An explicit time integration algorithm for linear and non-linear finite element analyses of dynamic and wave problems

2018 ◽  
Vol 36 (1) ◽  
pp. 161-177 ◽  
Author(s):  
Mi Zhao ◽  
Huifang Li ◽  
Shengtao Cao ◽  
Xiuli Du
Keyword(s):  
Finite Element ◽  
Time Integration ◽  
Single Step ◽  
Degree Of Freedom ◽  
Integration Algorithm ◽  
Explicit Time Integration ◽  
Content Type ◽  
Time Integration Algorithm ◽  
Non Linear ◽  
Explicit Time Integration Algorithm

Purpose The purpose of this paper is to propose a new explicit time integration algorithm for solution to the linear and non-linear finite element equations of structural dynamic and wave propagation problems. Design/methodology/approach The algorithm is completely explicit so that no linear equation system requires solving, if the mass matrix of the finite element equation is diagonal and whether the damping matrix does or not. The algorithm is a single-step method that has the simple starting and is applicable to the analysis with the variable time step size. The algorithm is second-order accurate and conditionally stable. Its numerical stability, dissipation and dispersion are analyzed for the dynamic single-degree-of-freedom equation. The stability of the multi-degrees-of-freedom non-proportional damping system can be evaluated directly by the stability theory on ordinary differential equation. Findings The performance of the proposed algorithm is demonstrated by several numerical examples including the linear single-degree-of-freedom problem, non-linear two-degree-of-freedom problem, wave propagation problem in two-dimensional layer and seismic elastoplastic analysis of high-rise structure. Originality/value A new single-step second-order accurate explicit time integration algorithm is proposed to solve the linear and non-linear dynamic finite element equations. The algorithm has advantages on the numerical stability and accuracy over the existing modified central difference method and Chung-Lee method though the theory and numerical analyses.

An Improved Time Integration Algorithm: A Collocation Time Finite Element Approach

2017 ◽  
Vol 17 (02) ◽  
pp. 1750024 ◽  
Author(s):  
Wooram Kim ◽  
J. N. Reddy
Keyword(s):  
Finite Element ◽  
Time Integration ◽  
Integration Algorithm ◽  
Finite Element Approach ◽  
New Family ◽  
Integration Schemes ◽  
Time Integration Algorithm ◽  
Time Integration Schemes ◽  
Element Approach ◽  
Integration Algorithms

A time collocation finite element approach is employed to develop one- and two-step time integration schemes with algorithmic dissipation control capability. The newly developed time integration schemes are combined to obtain a new family of time integration algorithms using the concept employed by Baig and Bathe. The newly developed algorithm can effectively control the algorithmic dissipation by relating the collocation parameters with the spectral radius in the high frequency limit. The new algorithm provides better accuracy compared with the generalized-[Formula: see text] method for highly dissipative cases and includes the Baig and Bathe method within its family as a special case.

scholarly journals A Robust and Efficient Composite Time Integration Algorithm for Nonlinear Structural Dynamic Analysis

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Lihong Zhang ◽  
Tianyun Liu ◽  
Qingbin Li
Keyword(s):  
Finite Element ◽  
Time Integration ◽  
Computational Cost ◽  
Nonlinear Problems ◽  
Central Difference ◽  
Integration Algorithm ◽  
Structural Dynamic ◽  
Nonlinear Structural ◽  
Time Integration Algorithm ◽  
Difference Formula

This paper presents a new robust and efficient time integration algorithm suitable for various complex nonlinear structural dynamic finite element problems. Based on the idea of composition, the three-point backward difference formula and a generalized central difference formula are combined to constitute the implicit algorithm. Theoretical analysis indicates that the composite algorithm is a single-solver algorithm with satisfactory accuracy, unconditional stability, and second-order convergence rate. Moreover, without any additional parameters, the composite algorithm maintains a symmetric effective stiffness matrix and the computational cost is the same as that of the trapezoidal rule. And more merits of the proposed algorithm are revealed through several representative finite element examples by comparing with analytical solutions or solutions provided by other numerical techniques. Results show that not only the linear stiff problem but also the nonlinear problems involving nonlinearities of geometry, contact, and material can be solved efficiently and successfully by this composite algorithm. Thus the prospect of its implementation in existing finite element codes can be foreseen.

Time Integration Method Based on Discrete Transfer Function

2016 ◽  
Vol 16 (05) ◽  
pp. 1550009 ◽  
Author(s):  
M. Rezaiee-Pajand ◽  
M. Hashemian
Keyword(s):  
Transfer Function ◽  
Time Integration ◽  
Integration Algorithm ◽  
Structural Dynamic ◽  
Time Integration Algorithm ◽  
Integration Techniques ◽  
Complex Structural ◽  
Nonlinear Dynamic Analyses ◽  
Explicit Time Integration Algorithm ◽  
Discrete Transfer Function

Complex structural dynamic problems are normally analyzed by finite element and numerical integration techniques. An explicit time integration algorithm with second-order accuracy and unconditional stability is presented for dynamic analysis. Utilizing weighted factors, the current displacement and velocity relations are defined in terms of the accelerations of two previous time steps. The concept of discrete transfer function and the pole mapping rule from the control theory are exploited to develop the new algorithm. Several linear and nonlinear dynamic analyses are performed to verify the efficiency of the method compared with the well-known Newmark method.

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