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Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon

Yıl 2020, Cilt: 32 Sayı: 1, 96 - 110, 31.03.2020
https://doi.org/10.7240/jeps.581959

Öz



Bu çalışmada,
ağsız yöntemler için radyal özelliğe sahip yeni bir temel fonksiyon
önerilmiştir. Önerilen fonksiyon, iki boyutta, dört farklı problemde, ağsız
yöntemlerde sıklıkla kullanılan Ters Multikuadrik ve Gauss fonksiyonlarıyla
birlikte test edilmiştir. Test problemlerinin üç tanesi 2. mertebeden
mühendislik problemlerini içerirken son test problemi 4. mertebeden bir
mühendislik problemi uygulaması olmuştur. 2. mertebeden test problemlerinde
farklı sınır koşulları ve problem türleri incelenmiştir. Yapılan sayısal deneyler,
önerilen fonksiyonun Ters Multikuadrik ve Gauss fonksiyonlarına kıyasla daha az
nokta sayılarında benzer mertebedeki hatalara ulaşabildiğini göstermiştir.
Ayrıca nokta sayısının artmasıyla aynı mertebedeki hatalar için
kullanılabilecek şekil/ölçek parametresinin (epsilon)
diğer iki fonksiyona kıyasla daha geniş bir aralıkta seçilebildiği
gösterilmiştir. Dolayısıyla, önerilen fonksiyon, ağsız yöntem uygulamalarında
bir alternatif olarak kullanılabilecektir.




Kaynakça

  • Altınkaynak, A., Gupta, M., Spalding, M. A., & Crabtree, S. L. (2011). Melting in a Single Screw Extruder: Experiments and 3D Finite Element Simulations. International Polymer Processing, 26(2), 182-196.
  • Uygun, M., & Kırkköprü, K. (2011). Katı yakıtlı roket motorlarında daimi olmayan akışların ikili zaman adımlaması yöntemi ile sayısal benzetimi. İTÜDERGİSİ/d, 8(2).
  • Kığılı, H. N. (2006). Tünel Üst Yapı Etkileşim Problemlerinin Sınır Elemanlar Yöntemiyle İncelenmesi. İTÜ Fen Bilimleri Enstitüsü.
  • Rodrigues, D., Belinha, J., Pires, F., Dinis, L., & Jorge, R. N. (2018). Homogenization technique for heterogeneous composite materials using meshless methods. Engineering Analysis with Boundary Elements, 92, 73-89.
  • Fasshauer, G., & McCourt, M. (2015). Kernel-based approximation methods using Matlab (Vol. 19): World Scientific Publishing Company.
  • Kansa, E. J. (1990b). Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers & Mathematics with Applications, 19(8), 147-161.
  • Kansa, E. J. (1990a). Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Computers & Mathematics with Applications, 19(8), 127-145.
  • Fornberg, B., & Flyer, N. (2015b). Solving PDEs with radial basis functions. Acta Numerica, 24, 215-258.
  • Fornberg, B., & Flyer, N. (2005). Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids. Advances in Computational Mathematics, 23(1-2), 5-20.
  • Fasshauer, G. E. (2007). Meshfree approximation methods with MATLAB (Vol. 6): World Scientific.
  • Fornberg, B., & Wright, G. (2004). Stable computation of multiquadric interpolants for all values of the shape parameter. Computers & Mathematics with Applications, 48(5-6), 853-867.
  • Larsson, E., Lehto, E., Heryudono, A., & Fornberg, B. (2013). Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions. SIAM Journal on Scientific Computing, 35(4), A2096-A2119.
  • Fasshauer, G. E. (1999). Solving differential equations with radial basis functions: multilevel methods and smoothing. Advances in Computational Mathematics, 11(2-3), 139-159.
  • Chou, C., Sun, C., Young, D., Sladek, J., & Sladek, V. (2015). Extrapolated local radial basis function collocation method for shallow water problems. Engineering Analysis with Boundary Elements, 50, 275-290.
  • Farahani, B. V., Berardo, J., Belinha, J., Ferreira, A., Tavares, P. J., & Moreira, P. (2017). On the optimal shape parameters of distinct versions of RBF meshless methods for the bending analysis of plates. Engineering Analysis with Boundary Elements, 84, 77-86.
  • Xia, C.-c., Jiang, T.-t., & Chen, W.-f. (2016). Particle Swarm Optimization of Aerodynamic Shapes With Nonuniform Shape Parameter–Based Radial Basis Function. Journal of Aerospace Engineering, 30(3), 04016089.
  • Biazar, J., & Hosami, M. (2017). An interval for the shape parameter in radial basis function approximation. Applied Mathematics and Computation, 315, 131-149.
  • Larsson, E., & Fornberg, B. (2003). A numerical study of some radial basis function based solution methods for elliptic PDEs. Computers & Mathematics with Applications, 46(5), 891-902.
  • Fornberg, B., & Flyer, N. (2015a). A primer on radial basis functions with applications to the geosciences: SIAM.
  • Li, J., Cheng, A. H.-D., & Chen, C.-S. (2003). A comparison of efficiency and error convergence of multiquadric collocation method and finite element method. Engineering Analysis with Boundary Elements, 27(3), 251-257.
  • Golbabai, A., Mohebianfar, E., & Rabiei, H. (2015). On the new variable shape parameter strategies for radial basis functions. Computational and Applied Mathematics, 34(2), 691-704.
  • Schaback, R. (2009). Solving the Laplace equation by meshless collocation using harmonic kernels. Advances in Computational Mathematics, 31(4), 457.
  • Leitao, V. M. (2001). A meshless method for Kirchhoff plate bending problems. International Journal for Numerical Methods in Engineering, 52(10), 1107-1130.

