Scaled Boundary Finite Element Method Coupled with Equilibrated Basis Functions for Heat Transfer Problems

Pirhaji Khouzani, Nazanin; Noormohammadi, Nima

doi:10.47176/jcme.42.1.1005

Scaled Boundary Finite Element Method Coupled with Equilibrated Basis Functions for Heat Transfer Problems

Document Type : Original Article

Authors

Department of Civil Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

10.47176/jcme.42.1.1005

Abstract

The Scaled Boundary Finite Element Method (SBFEM) discretizes only the boundary by using a technique for scaling the domain response onto its boundary. In this research, heat transfer problems in two-dimensional space are solved with a new approach based on combining the scaled boundary finite element method and the equilibrated basis functions. The SBFEM develops its relations in radial and circumferential coordinate systems, but only discretizes the boundary of the problem through development of a semi-analytical solution in radial direction. So the challenges of appropriate elemental grid for the solution domain, or the need for fundamental solutions of the equation, as usual in the finite element method or the boundary element method respectively, do not appear. In this research, after scaling the boundary in the scaled boundary finite element method and extracting the related equations, the equilibrated basis functions are used to approximate the semi-analytical solution in radial direction. After estimating the radial solution by the first kind Chebyshev polynomials, the weighted residual form of the governing equation is applied for approximately satisfaction. Finally, the unknown degrees of freedom of the boundary are derived, and there will be no need for the usual eigenvalue solution of the SBFEM. It will be shown that this approach benefits good accuracy and convergence rate.

Keywords

20.1001.1.22287698.1402.42.1.7.6

Main Subjects

Numerical Algorithms

References

Wolf, J. P., and Song, C. “The Scaled Boundary Finite-Element Method–A Primer: Derivations”, Computers & Structures, Vol. 78, issue. 1-3, pp. 191-210, 2000.
Wolf, J. P., “The Scaled Boundary Finite Element Method”, John Wiley & Sons, 2003.‏
Song, C., Ooi, E. T., and Natarajan, S., “A Review of the Scaled Boundary Finite Element Method for Two-Dimensional Linear Elastic Fracture Mechanics”, Engineering Fracture Mechanics, Vol. 187, pp. 45-73, 2018.‏
Song, C., and Wolf, J. P., “The Scaled Boundary Finite-Element Method–A Primer: Solution Procedures”, Computers & Structures, Vol. 78, issue. 1-3, pp. 211-225, 2000.
Bazyar, M. H., and Talebi, A., “Scaled Boundary Finite-Element Method for Solving Non-Homogeneous Anisotropic Heat Conduction Problems”, Applied Mathematical Modelling, Vol. 39, issue. 23-24, pp. 7583-7599, 2015.
Yang, Y., Zhang, Z., Feng, Y., Yu, Y., Wang, K., and Liang, L., “A Polygonal Scaled Boundary Finite Element Method for Solving Heat Conduction Problems”, arXiv preprint, arXiv:2106.12283, 2021.
He, Y., Yang, H., and Deeks, A. J., “On the Use of Cyclic Symmetry in SBFEM for Heat Transfer Problems”, International Journal of Heat and Mass Transfer, Vol. 71, pp. 98-105, 2014.
Yu, B., Hu, P., Saputra, A. A., and Gu, Y., “The Scaled Boundary Finite Element Method Based on the Hybrid Quadtree Mesh for Solving Transient Heat Conduction Problems”, Applied Mathematical Modelling, Vol. 89, pp. 541-571, 2021.
Mohasseb, S., Moradi, M., Sokhansefat, T., Kasaeian, A., and Mahian, O., “A Novel Approach to Solve Inverse Heat Conduction Problems: Coupling Scaled Boundary Finite Element Method to a Hybrid Optimization Algorithm”, Engineering Analysis with Boundary Elements, Vol. 84, pp. 206-212, 2017.
Deeks, A. J., and Wolf, J. P., “A Virtual Work Derivation of the Scaled Boundary Finite-Element Method for Elastostatics”, Computational Mechanics, Vol. 28, pp. 489-504, 2002. ‏
Adak, D., Pramod, A. L. N., Ooi, E. T., and Natarajan, S., “A Combined Virtual Element Method and the Scaled Boundary Finite Element Method for Linear Elastic Fracture Mechanics”, Engineering Analysis with Boundary Elements, Vol. 113, pp. 9-16, 2020.
Kita, E., and Kamiya, N., “Trefftz Method: An Overview”, Advances in Engineering software, Vol. 24, issue. 1-3, pp. 3-12, 1995. ‏
Fairweather, G., and Karageorghis, A., “The Method of Fundamental Solutions for Elliptic Boundary Value Problems”, Advances in Computational Mathematics, Vol. 9, pp. 69-95, 1998.‏
A. R. Boroomand, B., and Noormohammadi, N., “A Trefftz-Based Meshfree Local Method for Bending Analysis of Arbitrarily Shaped Laminated Composite and Isotropic Plates”, Engineering Analysis with Boundary Elements, Vol. 143, pp. 237-262, 2022.
Boroomand, B., and Noormohammadi. N., “Weakly Equilibrated Basis Functions for Elasticity Problems”, Engineering Analysis with Boundary Elements, Vol. 37, issue. 12, pp. 1712-1727, 2013.
Azizpooryan, M., Noormohammadi, N., and Boroomand, B., “Equilibrated Basis Functions for Static Analysis of In-plane Heterogeneous Laminated Composite Plates in Boundary and Meshfree Approaches”, Iranian Journal of Science and Technology – Transactions of Mechanical Engineering, Vol. 46, pp. 957–984, 2022.
Bateniparvar, O., and Noormohammadi. N., “An Enrichment Technique for Bending Analysis of In-plane Heterogeneous Thin Plates with Weak Singularities”, Engineering with Computers, 2022. https://doi.org/10.1007/s00366-022-01702-w
Noormohammadi, N., and Boroomand, B., “A Fictitious Domain Method Using Equilibrated Basis Functions for Harmonic and Bi-Harmonic Problems in Physics”, Journal of Computational Physics, Vol. 272, pp. 189-217, 2014.‏
Noormohammadi, N., and Boroomand, B., “A Domain Decomposition Approach Using Equilibrated Basis Functions: Special Reference to Structural Engineering Problems with Varying Material Properties”, Iranian Journal of Science and Technology – Transactions of Civil Engineering, 45, pp. 667-681, 2021. ‏
Kanjanakijkasem, W., “A Finite Element Method for Prediction of Unknown Boundary Conditions in Two-Dimensional Steady-State Heat Conduction Problems”, International Journal of Heat and Mass Transfer, Vol. 88, pp. 891-901, 2015.‏
Soleimani, S., Jalaal, M., Bararnia, H., Ghasemi, E., Ganji, D. D., and Mohammadi, F., “Local RBF-DQ Method for Two-Dimensional Transient Heat Conduction Problems”, International Communications in Heat and Mass Transfer, Vol. 37, issue. 9, pp. 1411-1418, 2010.
Zhang, Y. M., Liu, Z. Y., Chen, J. T., and Gu, Y., “A Novel Boundary Element Approach for Solving the Anisotropic Potential Problems”, Engineering Analysis with Boundary Elements, Vol. 35, issue. 11, pp. 1181-1189, 2011.