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

Yıl 2024, Cilt: 14 Sayı: 3, 1301 – 1321, 15.09.2024

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

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

• 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
• Aydın, E.S., (2022). Kayısı meyvesinin dondurarak kurutulmasının sayısal olarak incelenmesi için matematiksel bir model. Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi, 37 (1), 347-360. https://doi.org/10.17341/gazimmfd.791792
• Belytschko, T., Lu, Y.Y., ve Gu, L., (1994). Element-free Galerkin methods. The International Journal for Numerical Methods in Engineering, 37(2), 229-256. https://doi.org/10.1002/nme.1620370205.
• Boglietti, A., Cavagnino, A., Staton, D., Shanel, M., Mueller, M., ve Mejuto, C., (2009). Evolution and modern approaches for thermal analysis of electrical machines. IEEE Transactions on industrial electronics, 56(3), 871-882. https://doi.org/10.1109/TIE.2008.2011622
• Cao, J,, Cheng, P., ve Hong, F., (2008). A numerical analysis of forces imposed on particles in conventional dielectrophoresis in microchannels with interdigitated electrodes. Journal of Electrostatics, 66(11–12), 620-626. https://doi.org/10.1016/j.elstat.2008.09.003.
• Chakraverty, S., Mahato, N.R., Karunakar, P., ve Rao, T.D., (2019). Advanced numerical and semi-analytical methods for differential equations. John Wiley & Sons.
• Chen, Y., Eskandarian, A., Oskard, M., ve Lee, J.D., (2004). Meshless analysis of plasticity with application to crack growth problems. Theoretical and Applied Fracture Mechanics, 41(1–3), 83-94. https://doi.org/10.1016/j.tafmec.2003.11.024.
• Chong, Y.C., Goss, J., Popescu, M., Staton, D., Hawkin, D., ve Boglietti, A., (2019). Electromagnetic performance with and without considering the impact of rotation on convective cooling. The Journal of Engineering, 17, 3537-3541. https://doi.org/10.1049/joe.2018.8024
• Çengel ,Y.A., ve Ghajar, A.J., (2020). Isı ve Kütle Transferi. Palme Yayınevi. ISBN: 9786053552871
• Durak, B., (2020). Adi ve Kısmi Diferansiyel Denklemlerin Çözümlerinin Kollokasyon Yöntemiyle Bulunması. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 10(4), 1136-1143. https://doi.org/10.17714/gumusfenbil.681276
• Fallah, N.A., Bailey, C., Cross, M., Taylor, G.A., (2000). Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis. Applied Mathematical Modelling, 24(7), 439-455. https://doi.org/10.1016/S0307-904X(99)00047-5.
• Falzon, B.G., ve Muthu, N., (2018). 1.13-Micromechanical Modelling of Composite Materials Using the Element-Free Galerkin Approach. Editor(s): Beaumont PWR, Comprehensive Composite Materials II (s. 327-352). Zweben CH., Elsevier, ISBN 9780081005347. https://doi.org/10.1016/B978-0-12-803581-8.09888-X
• Fasshauer, G.E., (2007). Meshfree approximation methods with MATLAB (Interdisciplinary Mathematical Sciences: Volume 6). World Scientific.
• Fasshauer, G.E., ve McCourt, M.J., (2015). Kernel-based approximation methods using Matlab (Interdisciplinary Mathematical Sciences Vol 19). World Scientific Publishing Company.
• Fornberg, B., ve Flyer, N., (2015). Solving PDEs with radial basis functions. Acta Numerica, 24, 215-258. https://doi.org/10.1017/S0962492914000130
• Fornberg, B., ve Flyer, N., (2005). Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids. Advances in Computational Mathematics, 23, 5-20. https://doi.org/10.1007/s10444-004-1812-x
• Ghahfarokhi, P.S., Podgornovs, A., Kallaste, A., Marques Cardoso, A.J., Belahcen, A., Vaimann, T., Kudrjavtsev, O., Asad, B., ve Iqbal, M.N., (2022). Steady-State Thermal Modeling of Salient Pole Synchronous Generator. Energies, 15(24), 9460. https://doi.org/10.3390/en15249460
• Gingold, R.A., ve Monaghan, J.J., (1977). Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society, 181(3), 375–389. https://doi.org/10.1093/mnras/181.3.375
• Hahn, D.W., ve Özişik, M.N., (2012). Heat conduction. John Wiley & Sons.
• He, Y., Meng, Z., Xu, H., ve Zou, Y., (2020). A dynamic model of evaluating differential automatic method for solving plane problems based on BP neural network algorithm. Physica A: Statistical Mechanics and its Applications, 556(124845). https://doi.org/10.1016/j.physa.2020.124845.
• Herman, R. L., (2015). 6 Problems in Higher Dimensions. Introduction to partial differential equations. R. L. Herman. https://people.uncw.edu/hermanr/pde1/PDEbook/PDE_Main.pdf
• Jin, J-M., (1993). The Finite Element Method in Electromagnetics. Wiley. ISBN: 9780471586272
• Kansa, E.J., (1986). Application of Hardy’s multiquadric interpolation to hydrodynamics. Continuous System Simulation Languages, 1986, 111-117.
