Numerical integration schemes for evaluation of (hyper)singular integrals in 2D BEM

1998 ◽  
Vol 22 (1) ◽  
pp. 1-11 ◽  
Author(s):  
A. Aimi ◽  
A. Carini ◽  
M. Diligenti ◽  
G. Monegato
Keyword(s):  
Numerical Integration ◽  
Singular Integrals ◽  
Integration Schemes

scholarly journals On the Integration Schemes of Retrieving Impulse Response Functions from Transfer Functions

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
Kui Fu Chen ◽  
Yan Feng Li
Keyword(s):  
Numerical Integration ◽  
Transfer Functions ◽  
Sampling Interval ◽  
High Order ◽  
Impulse Response Functions ◽  
Inverse Laplace Transformation ◽  
High Order Scheme ◽  
Integration Schemes ◽  
Order Scheme ◽  
Better Than

The numerical inverse Laplace transformation (NILM) makes use of numerical integration. Generally, a high-order scheme of numerical integration renders high accuracy. However, surprisingly, this is not true for the NILM to the transfer function. Numerical examples show that the performance of higher-order schemes is no better than that of the trapezoidal scheme. In particular, the solutions from high-order scheme deviate from the exact one markedly over the rear portion of the period of interest. The underlying essence is examined. The deviation can be reduced by decreasing the frequency-sampling interval.

scholarly journals The Girsanov Linearization Method for Stochastically Driven Nonlinear Oscillators

2006 ◽  
Vol 74 (5) ◽  
pp. 885-897 ◽  
Author(s):  
Nilanjan Saha ◽  
D. Roy
Keyword(s):  
Numerical Integration ◽  
Nonlinear Oscillators ◽  
Linearization Method ◽  
Statistical Moments ◽  
Probability Measures ◽  
Numerical Accuracy ◽  
Girsanov Transformation ◽  
Nonlinear Part ◽  
Integration Schemes ◽  
Nikodym Derivative

For most practical purposes, the focus is often on obtaining statistical moments of the response of stochastically driven oscillators than on the determination of pathwise response histories. In the absence of analytical solutions of most nonlinear and higher-dimensional systems, Monte Carlo simulations with the aid of direct numerical integration remain the only viable route to estimate the statistical moments. Unfortunately, unlike the case of deterministic oscillators, available numerical integration schemes for stochastically driven oscillators have significantly poorer numerical accuracy. These schemes are generally derived through stochastic Taylor expansions and the limited accuracy results from difficulties in evaluating the multiple stochastic integrals. As a numerically superior and semi-analytic alternative, a weak linearization technique based on Girsanov transformation of probability measures is proposed for nonlinear oscillators driven by additive white-noise processes. The nonlinear part of the drift vector is appropriately decomposed and replaced, resulting in an exactly solvable linear system. The error in replacing the nonlinear terms is then corrected through the Radon-Nikodym derivative following a Girsanov transformation of probability measures. Since the Radon-Nikodym derivative is expressible in terms of a stochastic exponential of the linearized solution and computable with high accuracy, one can potentially achieve a remarkably high numerical accuracy. Although the Girsanov linearization method is applicable to a large class of oscillators, including those with nondifferentiable vector fields, the method is presently illustrated through applications to a few single- and multi-degree-of-freedom oscillators with polynomial nonlinearity.

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