روش معادلات مجزا براي حل مسايل دوبعدي الاستوديناميك در حوزه‌ي بسامد

Decoupled Equations Method for Solving Two-Dimensional Elastodynamic Problems in the Frequency Domain

در اين نوشتار، روش معادلات مجزا براي حل مسايل الاستوديناميك دوبعدي در حوزه‌ي بسامد با استفاده از تبديل سريع فوريه توسعه داده شده است. براي اين منظور، مرز فضاي مسيله با استفاده از المان هاي مرتبه‌ي بالاي غيرايزوپارامتريك ويژه گسسته‌سازي شده است. با استفاده از چندجمله‌يي هاي مرتبه‌ي بالاي چبيشف به عنوان توابع نگاشت ، توابع شكل ويژه، روش انتگرال گيري عددي كلنشا-كورتيس ، و همچنين روند توليد فرم انتگرالي با استفاده از روش باقيمانده‌هاي وزن‌دار ، ماتريس ضرايب در معادلات حاكم بر مسايل الاستوديناميك قطري شده است. اين به آن معناست كه معادله‌ي ديفرانسيل بسل حاكم براي هر درجه‌ي آزادي مستقل از ساير درجات آزادي در فضاي مسيله به دست آمده است. براي اولين بار در اين نوشتار، روش معادلات مجزا، كه قبلاً براي حل مسايل پتانسيل و الاستواستاتيك ارايه شده بود، براي حل مسايل الاستوديناميك دوبعدي در حوزه‌ي بسامد توسعه داده شده است. همچنين نتايج به دست‌آمده با استفاده از روش معادلات مجزا با نتايج ساير روش‌هاي عددي مقايسه شده است.

Mathematical models can be used to represent physical phenomena. However, mathematical models may not evaluate physical models sufficiently. Mathematical modeling is an important step in engineering analysis, and many numerical methods may be used for solving and modeling physical phenomena, particularly elastodynamic problems. These numerical methods have advantages and disadvantages. One of the disadvantages of these methods is that the differential equations are coupled. In this paper, a new semi-analytical method, called the Decoupled Equations Method, is developed for solving two-dimensional (2D) elastodynamic problems in the frequency domain. In the frequency domain approach, Fast Fourier Transform (FFT) is implemented to transform a time domain problem into a frequency domain one. Using specific non-isoparametric elements, the boundary of the problem domain is discretized. This new method is based upon a scaled boundary finite element method that has been developed for solving two and three dimensional engineering problems. By employing the advantages of numerical methods (such as SBFEM), and using higher-order Chebyshev mapping functions, special shape functions, the Clenshaw-Curtis quadrature rule, and implementing a weak form of the weighted residual method, coefficient matrices of governing differential equations for elastodynamic problems become diagonal. This fact results in a set of decoupled Bessel differential equations to be used for solving the whole system. This means that the governing Bessel differential equation for each degree of freedom (DOF) becomes independent from other DOFs of the domain. For each DOF, the Bessel differential equation is solved for a specific frequency. Finally, the time history of responses may be obtained by using Inverse Fast Fourier Transform (IFFT). The proposed shape functions have two specific characteristics: (a) The shape functions have a Kronecker Delta property, and (b) Their first derivatives are equal to zero at any given node. In this paper, 2D elastodynamic problems have been solved using the present method and compared with other numerical examples given in the literature and/or exact analytical solutions wherever available.