LSMPS Method for Solving Steady Heat Conduction Equation in Orthotropic Materials
Keywords:
anisotropic materials, heat conduction, LSMPS, meshless method, orthotropic
Abstract
Heat conduction in orthotropic materials has been an important topic in engineering. In this study, LSMPS method was used to solve steady heat conduction problem for orthotropic medium. Two methods were proposed in this paper, one using coordinate transformation and one without coordinate transformation. The numerical results were then compared to the analytical solution. The results from both methods showed good agreement with the analytical solution. The coordinate transformation method had slightly better accuracy, though it also took slightly longer computation time. However, the direct method, without coordinate transformation is more practical when dealing with non-orthogonality and non-homogeneity over the domain. In general, the proposed methods have been able to solve steady heat conduction problem for orthotropic medium.
References
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[2] Ozisik, M.N., Heat Conduction, Wiley, 1993.
[3] Chang, Y.P., Kang, C.S. & Chen, D.J., The Use of Fundamental Green’s Functions for the Solution of Problems of Heat Conduction in Anisotropic Media, International Journal of Heat and Mass Transfer, 16(10), pp. 1905-1918, October 1973.
[4] Ma, C.C. & Chang, S.W., Analytical Exact Solutions of Heat Conduction Problems for Anisotropic Multi-layered Media. International Journal of Heat and Mass Transfer, 47(8), pp. 1643-1655, April 2004.
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[6] Yan, G., Qingsong, H., Chuanzeng, Z. & Xiaoqiao, H., The Generalized Finite Difference Method for Long-time Transient Heat Conduction in 3D Anisotropic Composite Materials, Applied Mathematical Modelling, 71, pp. 316-330, July 2019.
[7] Van Es, B., Koren, B. & de Blank, H.J., Finite-volume Scheme for Anisotropic Diffusion, Journal of Computational Physics, 306, pp. 422-442, February 2016.
[8] Prestini, D., Filippini, G., Zdanski, P.S.B. & Vaz Jr, M., Fundamental Approach to Anisotropic Heat Conduction using the Element-Based Finite Volume Method, International Journal of Computation and Methodology, 71(4), pp. 327-345, April 2017.
[9] Bazyar, M.H. & Talebi, A., Scaled Boundary Finite-element Method for Solving Non-Homogeneous Anisotropic Heat Conduction Problems, Applied Mathematical Modelling, 39(23-24), pp. 7583-7599, December 2015.
[10] Annasabi, Z. & Erchiqui, F. 3D Hybrid Finite Elements for Anisotropic Heat Conduction in a Multi-material with Multiple Orientations of the Thermal Conductivity Tensors, International Journal of Heat and Mass Transfer, 158, September 2020.
[11] Sladek, J., Sladek, V., & Atluri, S. N., Meshless Local Petrov-Galerkin Method for Heat Conduction Problem in an Anisotropic Medium, Computer Modeling in Engineering and Sciences, 6(3), pp. 309-318, September 2004.
[12] Hidayat, M.I.P, Meshless Local B-spline Collocation Method for Heterogeneous Heat Conduction Problems, Engineering Analysis with Boundary Elements, 101, pp. 76-88, April 2019.
[13] Qiu, L., Wang, F. & Lin, J., A Meshless Singular Boundary Method for Transient Heat Conduction Problems in Layered Materials, Computers & Mathematics with Applications, 78(11), pp. 3544-3562, December 2019.
[14] Ud Din, Z., Ahsan, M., Ahmad, M., Khan, W., Mahmoud, E.E. & Abdel-Aty, A-H., Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media, Mathematics, 8(11), November 2020.
[15] Wang, C., Wang, F. & Gong, Y., Analysis of 2D Heat Conduction in Nonlinear Functionally Graded Materials using a Local Semi-Analytical Meshless Method, AIMS Mathematics, 6(11), pp. 12599-12618, September 2021.
[16] Chen, H. & Liu, D., Formulation of a Nonlocal Discrete Model for Anisotropic Heat Conduction Problems, International Journal of Thermal Sciences, 182(1), December 2022.
[17] Tamai, T. & Koshizuka, S., Least Square Moving Particle Semi-Implicit Method, Computational Particle Mechanics, 1(3), pp. 277-305, September 2014.
[18] Tanaka, M., Cardoso, R. & Bahai, H. Modification of the LSMPS Method for the Conservation of the Thermal Energy in Laser Irradiation Processes, International Journal for Numerical Methods in Engineering, 117(2), pp. 161-187, September 2018.
[19] Tanaka, M., Cardoso, R. & Bahai, H., Multi-resolution MPS Method, Journal of Computational Physics, 359, pp. 106-136, 2018.
