توسعه روش معادلات مجزا براي محاسبه فشار هيدروديناميكي مخزن سدهاي بتني وزني

Development of Decoupled Equations Methods for calculating hydrodynamic pressures on concrete gravity dams

معادله حاكم بر انتشار امواج هيدروديناميكي در مخازن سدها به هنگام وقوع زلزله، معادله‌ هلمهولتز است. در حل اين معادله براي رسيدن به توزيع واقعي فشار هيدروديناميكي روي بدنه سدها، لازم است شرايط مرزي مخزن منظور شود. اين شرايط عبارت اند از: مرز مشترك مخزن و سد (به-عنوان ناحيه‌ي آغاز تحريك مخزن)، جذب بخشي از انرژي امواج ياد شده به وسيله‌ي رسوبات كف، جذب بخش ديگري از انرژي امواج ياد شده به وسيله‌ي مرز دوردست، و تشكيل امواج سطحي در مخزن. هدف از اين پژوهش، مدل‌سازي پديده‌ فيزيكي ياد شده با استفاده از يك روش نيمه‌تحليلي جديد به نام روش معادلات مجزا، در حوزه فركانس است. در اين روش فقط مرزهاي مسيله با استفاده از المان‌هاي مرتبه بالاي غيرايزوپارامتريكِ ويژه گسسته‌سازي مي‌شود. با استفاده از چندجمله‌اي‌هاي مرتبه بالاي چبيشف به عنوان توابع نگاشت، توابع شكل ويژه، روش انتگرال‌‌گيري عددي كلنشا‌-كورتيس، و همچنين روند توليد فرم انتگرالي با استفاده از روش باقيمانده‌هاي وزن‌دار، ماتريس هاي ضرايب دستگاه معادلات حاكم، قطري مي‌شود. به عبارت ديگر، معادله ديفرانسيل حاكم براي هر درجه آزادي مستقل از ساير درجات آزادي در فضاي مسيله به دست مي‌آيد كه اين امر باعث كاهش قابل توجه حجم محاسبات نسبت به ساير روش‌هاي عددي مي‌شود. در اين پژوهش براي اولين بار به-كارگيري ابزارهاي ياد شده، معادلات شرايط مرزي مخزن سد در فضاي محلي روش حاضر استخراج شده و روند اعمال آنها در حل مسيله بيان مي‌شود. همچنين به منظور صحت‌سنجي، توزيع فشار هيدروديناميكي وارد بر بدنه يك سد وزني صلب محاسبه شده كه در مقايسه با حل تحليلي موجود از دقت قابل قبولي برخوردار است.

Dams as one of the most important structures are always exposed to various hazards such as earthquake. As dam failure may lead to financial damages and fatalities, it should be designed with most economical and accurate methods. An earthquake causes hydrodynamic pressure waves exerting on the dam. This is one of the important factors in design of dams that are always considered by consulting engineers. Helmholtz equation is the governing relation on the propagation of hydrodynamic pressure waves in dam reservoirs during an earthquake. In order to solve the Helmholtz equation to calculate hydrodynamic pressures on dams, the reservoir’s boundary conditions (BCs) should be taken exactly into account. The BCs include (a) the interface boundary of dam and reservoir (as initial zone of reservoir excitation), (b) bottom boundary (with partial absorption of wave energy by accumulated sediments), (c) upstream boundary (with radiation of another part of the wave energy from the reservoir), and (d) formation of surface waves in the upper boundary of the reservoir. The purpose of present study is to model the mentioned physical phenomena in the frequency domain, using a new semi-analytical method, called Decoupled Equations Method (DEM). In the DEM, only the domain boundaries are discretized by specific high-order non-isoparametric elements. The main features used for modeling of geometry and physics of the problem consists of: (1) high-order Chebyshev polynomials as mapping functions, (2) special shape functions of 2n_?+1 degree polynomials for (n_?+1)-node elements , (3) Clenshaw-Curtis quadrature, and (4) integral forms produced by weighted residual method. By using these features and their properties, coefficient matrices of the system of governing equations become diagonal. This means that the governing partial differential equation for each degree of freedom (DOF) becomes independent from other DOFs of the domain to be analyzed. Therefore, this reduction in space dimensions of the main problem may significantly reduce computational costs in comparison with other available numerical methods. In this study, for the first time in order to provide a solution by low costs to calculate the hydrodynamic pressure distribution on the gravity dams, the relations of reservoir’s BCs are derived in local coordinates by using of the DEM and, the process of applying derived equations is then expressed into the solution of Helmholtz equation. To verify this method, an example of this field is solved by using the DEM, where dam and its rigid foundation are excited by horizontal harmonic vibration. The obtained responses from the solution of this example indicates that the present method for modeling of the potential problems with natural boundary conditions under earthquake excitations, by considering propagation of hydrodynamic waves in the reservoir, show acceptable accuracy and feasibility in comparison with the available analytical solution. The results of the DEM should be developed for more general condition of dam-reservoir interaction, which include flexible concrete gravity dams with inclined dam-reservoir interaction boundary conditions along with partial absorption of wave energy by accumulated sediments. These features are being followed by the authors, and will be disseminated in new papers soon.