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روش پايداري توابع پايه اي و شعاعي و تفاضلات متناهي با هسته تركيبي

كارشناسي ارشد

رياضي كاربردي - آناليز عددي

1399/7/13

دكتر جليل رشيدي نيا

دكتر تورج نيك آزاد

رياضي

تحولات اخير اين امكان را فراهم آورده است كه با اتخاذ چارچوب شبكه اي مبتني بر تفاضلات متناهي(FD) بر مشكلات غلبه نماييم. چنين رويكردي ضرورت يك روند پياده سازي بدون شبكه را ايجاد مي كند و از آن به عنوان روش RBF-FD ذكر مي شود. در اين پايان نامه، ما يك رويكرد (RBF-FD) توابع پايه شعاعي مبتني بر تفاضلات متناهي گاوسي و مكعبي را گردآوري مي كنيم. اين هسته تركيبي RBF پايدار با يك هسته تركيبي ، توليد شده از طريق تركيب براي بهبود وضعيت ماتريس ضرايب دستگاه حاصله لازم است، در نتيجه، دستگاه خطي را مي توان با روش هاي مستقيم حل كرد كه منجر به كاهش قابل توجهي در هزينه محاسباتي در مقايسه با روش هاي استاندارد مي گردد. بر خلاف ساير روش ها ، روش ( RBF-FD) مقدار ويژه ي طيفي از ماتريسهاي مشتق صرف نظر از بي نظمي ، و اندازه استنسيل پايدار باقي مي ماند. به عنوان محك زدن در عمل، ما روش حاصله را روي معادله موج در فضاي دو بعدي به كار مي بريم. به منظور از بين بردن بازتاب هاي نادرست از مرزهاي در روند محاسباتي تمهيداتي در باره ي شرايط مرزي در نظر مي گيريم .

1399/09/07

Stability method of radial basis functions and finite differences with hybrid kernel

10/4/2020 12:00:00 AM

Recent developments have made it possible to overcome grid-based limitations of finite difference (FD) methods by adopting the kernel-based meshless framework using radial basis functions (RBFs). Such an approach provides a meshless implementation and is referred to as the radial basis-generated finite difference (RBF-FD) method. In this paper, we propose a stabilized RBF-FD approach with a hybrid kernel, generated through a hybridization of the Gaussian and cubic RBF. This hybrid kernel was found to improve the condition of the system matrix, consequently, the linear system can be solved with direct solvers which leads to a significant reduction in the computational cost as compared to standard RBF-FD methods coupled with present stable algorithms. Unlike other RBFFD approaches, the eigenvalue spectra of differentiation matrices were found to be stable irrespective of irregularity, and the size of the stencils. As an application, we solve the frequency-domain acoustic wave equation in a 2D half-space. In order to suppress spurious reflections from truncated computational boundaries, absorbing boundary conditions have been effectively implemented.direct solvers which leads to a significant reduction in the computational cost as compared to standard RBF-FD methods coupled with present stable algorithms. Unlike other RBFFD approaches, the eigenvalue spectra of differentiation matrices were found to be stable irrespective of irregularity, and the size of the stencils. As an application, we solve the frequency-domain acoustic wave equation in a 2D half-space. In order to suppress spurious reflections from truncated computational boundaries, absorbing boundary conditions have been effectively implemented.