Next Article in Journal
A Novel Approach for Outdoor Fall Detection Using Multidimensional Features from A Single Camera
Next Article in Special Issue
A Lattice-Based Group Authentication Scheme
Previous Article in Journal
Active Color Control in a Metasurface by Polarization Rotation
Previous Article in Special Issue
A Big Data and Time Series Analysis Technology-Based Multi-Agent System for Smart Tourism
Open AccessArticle

A Modified Polynomial Expansion Algorithm for Solving the Steady-State Allen-Cahn Equation for Heat Transfer in Thin Films

1
Department of Mechanical Engineering, National Chung Hsing University, Taichung 40227, Taiwan
2
Graduate Institute of Precision Manufacturing, National Chin-Yi University of Technology, Taichung 41170, Taiwan
*
Authors to whom correspondence should be addressed.
Appl. Sci. 2018, 8(6), 983; https://doi.org/10.3390/app8060983
Received: 20 April 2018 / Revised: 30 May 2018 / Accepted: 13 June 2018 / Published: 15 June 2018
(This article belongs to the Special Issue Selected Papers from the 2017 International Conference on Inventions)
Meshfree algorithms offer a convenient way of solving nonlinear steady-state problems in arbitrary plane areas surrounded by complicated boundary shapes. The simplest of these is the polynomial expansion approach. However, it is rarely utilized as a primary tool for this purpose because of its rather ill-conditioned behavior. A well behaved polynomial expansion algorithm is presented in this paper which can be more effectively used to solve the steady-state Allen-Cahn (AC) equation for heat transfer in thin films. In this method, modified polynomial expansion was used to cope with each iteration of the steady-state Allen-Cahn equation to produce nonlinear algebraic equations where multiple scales are automatically determined by the collocation points. These scales can largely decrease the condition number of the coefficient matrix in each nonlinear system, so that the iteration process converges very quickly. The numerical solutions were found to be accurate and stable against moderate noise to better than 7.5%. Computational results verified the method and showed the steady-state Allen-Cahn equation for heat transfer in thin films could easily be resolved for several arbitrary plane domains. View Full-Text
Keywords: steady-state Allen-Cahn equation; meshless approach; modified polynomial expansion; boundary value problems; heat transfer in thin films steady-state Allen-Cahn equation; meshless approach; modified polynomial expansion; boundary value problems; heat transfer in thin films
Show Figures

Graphical abstract

MDPI and ACS Style

Chang, C.-W.; Liu, C.-H.; Wang, C.-C. A Modified Polynomial Expansion Algorithm for Solving the Steady-State Allen-Cahn Equation for Heat Transfer in Thin Films. Appl. Sci. 2018, 8, 983. https://doi.org/10.3390/app8060983

AMA Style

Chang C-W, Liu C-H, Wang C-C. A Modified Polynomial Expansion Algorithm for Solving the Steady-State Allen-Cahn Equation for Heat Transfer in Thin Films. Applied Sciences. 2018; 8(6):983. https://doi.org/10.3390/app8060983

Chicago/Turabian Style

Chang, Chih-Wen; Liu, Chein-Hung; Wang, Cheng-Chi. 2018. "A Modified Polynomial Expansion Algorithm for Solving the Steady-State Allen-Cahn Equation for Heat Transfer in Thin Films" Appl. Sci. 8, no. 6: 983. https://doi.org/10.3390/app8060983

Find Other Styles
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Search more from Scilit
 
Search
Back to TopTop