A New Trigonometric Based Radial Basis Function for Meshless Method Applications

Yıl 2020, Cilt: 32 Sayı: 1, 96 - 110, 31.03.2020
https://doi.org/10.7240/jeps.581959

Öz



In this study, a new radial basis function
for meshless method is proposed. The proposed function was tested on four
different 2D problems along with the two well-known IMQ and Gauss functions.
Three of the test problems include 2nd order engineering problems
whereas the last test problem was a 4th order engineering problem. 2nd
order engineering problems were used to investigate the type of boundary
conditions and problems. Numerical experiments suggested that similar order of
error can be obtained using the proposed function with less number of nodes
compared to IMQ and Gauss functions. Besides that, with an increase on the
number of nodes, the range of shape/scale parameter (epsilon)
for the proposed function is broader
that that for the other two functions. Thus, the proposed function is a good
candidate for meshless method applications. 




Kaynakça

  • Altınkaynak, A., Gupta, M., Spalding, M. A., & Crabtree, S. L. (2011). Melting in a Single Screw Extruder: Experiments and 3D Finite Element Simulations. International Polymer Processing, 26(2), 182-196.
  • Uygun, M., & Kırkköprü, K. (2011). Katı yakıtlı roket motorlarında daimi olmayan akışların ikili zaman adımlaması yöntemi ile sayısal benzetimi. İTÜDERGİSİ/d, 8(2).
  • Kığılı, H. N. (2006). Tünel Üst Yapı Etkileşim Problemlerinin Sınır Elemanlar Yöntemiyle İncelenmesi. İTÜ Fen Bilimleri Enstitüsü.
  • Rodrigues, D., Belinha, J., Pires, F., Dinis, L., & Jorge, R. N. (2018). Homogenization technique for heterogeneous composite materials using meshless methods. Engineering Analysis with Boundary Elements, 92, 73-89.
  • Fasshauer, G., & McCourt, M. (2015). Kernel-based approximation methods using Matlab (Vol. 19): World Scientific Publishing Company.
  • Kansa, E. J. (1990b). Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers & Mathematics with Applications, 19(8), 147-161.
  • Kansa, E. J. (1990a). Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Computers & Mathematics with Applications, 19(8), 127-145.
  • Fornberg, B., & Flyer, N. (2015b). Solving PDEs with radial basis functions. Acta Numerica, 24, 215-258.
  • Fornberg, B., & Flyer, N. (2005). Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids. Advances in Computational Mathematics, 23(1-2), 5-20.
  • Fasshauer, G. E. (2007). Meshfree approximation methods with MATLAB (Vol. 6): World Scientific.
  • Fornberg, B., & Wright, G. (2004). Stable computation of multiquadric interpolants for all values of the shape parameter. Computers & Mathematics with Applications, 48(5-6), 853-867.
  • Larsson, E., Lehto, E., Heryudono, A., & Fornberg, B. (2013). Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions. SIAM Journal on Scientific Computing, 35(4), A2096-A2119.
  • Fasshauer, G. E. (1999). Solving differential equations with radial basis functions: multilevel methods and smoothing. Advances in Computational Mathematics, 11(2-3), 139-159.
  • Chou, C., Sun, C., Young, D., Sladek, J., & Sladek, V. (2015). Extrapolated local radial basis function collocation method for shallow water problems. Engineering Analysis with Boundary Elements, 50, 275-290.
