An exact numerical time integration of scalar equations for undamped structural systems

1984 ◽  
Vol 12 (1) ◽  
pp. 137-142 ◽  
Author(s):  
Jerzy Kujawski ◽  
Chandrakant S. Desai
Keyword(s):  
Time Integration ◽  
Structural Systems ◽  
Numerical Time Integration ◽  
Scalar Equations

scholarly journals Structural Reliability Sensitivities under Nonstationary Random Vibrations

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Rita Greco ◽  
Francesco Trentadue
Keyword(s):  
Structural Reliability ◽  
Structural Parameters ◽  
Time Integration ◽  
Structural Response ◽  
Integration Scheme ◽  
Structural Systems ◽  
Frame Building ◽  
Time Integration Scheme ◽  
Noise Input ◽  
The Time Domain

Response sensitivity evaluation is an important element in reliability evaluation and design optimization of structural systems. It has been widely studied under static and dynamic forcing conditions with deterministic input data. In this paper, structural response and reliability sensitivities are determined by means of the time domain covariance analysis in both classically and nonclassically damped linear structural systems. A time integration scheme is proposed for covariance sensitivity. A modulated, filtered, white noise input process is adopted to model the stochastic nonstationary loads. The method allows for the evaluation of sensitivity statistics of different quantities of dynamic response with respect to structural parameters. Finally, numerical examples are presented regarding a multistorey shear frame building.

DQFEM Analyses of Static and Dynamic Nonlinear Elastic-Plastic Problems Using a GSR-Based Accelerated Constant Stiffness Equilibrium Iteration Technique

2001 ◽  
Vol 123 (3) ◽  
pp. 310-317 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Finite Element ◽  
Numerical Technique ◽  
Time Integration ◽  
Differential Quadrature ◽  
Element Discretization ◽  
Constant Stiffness ◽  
Structural Problems ◽  
Nonlinear Structural ◽  
Numerical Time Integration ◽  
Nonlinear Elastic

An integrated numerical technique for static and dynamic nonlinear structural problems adopting the equilibrium iteration is proposed. The differential quadrature finite element method (DQFEM), which uses the differential quadrature (DQ) techniques to the finite element discretization, is used to analyze the static and dynamic nonlinear structural mechanics problems. Numerical time integration in conjunction with the use of equilibrium iteration is used to update the response history. The equilibrium iteration can be carried out by the accelerated iteration schemes. The global secant relaxation-based accelerated constant stiffness and diagonal stiffness-based predictor-corrector equilibrium iterations which are efficient and reliable are used for the numerical computations. Sample problems are analyzed. Numerical results demonstrate the algorithm.

Separate Time Integration Based on the Hilber, Hughes, Taylor Scheme for Flexible Bodies With a Large Number of Modes

2020 ◽  
Vol 15 (6) ◽  
Author(s):  
Wolfgang Witteveen ◽  
Florian Pichler
Keyword(s):  
Time Integration ◽  
Flexible Multibody Dynamics ◽  
Integration Scheme ◽  
Step Size ◽  
Body Contact ◽  
Time Integration Scheme ◽  
Core Idea ◽  
Numerical Time Integration ◽  
Taylor Scheme ◽  
Local Deformations

Abstract In the current development of flexible multibody dynamics, the efficient and accurate consideration of distributed and nonlinear forces is an active area of research. Examples are, forces due to body-body contact or due to elastohydrodynamics (EHD). This leads to many additional modes for representing the local deformations in the areas on which those forces act. Recent publications show that these can be several hundred to several thousand additional modes. A conventional, monolithic numerical time integration scheme would lead to unacceptable computing times. This paper presents a method for an efficient time integration of such systems. The core idea is to treat the equations associated with modes representing local deformations separately. Using the Newmark formulas, a fixed point iteration is proposed for these separated equations, which can always be stabilized with decreasing step size. The concluding examples underline this property, as well as the fact that the proposed method massively outperforms the conventional, monolithic time integration with increasing number of modes.

