ارائه تابع تخمین حداقل مربعات متحرک نگاشتی برای روش عددی بدون شبکه حداقل مربعات گسسته

نوع مقاله : مقاله پژوهشی

نویسندگان

1 دانشکده عمران

2 استادیار، مهندسی مکانیک، دانشگاه صنعتی امیرکبیر (پلی تکنیک)، تهران

3 مهندسی عمران، دانشگاه صنعتی امیرکبیر (پلی تکنیک)، تهران

چکیده

روش بدون شبکه حداقل مربعات گسسته کارایی مناسب خود را برای حل معادلات دیفرانسیلی مشتقات جزیی حاکم بر مسائل مهندسی نشان داده‌است. این روش بر پایه کمینه کردن تابعک حداقل مربعاتی استوار است. تابعک حداقل مربعاتی به صورت مجموع وزن‌داری از باقیمانده‌ی معادله دیفرانسیلی و شرایط مرزی حاکم تعریف شده‌است. معمولا از تابع تخمین حداقل مربعات متحرک (MLS)، برای ساختن توابع شکل در روش بدون شبکه حداقل مربعات گسسته استفاده می‌شود. هرچند با استفاده از این نوع تابع تخمین سازگاری مورد نیاز توابع تخمین ارضا می‌شود، اما روش در صورت تجمع و نزدیکی بیش از اندازه گره‌ها کارآیی مناسب خود را از دست می‌دهد. در این مطالعه مشکل مطرح شده، با استفاده از تابع تخمین نوینی که حداقل مربعات متحرک نگاشتی (MMLS) نامیده شده است، برطرف شده‌است. در این روش خوشه‌های گرهی مجتمع به یک آرایش گرهی استاندارد نگاشت می‌یابند؛ سپس تابع تخمین و مشتقات آن با در نظر گرفتن ملاحظاتی محاسبه می‌شوند. کارایی روش تخمین پیشنهادی MMLS برای برطرف کردن مشکل تابع تخمین MLS با تخمین توابع ریاضیاتی مورد ارزیابی قرار گرفته‌است. نتایج بدست آمده قابلیت روش پیشنهادی MMLS را جهت رفع مشکل نشان داده‌اند. تابع تخمین پیشنهادی در روش بدون حداقل مربعات گسسته مختلط استفاده شده و برای حل معادلات غیر خطی برگرز به کار گرفته شده‌است. نتایج بدست آمده کارایی و دقت بالای روش پیشنهادی را نشان می‌دهند.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Mapped Moving Least Squares Approximation Used in Mixed Discrete Least Squares Meshfree Method

نویسندگان [English]

  • Morteza Kolahdoozan 1
  • Ehsan Amani 2
  • Saeb Faraji 3
2 Department of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
3 Department of Civil and Environmental Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
چکیده [English]

The Mixed Least Squares Meshfree (MDLSM) method has shown its appropriate efficiency for solving Partial Differential Equations (PDEs) related to the engineering problems. The method is based on the minimizing the residual functional. The residual functional is defined as a summation of the weighted residuals on the governing PDEs and the boundaries. The Moving Least Squares (MLS) is usually applied in the MDLSM method for constructing the shape functions. Although the required consistency and compatibility for the approximation function are satisfied by the MLS, the method loses its appropriate efficiency when the nodal points cluster become too much. In the current study, the mentioned drawback is overcome using the novel approximation function called Mapped Moving Least Squares (MMLS). In this approach, the cluster of closed nodal was pointed maps to standard nodal distribution. Then the approximation function and its derivatives were computed incorporating some consideration. The efficiency of suggested MMLS for overcoming the drawback of MLS was evaluated by approximating the mathematical function. The obtained results showed the ability of suggested MMLS method to solve the drawback. The suggested approximation function was applied in MDLSM method, and used for solving the Burgers equations. Obtained results approved the efficiency of suggested method.

کلیدواژه‌ها [English]

  • Mapped Moving Least Squares (MMLS)
  • Moving Least Squares Meshfree (MLS)
  • Partial Differential Equations (PDEs)
  • Meshfree method
  • Discrete Least Squares Meshfree method (DLSM)

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نشریه مهندسی عمران امیرکبیر
دوره 51، شماره 4
مهر و آبان 1398
صفحه 805-816

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