## SEARCH

#### Keywords

structural dynamics Chebyshev polynomial of the first kind Crout decomposed method Gaussian quadrature method homogenized initial system method integral formula method inverse matrix calculation matrix exponential function numerical integration the Crout decomposed method time step integration method#### Institution

- Hunan University 3 (%)
- The University of Hong Kong 2 (%)
- Beijing Polytechnic University 1 (%)
- China Academy of Building Research 1 (%)

#### Author

- Wang, Mengfu [x] 3 (%)
- Au, F. T. K. 2 (%)
- F.T.K.Au F. T. K. Au 1 (%)
- MengfuWang Mengfu Wang 1 (%)
- Zhou, Xiyuan 1 (%)

#### Subject

- Civil Engineering 3 (%)
- Geosciences [x] 3 (%)
- Geotechnical Engineering 3 (%)
- Vibration, Dynamical Systems, Control 3 (%)

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## Modified precise time step integration method of structural dynamic analysis

### Earthquake Engineering and Engineering Vibration (2005-12-01) 4: 287-293 , December 01, 2005

The precise time step integration method proposed for linear time-invariant homogeneous dynamic systems can provide precise numerical results that approach an exact solution at the integration points. However, difficulty arises when the algorithm is used for non-homogeneous dynamic systems, due to the inverse matrix calculation and the simulation accuracy of the applied loading. By combining the Gaussian quadrature method and state space theory with the calculation technique of matrix exponential function in the precise time step integration method, a new modified precise time step integration method (e.g., an algorithm with an arbitrary order of accuracy) is proposed. In the new method, no inverse matrix calculation or simulation of the applied loading is needed, and the computing efficiency is improved. In particular, the proposed method is independent of the quality of the matrix *H*. If the matrix *H* is singular or nearly singular, the advantage of the method is remarkable. The numerical stability of the proposed algorithm is discussed and a numerical example is given to demonstrate the validity and efficiency of the algorithm.

## Precise integration method without inverse matrix calculation for structural dynamic equations

### Earthquake Engineering and Engineering Vibration (2007-03-01) 6: 57-64 , March 01, 2007

The precise integration method proposed for linear time-invariant homogeneous dynamic systems can provide accurate numerical results that approach an exact solution at integration points. However, difficulties arise when the algorithm is used for non-homogeneous dynamic systems due to the inverse matrix calculation required. In this paper, the structural dynamic equalibrium equations are converted into a special form, the inverse matrix calculation is replaced by the Crout decomposition method to solve the dynamic equilibrium equations, and the precise integration method without the inverse matrix calculation is obtained. The new algorithm enhances the present precise integration method by improving both the computational accuracy and efficiency. Two numerical examples are given to demonstrate the validity and efficiency of the proposed algorithm.

## Precise integration methods based on the Chebyshev polynomial of the first kind

### Earthquake Engineering and Engineering Vibration (2008-06-01) 7: 207-216 , June 01, 2008

This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method (IFM) and the homogenized initial system method (HISM). In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshev polynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, the recurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed method to solve linear algebraic equations. In this way, the IFM based on Chebyshev polynomial of the first kind is constructed. Transforming the non-homogenous initial system to the homogeneous dynamic system, and developing a special scheme without dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrix inversion operation. The accuracy of the time integration schemes is examined and compared with other commonly used schemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examples are presented to demonstrate the applicability of these new methods.