ترکیب روش المان محدود مرزی مقیاس‌شده و توابع پایه متعادل‌شده برای حل مسائل انتقال حرارت

نوع مقاله : مقاله پژوهشی

نویسندگان

دانشکده مهندسی عمران، دانشگاه صنعتی اصفهان، اصفهان، ایران

چکیده

روش اجزا محدود مرزی مقیاس‌شده با مقیاس نمودن پاسخ سطح المان به مرز آن، گسسته‌سازی را تنها به مرز دامنه محدود می‌سازد. در این پژوهش، حل مسائل انتقال حرارت در فضای دوبعدی با رویکردی جدید بر پایه روش اجزا محدود مرزی مقیاس‌شده به همراه روش توابع پایه متعادل‌شده مورد نظر قرار گرفته است. روش اجزا محدود مرزی مقیاس‌شده، با ارائه روابط در دستگاه مختصات حاوی مختصه‌های شعاعی و پیرامونی، و تنها با گسسته‌سازی مرز مسئله بر پایه توسعه حل نیمه‌تحلیلی در امتداد شعاعی، چالش‌های وابستگی به المان‌بندی مناسب در ناحیه حل و یا نیاز به حل‌های اساسی معادله، چنانکه به ترتیب در روش‌های اجزا محدود و اجزا مرزی معمول است، را مرتفع می‌سازد. در این پژوهش پس از مقیاس کردن مرز توسط روش اجزا محدود مرزی مقیاس‌‌شده و استخراج معادلات مربوطه، از روش توابع پایه متعادل‌شده برای تقریب تابع حل نیمه‌تحلیلی در امتداد شعاعی استفاده می‌شود؛ به این صورت که پس از تخمین بخش شعاعی تابع حل مسئله توسط توابع پایه اولیه از نوع چندجمله‌ای‌های چبی‌شف نوع اول، عملگر باقیمانده وزنی معادله بر آن اعمال می‌شود تا ارضای تقریبی آن تحقق یابد. در نهایت اقدام به برآورد ضرایب مجهول مجموعه پاسخ مسئله مرتبط با درجات آزادی مرزی مسئله می‌شود. بدین ترتیب نیازی به حل مسئله مقادیر ویژه در گام نهایی نخواهد بود. در نتایج عددی نشان داده خواهد شد که این رویکرد از دقت و همگرایی مطلوبی نیز برخوردار است. 

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Scaled Boundary Finite Element Method Coupled with Equilibrated Basis Functions for Heat Transfer Problems

نویسندگان [English]

  • Nazanin Pirhaji Khouzani
  • Nima Noormohammadi
Department of Civil Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran
چکیده [English]

The Scaled Boundary Finite Element Method (SBFEM) discretizes only the boundary by using a technique for scaling the domain response onto its boundary. In this research, heat transfer problems in two-dimensional space are solved with a new approach based on combining the scaled boundary finite element method and the equilibrated basis functions. The SBFEM develops its relations in radial and circumferential coordinate systems, but only discretizes the boundary of the problem through development of a semi-analytical solution in radial direction. So the challenges of appropriate elemental grid for the solution domain, or the need for fundamental solutions of the equation, as usual in the finite element method or the boundary element method respectively, do not appear. In this research, after scaling the boundary in the scaled boundary finite element method and extracting the related equations, the equilibrated basis functions are used to approximate the semi-analytical solution in radial direction. After estimating the radial solution by the first kind Chebyshev polynomials, the weighted residual form of the governing equation is applied for approximately satisfaction. Finally, the unknown degrees of freedom of the boundary are derived, and there will be no need for the usual eigenvalue solution of the SBFEM. It will be shown that this approach benefits good accuracy and convergence rate.

کلیدواژه‌ها [English]

  • scaled boundary finite element method
  • equilibrated basis functions
  • weighted residual integration

مراجع

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