IJE TRANSACTIONS A: Basics Vol. 31, No. 1 (January 2018) 50-57   

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Y. Karimi, S. Rashahmadi and R. Hasanzadeh
( Received: September 17, 2016 – Accepted in Revised Form: November 30, 2017 )

Abstract    The Newmark method is an effective method for numerical time integration in dynamic problems. The results of Newmark method are function of its parameters (β, γ and ∆t). In this paper, a stationary mode I dynamic crack problem is coded in extended finite element method )XFEM( framework in Matlab software and results are verified with analytical solution. This paper focuses on effects of main parameters in Newmark method for dynamic XFEM problems. Also use of the response surface method (RSM) a regression model is presented for estimating error of dynamic stress intensity factors (DSIF) with high validity according to results of analysis of variance (ANOVA). This work enables one to understand the effect of Newmark parameters on error of DSIFs and to find optimum β and γ for a determined number of time steps (N). This procedure is highly effective in order to manage the computational cost and enhance the accuracy at the desired domain. The effect of the considered parameters on error, is investigated using RSM in Minitab software and optimum state for minimization of errors is illustrated.


Keywords    Dynamic XFEM, Time integration, Newmark method, Response surface method, Error


چکیده    روش نیومارک روش مؤثری برای انتگرال­گیری زمانی عددی در مسائل دینامیکی است. نتایج روش نیومارک تابعی از پارامترهای آن هستند (β، γ و ∆t). در این تحقیق یک مسئله­ی ترک دینامیکی ایستای مود Iدر قالب XFEMو در نرم افزار Matlabکدنویسی شده است و نتایج با حل تحلیلی صحت سنجی شده­اند. تحقیق حاضر روی اثرات پارامترهای روش نیومارک در مسائل المان محدود توسعه یافته­ی دینامیکی(DXFEM) تمرکز دارد. همچنین با استفاده از روش سطح پاسخ (RSM) مدل رگرسیونی برای تخمین خطای ضرایب شدت تنش دینامیکی (DSIF) با دقت بالایی مطابق نتایج روش آنالیز واریانس (ANOVA) ارائه شده است. تحقیق ارائه شده محقق را قادر می­سازد که تأثیر پارامترهای نیومارک را روی خطای ضرایب شدت تنش دینامیکی دریابد و β و γ بهینه را برای تعداد گام­های زمانی معین بیابد. این رویه به منظور مدیریت هزینه­ی محاسبات و افزایش دقت در ناحیه­ی دلخواه بسیار مؤثر است. تاثیر پارامترهای در نظر گرفته شده روی خطا، با استفاده از روش RSM در نرم افزار Minitab بررسی شد و حالت بهینه برای کمینه کردن خطا نشان داده شد.


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