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Meshless method based on the local weak-forms for steady-state heat conduction problems. (English) Zbl 1144.80356
Summary: The meshless local Petrov-Galerkin (MLPG) method is applied to compute two steady-state heat conduction problems of irregular complex domain in 2D space. The essential boundary condition is enforced by the transformation method, and the MLS method is used for interpolation schemes. A numerical example that has analytical solution shows the present method can obtain desired accuracy and efficiency. Two cases in engineering with irregular boundary are computed to validate the approach by comparing the present method with the finite volume method (FVM) solutions obtained from a commercial CFD package FLUENT 6.3. The results show that the present method is in good agreement with FVM. It is expected that MLPG method (which is a truly meshless) is very promising in solving engineering heat conduction problems within irregular domains.

##### MSC:
 80A20 Heat and mass transfer, heat flow (MSC2010) 80M25 Other numerical methods (thermodynamics) (MSC2010)
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##### References:
 [1] Liu, G. R.; Liu, M. B.: Smoothed particle hydrodynamics: A meshfree particle method, (2003) · Zbl 1046.76001 [2] Lucy, L. B.: A numerical approach to testing of the fission hypothesis, Astron. J. 8, 1013-1024 (1977) [3] Monaghan, J. J.: Smoothed particle hydrodynamics, Annu. rev. Astron. astrophys 30, 543-574 (1992) [4] Nayroles, B.; Touzot, G.; Villon, P.: The diffuse approximations, CR acad. Sci. Paris sér. II 313, 133-138 (1991) · Zbl 0725.73085 [5] Nayroles, B.; Touzot, G.; Villon, P.: The diffuse element method, CR acad. Sci. Paris sér. II 313, 293-296 (1991) · Zbl 0753.65012 [6] Belytschko, T.; Lu, Y. Y.; Gu, L.: Element-free Galerkin methods, Int. J. Numer. methods eng. 37, 229-256 (1994) · Zbl 0796.73077 [7] Liu, W. K.; Jun, S.; Zhang, Y. F.: Reproducing kernel particle methods, Int. J. Numer. methods fluid 20, 1081-1106 (1995) · Zbl 0881.76072 [8] Onate, E.; Idelsohn, S.; Zienkiewicz, O.; Taylor, R. L.: A finite point method in computational mechanics application to convective transport and fluid flow, Int. J. Numer. methods eng. 39, 3839-3866 (1995) · Zbl 0884.76068 [9] Atluri, S. N.; Zhu, T.: A new meshless local Petrov – Galerkin (MLPG) approach in computational mechanics, Comput. mech. 24, 348-372 (1998) · Zbl 0932.76067 [10] Atluri, S. N.; Zhu, T.: A new meshless local Petrov – Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation, Comput. model. Simulat. eng. 3, 196-197 (1998) [11] Atluri, S. N.; Shen, S. P.: The meshless local Petrov – Galerkin (MLPG) method, (2002) · Zbl 1012.65116 [12] Sadat, H.; Couturier, S.: Performance and accuracy of a meshless method for laminar natural convection, Numer. heat transfer, part B 37, 455-467 (2000) [13] Sophy, T.; Sadat, H.; Prax, C.: A meshless formulation for three-dimensional lammiar natural convection, Numer. heat transfer, part B 41, 433-445 (2002) [14] Singh, I. V.; Jain, P. K.: Parallel meshless EFG solution for fluid flow problems, Numer. heat transfer, part B 48, 45-66 (2005) [15] Wu, Y. L.; Liu, G. R.; Gu, Y. T.: Application of meshless local Petrov – Galerkin (MLPG) approach to simulation of incompressible flow, Numer. heat transfer, part B 48, 459-475 (2005) [16] Cleary, P. W.; Monaghan, J. J.: Conduction modeling using smoothed particle hydrodynamics, J. comput. Phys 148, 27-264 (1999) · Zbl 0930.76069 [17] Chen, J. K.; Beraun, J. E.; Carney, T. C.: A corrective smoothed particle method for boundary value problems in heat conduction, Int. J. Numer. meth. Eng. 46, 231-252 (1999) · Zbl 0941.65104 [18] Singh, A.; Singh, I. V.; Prakash, R.: Numerical solution of temperature-dependent thermal conductivity problems using a meshless method, Numer. heat transfer, part A 50, 125-145 (2006) [19] Singh, I. V.; Sandeep, K.; Prakash, R.: Heat transfer analysis of two-dimensional fins using meshless element-free Galerkin method, Numer. heat transfer A 44, 73-84 (2003) [20] Singh, I. V.; Sandeep, K.; Prakash, R.: The element free Galerkin method in three-dimensional steady state heat conduction, Int. J. Comput. eng. Sci. 3, 291-303 (2002) [21] Singh, I. V.; Prakash, R.: The numerical solution of three-dimensional transient heat conduction problems using element free Galerkin method, Int. J. Heat tech. 21, 73-80 (2003) [22] Singh, I. V.: A numerical solution of composite heat transfer problems using meshless method, Int. J. Heat mass transfer 47, 2123-2138 (2004) · Zbl 1050.80006 [23] Liu, Y.; Zhang, X.; Liu, M. W.: A meshless method based on least-squares approach for steady-and unsteady state heat conduction problems, Numer. heat transfer 47, 257-275 (2005) [24] Tan, J. Y.; Liu, L. H.; Li, B. X.: Least-squares collocation meshless approach for coupled radiative and conductive heat transfer, Numer. heat transfer, part B 49, 179-195 (2006) [25] Sadat, H.; Dubus, N.; Gbahoue, L.; Sophy, T.: On the solution of heterogeneous heat conduction problems by a diffuse approximation meshless method, Numer. heat transfer, part B 50, 491-498 (2006) [26] Qian, L. F.; Batra, R. C.: Three dimensional transient heat conduction in a functionally graded thick plate with a high order plate theory and a meshless local Petrov Galerkin method, Comput. mech. 35, 214-226 (2005) · Zbl 1143.74321 [27] Sladek, J.; Sladek, V.; Atluri, S. N.: Meshless local Petrov – Galerkin method for heat conduction problem in an anisotropic medium, CMES: comput. Model. eng. Sci. 6, No. 3, 309-318 (2004) · Zbl 1084.80002 [28] Sladek, J.; Sladek, V.; Hellmich, Ch.; Eberhardsteiner, J.: Heat conduction analysis of 3-D axisymmetric and anisotropic FCM bodies by meshless local Petrov – Galerkin method, Comput. mech. 39, 323-333 (2007) · Zbl 1169.80001 [29] Lu, Y. Y.; Belytschko, T.; Gu, L.: A new implementation of the element free Galerkin method, Comput. methods appl. Mech. eng. 113, 397-414 (1994) · Zbl 0847.73064 [30] Liu, G. R.; Chen, X. L.; Reddy, J. N.: Buckling of symmetrically laminated composite plates using the element-free Galerkin method, Int. J. Struct. stabil. Dyn. 2, 281-294 (2002) · Zbl 1205.74162 [31] Atluri, S. N.; Kim, H.; Cho, J. Y.: A 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 [32] Liu, R. G.; Gu, Y. T.: An introduction to meshree methods and their programming, (2005) [33] Ozisik, M. Necati: Heat conduction, (1980) · Zbl 0855.65097 [34] Chen, C. R.; Hu, Y. D.; Zhao, C. Y.; Wang, Q. W.; Tao, W. Q.: Numerical study on heat transfer characteristic of conduction mud, J. eng. Thermophys. 22, No. 1, 101-103 (2001)
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