کاربرد روش توابع پایه-شعاعی چندربعی برای حل معادله هلمهولتز به‌منظور آنالیز امواج لرزه‌ای در مخازن سدهای صلب

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه مهدسی عمزان، دانشگاه قم

2 عضو هیات علمی دانشگاه قم

3 استاد دانشکده فنی دانشگاه تهران، گروه عمران

چکیده

هزینه بالای ساخت شبکه، نیاز به حل اساسی وابسته به شرایط مسئله، تکینگی، شبیه‌سازی کل میدان، نیاز به شبکه منظم از خطوط متقاطع و ... از برجسته‌ترین نقاط ضعف روش‌های عددی باشبکه پرکاربرد در حل مسائل مکانیک محیط‌های پیوسته می‌باشد. در این پژوهش، با هدف رفع برخی از این نواقص، روش بدون شبکه پایه-شعاعی چندربعی برای آنالیز دوبعدی امواج لرزهای در مخازن سدهای صلب توسعه داده شد. به این منظور، معادله هلمهولتز و شرایط مرزی مختلط حاکم بر مسئله با استفاده از تابع چندربعی در حوزه فرکانس بازتولید و روند حل آن ارائه گردید. نتایج نشان داد که استفاده از فرم اصلی و مختلط این تابع به ترتیب برای فرکانس‌های کمتر و بیشتر از فرکانس طبیعی مخزن، زمان محاسبات را بهینه می‌کند. همچنین برای تعیین مهمترین عامل در دقت و همگرایی روش مذکور یعنی پارامتر شکل بهینه، ابتدا ناکارآمدی برخی از روش‌های پرکاربرد پیشین به اثبات رسید سپس یک الگوریتم جدید و پرسرعت معرفی گردید. نتایج این پژوهش نشان داد که پارامتر شکل بهینه برحسب فرکانس‌های مختلف بارگذاری قابل فرمول‌بندی است. چنین ویژگی کاربرد روش چندربعی در این مسئله خاص را نسبت به سایر روش‌ها آسان‌تر و هزینه‌های محاسباتی آن را کمتر می‌کند. دقت بالای روش حاضر طی دو مثال مختلف به ترتیب با و بدون در نظر گرفتن اثر جذب رسوبات کف مخزن در مقایسه با حل‌های دقیق نشان داده شد که خطای ناچیز آن به دلیل تعریف یک تابع تخمین پیوسته دقیق در کل دامنه مسئله و نیز استفاده از یک الگوریتم کارا برای پیدا کردن پارامتر شکل بهینه می‌باشد.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Application of Multiquadric Radial Basis Function method for Helmholtz equation in seismic wave analysis for reservoir of rigid dams

نویسندگان [English]

  • Reza Babaee 1
  • Ehsan Jabbari 2
  • Morteza Eskandari-Ghadi 3
1 University of Qom
2 University of Qom
3 University of Tehran
چکیده [English]

 The high costs of mesh generation in mesh-dependent solution, weakness in capturing 
singularities, the need of modeling all over the domain, the need of problem dependent fundamental 
solutions, etc. are some of weaknesses in the common numerical mesh-dependent methods for solving 
continuum mechanics boundary value problems. In this study, aiming for eliminating some of these 
shortcomings, one of the well-known Radial Basis Functions (RBF) methods, Multiquadric (MQ), is 
developed for dynamic analysis of 2D reservoirs of rigid dams in frequency-domain. To this end, the 
Helmholtz equation and the governing complex boundary conditions are reproduced using MQ function 
in the frequency domain. The results show that with the use of real and complex forms of the MQ 
function, the computational time will be respectively optimized for frequencies smaller and larger than 
the natural frequency of the reservoir. Also, to determine the most important factors affecting both the 
accuracy and convergence of MQ method, first the inefficiency of some of the previously introduced 
methods is proved, and then a new high-speed algorithm is presented. It is shown that the optimal 
shape parameter for MQ method can be formulated in terms of the frequencies of seismic records. 
This advantage simplifies the application of MQ method in this particular problem and reduces the 
computational time, considerably. The high accuracy of the present method is shown in two different 
examples, where the effects of sediment absorption may either be considered or not. The high accuracy 
compared to the exact solutions achieved in this paper is due to a continuous estimation function defined 
all over the domain and also due to the simple algorithm used for finding the optimal shape parameter.

کلیدواژه‌ها [English]

  • Multiquadric Radial Basis Function (MQ-RBF)
  • Concrete Gravity Dam
  • Shape parameter
  • Frequency Domain
  • hydrodynamic pressure
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