كليدواژه :
روش سنتي رهايي پويا , ميرايي جنبشي , انرژي جنبشي , ميرايي متمركز
چكيده فارسي :
در تحليل هر نوع سازه اي نياز به حل يك دستگاه معادله هاي خطي يا ناخطي است. روش هاي گوناگون مستقيم و تكراري براي حل چنين دستگاهي وجود دارند. در شيوه هاي تكراري دقت يافتن پاسخ ها قابل تعيين است و با تكرار محاسبه مي توان خطاها را كاهش داد. يكي از فرآيندهاي ساده و توانمند اين دسته، فن رهايي پويا مي باشد. شيوه ي رهايي پويا با افزودن نيروهاي اينرسي و ميرايي ساختگي، سامانه ايستا را به فضاي پويا مي برند. با انتخاب مناسب عامل هاي ساختگي، سامانه پويا به پاسخ ايستا همگرا خواهد شد. در اين مقاله، نخست مروري از پژوهش هاي انجام شده براي تعيين عامل هاي رهايي پويا ارائه مي گردد. سپس، پايه هاي رياضي شيوه ي پيشنهادي مي آيد. در راهكار مزبور، همانند روش سنتي رهايي پويا عامل هاي جرم و گام زماني وجود دارد. ولي عامل ميرايي در گام هاي خاصي وارد مي گردد تا كارمايه جنبشي را به سوي صفر ببرد و پاسخ سامانه ي ايستا به دست آيد. با كاربرد عامل ميرايي به صورت متمركز در شمار محدودي از گام ها، حجم محاسبات كاهش مي يابد. همچنين نتيجه هاي عددي در تحليل قاب و خرپا نشانگر بهبود نرخ همگرايي است. افزون بر برتري هاي كاربردي، شيوه ي پيشنهادي دو راهكار متفاوت ميرايي سنتي و ميرايي جنبشي را در قالب يك روش ارائه مي دهد.
چكيده لاتين :
Solving a system of linear or non-linear equations is required to analyze any kind of structures. There are many ways to solve a system of equations. They can be classified as implicit and explicit techniques. The explicit methods eliminate round-off errors and use less memory. The dynamic relaxation method (DRM) is one of the most powerful and simplest explicit processes. The important point is that DRM does not require storage of the global stiffness matrix. It only uses the residual loads vector.
Utilizing the virtual masses, damping and time steps, the DRM converts a system of static equations to dynamic ones. The process is started by assuming an initial solution. The next steps are performed in such a way that the residual forces are decreased. The proper value of fictitious mass and time step guarantees the convergence of the proposed DR procedure. On the other hand, the convergence rate depends on the value of damping factor, which is calculated using the lowest eigenvalue of artificial dynamic system in the common dynamic relaxation method. It is evidence; the dynamic system oscillates when damping is zero. The convergence of DRM with zero damping factors is achieved by utilizing kinetic damping or -damping. In the kinetic dynamic relaxation process, velocities of the joints are set to zero when a fall occurs in the level of total kinetic energy of the structure. However, it is difficult to calculate the extreme point of kinetic energy. Considering the peak point at the mid-point of the previous time-step, when a fall down occur in kinetic energy, topping is suggested. The factor in the -damping method is time step ratio of two sequence steps. The time-step ratio can be calculated in such a way that the responses converge to exact solutions.
In this paper, a comprehensive review of dynamic relaxation algorithms is presented. Of these, the popular and kinetic damping DR methods are described in detail. Then, the new dynamic relaxation algorithm is proposed. In this procedure, the artificial mass and time steps are similar to the DR methods that have been recently introduced. However, the damping factor is different with these methods. Damping factor is calculated in some specified steps. In other words, damping is zero in most of the DR algorithm steps. Therefore, the total number of calculations is reduced. The concentrated damping is imposed when the value of total kinetic energy of system is at its peak point. Utilizing the proper values of concentrated damping factors, the kinetic energy converges to zero. The presented formulation shows the relation between common and kinetic dynamic relaxation processes, too. It should be noted that the procedures of minimizing the kinetic energy of proposed method and Topping algorithm are different. The kinetic technique requires more calculations. Finally, some benchmark problems of truss and frame structures are selected. The linear and geometric nonlinear analyses are performed. The numerical results also show that the convergence rate of the new DRM increases in most of the cases with respect to kinetic damping and also popular damping.
