توسعه روابطی برای تعیین متغیر شکل بهینه در روش عددی بدون شبکه چندربعی

نوع مقاله : مقاله پژوهشی

نویسندگان

گروه مهندسی عمران، دانشکده فنی و مهندسی، دانشگاه قم، قم، ایران

10.22091/cer.2025.12181.1598

چکیده

دقت و سرعت روش‌ پایه شعاعی چندربعی قویاً به روش تعیین متغیر شکل در آن وابسته است. هدف از پژوهش حاضر، جستجوی روابطی برای متغیر شکل بهینه در حل برخی معادلات دیفرانسیل جزئی متداول در مهندسی آب می‌باشد. به‌این‌منظور، ابتدا روند بازتولید و حل معادلات با روش چندربعی بیان می‌شود سپس متغیر شکل بهینه برحسب طول دامنه و تعداد مراکز محاسباتی مختلف به دست می‌آید. در ادامه، بر اساس نتایج حاصل‌شده، روابطی برای پارامتر شکل بهینه پیشنهاد می‌شود که حجم محاسبات را به‌طور چشم‌گیری کاهش می‌دهد. نتایج روابط استخراج‌شده با پاسخ‌های دقیق و روابط پیشین مقایسه گردیده و نشان داده شده است که برخلاف روابط پیشین از دقت بالایی برخوردار هستند به گونه‌ای که در مثال‌ها و معادلات مختلف خطای روابط معرفی شده در مقایسه با روابط موجود بسیار ناچیز و در مواردی در حد صفر است و در مقایسه با روشهای بهینه‌یابی، از آنجائیکه مقدار ضریب شکل بهینه با استفاده از یک رابطه ریاضی به دست می‌آید، سرعت اجرا به دلیل عدم نیاز به الگوریتم تکراری بهینه‌یابی، بسیار بیشتر است. همچنین وجود شرط پایداری در این الگوریتم‌ها مورد بررسی قرار گرفت و مشاهده گردید که علاوه بر وجود این شرط در معادلات پخش، پخش- انتقال و برگرز، بیشینه گام زمانی به مقدار ضریب پخش، سرعت و عدد رینولذر وابسته است.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Developing Relationships for Determining the Optimal Shape Parameter in the Multiquadric Meshless Method

نویسندگان [English]

  • Hanieh Talebi Kalan
  • Sara Mohsenzadeh Golafzani
  • Reza Babaee
  • Ehsan Jabbari
Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran
چکیده [English]

The accuracy and efficiency of the Multiquadric Radial Basis Function (MQ-RBF) method are highly sensitive to the choice of the shape parameter. This study aims to identify optimal relationships for determining the shape parameter in solving common partial differential equations (PDEs) encountered in water engineering. To this end, the procedure for reconstructing and solving PDEs using the MQ-RBF method is first outlined. Then, optimal values for the shape parameter are derived based on domain length and the number of computational centers. Based on these findings, empirical formulas for the optimal shape parameter are proposed, which significantly reduce computational cost. The performance of the proposed formulas is compared with exact solutions and existing empirical relations. Results show that, unlike previous approaches, the new formulas offer high accuracy, with negligible errors across different examples-sometimes approaching zero. Moreover, compared to optimization-based techniques, the proposed method dramatically improves computational speed by eliminating the need for iterative algorithms. The study also investigates stability conditions, showing that for diffusion, advection-diffusion, and Burgers’ equations, the maximum allowable time step depends on the diffusion coefficient, flow velocity, and Reynolds number.

کلیدواژه‌ها [English]

  • Multiquadric Meshless method
  • Radial Basis Function (RBF)
  • Shape parameter
  • Burgers' equation
  • Advection-diffusion equation
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