Numerical solutions for 2D unsteady Laplace-type problems of anisotropic functionally graded materials
Abstract
The time-dependent Laplace-type equation of variable coefficients for anisotropic inhomogeneous media is discussed in this paper. Numerical solutions to problems which are governed by the equation are sought by using a combined Laplace transform and boundary element method. The variable coefficients equation is transformed to a constant coefficients equation. The constant coefficients equation after being Laplace transformed is then written in a boundary-only integral equation involving a time-free fundamental solution. The boundary integral equation is therefore employed to find the numerical solutions using a standard boundary element method. Finally the numerical results are inversely transformed numerically using the Stehfest formula to obtain solutions in the time variable. Some problems of anisotropic functionally graded media are considered. The results show that the combined Laplace transform and boundary element method is accurate and easy to implement.
Keyword : anisotropic Laplace-type equation, variable coefficients, Laplace transform, boundary element method
How to Cite
Azis, M. I. (2022). Numerical solutions for 2D unsteady Laplace-type problems of anisotropic functionally graded materials. Mathematical Modelling and Analysis, 27(2), 303–321. https://doi.org/10.3846/mma.2022.14463
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A. Sutradhar and G.H. Paulino. The simple boundary element method for transient heat conduction in functionally graded materials. Computer Methods in Applied Mechanics and Engineering, 193:4511–4539, 2004. https://doi.org/10.1016/j.cma.2004.02.018
A. Sutradhar, G.H. Paulino and L.J. Gray. Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method. Engineering Analysis with Boundary Elements, 26(2):119– 132, 2002. https://doi.org/10.1016/S0955-7997(01)00090-X
R. Syam, Fahruddin, M.I. Azis and A. Hayat. Numerical solutions to anisotropic FGM BVPs governed by the modified Helmholtz type equation. IOP Conf. Ser.: Mater. Sci. Eng., 619:012061, 2019. https://doi.org/10.1088/1757899X/619/1/012061
W. Timpitak and N. Pochai. Numerical simulations to a one-dimensional groundwater pollution measurement model through heterogeneous soil. IAENG International Journal of Applied Mathematics, 50(3):558–565, 2020.
K. Yang, W.Z. Feng, J. Wang and X.W. Gao. RIBEM for 2D and 3D nonlinear heat conduction with temperature dependent conductivity. Engineering Analysis with Boundary Elements, 87:1–8, 2018. https://doi.org/10.1016/j.enganabound.2017.11.001
S. Abotula, A. Kidane, V.B. Chalivendra and A. Shukla. Dynamic curving cracks in functionally graded materials under thermo-mechanical loading. International Journal of Solids and Structures, 49:1637–1655, 2012. https://doi.org/10.1016/j.ijsolstr.2012.03.010
M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions: With Formulas, Graphs and Mathematical Tables. Dover Publications, Washington, 1972.
S.A. AL-Bayati and L.C. Wrobel. The dual reciprocity boundary element formulation for convection-diffusion-reaction problems with variable velocity field using different radial basis functions. International Journal of Mechanical Sciences, 145:367–377, 2018. https://doi.org/10.1016/j.ijmecsci.2018.07.003
S.A. AL-Bayati and L.C. Wrobel. A novel dual reciprocity boundary element formulation for two-dimensional transient convection–diffusion–reaction problems with variable velocity. Engineering Analysis with Boundary Elements, 94:60–68, 2018. https://doi.org/10.1016/j.enganabound.2018.06.001
M.I. Azis. BEM solutions to exponentially variable coefficient Helmholtz equation of anisotropic media. Journal of Physics: Conference Series, 1277:012036, 2019. https://doi.org/10.1088/1742-6596/1277/1/012036
M.I. Azis and D.L. Clements. Nonlinear transient heat conduction problems for a class of inhomogeneous anisotropic materials by BEM. Engineering Analysis with Boundary Elements, 32:1054–1060, 2008. https://doi.org/10.1016/j.enganabound.2007.04.007
M.I. Azis, R. Syam and S. Hamzah. BEM solutions to BVPs governed by the anisotropic modified Helmholtz equation for quadratically graded media. IOP Conference Series: Earth and Environmental Science, 279:012010, 2019. https://doi.org/10.1088/1755-1315/279/1/012010
Y. Chen and Q. Du. Some fast algorithms for exterior anisotropic problems in concave angle domains. IAENG International Journal of Applied Mathematics, 50(4):729–733, 2020.
