scholarly journals Eighth-Order Compact Finite Difference Scheme for 1D Heat Conduction Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Asma Yosaf ◽  
Shafiq Ur Rehman ◽  
Fayyaz Ahmad ◽  
Malik Zaka Ullah ◽  
Ali Saleh Alshomrani
Keyword(s):  
Heat Conduction ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Conditions ◽  
Difference Method ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Eighth Order ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method

The purpose of this paper is to develop a high-order compact finite difference method for solving one-dimensional (1D) heat conduction equation with Dirichlet and Neumann boundary conditions, respectively. A parameter is used for the direct implementation of Dirichlet and Neumann boundary conditions. The introduced parameter adjusts the position of the neighboring nodes very next to the boundary. In the case of Dirichlet boundary condition, we developed eighth-order compact finite difference method for the entire domain and fourth-order accurate proposal is presented for the Neumann boundary conditions. In the case of Dirichlet boundary conditions, the introduced parameter behaves like a free parameter and could take any value from its defined domain but for the Neumann boundary condition we obtained a particular value of the parameter. In both proposed compact finite difference methods, the order of accuracy is the same for all nodes. The time discretization is performed by using Crank-Nicholson finite difference method. The unconditional convergence of the proposed methods is presented. Finally, a set of 1D heat conduction equations is solved to show the validity and accuracy of our proposed methods.

Linear Heat Conduction Equation Based Filtering Iteration for Topology Optimization

2012 ◽  
Author(s):  
Qing-hai Zhao ◽  
Xiao-kai Chen ◽  
Yi Lin ◽  
Zheng-Dong Ma
Keyword(s):  
Topology Optimization ◽  
Heat Conduction ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Conditions ◽  
Difference Method ◽  
Heat Conduction Equation ◽  
Neumann Boundary ◽  
Filter Method ◽  
Stability Factor

This paper deals with an alternative approach to density and sensitivity filtering based on the solution of the linear heat conduction equation which is proposed for eliminating checkerboard patterns and mesh dependence in topology optimization problems. In order to guarantee the existence, uniqueness and stability of the solution of PDE, Neumann boundary conditions are introduced. With the help of the existing computational framework of FEM, boundary points have been extended to satisfy Neumann boundary conditions, and together with finite difference method to solve this initial boundary value. In order to guarantee the stability, stability factor is introduced to control the deviation for the solution of the finite difference method. Then the filtering technique is directly applied to the design variables and the design sensitivities, respectively. Especially, different from previous methods based on convolution operation, filtering iteration is employed to ensure the function to eliminate numerical instability. When the value of stability factor is changed at setting range, the number of times of filtering is manually corresponding set. At last, using different test examples in 2D show the advantage and effectiveness of filtering iteration of the new filter method in compared with previous filter method.

Sign in / Sign up

Export Citation Format

Share Document