Kansa Method for Unsteady Heat Flow in Nonhomogenous Material with a New Proposal of Finding the Good Value of RBF’s Shape Parameter
Abstract
:1. Introduction
2. Materials and Methods
2.1. Pseudospectral Formulation of Kansa Method for Initial-Value Problem of Heat Conduction
- Dirichlet boundary condition
- Neumann boundary condition
- Robin boundary condition
- for the Dirichlet boundary condition;
- for the Neumann boundary condition;
- for the Robin boundary condition.
2.2. The Algorithms for Finding the Good Value of Shape Parameter
Algorithm 1. Pseudocode describing the proposed condition algorithm. is the vector of collocation points coordinates, is a vector of the shape parameter values. The range of was selected empirically by trial and error. |
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Algorithm 2. Pseudocode describing the proposed modified Fasshauer algorithm. is the vector of collocation points coordinates, is a vector of the shape parameter values. The range of was selected empirically by trial and error. |
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2.3. Reference Solutions
- The analytical method for the steady-state;
- The finite difference method for the transient analysis.
2.4. Error Measures
- Mean percentage error calculated between the last time step solution of the Kansa method and the analytical steady-state solution
- Mean percentage error calculated between the Kansa method solution and the finite-difference method solution
- Mean percentage error calculated between the finite-difference method solution and the steady-state analytical solution
3. Numerical Results and Discussion
3.1. General Insights about Numerical Setup and Thermal Parameters Distributions
3.2. Discontinuous Distributions of Thermal Parameters, the Influence of Sharpness Parameter
- ;
- ;
- Thermal parameters distribution—discontinuous;
- Number of collocation points—80, 100, and 120;
- Boundary conditions type—Dirichlet & Dirichlet;
- Time step size—;
- Number of time steps—10.
3.3. Influence of Coefficient
- ;
- thermal parameters distribution—harmonic;
- the number of collocation points ;
- applied boundary conditions: Dirichlet on both boundaries;
- the size of the time step, ;
- the number of time steps—20.
- The measure is more important because it describes the quality of the solution in the transient state and not only at the last time step;
- The difference between and is small while the difference between and is significant;
- The value of depends on the numerical setup while does not.
3.4. The Algorithms for Finding the Good Value of the Shape Parameter
4. Conclusions
- The condition algorithm proposed in this work is a very reliable and precise algorithm for choosing a good shape parameter value for the considered class of problems. Particularly noteworthy is its ability to give shape parameter values that do not cause the interpolation matrix to be ill-conditioned. It is worth mentioning that the algorithm is more computationally expensive than classic algorithms such as the Fasshauer algorithm.
- The modified Fasshauer algorithm is an interesting alternative to the classic Fasshauer algorithm. It gives slightly greater reliability than Fasshauer’s algorithm; however, not as good as the condition algorithm. The accuracy of simulation results using it is slightly worse than using Fasshauer’s algorithm and the condition algorithm.
- The suggested value of for the considered class of problems due to the error measures and is 0;
- The selection of the optimal value of the sharpness parameter s is problematic. Based on the performed study, it is not possible to formulate a general selection rule, but some recommendations may be mentioned. The s value should be chosen so that the does not reach a tremendous value. It is worth noting that the higher the number of collocation points, the higher the optimal value of s.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Test Case | 1 | 2 | 3 | 4 |
---|---|---|---|---|
p | 0.5 | 0.4 | 0.3 | −0.5 |
Parameters distribution | Discontinuous | Harmonic | Exponential | Linear |
Collocation points number | 240 | 200 | 160 | 120 |
BC at left edge | Robin | Neumann | Dirichlet | Neumann |
BC at right edge | Robin | Robin | Robin | Dirichlet |
Time step size | 0.1 s | 0.5 s | 1 s | 2 s |
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Popczyk, O.; Dziatkiewicz, G. Kansa Method for Unsteady Heat Flow in Nonhomogenous Material with a New Proposal of Finding the Good Value of RBF’s Shape Parameter. Materials 2021, 14, 4178. https://doi.org/10.3390/ma14154178
Popczyk O, Dziatkiewicz G. Kansa Method for Unsteady Heat Flow in Nonhomogenous Material with a New Proposal of Finding the Good Value of RBF’s Shape Parameter. Materials. 2021; 14(15):4178. https://doi.org/10.3390/ma14154178
Chicago/Turabian StylePopczyk, Olaf, and Grzegorz Dziatkiewicz. 2021. "Kansa Method for Unsteady Heat Flow in Nonhomogenous Material with a New Proposal of Finding the Good Value of RBF’s Shape Parameter" Materials 14, no. 15: 4178. https://doi.org/10.3390/ma14154178