Z. Fu, J. Shi, W. Chen and L. Yang. Three-dimensional transient heat conduction analysis by boundary knot method. Mathematics and Computers in Simulation, 165:306–317, 2019. https://doi.org/10.1016/j.matcom.2018.11.025
S. Guo, J. Zhang, G. Li and F. Zhou. Three-dimensional transient heat conduction analysis by Laplace transformation and multiple reciprocity boundary face method. Engineering Analysis with Boundary Elements, 37:15, 2013. https://doi.org/10.1016/j.enganabound.2012.09.001
S. Hamzah, M.I. Azis and A.K. Amir. Numerical solutions to anisotropic BVPs for quadratically graded media governed by a Helmholtz equation. IOP Conf. Ser.: Mater. Sci. Eng., 619:012060, 2019. https://doi.org/10.1088/1757899X/619/1/012060
S. Hamzah, M.I. Azis, A. Haddade and E. Syamsuddin. On some examples of BEM solution to elasticity problems of isotropic functionally graded materials. IOP Conf. Ser.: Mater. Sci. Eng., 619:012018, 2019. https://doi.org/10.1088/1757-899X/619/1/012018
H. Hassanzadeh and H. Pooladi-Darvish. Comparison of different numerical Laplace inversion methods for engineering applications. Appl. Math. Comput., 189:1966–1981, 2007. https://doi.org/10.1016/j.amc.2006.12.072
St.N. Jabir, M.I. Azis, Z. Djafar and B. Nurwahyu. BEM solutions to a class of elliptic BVPs for anisotropic trigonometrically graded media. IOP Conf. Ser.: Mater. Sci. Eng., 619:012059, 2019. https://doi.org/10.1088/1757899X/619/1/012059
N. Lanafie, M.I. Azis and Fahruddin. Numerical solutions to BVPs governed by the anisotropic modified Helmholtz equation for trigonometrically graded media. IOP Conf. Ser.: Mater. Sci. Eng., 619:012058, 2019. https://doi.org/10.1088/1757-899X/619/1/012058
N. Lanafie, N. Ilyas, M.I. Azis and A.K. Amir. A class of variable coefficient elliptic equations solved using BEM. IOP Conf. Ser.: Mater. Sci. Eng., 619:012025, 2019. https://doi.org/10.1088/1757-899X/619/1/012025
Q. Li, S. Chen and X. Luo. Steady heat conduction analyses using an interpolating element-free Galerkin scaled boundary method. Applied Mathematics and Computation, 300:103–115, 2017. https://doi.org/10.1016/j.amc.2016.12.007
N. Noda, N. Sumi and M. Ohmichi. Analysis of transient plane thermal stresses in functionally graded orthotropic strip. Journal of Thermal Stresses, 41:1225– 1243, 2018. https://doi.org/10.1080/01495739.2018.1505445
B. Nurwahyu, B. Abdullah, A. Massinai and M.I. Azis. Numerical solutions for BVPs governed by a Helmholtz equation of anisotropic FGM. IOP Conf. Ser.: Earth Environ. Sci., 279:012008, 2019. https://doi.org/10.1088/17551315/279/1/012008
J. Ravnik and J. Tibuat. Fast boundary-domain integral method for unsteady convection-diffusion equation with variable diffusivity using the modified Helmholtz fundamental solution. Numerical Algorithms, 82:1441–1466, 2019. https://doi.org/10.1007/s11075-019-00664-3
J. Ravnik and L. Skerget.ˇ A gradient free integral equation for diffusion–convection equation with variable coefficient and velocity. Engineering Analysis with Boundary Elements, 37:683–690, 2013. https://doi.org/10.1016/j.enganabound.2013.01.012
J. Ravnik and L. Skerget.ˇ Integral equation formulation of an unsteady diffusion–convection equation with variable coefficient and velocity. Computers and Mathematics with Applications, 66:2477–2488, 2014. https://doi.org/10.1016/j.camwa.2013.09.021
S.Y. Reutskiy. A meshless radial basis function method for 2D steadystate heat conduction problems in anisotropic and inhomogeneous media. Engineering Analysis with Boundary Elements, 66:1–11, 2016. https://doi.org/10.1016/j.enganabound.2016.01.013
N. Samec and L. Skerget. Integral formulation of a diffusive–convective trans-ˇ port equation for reacting flows. Engineering Analysis with Boundary Elements, 28:1055–1060, 2004. https://doi.org/10.1016/j.enganabound.2004.02.005
Y.C. Shiah, Y-C. Chaing and T. Matsumoto. Analytical transformation of volume integral for the time-stepping BEM analysis of 2D transient heat conduction in anisotropic media. Engineering Analysis with Boundary Elements, 64:101– 110, 2016. https://doi.org/10.1016/j.enganabound.2015.12.008
A. Sutradhar and G.H. Paulino. The simple boundary element method for transient heat conduction in functionally graded materials. Computer Methods in Applied Mechanics and Engineering, 193:4511–4539, 2004. https://doi.org/10.1016/j.cma.2004.02.018
A. Sutradhar, G.H. Paulino and L.J. Gray. Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method. Engineering Analysis with Boundary Elements, 26(2):119– 132, 2002. https://doi.org/10.1016/S0955-7997(01)00090-X
R. Syam, Fahruddin, M.I. Azis and A. Hayat. Numerical solutions to anisotropic FGM BVPs governed by the modified Helmholtz type equation. IOP Conf. Ser.: Mater. Sci. Eng., 619:012061, 2019. https://doi.org/10.1088/1757899X/619/1/012061
W. Timpitak and N. Pochai. Numerical simulations to a one-dimensional groundwater pollution measurement model through heterogeneous soil. IAENG International Journal of Applied Mathematics, 50(3):558–565, 2020.
K. Yang, W.Z. Feng, J. Wang and X.W. Gao. RIBEM for 2D and 3D nonlinear heat conduction with temperature dependent conductivity. Engineering Analysis with Boundary Elements, 87:1–8, 2018. https://doi.org/10.1016/j.enganabound.2017.11.001