Radyal Baz Fonksiyonu (RBF) kullanan Ağsız (Meshless) Çözüm Yöntemlerinde Şekil Parametresi ve Merkez Nokta Sayısının Çözüme Etkisi

Year 2024, Volume: 14 Issue: 3, 1301 - 1321, 15.09.2024

Hüseyin Yıldız , Hasan Ömür Özer , Birkan Durak , Erol Uzal

https://doi.org/10.31466/kfbd.1455017

Abstract

yere sahiptir. Fiziksel olaylar, belirli sınır şartları sağlayan diferansiyel denklem sistemleri ile matematiksel olarak modellenebilir. Genellikle denklem sisteminin analitik çözümünü bulmak mümkün olmaz. Bu nedenle çeşitli sayısal yöntemler geliştirilmiştir. Günümüzde en çok kullanılan sayısal çözüm yöntemlerinden ikisi Sonlu Elemanlar Yöntemi (SEY) ve Sonlu Farklar Yöntemi (SFY)’dir. Bu yöntemlerde çözüm alanı ağ adı verilen küçük parçalara (bölgelere) ayrılarak hesaplamalar yapılır. Ağ örme işlemi oldukça karmaşık ve uzun zaman alan bir işlemdir. Kırılma mekaniği ve hareketli sistemlerin modellenmesinde her hesaplama sonrası ağın yenilenmesi gereklidir. Araştırmacılar, özellikle 20. yüzyılın sonlarında bu zorlukların üstesinden gelmek için ağsız çözüm yöntemleri geliştirdiler. Çözüm alanına düzenli veya düzensiz örnekleme noktaları yerleştiren ağsız çözüm teknikleri için uygun bir temel fonksiyon ailesi de gereklidir. Önerilen baz fonksiyon ailesi, diferansiyel denklem sistemini ve sınır şartlarını sağlayacak şekil katsayıları ile temsil edilir. Bu çalışmada radyal baz fonksiyon (RBF) kullanan ağsız çözüm yöntemi bir boyutlu ve iki boyutlu ısı geçiş problemlerine uygulanmıştır. İncelenen problemlerde merkez noktaların ve şekil katsayısının benzetim sonuçlarına etkisi incelenmiştir. Bulgular, kontrol (kollokasyon) noktalarının sayısının doğrudan çözümün kararlılığıyla ilişkili olduğunu ve kontrol nokta sayısının merkez nokta sayısından fazla olduğunda kararlılığa katkıda bulunduğunu göstermektedir. Şekil yapısının uygun çözümü için merkez nokta değişikliklerinin büyüklüğünde bir artışın gerekli olduğu gözlemlenmiştir. Bu çalışmanın sonuçları, şekil katsayısı arttıkça doğru bir çözüme ulaşmak için merkez nokta sayısının ve yineleme sayısının da arttırılması gerektiğini göstermektedir.

Keywords

Kollokasyon yöntemi, Radyal baz fonksiyon, Sayısal çözüm, Şekil fonksiyonu

Effect of the Shape Parameter and the Number of Center Points on the Solution in Meshless Solution Methods Using Radial Basis Function (RBF)

Abstract

Ordinary differential equations (ODE) and partial differential equations (PDE) have an important role in solving engineering and physics problems. Physical phenomena may be represented mathematically using differential equation systems under certain boundary conditions. However, it is usually not possible to find an analytical solution to the resulting system of equations. Therefore, various numerical methods have been developed. Currently, the Finite Element Method (FEM) and the Finite Difference Method (FED) are the most widely used numerical solution techniques. In these methodologies, computations are carried out by dividing the solution space into small pieces called mesh. This procedure is a highly complex and time-consuming process. Due to coupled calculation points within the mesh, it is necessary to update the mesh after each calculation when modelling fracture mechanics and dynamic systems. Researchers developed meshless solution methods to address these challenges, especially in the late 20th century. For meshless solution techniques that place regular or irregular sampling points in the solution domain, a suitable family of basis functions is also required. This family is expressed by the shape coefficients to satisfy the system of differential equations and boundary conditions. By considering one and two dimensional heat transfer problems, this study investigated the effect of center points and shape coefficients on the simulation results in the meshless solution method using Radial Basis Functions (RBF). The results show that the number of control (collocation) points is directly related to the stability of the solution and contributes to stability when the number of control points is greater than the number of center points. It has been observed that an increase in the magnitude of the center point changes is necessary for the optimal solution of the shape structure. The results of this study demonstrate that as the shape coefficient increases, the number of centre points and the number of iterations must also be increased to achieve an accurate solution.

Keywords

Collocation method, Radial basis function, Numerical solution, Shape function

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