• Kansa, E.J., (1990). Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Computers & Mathematics with applications, 19(8-9), 127-145. https://doi.org/10.1016/0898-1221(90)90270-T
• Karakoc, S.B.G., (2018). Kollokasyon Sonlu Eleman Yöntemi ile MKdV Denkleminin Sayısal Çözümleri. Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B – Teorik Bilimler, 6(2), 206-218. https://doi:10.20290/aubtdb.420247
• Li, J., Wang, G., Zhan, J., Liu, S., Guan, Y., Naceur, H., Coutellier, D., ve Lin, J., (2021). Meshless SPH analysis for transient heat conduction in the functionally graded structures. Composites Communications, 24(100664). https://doi.org/10.1016/j.coco.2021.100664.
• Libersky, L.D., Petschek, A.G., Carney, T.C., Hipp, J.R., ve Allahdadi, F.A., (1993). High Strain Lagrangian Hydrodynamics: A Three-Dimensional SPH Code for Dynamic Material Response. Journal of Computational Physics, 109(1), 67-75. https://doi.org/10.1006/jcph.1993.1199.
• Mugglestone, J., Pickering, S.J., ve Lampard, D., (1999). Effect of geometric changes on the flow and heat transfer in the end region of a TEFC induction motor. Ninth International Conference on Electrical Machines and Drives (s. 40-44). Canterbury, UK. https://doi.org/10.1049/cp:19990987
• Narimani, N., ve Dehghan, M., (2022). A direct RBF-PU method for simulating the infiltration of cytotoxic T-lymphocytes into the tumor microenvironment. Communications in Nonlinear Science and Numerical Simulation, 114(106616). https://doi.org/10.1016/j.cnsns.2022.106616.
• Nayroles, B., Touzot, G., ve Villon, P., (1992). Generalizing the finite element method: Diffuse approximation and diffuse elements. Computational Mechanics, 10, 307-318.
• Pandey, A., Madduri, B., Perng, C.Y., Srinivasan, C., ve Dhar, S., (2022). Multiphase Flow and Heat Transfer in an Electric Motor. ASME International Mechanical Engineering Congress and Exposition (s. 1-17). Columbus, Ohio, USA. https://doi.org/10.1115/IMECE2022-96801
• Papini, L,, ve Gerada, C., (2014). Thermal-electromagnetic analysis of solid rotor induction machine. 7th IET International Conference on Power Electronics, Machines and Drives (PEMD 2014) (s. 1-6). Manchester, UK. https://doi.org/10.1049/cp.2014.0462
• Pekedis, M., ve Yıldız, H., (2010). Ağsız Yöntemler ve Sınıflandırılması. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi, 16(1), 1-9. https://dergipark.org.tr/tr/pub/pajes/issue/20507/218301
• Pickering, S.J., Lampard, D., ve Shanel, M., (2001). Modelling ventilation and cooling of the rotors of salient pole machines. (IEMDC) IEEE International Electric Machines and Drives Conference (s. 806-808). Cambridge, MA, USA.
• Stach. S., (2014). 11- Modelling fracture processes in orthopaedic implants,. Editor(s): Zhongmin, J., Computational Modelling of Biomechanics and Biotribology in the Musculoskeletal System Biomaterials and Tissues (s. 331-368). Woodhead Publishing, ISBN 978-0-85709-661-6.
• Staton, D., Pickering, S.J., ve Lampard, D., (2001). Recent advancement in the thermal design of electric motors. SMMA 2001 Fall Technology Conference (s. 1-11). Durham, North Carolina, USA.
• Tanbay, T. (2019). Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. Journal of Innovative Science and Engineering, 3(1), 23-31. https://doi.org/10.38088/jise.570328
• Wang, H., ve Qin, Q.H., (2019). Chapter 1 – Overview of meshless methods. Editor(s): Wang, H., Qin, Q.H., Methods of Fundamental Solutions in Solid Mechanics (s. 3-51). Elsevier. ISBN 9780128182833, https://doi.org/10.1016/C2018-0-03684-1
• Wang, H., ve Qin, Q.H., (2019). Chapter 5 – Meshless analysis for thin plate bending problems. Editor(s): Wang, H., Qin, Q.H., Methods of Fundamental Solutions in Solid Mechanics (s. 127-142). Elsevier, ISBN 9780128182833, https://doi.org/10.1016/C2018-0-03684-1
• Yıldız, H., Korkmaz Can, N., Ozguney, O.C., ve Yagiz, N., (2020). Sliding mode control of a line following robot, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 42(561), 1-13. https://doi.org/10.1007/s40430-020-02645-3
• Zarin, R., (2022). Numerical study of a nonlinear COVID-19 pandemic model by finite difference and meshless methods. Partial Differential Equations in Applied Mathematics, 6(100460). https://doi.org/10.1016/j.padiff.2022.100460.
• Zhong, R., Wang, Q., Hu, S., Qin, B., ve Shuai, C., (2023). Meshless analysis for modal properties and stochastic responses of heated laminated rectangular/sectorial plates in supersonic airflow. European Journal of Mechanics – A/Solids, 98(104872). https://doi.org/10.1016/j.euromechsol.2022.104872.
• Zienkiewicz, O.C., Taylor, R.L., ve Zhu, J.Z., (2005). The Finite Element Method: Its Basis and Fundamentals. 6th Edition. Butterworth-Heinemann.

