Ağsız yöntem uygulamalarında kullanılması için yeni radyal temel fonksiyonlar önerilmesi ve önerilen fonksiyonların karakteristik davranışlarının belirlenmesi
Recommending new radial basis functions for using in meshless method applications and determining the characteristic behaviors of the recommended functions
Yazar: AHMET CAN BİLGEN
Danışman: DOÇ. DR. ATAKAN ALTINKAYNAK
İstanbul Teknik Üniversitesi / Lisansüstü Eğitim Enstitüsü / Makine Mühendisliği Ana Bilim Dalı / Katı Cisimlerin Mekaniği Bilim Dalı
Derece: Yüksek Lisans, 2022
Konu: Makine Mühendisliği = Mechanical Engineering
Dizin: Ağsız yöntemler = Meshless methods ; Radyal baz fonksiyonları = Radial basis functions ; Sonlu farklar yöntemi = Finite differences method

Özet:
Ağsız yöntemler, ilk kez 1970'li yıllarda ortaya çıkmış ve yaklaşık elli yıldır mühendislik problemlerin sayısal çözümü için kullanılan bir yöntemdir. Ağsız yöntemler, çözüm bölgesine ağ yapısı oluşturmak yerine tanımlı düğüm noktaları oluşturarak sonuçlar elde etmeyi amaçlar. Ağ yapısı gerektirmemesi, büyük deformasyon problemlerini daha sağlam bir şekilde ele alabilmesi, yüksek dereceli sürekli şekil fonksiyonlarına ve yerel olmayan enterpolasyon karakterlerine hassasiyeti olmaması ve 3 boyutlu yapılar için uygun olması, bu yöntemin avantajları olarak sayılabilir. Noktasal radyal temel fonksiyon yöntemi, ağsız yönteme dayanmaktadır. Kısmı diferansiyel problemlerin çözümü için ağ yapısının olmaması ve programlamadaki esneklikten dolayı karmaşık ve düzensiz geometrilerde bile çekici bir çözüm tekniğidir. Hem sınır koşulları hem de diferansiyel denklemi sağlamak için çözüm bölgesindeki noktalar arasındaki uzaklığa bağlı olarak radyal temel fonksiyonlar ile yaklaşık olarak elde edilen özel çözüm, aynı zamanda problemdeki denklemin çözümü olur. Birden fazla radyal temel fonksiyon türü vardır. Problem yaklaşımında doğruluk ve yakınsaması için radyal temel fonksiyonların seçimi önem arz etmektedir. Bu tez çalışmasında, ağsız yöntemlerde kullanılması için radyal özelliğe sahip yeni temel fonksiyonlar önerisinde bulunulmuştur. Önerilen radyal temel fonksiyonları, dört farklı mühendislik sayısal probleminin çözümü için kullanılmıştır. Elde edilen sonuçlar ile ağsız yöntemlerde sıklıkla kullanılan Gauss ve Ters Multikuadrik radyal temel fonksiyonları ile elde edilen sonuçlar karşılaştırılarak, radyal özelliğe sahip yeni temel fonksiyonların karakterislik davranışları incelenmiştir. Yapılan sayısal deneylerin sonucunda, önerilen Sinüs, Tanh, Sech ve Sech Seri Açılımı radyal temel fonksiyonlarının Gauss ve Ters Multikuadrarik fonksiyonlara kıyasla daha az nokta sayısın daha düşük hata mertebeleri elde ettiği görülmüştür. Ayrıca, Sinüs 5.Seri Açılım radyal temel fonksiyonun dışındaki diğer önerilen fonksiyonlar ise Ters Multikuadrik ve Gauss fonksiyolarının farklı nokta sayıları ile elde edilen hata değerlerinde göstermiş oldukları davranışa benzer davranışlar gösterdiği gözlemlenmiştir. Sinüs, Sinüs Seri Açılımı, Tanh Seri Açılımı ve Sech Seri Açılımı radyal temel fonksiyonlarının, Gauss ve Ters Multikuadrik fonksiyonlara kıyasla, nokta sayısında artışı ile aynı mertebedeki hata değerleri elde edilebilmesi için daha geniş bir aralıkta kullanılabilecek şekil parametresi seçilebileceğini göstermişlerdir. Ayrıca, Tanh ve Sech radyal temel fonksiyonları ise Gauss ve Ters Multikuadrik fonksiyonlar ile farklı şekil parametre değerlerinde elde ettiği hata değerlerine benzer değerler elde ettikleri gözlemlenmiştir. Akış probleminde ise, şekil parametresinin hata değeri üzerinde göstermiş olduğu etki, Sinüs 5.Seri Açılım radyal temel fonksiyonu dışında önerilen diğer fonksiyonlar ile Gauss ve Ters Multikuadrik fonksiyonlarla benzer davranışları göstermişlerdir. Bu sonuçlar neticesinde, önerilen Sinüs 5.Seri Açılım radyal temel fonksiyonu dışındaki diğer önerilen tüm fonksiyonlar, Gauss ve Ters Multikuadrik fonksiyonları yerine ağsız yöntem uygulamalarında alternatif olarak kullanılabileceğini göstermişlerdir.

