A Note on the DQ Analysis of Anisotropic Plates

Recently, Bert, Wang and Striz [1, 2] applied the differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods to analyze static and dynamic behaviors of anisotropic plates. Their studies showed that the methods were conceptually simple and computationally efficient in comparison to other numerical techniques. Based on some recent work by the present author [3, 4], the purpose of this note is to further simplify the formulation effort and improve computing efficiency in applying the DQ and HDQ methods for these cases

Efficient frequency averaging techniques for noise and vibration simulations

In the latest decades, noise and vibration characteristics of products have been growing in importance, driven by market expectations and tightening regulations. Accordingly, CAE tools have become irreplaceable in assisting acousticians through the design process, and their accuracy and efficiency are essential to model the behavior of complex engineering systems.This dissertation aims at increasing the computational efficiency of deterministic simulation techniques for steady-state noise and vibration problems. In particular, the focus is on the efficient evaluation of weighted frequency integrals. Classic approaches make use of numerical quadrature to evaluate a frequency integral. However, the response of a vibrating system is commonly a highly oscillating function of frequency, and a large number of sampling points might be required to achieve an accurate integration. As an alternative, the residue theorem is proposed to compute the weighted integrals. The refined integration over real frequencies is replaced by a few computations at complex frequencies, with a consequent increased accuracy and computational efficiency.A weighted integral is first evaluated to compute the band-averaged input power into a vibrating system. The ideal rectangular weighting function is approximated by using the square magnitude of a Butterworth filter. Applying the residue theorem, the integral can be evaluated by computing the system response at a few points in the complex frequency plane. These points are some of the filter poles, equal in number to the order of the filter. This allows for an efficient integration, regardless of the bandwidth of analysis. Such a result is successively generalized. The band integral is computed by moving the path of integration to the complex frequency plane and applying efficient quadrature schemes. Due to the smoothness of the integrand in the complex frequency plane, the accuracy and efficiency of the technique are further increased. Moreover, it is shown that using numerical quadrature in the complex plane indirectly leads to the definition of a novel family of weighting functions over the real frequency domain.The proposed techniques for the evaluation of the band-averaged input power are accurate, efficient, easy to implement and can be employed within any classic deterministic framework. Applications allow assessing the effectiveness of the methodology for complex geometries and frequency dependent properties, and in combination with optimization schemes.When the order of the Butterworth filter is one, its square magnitude corresponds to the Lorentzian function, which allows computing the weighted average over a wide frequency range by evaluating the response at a single complex frequency. However, due to its bell-shape, the Lorentzian is not suitable to evaluate band values, and its use as a weighting function is investigated in three different ways. The ensemble mean input power is estimated by using the Lorentzian-weighted frequency averaging. The same procedure is used to evaluate the direct field dynamic stiffness of a component. Finally, the Lorentzian is used as a mass-frequency density function within the Fuzzy Structure Theory.status: publishe

Vibration of an initially stressed rectangular plate due to an accelerated traveling mass

AbstractThis paper presents a combined application of the Ritz method, the Differential Quadrature (DQ) method, and the Integral Quadrature (IQ) method to vibration problem of rectangular plates subjected to accelerated traveling masses. In this study, the Ritz method with beam eigenfunctions is first used to discretize the spatial partial derivatives with respect to a co-ordinate direction of the plate. The DQ and IQ methods are then employed to analogize the resultant system of partial differential equations. The resulting system of ordinary differential equations is then solved by using the Newmark time integration scheme. The mixed scheme combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The accuracy of the proposed method is demonstrated by comparing the calculated results with those of the existing literature. It is shown that highly accurate results can be obtained using a small number of Ritz terms and DQM sampling points. Finally, the effects of following parameters having something to do with the title problem are investigated: moving load speed and acceleration, and transverse inertia of the moving load. Numerical results show that all the above-mentioned parameters have significant effects on the transient response of such structures under traveling dynamic loads

Direct and Inverse Computational Methods for Electromagnetic Scattering in Biological Diagnostics

Scattering theory has had a major roll in twentieth century mathematical physics. Mathematical modeling and algorithms of direct,- and inverse electromagnetic scattering formulation due to biological tissues are investigated. The algorithms are used for a model based illustration technique within the microwave range. A number of methods is given to solve the inverse electromagnetic scattering problem in which the nonlinear and ill-posed nature of the problem are acknowledged.Comment: 61 pages, 5 figure