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

Yıl 2024, Cilt: 14 Sayı: 3, 1301 – 1321, 15.09.2024

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

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

• 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
• Aydın, E.S., (2022). Kayısı meyvesinin dondurarak kurutulmasının sayısal olarak incelenmesi için matematiksel bir model. Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi, 37 (1), 347-360. https://doi.org/10.17341/gazimmfd.791792
• Belytschko, T., Lu, Y.Y., ve Gu, L., (1994). Element-free Galerkin methods. The International Journal for Numerical Methods in Engineering, 37(2), 229-256. https://doi.org/10.1002/nme.1620370205.
• Boglietti, A., Cavagnino, A., Staton, D., Shanel, M., Mueller, M., ve Mejuto, C., (2009). Evolution and modern approaches for thermal analysis of electrical machines. IEEE Transactions on industrial electronics, 56(3), 871-882. https://doi.org/10.1109/TIE.2008.2011622
• Cao, J,, Cheng, P., ve Hong, F., (2008). A numerical analysis of forces imposed on particles in conventional dielectrophoresis in microchannels with interdigitated electrodes. Journal of Electrostatics, 66(11–12), 620-626. https://doi.org/10.1016/j.elstat.2008.09.003.
• Chakraverty, S., Mahato, N.R., Karunakar, P., ve Rao, T.D., (2019). Advanced numerical and semi-analytical methods for differential equations. John Wiley & Sons.
• Chen, Y., Eskandarian, A., Oskard, M., ve Lee, J.D., (2004). Meshless analysis of plasticity with application to crack growth problems. Theoretical and Applied Fracture Mechanics, 41(1–3), 83-94. https://doi.org/10.1016/j.tafmec.2003.11.024.
• Chong, Y.C., Goss, J., Popescu, M., Staton, D., Hawkin, D., ve Boglietti, A., (2019). Electromagnetic performance with and without considering the impact of rotation on convective cooling. The Journal of Engineering, 17, 3537-3541. https://doi.org/10.1049/joe.2018.8024
• Çengel ,Y.A., ve Ghajar, A.J., (2020). Isı ve Kütle Transferi. Palme Yayınevi. ISBN: 9786053552871
• Durak, B., (2020). Adi ve Kısmi Diferansiyel Denklemlerin Çözümlerinin Kollokasyon Yöntemiyle Bulunması. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 10(4), 1136-1143. https://doi.org/10.17714/gumusfenbil.681276
• Fallah, N.A., Bailey, C., Cross, M., Taylor, G.A., (2000). Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis. Applied Mathematical Modelling, 24(7), 439-455. https://doi.org/10.1016/S0307-904X(99)00047-5.
• Falzon, B.G., ve Muthu, N., (2018). 1.13-Micromechanical Modelling of Composite Materials Using the Element-Free Galerkin Approach. Editor(s): Beaumont PWR, Comprehensive Composite Materials II (s. 327-352). Zweben CH., Elsevier, ISBN 9780081005347. https://doi.org/10.1016/B978-0-12-803581-8.09888-X
• Fasshauer, G.E., (2007). Meshfree approximation methods with MATLAB (Interdisciplinary Mathematical Sciences: Volume 6). World Scientific.
• Fasshauer, G.E., ve McCourt, M.J., (2015). Kernel-based approximation methods using Matlab (Interdisciplinary Mathematical Sciences Vol 19). World Scientific Publishing Company.
