Computational Aspects of Time Integration Procedures in Structural Dynamics—Part 2: Error Propagation

1978 ◽  
Vol 45 (3) ◽  
pp. 603-611 ◽  
Author(s):  
K. C. Park ◽  
C. A. Felippa
Keyword(s):  
Structural Dynamics ◽  
Error Propagation ◽  
Error Control ◽  
Time Integration ◽  
Trapezoidal Rule ◽  
Computational Error ◽  
Asymptotic Error ◽  
Average Acceleration ◽  
Computational Aspects ◽  
Acceleration Method

The propagation of computational error in the direct time integration of the equations of structural dynamics is investigated. Asymptotic error propagation equations corresponding to the computational paths presented in Part 1 are derived and verified by means of numerical experiments. It is shown that there exists an implementation form that achieves optimum error control when used in conjunction with one-derivative methods. No such form is found for two-derivative methods. A numerical beating phenomenon is observed for certain implementations of the average acceleration method and the trapezoidal rule, which from an error propagation standpoint, is highly undesirable.

Computational Aspects of Time Integration Procedures in Structural Dynamics—Part 1: Implementation

1978 ◽  
Vol 45 (3) ◽  
pp. 595-602 ◽  
Author(s):  
C. A. Felippa ◽  
K. C. Park
Keyword(s):  
Structural Dynamics ◽  
Error Propagation ◽  
Equations Of Motion ◽  
Time Integration ◽  
Computational Effort ◽  
Computational Error ◽  
Unified Approach ◽  
Time Step ◽  
Computational Aspects ◽  
Specific Implementation

A unified approach for the implementation of direct time integration procedures in structural dynamics is presented. Two key performance assessment factors are considered, viz., computational effort and error propagation. It is shown that these factors are strongly affected by details in the reduction of the second-order equations of motion to a system of first-order equations, and by the computational path followed at each time step. Part 1 is primarily devoted to the study of the organization of the computational process. Specific implementation forms derivable from the unified approach are studied in detail, and rated accordingly. An analysis of the computational error propagation characteristics of these implementations is presented in Part 2.

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.

A New Unconditionally Stable Time Integration Method for Analysis of Nonlinear Structural Dynamics

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
Ali Akbar Gholampour ◽  
Mehdi Ghassemieh ◽  
Mahdi Karimi-Rad
Keyword(s):  
Time Integration ◽  
Second Order ◽  
Numerical Dispersion ◽  
Computational Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Similar Time ◽  
Unconditionally Stable ◽  
Average Acceleration

A new time integration scheme is presented for solving the differential equation of motion with nonlinear stiffness. In this new implicit method, it is assumed that the acceleration varies quadratically within each time step. By increasing the order of acceleration, more terms of the Taylor series are used, which are expected to have responses with better accuracy than the classical methods. By considering this assumption and employing two parameters δ and α, a new family of unconditionally stable schemes is obtained. The order of accuracy, numerical dissipation, and numerical dispersion are used to measure the accuracy of the proposed method. Second order accuracy is achieved for all values of δ and α. The proposed method presents less dissipation at the lower modes in comparison with Newmark's average acceleration, Wilson-θ, and generalized-α methods. Moreover, this second order accurate method can control numerical damping in the higher modes. The numerical dispersion of the proposed method is compared with three unconditionally stable methods, namely, Newmark's average acceleration, Wilson-θ, and generalized-α methods. Furthermore, the overshooting effect of the proposed method is compared with these methods. By evaluating the computational time for analysis with similar time step duration, the proposed method is shown to be faster in comparison with the other methods.

Internal Resonance of a Beam/Traveling-Mass System

2000 ◽  
Author(s):  
S. Siddiqui ◽  
F. Golnaraghi ◽  
G. R. Heppler
Keyword(s):  
Differential Equations ◽  
Nonlinear Interaction ◽  
Internal Resonance ◽  
Series Expansions ◽  
Nonlinear Ordinary Differential Equations ◽  
Mass System ◽  
Weighted Residuals ◽  
Average Acceleration ◽  
Acceleration Method ◽  
Traveling Mass

Abstract The dynamics of a cantilever beam undergoing large amplitude oscillations while experiencing a nonlinear interaction with an attached lumped spring/mass system are investigated. The resulting partial differential equations contain trigonometric non-linearities that are treated using methods of weighted residuals and do not require series expansions about the equilibrium positions. The resulting nonlinear ordinary differential equations are solved using a formulation of the average acceleration method.

A Comparative Study of Implicit and Explicit Composite Time Integration Schemes

2020 ◽  
Vol 20 (13) ◽  
pp. 2041003 ◽  
Author(s):  
Wooram Kim ◽  
J. N. Reddy
Keyword(s):  
Time Integration ◽  
Point Of View ◽  
Structural Problems ◽  
Integration Schemes ◽  
Time Integration Schemes ◽  
Nonlinear Transient ◽  
Computational Aspects ◽  
Similarities And Differences ◽  
Implicit And Explicit

In this paper, a number of recently proposed implicit and explicit composite time integration schemes are reviewed and critically evaluated. To give suitable guidelines of using them in practical transient analyses of structural problems, numerical performances of these schemes are compared through illustrative examples. Meaningful insights into computational aspects of the composite schemes are also provided. In the discussion, the role of the splitting ratio of the recent composite schemes is also investigated through a different point of view, and similarities and differences of various composite schemes are also studied. It is shown that the explicit composite scheme proposed recently by the authors can noticeably increase the efficiency and the accuracy of linear and nonlinear transient analyses when compared with other well-known composite schemes.

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