Partial differential equations (PDE) and ordinary differential equations (ODE) frequently used to state motions of structural members. Two solution methods are available for the solution of such equations. They include the exact solution technique resulting from modal decomposition method and the numerical solutions that result from direct step by step time integration algorithms. Even though modal decomposition method can be applied in solving problems relating to multi-degree of freedom, its use is limited especially for large, non-linear and complex systems. Such systems require step by step direct time integration algorithms.
Despite having several steps by step methods for time integration algorithms, most of the functional analysis still uses the second-order implicit time integration algorithms. It is preferred due to some of its advantages such as being less complex and easy to implement on computers. For instance, higher-order accurate algorithms become more complex as the accuracy order increases. Second-order algorithms can also be applied directly to the ODE equations without necessarily rearranging the equations.
However, with the increasing complexity of the structures, there has been a great need to enhance the performance of the second-order algorithms in analyzing the dynamic problems in the structures. As a result, modification of the second-order algorithms gave birth to the generalized-α method. The method has been successful for effective control of the algorithmic dissipation while at the same time retaining the accuracy of the algorithms even in dissipative conditions. However, the generalized-α method has several drawbacks that hinder its full performance. For example, application of the generalized- α method to nonlinear systems is not intuitive, and the numerical damping becomes too excessive when highly dissipative cases are used.
Associate Professor Wooram Kim and Assistant Professor Su Yeon Choi at Korea Army Academy, Department of Mechanical Engineering in the Republic of Korea developed a generalized composite time integration algorithm based on the weighted residual method. This was in a bid to improve the performance of the composite collocation algorithm and effectively control the algorithmic dissipation such as the one for the generalized-α method. Their work is currently published in the research journal, Computers and Structures.
The authors observed that the developed time integration algorithm had a computational structure that was almost identical to adopted in that the Bathe method. It was capable of controlling the algorithmic dissipations, especially at high frequencies through optimization of the weighting parameters.
According to Professors Kim and Choi, the developed generalized composite algorithm has several advantages as compared to its counterparts such as the Bathe method. It is capable of controlling a full range of numerical dissipation. The computational effort is also reduced by the fact that it requires only one construction and inverse of the effective stiffness matrix in linear cases regardless the level of numerical dissipation. Generally, generalized composite algorithms can provide accurate solutions as can maximize or minimize the level of numerical dissipations. This capability gives more flexibility to users and numerical solutions can be improved by using the optimized level of numerical dissipation.
Kim, W., & Choi, S. (2018). An improved implicit time integration algorithm: The generalized composite time integration algorithm. Computers & Structures, 196, 341-354.Go To Computers & Structures