• Fornberg, B., ve Flyer, N., (2015). Solving PDEs with radial basis functions. Acta Numerica, 24, 215-258. https://doi.org/10.1017/S0962492914000130
• Fornberg, B., ve Flyer, N., (2005). Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids. Advances in Computational Mathematics, 23, 5-20. https://doi.org/10.1007/s10444-004-1812-x
• Ghahfarokhi, P.S., Podgornovs, A., Kallaste, A., Marques Cardoso, A.J., Belahcen, A., Vaimann, T., Kudrjavtsev, O., Asad, B., ve Iqbal, M.N., (2022). Steady-State Thermal Modeling of Salient Pole Synchronous Generator. Energies, 15(24), 9460. https://doi.org/10.3390/en15249460
• Gingold, R.A., ve Monaghan, J.J., (1977). Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society, 181(3), 375–389. https://doi.org/10.1093/mnras/181.3.375
• Hahn, D.W., ve Özişik, M.N., (2012). Heat conduction. John Wiley & Sons.
• He, Y., Meng, Z., Xu, H., ve Zou, Y., (2020). A dynamic model of evaluating differential automatic method for solving plane problems based on BP neural network algorithm. Physica A: Statistical Mechanics and its Applications, 556(124845). https://doi.org/10.1016/j.physa.2020.124845.
• Herman, R. L., (2015). 6 Problems in Higher Dimensions. Introduction to partial differential equations. R. L. Herman. https://people.uncw.edu/hermanr/pde1/PDEbook/PDE_Main.pdf
• Jin, J-M., (1993). The Finite Element Method in Electromagnetics. Wiley. ISBN: 9780471586272
• Kansa, E.J., (1986). Application of Hardy’s multiquadric interpolation to hydrodynamics. Continuous System Simulation Languages, 1986, 111-117.
• Kansa, E.J., (1990). Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Computers & Mathematics with applications, 19(8-9), 127-145. https://doi.org/10.1016/0898-1221(90)90270-T
• Karakoc, S.B.G., (2018). Kollokasyon Sonlu Eleman Yöntemi ile MKdV Denkleminin Sayısal Çözümleri. Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B – Teorik Bilimler, 6(2), 206-218. https://doi:10.20290/aubtdb.420247
• Li, J., Wang, G., Zhan, J., Liu, S., Guan, Y., Naceur, H., Coutellier, D., ve Lin, J., (2021). Meshless SPH analysis for transient heat conduction in the functionally graded structures. Composites Communications, 24(100664). https://doi.org/10.1016/j.coco.2021.100664.
• Libersky, L.D., Petschek, A.G., Carney, T.C., Hipp, J.R., ve Allahdadi, F.A., (1993). High Strain Lagrangian Hydrodynamics: A Three-Dimensional SPH Code for Dynamic Material Response. Journal of Computational Physics, 109(1), 67-75. https://doi.org/10.1006/jcph.1993.1199.
• Mugglestone, J., Pickering, S.J., ve Lampard, D., (1999). Effect of geometric changes on the flow and heat transfer in the end region of a TEFC induction motor. Ninth International Conference on Electrical Machines and Drives (s. 40-44). Canterbury, UK. https://doi.org/10.1049/cp:19990987
• Narimani, N., ve Dehghan, M., (2022). A direct RBF-PU method for simulating the infiltration of cytotoxic T-lymphocytes into the tumor microenvironment. Communications in Nonlinear Science and Numerical Simulation, 114(106616). https://doi.org/10.1016/j.cnsns.2022.106616.
• Nayroles, B., Touzot, G., ve Villon, P., (1992). Generalizing the finite element method: Diffuse approximation and diffuse elements. Computational Mechanics, 10, 307-318.
