روش جدید برای ریز‌کردن محلی شبکه در روش اجزای محدود به کمک توابع شکل جایگزین

نوع مقاله : مقاله پژوهشی

نویسندگان

دانشکده مهندسی عمران و محیط زیست، دانشگاه صنعتی امیرکبیر

چکیده

بر خلاف شبکه‌های متشکل از المان‌های مثلثی/ چهاروجهی در روش اجزای محدود دو/سه بعدی، ریز‌کردن محلی شبکه‌های متشکل از المان‌های چهارضلعی/ شش وجهی با حفظ سازگاری دشوار بوده و معمولا منجر به اعوجاج شدید المان‌ها می‌گردد. روش شناخته­شده و پرکاربرد برای برطرف کردن این مشکل، ریزکردن محلی شبکه بر اساس المان‌های انتقالی و گره‌های معلق است. نکته‌ی کلیدی در این روش، اعمال پیوستگی تغییر مکان در مرز المان‌های انتقالی در حضور گره‌های معلق است. المان‌های انتقالی معرفی‌شده در ادبیات فنی با توجه به نحوه‌ی قرارگیری در شبکه رابطه­بندی‌های مختلف داشته و نیز محدود به یک تعداد مشخص از گره‌های معلق بر مرز المان می‌باشند. بنابراین پیاده‌سازی آن‌ها برای یک حالت بسیار عمومی پیچیده است. در این مقاله یک المان انتقالی جدید بر اساس توابع شکل جایگزین معرفی شده است که با یک رابطه‌بندی واحد حالات مختلف قرارگیری المان انتقالی در شبکه و نیز تعداد دلخواه گره‌های معلق را پشتیبانی می‌کند. هم چنین اثبات تحلیلی به منظور نشان دادن حفظ کامل شرایط پیوستگی و تقسیم جزء واحد در روش پیشنهادی هنگام ریزکردن محلی شبکه به کمک توابع شکل جایگزین ارائه شده است. در نهایت با حل مثال‌های عددی دو و سه‌بعدی مقایسه‌ بین دقت و همگرایی روش پیشنهادی و روش‌های موجود در پیشینه فنی صورت گرفته است.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

A novel technique for local mesh refinement in the finite element method based on the alternative shape functions

نویسندگان [English]

  • Nasrin Kheirkhah Barzaki
  • Alireza Sadeghirad
Department of Civil and Environmental Engineering, Amirkabir University of Technology
چکیده [English]

Unlike triangular/tetrahedral elements used in the finite element method for two/three-dimensional problems, local refinement of meshes composed of quadrilateral/hexahedral elements while maintaining the compatibility is challenging, and often results in severe distortion of the elements. A well-known and widely used approach to address this issue is the local mesh refinement based on transitional elements with hanging nodes. The key point in this method is enforcing the displacement continuity at the transitional element boundaries in the presence of hanging nodes. Transitional elements introduced in the literature employ various formulations depending on their placement within the mesh, and are also constrained by a maximum number of hanging nodes. along 
the element boundary. Therefore, their implementation for a general case is quite complicated. This paper presents a novel transitional element based on alternative shape functions, which offers a unified formulation for different placements of transitional elements in the mesh, applicable to any number of hanging nodes. Additionally, an analytical proof is provided to demonstrate the continuity and partition of unity properties in the proposed method, used in local mesh refinement. Finally, numerical examples in two and three dimensions are simulated to compare the accuracy and convergence of the proposed method against the existing methods in the literature.

کلیدواژه‌ها [English]

  • Finite element method
  • Local mesh refinement
  • Hanging nodes
  • Transition elements
  • Alternative shape functions

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