[2] Ozisik, M.N., Heat Conduction, Wiley, 1993.
[3] Chang, Y.P., Kang, C.S. & Chen, D.J., The Use of Fundamental Green’s Functions for the Solution of Problems of Heat Conduction in Anisotropic Media, International Journal of Heat and Mass Transfer, 16(10), pp. 1905-1918, October 1973.
[4] Ma, C.C. & Chang, S.W., Analytical Exact Solutions of Heat Conduction Problems for Anisotropic Multi-layered Media. International Journal of Heat and Mass Transfer, 47(8), pp. 1643-1655, April 2004.
[5] Sameti, M., Astaraie, F.R., Pourfayaz, F. & Kasaeian, A., Analytical and FDM Solutions for Anisotropic Heat Conduction in an Orthotropic Rectangular, American Journal of Numerical Analysis, 2(2), pp. 65-68, March 2014.
[6] Yan, G., Qingsong, H., Chuanzeng, Z. & Xiaoqiao, H., The Generalized Finite Difference Method for Long-time Transient Heat Conduction in 3D Anisotropic Composite Materials, Applied Mathematical Modelling, 71, pp. 316-330, July 2019.
[7] Van Es, B., Koren, B. & de Blank, H.J., Finite-volume Scheme for Anisotropic Diffusion, Journal of Computational Physics, 306, pp. 422-442, February 2016.
[8] Prestini, D., Filippini, G., Zdanski, P.S.B. & Vaz Jr, M., Fundamental Approach to Anisotropic Heat Conduction using the Element-Based Finite Volume Method, International Journal of Computation and Methodology, 71(4), pp. 327-345, April 2017.
[9] Bazyar, M.H. & Talebi, A., Scaled Boundary Finite-element Method for Solving Non-Homogeneous Anisotropic Heat Conduction Problems, Applied Mathematical Modelling, 39(23-24), pp. 7583-7599, December 2015.
[10] Annasabi, Z. & Erchiqui, F. 3D Hybrid Finite Elements for Anisotropic Heat Conduction in a Multi-material with Multiple Orientations of the Thermal Conductivity Tensors, International Journal of Heat and Mass Transfer, 158, September 2020.
[11] Sladek, J., Sladek, V., & Atluri, S. N., Meshless Local Petrov-Galerkin Method for Heat Conduction Problem in an Anisotropic Medium, Computer Modeling in Engineering and Sciences, 6(3), pp. 309-318, September 2004.
[12] Hidayat, M.I.P, Meshless Local B-spline Collocation Method for Heterogeneous Heat Conduction Problems, Engineering Analysis with Boundary Elements, 101, pp. 76-88, April 2019.
[13] Qiu, L., Wang, F. & Lin, J., A Meshless Singular Boundary Method for Transient Heat Conduction Problems in Layered Materials, Computers & Mathematics with Applications, 78(11), pp. 3544-3562, December 2019.
[14] Ud Din, Z., Ahsan, M., Ahmad, M., Khan, W., Mahmoud, E.E. & Abdel-Aty, A-H., Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media, Mathematics, 8(11), November 2020.
[15] Wang, C., Wang, F. & Gong, Y., Analysis of 2D Heat Conduction in Nonlinear Functionally Graded Materials using a Local Semi-Analytical Meshless Method, AIMS Mathematics, 6(11), pp. 12599-12618, September 2021.
[16] Chen, H. & Liu, D., Formulation of a Nonlocal Discrete Model for Anisotropic Heat Conduction Problems, International Journal of Thermal Sciences, 182(1), December 2022.
[17] Tamai, T. & Koshizuka, S., Least Square Moving Particle Semi-Implicit Method, Computational Particle Mechanics, 1(3), pp. 277-305, September 2014.
[18] Tanaka, M., Cardoso, R. & Bahai, H. Modification of the LSMPS Method for the Conservation of the Thermal Energy in Laser Irradiation Processes, International Journal for Numerical Methods in Engineering, 117(2), pp. 161-187, September 2018.
[19] Tanaka, M., Cardoso, R. & Bahai, H., Multi-resolution MPS Method, Journal of Computational Physics, 359, pp. 106-136, 2018.
Published
2023-09-30
How to Cite
Nugroho, M. A., Palar, P. S., Mu’min, G. F., & Zuhal, L. R. (2023). LSMPS Method for Solving Steady Heat Conduction Equation in Orthotropic Materials. ITB Graduate School Conference, 3(1), 55-65. Retrieved from https://gcs.itb.ac.id/proceeding-igsc/index.php/igsc/article/view/132
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