Numerical Integrators for Stiff Problems



Key words: Numerical integrators, stiff dynamic problems, structural dynamic problems, dual reciprocity BEM (DRBEM), differential quadrature method (DQM), precise integration method.

 

We presented the DQM approximate formulas in matrix form for initial value problem and first applied this method to approximate temporal derivative successfully. Two new approaches were presented to apply the multiple initial conditions for initial value problems of two order or above by analogy with the techniques applying multiple boundary conditions. It was noted that the formulations of initial value problems can be expressed as the Lyapunov algebraic matrix equation. Several fast algorithms in the solution of the Lyapunov matrix equations are applied to reduce the computing effort and storage requirements by an order of N^3 and N^2, respectively, where N is the number of interior grid points. Consequently, the DQM requires comparable computational effort in the solution of linear dynamic problems as the existing multistep and single step methods such as the Newmark and Gear methods, etc. while its high order of computational accuracy is maintained. Numerical experiments were done in structural dynamics and stiff dynamic systems. It should be pointed out that the method is unconditional stability, namely, A-stable.

In conjunction with the DRBEM discretization in space, the predominant numerical procedures currently used for time integration in analysis of elastodynamic problems are finite difference approximations such as the Newmark,Houbolt and Wilson methods, which is similar to the situation in the FEM. In compound sources of errors due to approximate representations of spatial and time derivatives, it is seen that the accuracy and computing efficiency of the resulting solutions depend greatly on the proper choice of time-marching schemes. Our study places emphasis on a comprehensive comparative study of various standard time integrators in the context of the DRBEM formulation of elastodynamic problems. Although the Houbolt method seems predominant currently in the solution of the DRBEM formulation of elastodynamics problems, we concluded throught numerical experiments and analysis that the Newmark method should be in general preferred in comparison with the Wilson and Houbolt methods.

Also, for the cases where response is primarily dominated by low andintermediate frequency modes, the DQM in time approximation exhibits an impressive advantage in the solution accuracy. For the systems on which high modes have important affect such as plate-impact problems, the damped Newmark method is preferred. The precise integration method, which was recently introduced by Zhong and Williams and in fact equivalent to the exponential matrix approach, was also investigated. We found that the method produced nearly the same accuracy as the DQM but required far more computational effort. In addition, the method lacks the numerical damping and is therefore inefficient for the impact problems.


Related Publications

 

Ph.D. dissertation:

Differential Quadrature Method and Its Applications to Engineering Problems - Applying special matrix product to nonlinear computations (in English, defense 1997).

International Journals

  1. M. Tanaka and W. Chen, Dual reciprocity BEM applied to transient elastodynamic problems with differential quadrature method in time, Computer Methods in Applied Mechanics and Engineering, 190, 2331-2347, 2001.

 

  1. M. Tanaka and W. Chen, Coupling dual reciprocity BEM and differential quadrature method for time-dependent diffusion problems, Applied Mathematical Modelling, 25(3), 257-268, 2001.

 

  1. W. Chen and M. Tanaka, A study on time schemes for DRBEM analysis of scalar impact wave, Comput. Mech. 28, 331-338, 2002.

Proceedings

  1. M. Tanaka and W. Chen, "Transient diffusion problems by a combined use of dual reciprocity and differential quadrature method", Proceedings of Computational Engineering Conference, Tokyo, Japan, Vol.4, 970-974,1999.

 

  1. M. Tanaka and W. Chen, "A combined use of BEM and differential quadrature method for transient elastodynamic problems", Tokyo, Japan, BETC-99, 13-18, 1999.

 

  1. M. Tanaka and W. Chen, "Combination of dual reciprocity BEM in space and differential quadrature method in time for elastodynamic problems", Proc.of the 12th JSME Comput. Mech. Conference, 415-416, Nov. 1999, Tokyo, Japan.

 

  1. M. Tanaka and W. Chen, "On highly accurate time integration scheme for the DRBEM analysis plate-impact responses", The 16th Japan National Symposium on BEM (JASCOME), 45-50, Dec. 1999, Matsuyama, Japan.

 

  1. M. Tanaka and W. Chen, "A Combined Method of Solution by DRBEM and DQM for the Forced Vibration Problem of Damped Elastic Plates", IABEM 2000, 233-236, July, 2000, Brescia, Italy.

 

  1. M. Tanaka and W. Chen, "Dual reciprocity BEM solution of an inverse heat conduction boundary problem", Proceedings of JSME conference on thermoelastic dynamic problems, Aug. 2000, Nagoya, Japan.

 

  1. W. Chen and M. Tanaka, Solution of some inverse heat conduction problems by the dynamic programming filter and BEM, International Symposium on Inverse Problems in Engineering Mechanics 2001, 23-28, Feb. 2001, Nagano, Japan.




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Last updated 08/07/2002