0
Research Papers: Conduction

Meshless Local B-Spline Collocation Method for Two-Dimensional Heat Conduction Problems With Nonhomogenous and Time-Dependent Heat Sources

[+] Author and Article Information
Mas Irfan P. Hidayat

Department of Materials and
Metallurgical Engineering,
Institut Teknologi Sepuluh Nopember,
Kampus ITS Keputih Sukolilo,
Surabaya 60111, East Java, Indonesia
e-mail: irfan@mat-eng.its.ac.id

Bambang Ariwahjoedi

Department of Fundamental and Applied Science,
Universiti Teknologi Petronas,
Bandar Seri Iskandar,
Tronoh 31750, Perak Darul Ridzuan, Malaysia
e-mail: ariwahjoedi@gmail.com

Setyamartana Parman

Department of Mechanical Engineering,
Universiti Teknologi Petronas,
Bandar Seri Iskandar,
Tronoh 31750, Perak Darul Ridzuan, Malaysia
e-mail: setyamartana@petronas.com.my

T. V. V. L. N. Rao

Department of Mechanical-
Mechatronics Engineering,
The LNM Institute of Information Technology,
Jaipur 302031, Rajasthan, India
e-mail: tvvlnrao@gmail.com

1Corresponding author.

2Not affiliated with Universiti Petronas as of September 2016.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 24, 2015; final manuscript received January 19, 2017; published online April 4, 2017. Assoc. Editor: P. K. Das.

J. Heat Transfer 139(7), 071302 (Apr 04, 2017) (11 pages) Paper No: HT-15-1067; doi: 10.1115/1.4036003 History: Received January 24, 2015; Revised January 19, 2017

