یک الگوریتم انطباقی جدید برای توزیع بهینه مراکز محاسباتی در روش بدون شبکه چندربعی

نوع مقاله : مقاله پژوهشی

نویسندگان

1 دانش‌آموخته کارشناسی‌ارشد گروه مهندسی عمران، دانشکده فنی و مهندسی، دانشگاه قم، قم، ایران.

2 دانش‌آموخته دکتری گروه مهندسی عمران، دانشکده فنی و مهندسی، دانشگاه قم، قم، ایران.

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

چکیده

مراکز محاسباتی در روش توابع پایه شعاعی بدون شبکه چندربعی به علت عدم ارتباط هندسی و فیزیکی با یکدیگر دارای قابلیت انطباق‌پذیری بالا و لذا مناسب برای اعمال توزیع‌های انطباقی هستند. در این پژوهش، برای اولین بار یک الگوریتم انطباقی جدید بر مبنای تغییرات (گرادیان) متغیرهای فیزیکی مسئله با هدف ایجاد یک توزیع بهینه پیشنهاد شده است. توزیع انطباقی این الگوریتم، دقت و سرعت روش چندربعی را نسبت به توزیع یکنواخت در مسائل ناپایا به میزان قابل توجهی بهبود می‌بخشد. در این رویکرد نواحی با تغییرات فیزیکی کم و زیاد در گام زمانی معلوم شناسایی می‌شوند و تعداد مراکز محاسباتی به ترتیب در آن‌ها کاهش و افزایش می‌یابد. بنابراین نیاز به یک توزیع فشرده یکنواخت در سراسر میدان به منظور کاهش خطا در نواحی با تغییرات فیزیکی زیاد از بین می‌رود. در مواجهه با دیگر چالش مهم روش چندربعی یعنی تعیین متغیر شکل بهینه نیز یک روش ساده و کارآمد به‌گونه‌ای معرفی می‌شود که به بهینه‌سازی متغیر شکل در هر گام زمانی نیاز نباشد و هزینه‌های محاسباتی کنترل گردد. در پایان، کارایی روش پیشنهادی با ارائه مثال‌هایی از معادلات پخش، جابجایی و پخش-جابجایی برای مقایسه با توزیع‌های یکنواخت و سنجش میزان سرعت و دقت آن‌ها با حل دقیق نشان داده می‌شود.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

A New Adaptive Algorithm for the Optimal Distribution of Computational Centers in the Meshless Multiquadric Method

نویسندگان [English]

  • Samira MohammadAlian 1
  • Reza Babaee 2
  • Ehsan Jabbari 3
1 Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran.
2 Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran.
3 Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran
چکیده [English]

The computational centers in the multiquadric radial basis functions meshless method have high adaptability considering the lack of geometric and physical connection between the centers. In this research, a new adaptive algorithm is proposed based on the gradients of the physical variables of the problem with the aim of creating an optimal distribution. The resulted adaptive distribution generated by this algorithm improves significantly the accuracy and speed of the multiquadric method compared to the uniform distribution in steady and unsteady problems. In this approach, firstly, the domains with low and high physical variations are identified in a known time step, then the number of computational centers decreases and increases in these areas, respectively. Thus, the centers will be distributed more compact where needed and will be eliminated where not. Facing another important challenge of the multiquadric method, i.e. determining the optimal shape parameter, a simple and efficient method is introduced in such a way that there is no need to optimize the shape parameter at each time step and the computational costs are controlled. Finally, the effectiveness of the proposed method is shown by solving examples of diffusion, convection and convection-diffusion equations. The results are compared to their uniform distributions by measuring their efficiency and to the exact solution by evaluating the accuracy.

کلیدواژه‌ها [English]

  • Meshless Method
  • Radial Basis Functions
  • Multiquadric -Method
  • Adaptive Algorithm
  • Gradient

مراجع

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