Abstract:
Meshless methods first emerged in the 1970s and have been used for the numerical solution of engineering problems for about fifty years. Meshless methods use a set of nodes scattered within the solution region space of the problem, as well as a set of nodes scattered at the boundaries of the solution region space, to represent the problem domain and its boundaries. These scattered sets of nodes do not form a mesh, that is, it does not require any prior knowledge of the relationship between nodes for the interpolation or approximation of the unknown functions of the field variables. The advantages of this method are that it does not require a mesh structure, can handle large deformation problems more robustly, is not sensitive to high-order continuous shape functions and non-local interpolation characters, and is suitable for 3-dimensional structures. The collocation radial basis function method is based on the meshless method. It is an attractive solution technique for partial differential problems even in complex and irregular geometries due to the lack of mesh structure and flexibility in programming. In order to provide both the boundary conditions and the differential equation, the particular solution obtained with the radial basic functions depending on the distance between the points in the solution region is also the solution of the equation in the problem. Factors such as boundary conditions, number of points, problem type, radial basis function type affect the characteristics of the collocation matrix. As a result of these influences, bad conditioning of this matrix can be caused and instabilities may arise in the solution due to this situation. There are multiple types of radial basis functions. The selection of radial basis functions is important for the accuracy and convergence of the problem approach. In this thesis, a new basic function with radial property is proposed for use in meshless methods. The proposed radial basis functions and Gaussian and Inverse Multiquadric radial basis functions, which are frequently used in meshless methods, are used to solve the Poisson problems with different boundary conditions, the general elliptic problem and the 2D square cavity problem. In the Poisson and general elliptical problems, the square error values are calculated for different shape parameter and point number values. In addition, the behavior of the error obtained with the radial basis functions in the solution region is examined. In the 2D square cavity problem, the square error values were calculated for different shape parameter values, and as a result of these values, the optimum shape parameter value was found for each radial basis function. By using the shape parameter values found, the values of the velocity profiles at the center line of the square cavity were calculated. By comparing the results of the proposed radial basis functions with the results of Gaussian and Inverse Multiquadric radial basis functions, the characteristics of the proposed radial basis functions are investigated. In the first Poisson problem, only the Dirichlet boundary condition is applied at the boundaries of the solution region. Square error values were calculated for different shape parameter values in 225 and 625 point numbers, uniform and Halton point distribution type. When the results are examined, it is seen that the proposed Sine and Sine Series Expansion radial basis functions obtain values similar to the square error values obtained with the Gaussian and Inverse Multiquadric radial basis functions in a wider range of shape parameters. In the low shape parameter, instabilities appeared due to poor conditioning of the collocation matrix. At high shape parameter values, a lower error value was obtained with the proposed Tanh Series Expansion, Sine Series Expansion and Sech Serial Expansion radial basis functions, compared to the square error value obtained with the Gaussian and Inverse Multiquadric radial fundamental functions. In Halton and uniform point distribution type, the square error value was calculated for different number of points. When the results were examined, it was observed that with the proposed Sine radial basis function, a lower error value was obtained with a lower number of points. With the proposed Sine 5th Series Expansion, very high error values were obtained compared to the square error value obtained with all radial basis functions at high point numbers. The behavior of the square error value with the increase in the number of points is almost similar for all radial basis functions. The behavior of the error obtained with radial basis functions in the solution region of 81 point number, Halton and uniform point distribution type is investigated. In the uniform point distribution type, it has been observed that the error generally increases close to the boundaries. In the Halton point distribution, it is observed that the error obtained with the proposed Sech and Inverse Multiquadric radial basis functions increases close to the corners. It has been observed that in other radial basis function types, the error is distributed over the entire solution region and occurs in the form of hills. In the second Poisson problem, Neumann and Dirichlet boundary conditions are applied at the boundaries of the solution region. Square error values were calculated for different shape parameter values in 225 and 625 point numbers, uniform and Halton point distribution type. When the results were examined, it was observed that the square error was affected by the promlem and boundary conditions, apart from the number of points and the point distribution type. It has been observed that the proposed Sine Series Expansion radial basis functions have similar values to the square error values obtained with the Gaussian and Inverse Multiquadric radial basis functions in a wider range of shape parameters. In addition to low shape parameter values, instabilities appeared at high shape parameter values. This is thought to be due to the application of the Neumann boundary condition. At high shape parameter values, a lower error value was obtained with the proposed Tanh Series Expansion, Sine Series Expansion and Sech Serial Expansion radial basis functions, as in the first Poisson problem, compared to the square error value obtained with Gaussian and Inverse Multiquadric radial basis functions. In Halton and uniform point distribution type, the square error value was calculated for different number of points. When the results were examined, it was observed that in the halton point distribution type, a lower error value was obtained with the proposed Sech radial basis function at a lower number of points. With the proposed Sine 5th Series Expansion, very high error values were obtained at high point numbers compared to the square error value obtained with all radial basis functions. In addition, higher square error was obtained with the radial basis functions of Sine and Sine Series Expansion compared to the first Poisson problem with high number of points. The behavior of the error obtained with radial basis functions in the