This paper presents a new approach of meshless local B-spline based finite difference (FD) method for transient 2D heat conduction problems with nonhomogenous and time-dependent heat sources. In this method, any governing equations are discretized by B-spline approximation which is implemented as a generalized FD technique using local B-spline collocation scheme. The key aspect of the method is that any derivative is stated as neighboring nodal values based on B-spline interpolants. The set of neighboring nodes is allowed to be randomly distributed. This allows enhanced flexibility to be obtained in the simulation. The method is truly meshless as no mesh connectivity is required for field variable approximation or integration. Galerkin implicit scheme is employed for time integration. Several transient 2D heat conduction problems with nonuniform heat sources in arbitrary complex geometries are examined to show the efficacy of the method. Comparison of the simulation results with solutions from other numerical methods in the literature is given. Good agreement with reference numerical methods is obtained. The method is shown to be simple and accurate for the time-dependent problems.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Griebel, M. , and Schweitzer, M. A. , 2000, “ A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic, and Hyperbolic PDEs,” SIAM J. Sci. Comput., 22(3), pp. 853–890. [CrossRef]
Gingold, R. A. , and Monaghan, J. J. , 1977, “ Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars,” Mon. Not. R. Astron. Soc., 181(3), pp. 375–389. [CrossRef]
Lucy, L. B. , 1977, “ A Numerical Approach to the Testing of the Fission Hypothesis,” Astron. J., 82, pp. 1013–1024. [CrossRef]
Nayroles, B. , Touzot, G. , and Villon, P. , 1992, “ Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements,” Comput. Mech., 10(5), pp. 307–318. [CrossRef]
Belytschko, T. , Lu, Y. Y. , and Gu, L. , 1994, “ Element-Free Galerkin Method,” Int. J. Numer. Methods Eng., 37(2), pp. 229–256. [CrossRef]
Liu, W. K. , Jun, S. , and Zhang, Y. F. , 1995, “ Reproducing Kernel Particle Methods,” Int. J. Numer. Methods Fluids, 20(8–9), pp. 1081–1106. [CrossRef]
Atluri, S. N. , and Zhu, T. , 1998, “ A New Meshless Local Petrov–Galerkin (MLPG) Approach in Computational Mechanics,” Comput. Mech., 22(2), pp. 117–127. [CrossRef]
Wang, H. , Qin, Q. H. , and Kang, Y. L. , 2006, “ A Meshless Model for Transient Heat Conduction in Functionally Graded Materials,” Comput. Mech., 38(1), pp. 51–60. [CrossRef]
Wu, X. H. , Shen, S. P. , and Tao, W. Q. , 2007, “ Meshless Local Petrov–Galerkin Collocation Method for Two-Dimensional Heat Conduction Problems,” CMES, 22(1), pp. 65–76.
Vishwakarma, V. , Das, A. K. , and Das, P. K. , 2011, “ Steady State Conduction Through 2D Irregular Bodies by Smoothed Particle Hydrodynamics,” Int. J. Heat Mass Transfer, 54(1–3), pp. 314–325. [CrossRef]
Soleimani, S. , Jalaal, M. , Bararnia, H. , Ghasemi, E. , Ganji, D. D. , and Mohammadi, F. , 2010, “ Local RBF-DQ Method for Two-Dimensional Transient Heat Conduction Problem,” Int. Commun. Heat Mass Transfer, 37(9), pp. 1411–1418. [CrossRef]
Chen, L. , and Liew, K. M. , 2011, “ A Local Petrov–Galerkin Approach With Moving Kriging Interpolation for Solving Transient Heat Conduction Problems,” Comput. Mech., 47(4), pp. 455–467. [CrossRef]
Li, Q. H. , Chen, S. S. , and Kou, G. X. , 2011, “ Transient Heat Conduction Analysis Using the MLPG Method and Modified Precise Time Step Integration Method,” J. Comput. Phys., 230(7), pp. 2736–2750. [CrossRef]
Varanasi, C. , Murthy, J. Y. , and Mathur, S. , 2010, “ A Meshless Finite Difference Method for Conjugate Heat Conduction Problems,” ASME J. Heat Transfer, 132(8), p. 081303. [CrossRef]
Pepper, D. W. , Wang, X. , and Carrington, D. B. , 2013, “ A Meshless Method for Modelling Convective Heat Transfer,” ASME J. Heat Transfer, 135(1), p. 011003. [CrossRef]
Divo, E. , and Kassab, A. J. , 2007, “ An Efficient Localized Radial Basis Function Meshless Method for Fluid Flow and Conjugate Heat Transfer,” ASME J. Heat Transfer, 129(2), pp. 124–136. [CrossRef]
Singh, A. , Singh, I. V. , and Prakash, R. , 2007, “ Meshless Element Free Galerkin Method for Unsteady Nonlinear Heat Transfer Problems,” Int. J. Heat Mass Transfer, 50(5–6), pp. 1212–1219. [CrossRef]
Chantasiriwan, S. , 2007, “ Multiquadric Collocation Method for Time-Dependent Heat Conduction Problems With Temperature-Dependent Thermal Properties,” ASME J. Heat Transfer, 129(2), pp. 109–113. [CrossRef]
Sladek, J. , Sladek, V. , Tan, C. L. , and Atluri, S. N. , 2008, “ Analysis of Transient Heat Conduction in 3D Anisotropic Functionally Graded Solids, by the MLPG Method,” CMES, 32(3), pp. 161–174.
Vishwakarma, V. , Das, A. K. , and Das, P. K. , 2011, “ Analysis of Non-Fourier Heat Conduction Using Smoothed Particle Hydrodynamics,” Appl. Therm. Eng., 31(14–15), pp. 2963–2970. [CrossRef]
Sikarudi, M. A. E. , and Nikseresht, A. H. , 2016, “ Neumann and Robin Boundary Conditions for Heat Conduction Modeling Using Smoothed Particle Hydrodynamics,” Comput. Phys. Commun., 198, pp. 1–11. [CrossRef]
Zhang, X. , Zhang, P. , and Zhang, L. , 2013, “ An Improved Meshless Method With Almost Interpolation Property for Isotropic Heat Conduction Problems,” Eng. Anal. Boundary Elem., 37(5), pp. 850–859. [CrossRef]
Hidayat, M. I. P. , Wahjoedi, B. A. , Parman, S. , and Yusoff, P. S. M. M. , 2014, “ Meshless Local B-Spline-FD Method and Its Application for 2D Heat Conduction Problems With Spatially Varying Thermal Conductivity,” Appl. Math. Comput., 242, pp. 236–254.
Sun, Y. S. , and Li, B. W. , 2010, “ Spectral Collocation Method for Transient Combined Radiation and Conduction in an Anisotropic Scattering Slab With Graded Index,” ASME J. Heat Transfer, 132(5), p. 052701. [CrossRef]
Ma, J. , Sun, Y. S. , and Li, B. W. , 2014, “ Completely Spectral Collocation Solution of Radiative Heat Transfer in an Anisotropic Scattering Slab With a Graded Index Medium,” ASME J. Heat Transfer, 136(1), p. 012701. [CrossRef]
Liu, G. R. , 2009, Meshfree Methods: Moving Beyond the Finite Element Method, 2nd ed., CRC Press, Boca Raton, FL.
Bornemann, P. B. , and Cirak, F. , 2013, “ A Subdivision-Based Implementation of the Hierarchical b-Spline Finite Element Method,” Comput. Methods Appl. Mech. Eng., 253, pp. 584–598. [CrossRef]
Kansa, E. J. , 1990, “ Multiquadric—A Scattered Data Approximation Scheme With Applications to Computational Fluid Dynamics II,” Comput. Math. Appl., 19(8–9), pp. 147–161. [CrossRef]
Cooper, K. D. , 1993, “ Domain-Imbedding Alternating Direction Method for Linear Elliptic Equations on Irregular Regions Using Collocation,” Numer. Methods Partial Differ. Equations, 9(1), pp. 93–106. [CrossRef]
Sun, W. W. , Wu, J. M. , and Zhang, X. P. , 2007, “ Nonconforming Spline Collocation Methods in Irregular Domains,” Numer. Methods Partial Differ. Equations, 23(6), pp. 1509–1529. [CrossRef]
Kundu, B. , and Das, P. K. , 2005, “ Optimum Profile of Thin Fins With Volumetric Heat Generation: A Unified Approach,” ASME J. Heat Transfer, 127(8), pp. 945–948. [CrossRef]
Bartels, R. H. , Beatty, J. C. , and Barsky, B. A. , 1987, An Introduction to Splines for Use in Computer Graphics and Geometric Modeling, Morgan Kaufmann Publishers, San Mateo, CA.
de Boor, C. , 1972, “ On Calculating With B-Splines,” J. Approximation Theory, 6(1), pp. 50–62. [CrossRef]
Cox, M. , 1972, “ The Numerical Evaluation of B-Spline,” J. Inst. Math. Appl., 10(2), pp. 134–149. [CrossRef]
de Boor, C. , 2001, A Practical Guide to Splines, Revised ed., Springer, New York.
Thakur, H. , Singh, K. M. , and Sahoo, P. K. , 2010, “ Meshless Local Petrov–Galerkin Method for Nonlinear Heat Conduction Problems,” Numer. Heat Transfer, Part B, 56(5), pp. 393–410. [CrossRef]
Hematiyan, M. R. , Mohammadi, M. , Marin, L. , and Khosravifard, A. , 2011, “ Boundary Element Analysis of Uncoupled Transient Thermo-Elastic Problems With Time- and Space-Dependent Heat Sources,” Appl. Math. Comput., 218(5), pp. 1862–1882.
Mohammadi, M. , Hematiyan, M. R. , and Marin, L. , 2010, “ Boundary Element Analysis of Nonlinear Transient Heat Conduction Problems Involving Non-Homogenous and Nonlinear Heat Sources Using Time-Dependent Fundamental Solutions,” Eng. Anal. Boundary Elem., 34(7), pp. 655–665. [CrossRef]
Touloukian, Y. S. , 1976, Thermophysical Properties of High Temperature Solid Materials, Macmillan, New York.
Khosravifard, A. , Hematiyan, M. R. , and Marin, L. , 2011, “ Nonlinear Transient Heat Conduction Analysis of Functionally Graded Materials in the Presence of Heat Sources Using an Improved Meshless Radial Point Interpolation Method,” Appl. Math. Modell., 35(9), pp. 4157–4174. [CrossRef]
Shu, C. , Ding, H. , and Yeo, K. S. , 2003, “ 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(7–8), pp. 941–954. [CrossRef]
Shan, Y. Y. , Shu, C. , and Qin, N. , 2009, “ Multiquadric Finite Difference (MQ-FD) Method and Its Application,” Adv. Appl. Math. Mech., 1(5), pp. 615–638.
Hidayat, M. I. P. , Ariwahjoedi, B. , and Parman, S. , 2015, “ A New Meshless Local B-Spline Basis Functions-FD Method for Two-Dimensional Heat Conduction Problems,” Int. J. Numer. Methods Heat Fluid Flow, 25(2), pp. 225–251. [CrossRef]
Sarra, S. A. , 2006, “ Integrated Multiquadric Radial Basis Function Approximation Methods,” Comput. Math. Appl., 51(8), pp. 1283–1296. [CrossRef]
Hidayat, M. I. P. , Ariwahjoedi, B. , and Parman, S. , 2015, “ B-Spline Collocation With Domain Decomposition Method and Its Application for Singularly Perturbed Convection-Diffusion Problems,” Recent Trends in Physics of Material Science and Technology, F. L. Gaol , K. Shrivastava , and J. Akhtar , eds., Springer, Berlin.

