استفاده از تابع وزن خطا در حل عددی مسائل غیرمحلی با استفاده از روش اجزای محدود

نوع مقاله : مقاله پژوهشی

نویسندگان

1 دانشکده مهندسی مکانیک. دانشگاه صنعتی اصفهان

2 دانشکده مهندسی مکانیک، دانشگاه صنعتی اصفهان

چکیده

تابع‌های مختلفی به عنوان تابع وزن در حل عددی مسائل غیرمحلی استفاده می‌شوند که تابع خطا یکی از آن‌ها است. مقدار تابع خطا در یک نقطه، وابسته به پارامتر غیرمحلی و فاصله تا نقطه هدف است. برای کاهش هزینه محاسباتی در حل عددی، اثر متقابل نقاطی که فاصله‌شان بیشتر از یک شعاع مشخص (شعاع برهم‌کنش) است، معمولا در نظر گرفته نمی‌شود. بنابراین، با هدف تعیین شعاع برهم‌کنش بهینه، یک ارتباط بین این شعاع و پارامتر غیرمحلی در تابع خطا به دست آمد. با توجه به کاربرد گسترده روش اجزای محدود در حل مسائل غیرمحلی و کاربرد مقدار انتگرال تابع وزن در کرنل‌های دوفازی، تاثیر اندازه المان‌ در مقدار این انتگرال بررسی شد. یافته‌ها نشان می‌دهد که به دلیل گسسته‌سازی، با افزایش نسبت اندازه المان به پارامتر غیرمحلی، خطا در محاسبه مقدار انتگرال تابع وزن افزایش می‌یابد. این موضوع می‌تواند منجر به مقدار انتگرال بیش از یک شود و در نتیجه شرط بهنجارشدن تابع وزن را نقض کند.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

On the Use of Error Weight Function in the Numerical Solution of Nonlocal Problems Using the Finite Element Method

نویسندگان [English]

  • Ali Gerami 1
  • mohammad silani 2
  • Mahdi Javanbakht 1
1 Department of Mechanical Engineering, Isfahan University of Technology
2 Department of Mechanical Engineering, Isfahan University of Technology
چکیده [English]

Various functions can serve as weight functions in the numerical solution of nonlocal problems, including the error function. The value of the error function at a given point depends on the nonlocal parameter and the distance from the target point. To reduce computational costs in numerical procedures, interactions are often neglected between the points with a distance above a specific radius (interaction radius). Consequently, a relationship between the radius and the nonlocal parameter in the error function is established to determine the optimal interaction radius. Due to the widespread utilization of the finite element method in the solution of non-local problems, and considering application of the integral value of the weight function in two-phase kernels, the effect of element size on the calculation of the integral value is investigated. The findings indicate that, due to discretization, the error in calculating the integral value of the weight function increases as the ratio of the element size to the nonlocal parameter grows. This can result in an integral value exceeding 1, thereby violating the normalization condition of the weight function.

کلیدواژه‌ها [English]

  • Nonlocal theory
  • Nonlocal weight function
  • Nonlocal kernel
  • Error function
  • Integral of weight function

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