solution region of 81 point number, Halton and uniform point distribution type is investigated. In the uniform point distribution type, it has been observed that the error increases close to the boundary where the Neumann boundary condition is defined in all other radial basis functions except the proposed Sine 5th Series Expansion radial basis function due to the effect of Neumann boundary condition. This situation is eliminated in the Halton point distribution type. It has been observed that the error obtained with the Inverse Multiquadric radial basis function increases close to the corners and boundaries in the solution region. The behavior of the error obtained with the proposed Tanh and Sech radial basis functions in the solution region is similar to the behavior of the error obtained with the Inverse Multiquadric radial basis function in the solution region. The behavior in the solution region of the error obtained with other types of radial basis functions exhibited the same behavior as in the first problem. In the general elliptic problem, only the Dirichlet boundary condition is applied at the boundaries of the solution region. Square error values were calculated for different shape parameter values in 225 and 625 point numbers, uniform and Halton point distribution type. When the results are examined, since the order of the analytical solution function in this problem is 10 times lower than in the first two Poisson problems, the square error values obtained have decreased approximately 10 times. It has been observed that the proposed Sech Series Expansion radial basis functions obtain values similar to the square error values obtained with Gaussian and Inverse Multiquadric radial basis functions in a wider range of shape parameters. It has been observed that the squared error value obtained with the radial basis function of the proposed Sine Series Expansion is affected by the $a$ and $b$ functions defined in the general elitist problem. As a result, according to the results in the first two Poisson problems, a very high square error value was obtained at high shape parameter values. Other proposed radial basis functions have behaved similarly to those of the first two Poisson problems. In Halton and uniform point distribution type, the square error value was calculated for different number of points. When the results were examined, it was observed that in the halton point distribution type, a lower error value was obtained with the proposed Sech Series Expansion radial basis function at a lower number of points. With the proposed Sine 5th Series Expansion and Sine Series Expansion radial basis functions, very high error values were obtained in high point numbers compared to the square error values obtained with all radial basic functions. The behavior of the error obtained with radial basis functions in the solution region of 81 point number, Halton and uniform point distribution type is investigated. In the uniform point distribution type, it has been observed that the error solution region is high in all other radial basis functions except the proposed Sine 5th Series Expansion radial basis function. This situation is eliminated in the Halton point distribution type. It has been observed that the error obtained with the Inverse Multiquadric radial basis function increases close to the corners and boundaries in the solution region. The behavior of the error obtained with the proposed Sech radial basis functions in the solution region is similar to the behavior of the error obtained with the Inverse Multiquadric radial basis function in the solution region. The behavior in the solution region of the error obtained with other types of radial basis functions exhibited the same behavior as in the first problem. In the 2D square cavity problem, only the Dirichlet boundary condition is applied at the boundaries of the solution region. For the solution of the problem, the vorticity transport equation is derived from the Navier-Stokes equations. A total of 121 points in the solution region were used for the solution of the problem, 81 points in the solution region and 40 points at the borders of the solution region. The points are positioned in the solution region using the uniform point distribution type. The coefficients required to find the velocity components and vorticity values in the solution region were obtained using the iteration technique. The square error value was calculated for different shape parameter values in different velocity components. When the results are examined, it has been observed that the square error value obtained with all radial basis functions except the proposed Sine 5th Series Expansion radial basis function is very sensitively affected in the shape parameter. In all cases, almost the same square error value was obtained for all shape parameter values with the proposed Sine 5th Series Expansion radial basis function. It has been observed that the square error values obtained for the y-direction component of the velocity are lower than the square error values obtained for the x-direction component of the velocity. While the lowest squared error values obtained for the y-direction component of the velocity were obtained with the Inverse Mulquadric radial basis function, the lowest square error value obtained for the x-direction component of the velocity was obtained with the Gauss radial basis function. In the center line of the square cavity, the optimum shape parameter value required to determine the values of the velocity components was determined as a result of the square error values calculated for different shape parameter values. In the center line of the square cavity, the optimum shape parameter value required to determine the values of the velocity components was determined as a result of the square error values calculated for different shape parameter values. When the values of the velocity component in the x-direction are examined, the difference between the values of the velocity component in the x-direction and the reference values obtained with the radial basis functions of Sine Series Expansion, Sine 5th Series Expansion, Sech Series Expansion and Tanh Series Expansion is higher than the other radial basis functions. The closest values to the reference values were obtained with the Gaussian radial basis function. When the values of the velocity component in the y direction are examined, at some points, the difference between the y-direction component values of the velocity obtained with the radial basis functions of the Sine and the reference values is higher than the difference between the other radial basis functions. In all cases, at the boundary points, the closest values of the velocity components at the boundary were obtained with the proposed Sine, Sech and Tanh, and Gaussian and Inverse Multiquadric radial basis functions. When the results are examined in general, it is seen that other proposed radial basis functions other than the proposed Sine 5th Series Expansion can be used instead of Gaussian and Inverse Multiquadric radial basis functions in meshless method applications.