysis in functionally graded materials. Eng. Anal. Bound. Elem. 36, 511–519 (2012)
Mirzaei, D., Schaback, R.: Direct Meshless Local Petrov-Galerkin (DMLPG) method: a generalized MLS approximation. Appl. Numer. Math. 68, 73–82 (2013). doi:10.1016/j.apnum.2013.01.002
Mirzaei, D., Schaback, R., Dehghan, M.: On generalized moving least squares and diffuse derivatives. IMA J. Numer. Anal. 32, 983–1000 (2012)
Sladek, J., Sladek, V., Atluri, S.: Meshless local Petrov-Galerkin method for heat conduction problem in an anisotropic medium. CMES–Comput. Model. Eng. Sci. 6, 309–318 (2004)
Sladek, J., Sladek, V., Hellmich, C., Eberhardsteiner, J.: Heat conduction analysis of 3-D axisymmetric and anisotropic FGM bodies by meshless local Petrov-Galerkin method. Comput. Mech. 39, 223–233 (2007)
Sladek, J., Sladek, V., Zhang, C.: Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method. Comput. Mater. Sci. 28, 494–504 (2003)
Sladek, V., Sladek, J., Tanaka, M., Zhang, C.: Transient heat conduction in anisotropic and functionally graded media by local integral equations. Eng. Anal. Bound. Elem. 29, 1047–1065 (2005)
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)
Wendland, H.: Scattered data approximation. Cambridge University Press, Cambridge (2005)
Zhu, T., Zhang, J., Atluri, S.: A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Comput. Mech. 21, 223–235 (1998)
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Mirzaei, D., Schaback, R. Solving heat conduction problems by the Direct Meshless Local Petrov-Galerkin (DMLPG) method. Numer Algor 65, 275–291 (2014). https://doi.org/10.1007/s11075-013-9711-1
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DOI: https://doi.org/10.1007/s11075-013-9711-1