Effective Higher-Order Time Integration Algorithms for the Analysis of Linear Structural Dynamics

2017 ◽  
Vol 84 (7) ◽  
Author(s):  
Wooram Kim ◽  
J. N. Reddy
Keyword(s):  
Structural Dynamics ◽  
Displacement Vector ◽  
Time Integration ◽  
Higher Order ◽  
Time Interval ◽  
Weighted Residual Method ◽  
Residual Vector ◽  
Integral Forms ◽  
Integration Algorithms ◽  
Hermite Approximation

For the development of a new family of higher-order time integration algorithms for structural dynamics problems, the displacement vector is approximated over a typical time interval using the pth-degree Hermite interpolation functions in time. The residual vector is defined by substituting the approximated displacement vector into the equation of structural dynamics. The modified weighted-residual method is applied to the residual vector. The weight parameters are used to restate the integral forms of the weighted-residual statements in algebraic forms, and then, these parameters are optimized by using the single-degree-of-freedom problem and its exact solution to achieve improved accuracy and unconditional stability. As a result of the pth-degree Hermite approximation of the displacement vector, pth-order (for dissipative cases) and (p + 1)st-order (for the nondissipative case) accurate algorithms with dissipation control capabilities are obtained. Numerical examples are used to illustrate performances of the newly developed algorithms.

A New Family of Higher-Order Time Integration Algorithms for the Analysis of Structural Dynamics

2017 ◽  
Vol 84 (7) ◽  
Author(s):  
Wooram Kim ◽  
J. N. Reddy
Keyword(s):  
Structural Dynamics ◽  
Time Integration ◽  
Higher Order ◽  
Weighted Residual Method ◽  
New Family ◽  
Equal Degree ◽  
Fundamental Limitations ◽  
Mixed Formulations ◽  
Dependent Variables ◽  
Integration Algorithms

For the development of a new family of implicit higher-order time integration algorithms, mixed formulations that include three time-dependent variables (i.e., the displacement, velocity, and acceleration vectors) are developed. Equal degree Lagrange type interpolation functions in time are used to approximate the dependent variables in the mixed formulations, and the time finite element method and the modified weighted-residual method are applied to the velocity–displacement and velocity–acceleration relations of the mixed formulations. Weight parameters are introduced and optimized to achieve preferable attributes of the time integration algorithms. Specific problems of structural dynamics are used in the numerical examples to discuss some fundamental limitations of the well-known second-order accurate algorithms as well as to demonstrate advantages of using the developed higher-order algorithms.

A Comparative Study of Two Families of Higher-Order Accurate Time Integration Algorithms

2019 ◽  
Vol 17 (08) ◽  
pp. 1950048 ◽  
Author(s):  
Wooram Kim ◽  
Jin Ho Lee
Keyword(s):  
Structural Dynamics ◽  
Nonlinear Dynamic ◽  
Time Integration ◽  
Nonlinear Problems ◽  
Higher Order ◽  
Dynamic Problems ◽  
The Past ◽  
Numerical Tests ◽  
Integration Algorithms ◽  
Computational Structures

Two families of higher-order accurate time integration algorithms are numerically tested by using various nonlinear problems of structural dynamics, and the numerical results obtained from them are compared. To be specific, the higher-order algorithms of Kim and Reddy and the higher-order algorithms of Fung are used for this study. In linear analyses, these two different families of higher-order algorithms do not present noticeable differences. However, performances of these algorithms are quite different when they are applied to various nonlinear dynamic problems. For the numerical tests, well-known nonlinear problems are selected from the past studies. For the completeness, the two families of algorithms are briefly reviewed, and their advantageous computational structures are also explained.

Construction of Higher-Order Accurate Time-Step Integration Algorithms by Equal-Order Polynomial Projection

2005 ◽  
Vol 11 (1) ◽  
pp. 19-49 ◽  
Author(s):  
T. C. Fung
Keyword(s):  
Integral Relation ◽  
Higher Order ◽  
Numerical Results ◽  
Numerical Dissipation ◽  
Time Interval ◽  
Time Step ◽  
Higher Order Terms ◽  
Optimal Polynomial ◽  
Integration Algorithms ◽  
New Framework

This paper presents a new framework to construct higher-order accurate time-step integration algorithms based on weakly enforcing the differential/integral relation. The dependent variable and its time derivatives are assumed to be polynomials of equal order. A differential equation is then transformed into an algebraic equation directly. The main issue is how to approximate the integral of a polynomial by another polynomial of the same degree. Various methods to determine the optimal representation (or projection) are considered. It is shown that to reproduce numerical results equivalent to the Padé or generalized Padé approximations, the coefficients of the optimal polynomial representation are related to the weighting parameters derived previously for time-step integration algorithms with predetermined coefficients. A special feature for the present formulation is that the same procedure can be used to solve first-, second-, and even higher-order non-homogeneous initial value problems in a unified manner. The resultant algorithms are higher-order accurate and unconditionally A-stable with controllable numerical dissipation. It is also shown that for the numerical results to maintain higher-order accuracy at the end of a time interval, the higher-order terms in the excitation have to be projected as polynomials of lower degree within the present framework as well. Numerical examples are given to illustrate the validity of the present formulations.

A Cross Weighted-Residual Time Integration Scheme for Structural Dynamics

2014 ◽  
Vol 14 (06) ◽  
pp. 1450023 ◽  
Author(s):  
Wooram Kim ◽  
Sang-Shin Park ◽  
J. N. Reddy
Keyword(s):  
Structural Dynamics ◽  
Linear Time ◽  
Time Integration ◽  
Velocity Fields ◽  
Inner Product ◽  
Integration Scheme ◽  
Time Interval ◽  
Element Discretization ◽  
Integration Schemes ◽  
Time Integration Scheme

In this article, we develop a novel stable time integration scheme for the transient analysis of structural dynamics problems. A second-order (in time) differential operator equation (e.g. obtained after finite element discretization in space) is written as a pair of first-order equations in terms of displacements and velocities. Then the solution is sought by minimizing the inner product of the residuals in the two equations (an unconventional approach) over typical time interval to obtain a symmetric set of algebraic equations involving displacements and velocities at two subsequent intervals. The new time integration scheme is termed the cross weighted-residual (CWR) time integration scheme because each of the two residuals takes the other one as a weight function in the minimization. The CWR time integration scheme is developed by using a uniform linear time approximation of the displacement and velocity fields to yield only a single step time integration scheme, which is comparable to the Newmark family of time integration scheme. A reduced integration technique is used to prevent velocity locking, which is caused by linear approximation of both the displacement and velocity fields. For the verification of the consistency and the stability, the CWR time integration scheme is tested with single-degree as well as multi-degree of freedom problems. The scheme performs extremely well compared with those of the well-known Newmark family of time integration schemes.

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