Precise integration method without inverse matrix calculation for structural dynamic equations

There were many issues in numerical methods of heat transfer problems such as instability at a big time step length or grid size and no-existence of inverse matrix by time-precise integration method. For sake of avoiding instability and calculating an inverse matrix, a coupling method was put forward based on EFGM and IDPIM. Formulae were deduced according to EFGM and IDPIM. Results show that the coupling method has a higher accuracy and its stability is small subjected to the time step length or grid size, and is to deserve to be popularized.

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Precise Integration Method for Solving Noncooperative LQ Differential Game

Mathematical Problems in Engineering ◽

10.1155/2013/713725 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9

Author(s):

Hai-Jun Peng ◽

Sheng Zhang ◽

Zhi-Gang Wu ◽

Biao-Song Chen

Keyword(s):

Differential Equation ◽

Differential Game ◽

Integration Method ◽

Transformation Method ◽

Linear Quadratic ◽

Riccati Differential Equation ◽

Precise Integration ◽

Precise Integration Method ◽

Direct Integration Method ◽

Two Parameters

The key of solving the noncooperative linear quadratic (LQ) differential game is to solve the coupled matrix Riccati differential equation. The precise integration method based on the adaptive choosing of the two parameters is expanded from the traditional symmetric Riccati differential equation to the coupled asymmetric Riccati differential equation in this paper. The proposed expanded precise integration method can overcome the difficulty of the singularity point and the ill-conditioned matrix in the solving of coupled asymmetric Riccati differential equation. The numerical examples show that the expanded precise integration method gives more stable and accurate numerical results than the “direct integration method” and the “linear transformation method”.

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Precise integration method for solving the seepage‐stress coupling problem in rock mass with singular possibilities of the coupling matrix

International Journal for Numerical and Analytical Methods in Geomechanics ◽

10.1002/nag.3083 ◽

2020 ◽

Vol 44 (11) ◽

pp. 1587-1600

Author(s):

Zhen Liu ◽

Weihua Ming ◽

Zuolei Du ◽

Cuiying Zhou

Keyword(s):

Rock Mass ◽

Integration Method ◽

Coupling Matrix ◽

Precise Integration ◽

Coupling Problem ◽

Precise Integration Method

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Extension of the Wittrick-Williams Algorithm to Mixed Variable Systems

Journal of Vibration and Acoustics ◽

10.1115/1.2889728 ◽

1997 ◽

Vol 119 (3) ◽

pp. 334-340 ◽

Cited By ~ 32

Author(s):

Zhong Wanxie ◽

F. W. Williams ◽

P. N. Bennett

Keyword(s):

Timoshenko Beam ◽

Stiffness Matrix ◽

Integration Method ◽

Dynamic Stiffness ◽

Integration Algorithm ◽

Precise Integration ◽

Matrix Computations ◽

Precise Integration Method ◽

Count Method ◽

Mixed Variable

A precise integration algorithm has recently been proposed by Zhong (1994) for dynamic stiffness matrix computations, but he did not give a corresponding eigenvalue count method. The Wittrick-Williams algorithm gives an eigenvalue count method for pure displacement formulations, but the precise integration method uses a mixed variable formulation. Therefore the Wittrick-Williams method is extended in this paper to give the eigenvalue count needed by the precise integration method and by other methods involving mixed variable formulations. A simple Timoshenko beam example is included.

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