• Pandey, A., Madduri, B., Perng, C.Y., Srinivasan, C., ve Dhar, S., (2022). Multiphase Flow and Heat Transfer in an Electric Motor. ASME International Mechanical Engineering Congress and Exposition (s. 1-17). Columbus, Ohio, USA. https://doi.org/10.1115/IMECE2022-96801
• Papini, L,, ve Gerada, C., (2014). Thermal-electromagnetic analysis of solid rotor induction machine. 7th IET International Conference on Power Electronics, Machines and Drives (PEMD 2014) (s. 1-6). Manchester, UK. https://doi.org/10.1049/cp.2014.0462
• Pekedis, M., ve Yıldız, H., (2010). Ağsız Yöntemler ve Sınıflandırılması. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi, 16(1), 1-9. https://dergipark.org.tr/tr/pub/pajes/issue/20507/218301
• Pickering, S.J., Lampard, D., ve Shanel, M., (2001). Modelling ventilation and cooling of the rotors of salient pole machines. (IEMDC) IEEE International Electric Machines and Drives Conference (s. 806-808). Cambridge, MA, USA.
• Stach. S., (2014). 11- Modelling fracture processes in orthopaedic implants,. Editor(s): Zhongmin, J., Computational Modelling of Biomechanics and Biotribology in the Musculoskeletal System Biomaterials and Tissues (s. 331-368). Woodhead Publishing, ISBN 978-0-85709-661-6.
• Staton, D., Pickering, S.J., ve Lampard, D., (2001). Recent advancement in the thermal design of electric motors. SMMA 2001 Fall Technology Conference (s. 1-11). Durham, North Carolina, USA.
• Tanbay, T. (2019). Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. Journal of Innovative Science and Engineering, 3(1), 23-31. https://doi.org/10.38088/jise.570328
• Wang, H., ve Qin, Q.H., (2019). Chapter 1 – Overview of meshless methods. Editor(s): Wang, H., Qin, Q.H., Methods of Fundamental Solutions in Solid Mechanics (s. 3-51). Elsevier. ISBN 9780128182833, https://doi.org/10.1016/C2018-0-03684-1
• Wang, H., ve Qin, Q.H., (2019). Chapter 5 – Meshless analysis for thin plate bending problems. Editor(s): Wang, H., Qin, Q.H., Methods of Fundamental Solutions in Solid Mechanics (s. 127-142). Elsevier, ISBN 9780128182833, https://doi.org/10.1016/C2018-0-03684-1
• Yıldız, H., Korkmaz Can, N., Ozguney, O.C., ve Yagiz, N., (2020). Sliding mode control of a line following robot, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 42(561), 1-13. https://doi.org/10.1007/s40430-020-02645-3
• Zarin, R., (2022). Numerical study of a nonlinear COVID-19 pandemic model by finite difference and meshless methods. Partial Differential Equations in Applied Mathematics, 6(100460). https://doi.org/10.1016/j.padiff.2022.100460.
• Zhong, R., Wang, Q., Hu, S., Qin, B., ve Shuai, C., (2023). Meshless analysis for modal properties and stochastic responses of heated laminated rectangular/sectorial plates in supersonic airflow. European Journal of Mechanics – A/Solids, 98(104872). https://doi.org/10.1016/j.euromechsol.2022.104872.
• Zienkiewicz, O.C., Taylor, R.L., ve Zhu, J.Z., (2005). The Finite Element Method: Its Basis and Fundamentals. 6th Edition. Butterworth-Heinemann.

Toplam 43 adet kaynakça vardır.

Ayrıntılar

Kaynak Göster

Download or read online: Click here