Combined Rayleigh–Taylor and Kelvin–Helmholtz instabilities on an annular liquid sheet

2016 ◽  
Vol 812 ◽  
pp. 152-177 ◽  
Author(s):  
M. Vadivukkarasan ◽  
Mahesh V. Panchagnula
Keyword(s):  
Total Energy ◽  
Time Integration ◽  
Three Dimensional ◽  
Liquid Sheet ◽  
Length Scale ◽  
Time Varying ◽  
Varying Coefficients ◽  
Flow Approximation ◽  
Time Varying Coefficients ◽  
Numerical Time Integration

This paper describes the three-dimensional destabilization characteristics of an annular liquid sheet when subjected to the combined action of Rayleigh–Taylor (RT) and Kelvin–Helmholtz (KH) instability mechanisms. The stability characteristics are studied using temporal linear stability analysis and by assuming that the fluids are incompressible, immiscible and inviscid. Surface tension is also taken into account at both the interfaces. Linearized equations governing the growth of instability amplitude have been derived. These equations involve time-varying coefficients and have been analysed using two approaches – direct numerical time integration and frozen-flow approximation. From the direct numerical time integration, we show that the time-varying coefficients evolve on a slow time scale in comparison with the amplitude growth. Therefore, we justify the use of the frozen-flow approximation and derive a closed-form dispersion relation from the appropriate governing equations and boundary conditions. The effect of flow conditions and fluid properties is investigated by introducing dimensionless numbers such as Bond number ($Bo$), inner and outer Weber numbers ($We_{i}$, $We_{o}$) and inner and outer density ratios ($Q_{i}$, $Q_{o}$). We show that four instability modes are possible – Taylor, sinuous, flute and helical. It is observed that the choice of instability mode is influenced by a combination of both $Bo$ as well as $We_{i}$ and $We_{o}$. However, the instability length scale calculated from the most unstable wavenumbers is primarily a function of $Bo$. We show a regime map in the $Bo,We_{i},We_{o}$ parameter space to identify regions where the system is susceptible to three-dimensional helical modes. Finally, we show an optimal partitioning of a given total energy ($\unicode[STIX]{x1D701}$) into acceleration-induced and shear-induced instability mechanisms in order to achieve a minimum instability length scale (${\mathcal{L}}_{m}^{\ast }$). We show that it is beneficial to introduce at least 90 % of the total energy into acceleration induced RT instability mechanism. In addition, we show that when the RT mechanism is invoked to destabilize an annular liquid sheet, ${\mathcal{L}}_{m}^{\ast }\sim \unicode[STIX]{x1D701}^{-3/5}$.

Numerical Integration of Some Stiff Constitutive Models of Inelastic Deformation

1980 ◽  
Vol 102 (1) ◽  
pp. 92-96 ◽  
Author(s):  
Virendra Kumar ◽  
Mahesh Morjaria ◽  
Subrata Mukherjee
Keyword(s):  
Numerical Integration ◽  
Inelastic Deformation ◽  
Time Integration ◽  
Constitutive Models ◽  
Integration Scheme ◽  
Uniaxial Deformation ◽  
Time Step ◽  
One Step ◽  
Numerical Time Integration ◽  
Step Control

Several strategies for numerical time-integration of some stiff constitutive models of inelastic deformation are presented in this paper. Numerical results and comparisons are presented for the integration of one such model for the case of uniaxial deformation under various prescribed histories of stress or strain. A simple one step Euler type integration scheme with automatic time-step control, which can be easily adapted to the solution of multiaxial boundary value problems, appears promising.

scholarly journals Dynamics of gravity–capillary solitary waves in deep water

2012 ◽  
Vol 708 ◽  
pp. 480-501 ◽  
Author(s):  
Zhan Wang ◽  
Paul A. Milewski
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Periodic Structures ◽  
Time Integration ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Full Potential ◽  
Numerical Time Integration ◽  
The Stability ◽  
Nonlinear Focusing

AbstractThe dynamics of solitary gravity–capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time-dependent solutions, we simplify the full potential flow problem by using surface variables and taking a particular cubic truncation possessing a Hamiltonian with desirable properties. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a two-dimensional fluid domain, and with higher-order truncations in three dimensions. Fully localized solitary waves are then computed in the three-dimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. There are many solitary wave branches, indexed by their finite energy as their amplitude tends to zero. The dynamics of the solitary waves is complex, involving nonlinear focusing of wavepackets, quasi-elastic collisions, and the generation of propagating, spatially localized, time-periodic structures akin to breathers.

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