A weighted residual development of a time-stepping algorithm for structural dynamics using two general weight functions

1998 ◽  
Vol 42 (1) ◽  
pp. 93-103 ◽  
Author(s):  
B. W. Golley
Keyword(s):  
Structural Dynamics ◽  
Weight Functions ◽  
Time Stepping ◽  
General Weight

Application of Weight Functions in Nonlinear Analysis of Structural Dynamics Problems

2016 ◽  
Vol 13 (01) ◽  
pp. 1650005 ◽  
Author(s):  
M. Ghassemieh ◽  
A. A. Gholampour ◽  
S. R. Massah
Keyword(s):  
Weight Function ◽  
Structural Dynamics ◽  
Critical Time ◽  
Weight Functions ◽  
Time Step ◽  
Weighted Function ◽  
Nonlinear Structural Dynamics ◽  
Step Duration ◽  
Dynamics Problems ◽  
Critical Time Step

This paper presents a weighted residual method with several weight functions for solving differential equation of motion in nonlinear structural dynamics problems. Order of variation of acceleration is assumed to be quadratic in each time step in which polynomial of displacement would contain five unknown coefficients. Five equations are required for determination of these coefficients in each time step. These equations are obtained from initial conditions, satisfying equation of motions at both ends, and weighted residual integration. In this study, four procedures are considered for weight function to be used in the weighted residual integration as; unit weight function, Petrov–Galerkin’s weight function, least square weight function, and collocation weight function. Due to higher order of acceleration in the proposed method, the results indicate better and more accurate responses. Among the tested functions, the unit weighted function method demonstrated to be non-dissipative and its numerical dispersion showed to be clearly less than the common Newmark’s linear acceleration method. Also critical time step duration in stability investigation for weighted function procedure showed to be larger than the critical time step duration obtained by other methods used in the nonlinear structural dynamics problems.

Determination of Weight Functions for Cracks Under Mode-II Loading

2006 ◽  
Author(s):  
H. L. Li ◽  
X. Wang ◽  
R. Bell
Keyword(s):  
Weight Functions ◽  
Mode Ii ◽  
Unknown Parameters ◽  
Cracked Plate ◽  
Reference Stress ◽  
General Weight ◽  
Function Expression ◽  
Mode Ii Loading ◽  
General Weight Function

For cracks under mode-I loading, it has been demonstrated that a general weight function expression with three unknown parameters can be used to approximate a variety of crack configurations under mode-I loading. For a given crack geometry, the unknown parameters can be determined from reference stress intensity factors (SIFs) together with characteristic properties of the weight functions. It is demonstrated in this paper that a general weight function expression also exists for cracks under mode II loading. The determination of weight functions for cracks in mode II can then also be conducted using reference stress intensity factors (SIFs) together with characteristic properties of the weight functions. This method is used to obtain the mode II weight functions for test specimens including single edge cracked plate, internal center cracked plate and double edge cracked plate. These derived weight functions were further used to calculate the SIFs for the above cracks subjected to several linear and non-linear shear loads and were compared to available SIF solutions.

IMPROVING STABILITY IN THE TIME-STEPPING ANALYSIS OF STRUCTURAL NONLINEAR DYNAMICS

2008 ◽  
Vol 08 (02) ◽  
pp. 257-270 ◽  
Author(s):  
S. LOPEZ ◽  
K. RUSSO
Keyword(s):  
Structural Dynamics ◽  
Time Integration ◽  
Integration Step ◽  
Two Dimensional ◽  
Motion Equations ◽  
Time Stepping ◽  
Nonlinear Structural Dynamics ◽  
Nonlinear Form ◽  
Newton Iterations ◽  
Discrete Motion

A change in the representation of discrete motion equations for nonlinear structural dynamics of two-dimensional bodies is developed. The objective is to write the motion equation in a less nonlinear form. This leads to a significant increase in the range of stability of the time integration process and a reduction in the number of Newton iterations required in the time integration step.

A review of automatic time-stepping strategies on numerical time integration for structural dynamics analysis

2014 ◽  
Vol 80 ◽  
pp. 118-136 ◽  
Author(s):  
Diogo Folador Rossi ◽  
Walnório Graça Ferreira ◽  
Webe João Mansur ◽  
Adenilcia Fernanda Grobério Calenzani
Keyword(s):  
Structural Dynamics ◽  
Time Integration ◽  
Dynamics Analysis ◽  
Time Stepping ◽  
Numerical Time Integration

On Weighted Norm Inequalities for Fractional and Singular Integrals

1971 ◽  
Vol 23 (5) ◽  
pp. 907-928 ◽  
Author(s):  
T. Walsh
Keyword(s):  
Singular Integrals ◽  
Integrable Function ◽  
Sufficient Conditions ◽  
Weight Functions ◽  
Weighted Norm ◽  
Variable Kernel ◽  
Norm Inequalities ◽  
General Weight ◽  
Locally Integrable Function ◽  
Necessary And Sufficient

In a recent paper [12] Muckenhoupt and Wheeden have established necessary and sufficient conditions for the validity of norm inequalities of the form ‖ |x|αTƒ ‖q ≦ C‖ |x|αƒ ‖p, where Tƒ denotes a Calderón and Zygmund singular integral of ƒ or a fractional integral with variable kernel. The purpose of the present paper is to prove, by somewhat different methods, similar inequalities for more general weight functions.In what follows, for p ≧ 1, p′ is the exponent conjugate to p, given by l/p + l/p′ = 1. Ω will always denote a locally integrable function on Rn which is homogeneous of degree 0, Ω∼ will denote a measurable function on Rn × Rn such that for each x ∈ Rn, Ω∼(x, .) is locally integrable and homogeneous of degree 0.

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