Trilinear Finite Element Solution of Three Dimensional Heat Conduction Partial Differential Equations

  • Authors

    • Erwin Sulaeman
    • S. M. Afzal Hoq
    • Abdurahim Okhunov
    • Marwan Badran
    2018-12-09
    https://doi.org/10.14419/ijet.v7i4.36.28146
  • Trilinear finite element method, hexahedron finite element, Galerkin method, 3D-Laplace equation, error analysis, heat conduction.
  • Solution of partial differential equations (PDEs) of three dimensional steady state heat conduction and its error analysis are elaborated in the present paper by using a Trilinear Galerkin Finite Element method (TGFEM). An eight-node hexahedron element model is developed for the TGFEM based on a trilinear basis function where physical domain is meshed by structured grid.  The stiffness matrix of the hexahedron element is formulated by using a direct integration scheme without the necessity to use the Jacobian matrix.  To check the accuracy of the established scheme, comparisons of the results using error analysis between the present TGFEM and exact solution is conducted for various number of the elements.  For this purpose, analytical solution is derived in detailed for a particular heat conduction problem.  The comparison shows promising result where its convergence is approximately O(h²) for matrix norms L1, L2 and LÂ¥.

     

     


  • References

    1. [1] Richter T. & Vexler B (2013).Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems. Numerische Mathematik. 124(1), 151–182.

      [2] Yong WK & Hyochoong B (2000). The Finite Element method using Matlab. CRC Press, Boca Raton London.

      [3] Antonopoulou DC & Plexousaki M (2010), Discontinuous Galerkin methods for the linear Schrödinger equation in non-cylindrical domains. Numerische Mathematik 115(4), 585-608.

      [4] Dow JO (1999) A Unified Approach to The Finite Element Method and Error Analysis Procedures, Academic Pers, London.

      [5] Schotzau D & Schwab C (2000), Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method. SIAM Journal on Numerical Analysis 38(3), 837-75.

      [6] Dongfang Li & Chengjian Z. (2014), Error estimates of discontinuous Galerkin methods for delay differential equations. Applied Numerical Mathematics 82, 1–10.

      [7] Manas, KD., Babuska, IM., & Oden, JT. (2001) Solution of stochastic partial differential equations using Galerkin finite element method. Compt. Methods Appl. Mech. Eng. 190, 6359–72.

      [8] Gil S, Saleta ME & Tobia. (2002), Experimental Study of the Neumann and Dirichlet Boundary Conditions in Two-Dimensional Electrostatic Problems. American Journal of Physics, 70 (12), 1208-1213.

      [9] Burden RL & Faires JD (2001). Numerical Analysis. 9th ed., Brooks/Cole, Boston.

      [10] Hoq SMA, Sulaeman E & Okhunov A (2016), Error Analysis of Heat Conduction Partial Differential Equations using Galerkin’s Finite Element Method. Indian Journal of Science and Technology 9 (36).

      [11] Ainsworth M & Oden JT(1997), A posterior error estimation in finite element method, Computer Methods in Applied Mechanics and Engineering 142 (1–2), 1-88.

      [12] Lijun Yi. (2015) An -error Estimate for the h-p Version Continuous Petrov-Galerkin Method for Nonlinear Initial Value Problems. East Asian Journal on Applied Mathematics East Asian Journal on Applied Mathematics 5(4), 301-311.

      [13] Kreyszig, E. (2011) Advanced Engineering Mathematics. 10th edn. John Wiley & Sons: New York.

  • Downloads

  • How to Cite

    Sulaeman, E., M. Afzal Hoq, S., Okhunov, A., & Badran, M. (2018). Trilinear Finite Element Solution of Three Dimensional Heat Conduction Partial Differential Equations. International Journal of Engineering & Technology, 7(4.36), 379-384. https://doi.org/10.14419/ijet.v7i4.36.28146