فرمول‌بندی و کاربرد المان‌های هنکل کروی در مدلسازی عددی مسائل پتانسیل به کمک روش المان مرزی

نوع مقاله : پژوهشی اصیل (کامل)

نویسندگان
صالح حمزهء جواران 1
سعید شجاعی 2
1 استادیار، بخش مهندسی عمران، دانشکده فنی و مهندسی، دانشگاه شهید باهنر کرمان، کرمان، ایران
2 دانشیار، بخش مهندسی عمران، دانشکده فنی و مهندسی، دانشگاه شهید باهنر کرمان، کرمان، ایران
چکیده
در این مقاله، یک آنالیز المان مرزی جدیدی برای مدلسازی مسائل دو بعدی پتانسیل پیشنهاد شده است. روش المان مرزی بر مبنای المان‌های هنکل کروی به منظور تقریب متغیرهای حالت معادلات دیفرانسیل پواسون و لاپلاس (پتانسیل‌ها و شارها)، بازفرمول‌بندی شده است. با استفاده از غنی‌سازی توابع پایه‌ی شعاعی هنکل کروی، توابع انترپولاسیون روش المان مرزی حاصل شده‌اند. بدین منظور، به بسط تابعی‌ای که در آن فقط از تقریب توابع پایه‌ی شعاعی هنکل کروی استفاده می‌شود، ترم‌های چندجمله‌ای الحاق می‌شود. از جمله خواص منحصر بفرد انترپولاسیون پیشنهادی می­توان به مشارکت میدان توابع نوع اول و دوم بسل در فضای مختلط علاوه بر اغنای میدان توابع چندجمله‌ای، بر خلاف توابع کلاسیک لاگرانژ که فقط توابع چندجمله­ای را اغنا می­کنند، اشاره کرد. بعلاوه توابع شکل پیشنهادی از خاصیت قطعه قطعه پیوسته از مرتبه بینهایت سود می‌برند که این امر برای توابع شکل کلاسیک لاگرانژ که دارای مرتبه پیوستگی محدودی هستند، وجود ندارد. تابع هنکل کروی نوع اول دارای سینگولاریتی قوی در قسمت موهومی خود، تابع نیومن کروی، می‌باشد که این مطلب عدم وجود حد برای میل نرم اقلیدس به سمت صفر را در بر دارد. در ادامه برای رفع سینگولاریتی از ترم اضافی با توان استفاده شده است. پس از رفع سینگولاریتی، حالت حدیِ انطباق نقطه‌ی چشمه و گره‌ی مرزی محاسبه شده است. برای نشان دادن کارایی و دقت روش حاضر، چند مثال عددی در نظر گرفته شده است و نتایج حاصل با نتایج حل تحلیلی و نتایج توابع شکل کلاسیک لاگرانژ مقایسه شده است. نتایج این مقایسه ‌ها حاکی از دقت بسیار بالاتر روش پیشنهادی می‌باشد.

کلیدواژه‌ها

موضوعات

عنوان مقاله English

Formulation and Application of Spherical Hankel Elements in Numerical Modelling of Potential Problems using Boundary Element Method

نویسندگان English

Saleh Hamzehei-Javaran 1
Saeed Shojaee 2
1 Assistant Professor, Civil Engineering Department, Faculty of Engineering, Shahid Bahonar University of Kerman, Kerman, Iran
2 Associate Professor, Civil Engineering Department, Faculty of Engineering, Shahid Bahonar University of Kerman, Kerman, Iran
چکیده English

In this paper, a new boundary element analysis for the modeling of two-dimensional potential problems is proposed. The boundary element method is reformulated here based on spherical Hankel elements for the purpose of approximation of the state variables of the Poisson and Laplace differential equations (potentials and fluxes). Spherical Hankel function is obtained by combing Bessel function of the first (similar to J-Bessel ones) and second (also called Neumann functions) kind so that the properties of both mentioned functions will be combined and result in a robust interpolation tool. The interpolation functions of the boundary element method are obtained using the enrichment of the spherical Hankel radial basis functions. To this end, the expansion of a function in which only the spherical Hankel radial basis functions approximations are used have been given polynomial terms. Generally, radial basis function (RBF) is an efficient tool in finding the solution of non-homogeneous partial differential equations. Its main idea is the expansion of non-homogeneous term by its values in interpolation nodes, based on Euclidean norm that leads to obtaining a particular solution. Although the J-Bessel RBF contains the features of the first kind of Bessel function, it usually cannot represent the full properties of a physical phenomenon. Therefore, using the combination of the first and second kind of Bessel function in complex space (Hankel function) may lead to more accurate and robust results. In other words, the solution of Bessel equation can be referred as a prominent usage of both first and second kind of Bessel, which shows that using them together may result in more accuracy and robustness. The aforementioned discussion brings this matter to mind whether it is possible to present RBFs that benefit from both Bessel functions of the first and second kind. Therefore, by the idea of combining spherical Hankel in imaginary space, enrichment of them for a three-node element in the natural coordinate system is explained in this paper. Moreover, the algebraic manipulations and formulations are reduced because of profiting from the advantages of complex number space in functional space. It is also possible for the proposed shape function to satisfy both Bessel function fields and polynomial functions, unlike classic Lagrange shape functions that only satisfy the polynomial function fields. Moreover, the proposed shape functions benefit from the infinite piecewise continuous property, which does not exist in the classic Lagrange shape functions with limited continuity. The spherical Hankel function of the first kind has a strong singularity in its imaginary part, the spherical Neumann function. This issue results in the fact that when the Euclidean norm tends to zero, the limit does not exist. In the following, an extra term with power is applied to remove this singularity. After the elimination of the singularity, the limit state of coinciding source point and field point is calculated. In the end, to demonstrate the accuracy and efficiency of the proposed shape functions, several numerical examples are solved and compared with the analytical results as well as those obtained by classic Lagrange shape functions. The numerical results show that the proposed Hankel shape functions represent more accurate solutions, using fewer degrees of freedom, in comparison with classic Lagrange shape functions.

کلیدواژه‌ها English

Spherical Hankel elements
Boundary element method
2D potential problems
Poisson's and Laplace's equations
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مهندسی عمران مدرس
دوره 19، شماره 5
آذر و دی 1398
صفحه 85-96