Figures

Grahic Jump Location
Fig. 1

Classical spline to draw a smooth curve through a set of points before computer modeling

Grahic Jump Location
Fig. 2

A global domain and nodes of subset in the support domain of center point

Grahic Jump Location
Fig. 3

Geometry and boundary conditions for transient heat conduction of case I

Grahic Jump Location
Fig. 4

Simulation results of temperature histories at points A and B for thermal problem of case I with respect to (a) B-spline order and (b) number of supporting nodes

Grahic Jump Location
Fig. 5

Simulation results of temperature histories at points A and B for thermal problem of case I with respect to (a) number of collocation points and (b) time-steps

Grahic Jump Location
Fig. 6

Geometry and boundary conditions for transient heat conduction of case II

Grahic Jump Location
Fig. 7

Simulation results of temperature histories at points A and B for thermal problem of case II for several time-steps

Grahic Jump Location
Fig. 8

Temperature profiles through the points: (a) A (line y = 0.1) and (b) B (line x = 0) with respect to number of nodes (μ = 5, ns = 10, and Δt = 25 s)

Grahic Jump Location
Fig. 9

Domain of FGM made of ZrO2 and Ti–6Al–4V and its boundary conditions of case III

Grahic Jump Location
Fig. 10

Simulation results of: (a) temperature histories at points A and B and (b) temperature profiles for thermal problem of case III with respect to number of nodes (μ = 5, ns = 7, and Δt = 20 s)

Grahic Jump Location
Fig. 11

Simulation results of (a) temperature histories at points A and B and (b) temperature profiles for thermal problem of case III with respect to time-steps (μ = 5, ns = 7971 nodes)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In