×

A meshless analysis of three-dimensional transient heat conduction problems. (English) Zbl 1245.80010

Summary: We consider a numerical modeling of a three-dimensional transient heat conduction problem. The modeling is carried out using a meshless reproducing kernel particle (RKPM) method. In the mathematical formulation, a variational method is employed to derive the discrete equations. The essential boundary conditions of the formulated problems are enforced by the penalty method. Compared with numerical methods based on meshes, the RKPM needs only scattered nodes, rather than having to mesh the domain of the problem. An error analysis of the RKPM for three-dimensional transient heat conduction problem is also presented in this paper. In order to demonstrate the applicability of the proposed solution procedures, numerical experiments are carried out for a few selected three-dimensional transient heat conduction problems.

MSC:

80M25 Other numerical methods (thermodynamics) (MSC2010)
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Donea, J.; Giuliani, S., Finite element analysis of steady-state nonlinear heat transfer problems, Nucl eng des, 30, 205-213, (1974)
[2] Bathe, K.J.; Khoshgoftaar, M.R., Finite element formulation and solution of nonlinear heat transfer, Nucl eng des, 51, 389-401, (1979)
[3] Skerget, P.; Alujevic, A., Boundary element method for nonlinear transient heat transfer of reactor solids with convection and radiation on surfaces, Nucl eng des, 76, 47-54, (1983)
[4] ()
[5] Chen, W.; Fu, Z.J.; Jin, B.T., A truly boundary-only meshfree method for inhomogeneous problems based on recursive composite multiple reciprocity technique, Eng anal boundary elem, 34, 196-205, (2010) · Zbl 1244.65218
[6] Belytschko, T.; Krongauz, Y.; Organ, D., Meshless methods: an overview and recent developments, Comput meth appl mech eng, 139, 3-47, (1996) · Zbl 0891.73075
[7] Monaghan, J.J., An introduction to SPH, Comput phys commun, 48, 89-96, (1988) · Zbl 0673.76089
[8] Chen, W., New RBF collocation methods and kernel RBF with applications, Meshfree methods for partial differential equations, (2000), Springer Verlag, pp. 175-186
[9] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int J numer methods eng, 37, 229-256, (1994) · Zbl 0796.73077
[10] Liu, W.K.; Jun, S.; Zhang, Y.F., Reproducing kernel particle methods, Int J numer methods eng, 20, 1081-1106, (1995) · Zbl 0881.76072
[11] Atluri, S.N.; Zhu, T., A new meshless local petrov – galerkin (MLPG) approach in computational mechanics, Comput mech, 22, 2, 117-127, (1998) · Zbl 0932.76067
[12] Liew, K.M.; Ng, T.Y.; Zhao, X.; Reddy, J.N., Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells, Computer methods appl mech eng, 191, 37-38, 4141-4157, (2002) · Zbl 1083.74609
[13] Liew, K.M.; Wu, H.Y.; Ng, T.Y., Meshless method for modeling of human proximal femur: treatment of nonconvex boundaries and stress analysis, Comput mech, 28, 5, 390-400, (2002) · Zbl 1038.74053
[14] Liew, K.M.; Wu, Y.C.; Zou, G.P.; Ng, T.Y., Elasto-plasticity revisited: numerical analysis via reproducing kernel particle method and parametric quadratic programming, Int J numer methods eng, 55, 6, 669-683, (2002) · Zbl 1033.74050
[15] Liew, K.M.; Lim, H.K.; Tan, M.J.; He, X.Q., Analysis of laminated composite beams and plates with piezoelectric patches using the element-free Galerkin method, Comput mech, 29, 6, 486-497, (2002) · Zbl 1146.74370
[16] Liew, K.M.; Chen, X.L.; Reddy, J.N., Mesh-free radial basis function method for buckling analysis of non-uniformly loaded arbitrarily shaped shear deformable plates, Comput methods appl mech eng, 193, 3-5, 205-224, (2004) · Zbl 1075.74700
[17] Chen, C.S., A numerical method for heat transfer problems using collocation and radial basis functions, Int J numer methods eng, 42, 1263-1278, (1998) · Zbl 0907.65095
[18] Singh, I.V.; Meshless, E.F.G., Method in three-dimensional heat transfer problems: a numerical comparison, cost and error analysis, Numer heat transfer part A, 46, 199-220, (2004)
[19] Arefmanesh, A.; Najafi, M.; Abdi, H., A meshless local Petrov-Galerkin method for fluid dynamics and heat transfer applications, J fluids eng, 127, 647-655, (2005)
[20] Batra, R.C.; Porfiri, M.; Spinello, D., Treatment of material discontinuity in two meshless local Petrov-Galerkin (MLPG) formulations of axisymmetric transient heat conduction, Int J numer methods eng, 61, 2461-2479, (2004) · Zbl 1075.80001
[21] Liu, Y.; Zhang, X.; Lu, M.W., Meshless least-squares method for solving the steady-state heat conduction equation, Tsinghua sci tech, 10, 1, 61-66, (2005) · Zbl 1092.65101
[22] Liu, Y.; Zhang, X.; Lu, M.W., A meshless method based on least-squares approach for steady and unsteady state heat conduction problems, Numer heat transfer, 47, 3, 257-275, (2005)
[23] Sladek, J.; Sladek, V.; Atluri, S.N., A pure contour formulation for the meshless local boundary integral equation method in thermoelasticity, Comp model eng sci, 2, 4, 423-433, (2001) · Zbl 1060.74069
[24] Sladek, J.; Sladek, V.; Hellmich, C., Heat conduction analysis of 3-D axisymmetric and anisotropic FGM bodies by meshless local petrov – galerkin method, Comput mech, 38, 157-167, (2006)
[25] Cheng, R.; Liew, K.M., The reproducing kernel particle method for two-dimensional unsteady heat conduction problems, Comput mech, 45, 1-10, (2009) · Zbl 1398.74065
[26] Han, W.M.; Meng, X.P., Error analysis of the reproducing kernel particle method, Comput methods appl mech eng, 190, 6157-6181, (2001) · Zbl 0992.65119
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.