Dynamic relaxation with concentrated damping

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 powerful and simple explicit processes. The important point is that the DRM does not require to storage the global stiffness matrix. It just uses the residual loads vector.
Utilizing the virtual masses, damping and time steps, the DRM convert a system of static equations to dynamic ones. The process is started by assuming an initial solution. The next steps are done 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 is dependent on 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 utilizing kinetic damping or -damping. In the kinetic dynamic relaxation process, the velocities of the joints are set to zero when a fall in the level of total kinetic energy of the structure occurs. However, it is difficult to calculate the extreme point of kinetic energy. Topping suggested assuming the peak point at the mid-point of the previous time-step, when a fall down in kinetic energy is occurred. 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 the most step of DR algorithm. 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; the procedures of minimizing the kinetic energy of proposed method and Topping algorithm are different. The kinetic technique is required 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 the majority of cases with respect to kinetic damping and also popular damping.