An optimal family of controllable numerically dissipative time integration algorithms for structural dynamics

2001 ◽  
Author(s):  
Xiangmin Zhou ◽  
Kumar Tamma ◽  
Desong Sha
Keyword(s):  
Structural Dynamics ◽  
Time Integration ◽  
Integration 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.

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 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.

scholarly journals Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions

Materials ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2288
Author(s):  
Hongming Luo ◽  
Guanhua Sun
Keyword(s):  
Structural Dynamics ◽  
Interpolation Method ◽  
Mass Matrix ◽  
Time Integration ◽  
Partition Of Unity ◽  
Implicit Time ◽  
Lumped Mass ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Consistent Mass Matrix

The PU (partition-of-unity) based FE-RPIM QUAD4 (4-node quadrilateral) element was proposed for statics problems. In this element, hybrid shape functions are constructed through multiplying QUAD4 shape function with radial point interpolation method (RPIM). In the present work, the FE-RPIM QUAD4 element is further applied for structural dynamics. Numerical examples regarding to free and forced vibration analyses are presented. The numerical results show that: (1) If CMM (consistent mass matrix) is employed, the FE-RPIM QUAD4 element has better performance than QUAD4 element under both regular and distorted meshes; (2) The DLMM (diagonally lumped mass matrix) can supersede the CMM in the context of the FE-RPIM QUAD4 element even for the scheme of implicit time integration.

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