کاربرد تابع پایه-شعاعی چندربعی در حل مسائل تراوش با الگوریتمی جدید برای بهینه‌سازی پارامتر شکل

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه مهندسی عمران، دانشگاه قم

2 گروه مهدسی عمزان، دانشگاه قم

3 عضو هیات علمی دانشگاه قم

چکیده

دقت روش بدون شبکه چندربعی کاملا به انتخاب پارامتر شکل آن وابسته است. هدف از پژوهش حاضر، پیشنهاد یک الگوریتم نوین برای تعیین پارامتر شکل بهینه است، به‌طوری‌که برخی از مشکلات پیشین شامل؛ وابسته بودن به تعداد نقاط محاسباتی و یک حل دقیق از مسئله، هزینه بالا و دقت پایین محاسبات، تجربی بودن، همگرا شدن روش‌های بهینه‌سازی کلاسیک به نقاط بهینه محلی و دیگر موارد را برطرف نماید. به این منظور ضمن معرفی یک تابع هدف جدید، از الگوریتم ژنتیک استفاده شد و برای سرعت بخشیدن به روند حل آن، حد پایین پارامتر شکل؛ کمینه شعاع نقاط محاسباتی با شرط عدم تکینگی ماتریس ضرایب و حد بالای آن؛ بیشینه شعاع پیشنهاد گردید. الگوریتم مذکور از چهار مرحله تشکیل می‌شود: 1 )تولید پارامتر شکل توسط الگوریتم ژنتیک در بازه پیشنهادشده، 2 )تشکیل تابع چندربعی با تعداد پایینی از نقاط محاسباتی، 3 )تشکیل تابع چندربعی با تعداد بالایی از نقاط محاسباتی و 4 )کمینه‌سازی اختلاف جواب توابع به‌دست آمده از دو مرحله قبل. در الگوریتم فرا ابتکاری حاضر، توزیع‌های یکنواخت و غیریکنواخت منظم از نقاط محاسباتی با موفقیت به‌کار رفت و نشان داده شد که با آن می‌توان به پارامتر شکلی بهینه و ثابت و نیز مستقل از تعداد نقاط محاسباتی برای هندسه‌های دلخواه دست یافت. برای صحت‌سنجی نیز مسائلی همگن، ناهمگن و ناهمسان از پدیده تراوش حل شد و از فن تجزیه دامنه برای کاهش حجم محاسبات و شبیه‌سازی راحت‌تر میدان‌های ناهمگن و پیچیده استفاده گردید. مقایسه نتایج با سایر حل‌های دقیق و عددی، توانایی و دقت بالای الگوریتم پیشنهادی را نشان می‌دهد.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Application of MQ-RBF method for solving seepage problems with a new algorithm for optimization of the shape parameter

نویسندگان [English]

  • Masoumeh Koushki 1
  • Reza Babaee 2
  • Ehsan Jabbari 3
1 University of Qom
2 University of Qom
3 University of Qom
چکیده [English]

The accuracy of the meshless method, Multiquadric, depends completely on the choice of its optimal shape parameter. The purpose of this research is proposing a new algorithm for determining the optimal shape parameter. It resolves some of the previous difficulties, such as depending on the number of computational nodes or an exact solution of the problem, high cost and low accuracy of calculations, being experimental, convergence of classical optimization methods to local optimal points and so on. For this purpose, in addition to introducing a new objective function, Genetic Algorithm(GA) was used and for speeding up the process of its solution, lower bound and upper bound of the shape parameter are suggested as minimum (when the coefficient matrix is not singular) and maximum radius of computational nodes, respectively. The algorithm consists of four steps: 1) producing initial shape parameters by GA in the proposed range, 2) introducing the MQ function with a few numbers of computational points, 3) introducing the MQ function with a large number of computational points, and 4) minimizing the difference between solutions of two functions obtained from the two preceding steps. In the meta-heuristic algorithm, uniform and non-uniform regular distributions of computational nodes have been successfully applied and it was shown that with this approach, an optimal constant shape parameter independent of the number of computational points could be obtained for arbitrary geometries. For verification, examples of homogeneous, inhomogeneous and anisotropic types of the seepage phenomena were solved so that domain decomposition technique was used for inhomogeneous problems and complex geometries. A comparison of results with other exact and numerical solutions showed the high ability and accuracy of the proposed algorithm.

کلیدواژه‌ها [English]

  • Shape parameter
  • Multiquadric method
  • Radial Basis Function
  • Genetic Algorithm
  • Seepage

مراجع

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نشریه مهندسی عمران امیرکبیر
دوره 52، شماره 4
تیر 1399
صفحه 1009-1024

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