  • Farahani, B. V., Berardo, J., Belinha, J., Ferreira, A., Tavares, P. J., & Moreira, P. (2017). On the optimal shape parameters of distinct versions of RBF meshless methods for the bending analysis of plates. Engineering Analysis with Boundary Elements, 84, 77-86.
  • Xia, C.-c., Jiang, T.-t., & Chen, W.-f. (2016). Particle Swarm Optimization of Aerodynamic Shapes With Nonuniform Shape Parameter–Based Radial Basis Function. Journal of Aerospace Engineering, 30(3), 04016089.
  • Biazar, J., & Hosami, M. (2017). An interval for the shape parameter in radial basis function approximation. Applied Mathematics and Computation, 315, 131-149.
  • Larsson, E., & Fornberg, B. (2003). A numerical study of some radial basis function based solution methods for elliptic PDEs. Computers & Mathematics with Applications, 46(5), 891-902.
  • Fornberg, B., & Flyer, N. (2015a). A primer on radial basis functions with applications to the geosciences: SIAM.
  • Li, J., Cheng, A. H.-D., & Chen, C.-S. (2003). A comparison of efficiency and error convergence of multiquadric collocation method and finite element method. Engineering Analysis with Boundary Elements, 27(3), 251-257.
  • Golbabai, A., Mohebianfar, E., & Rabiei, H. (2015). On the new variable shape parameter strategies for radial basis functions. Computational and Applied Mathematics, 34(2), 691-704.
  • Schaback, R. (2009). Solving the Laplace equation by meshless collocation using harmonic kernels. Advances in Computational Mathematics, 31(4), 457.
  • Leitao, V. M. (2001). A meshless method for Kirchhoff plate bending problems. International Journal for Numerical Methods in Engineering, 52(10), 1107-1130.
Toplam 23 adet kaynakça vardır.

Ayrıntılar

Birincil Dil Türkçe
Bölüm Araştırma Makaleleri
Yazarlar

Atakan Altınkaynak 0000-0003-3971-3641

Yayımlanma Tarihi 31 Mart 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 32 Sayı: 1

Kaynak Göster

APA Altınkaynak, A. (2020). Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon. International Journal of Advances in Engineering and Pure Sciences, 32(1), 96-110. https://doi.org/10.7240/jeps.581959
AMA Altınkaynak A. Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon. JEPS. Mart 2020;32(1):96-110. doi:10.7240/jeps.581959
Chicago Altınkaynak, Atakan. “Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon”. International Journal of Advances in Engineering and Pure Sciences 32, sy. 1 (Mart 2020): 96-110. https://doi.org/10.7240/jeps.581959.
EndNote Altınkaynak A (01 Mart 2020) Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon. International Journal of Advances in Engineering and Pure Sciences 32 1 96–110.
IEEE A. Altınkaynak, “Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon”, JEPS, c. 32, sy. 1, ss. 96–110, 2020, doi: 10.7240/jeps.581959.
ISNAD Altınkaynak, Atakan. “Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon”. International Journal of Advances in Engineering and Pure Sciences 32/1 (Mart 2020), 96-110. https://doi.org/10.7240/jeps.581959.
JAMA Altınkaynak A. Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon. JEPS. 2020;32:96–110.
MLA Altınkaynak, Atakan. “Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon”. International Journal of Advances in Engineering and Pure Sciences, c. 32, sy. 1, 2020, ss. 96-110, doi:10.7240/jeps.581959.
Vancouver Altınkaynak A. Ağsız Yöntem Uygulamaları için Trigonometri Tabanlı Radyal Özelliğe Sahip Yeni Bir Temel Fonksiyon. JEPS. 2020;32(1):96-110.