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Second-order Derivative (second-order + derivative)
Selected AbstractsAn improved meshless collocation method for elastostatic and elastodynamic problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2008P. H. Wen Abstract Meshless methods for solving differential equations have become a promising alternative to the finite element and boundary element methods. In this paper, an improved meshless collocation method is presented for use with either moving least square (MLS) or compactly supported radial basis functions (RBFs). A new technique referred to as an indirect derivative method is developed and compared with the direct derivative technique used for evaluation of second-order derivatives and higher-order derivatives of the MLS and RBF shape functions at the field point. As the derivatives are obtained from a local approximation (MLS or compact support RBFs), the new method is computationally economical and efficient. Neither the connectivity of mesh in the domain/boundary nor integrations with fundamental/particular solutions is required in this approach. The accuracy of the two techniques to determine the second-order derivative of shape function is assessed. The applications of meshless method to two-dimensional elastostatic and elastodynamic problems have been presented and comparisons have been made with benchmark analytical solutions. Copyright © 2007 John Wiley & Sons, Ltd. [source] On the applicability of the HSAB principle through the use of improved computational schemes for chemical hardness evaluationJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 7 2004Mihai V. Putz Abstract Finite difference schemes, named Compact Finite Difference Schemes with Spectral-like Resolution, have been used for a less crude approximation of the analytical hardness definition as the second-order derivative of the energy with respect to the electron number. The improved computational schemes, at different levels of theory, have been used to calculate global hardness values of some probe bases, traditionally classified as hard and soft on the basis of their chemical behavior, and to investigate the quantitative applicability of the HSAB principle. Exchange acid-base reactions have been used to test the HSAB principle assuming the reaction energies as a measure of the stabilization of product adducts. © 2004 Wiley Periodicals, Inc. J Comput Chem 25: 994,1003, 2004 [source] A convergent three-level finite difference scheme for solving a dual-phase-lagging heat transport equation in spherical coordinatesNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2004Weizhong Dai Abstract Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation since a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in spherical coordinates and develop a three-level finite difference scheme for solving the heat transport equation in a microsphere. It is shown that the scheme is convergent, which implies that the scheme is unconditionally stable. Results show that the numerical solution converges to the exact solution. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 60,71, 2004. [source] Assessment of embryonic growth in chicken eggs by means of visible transmission spectroscopyBIOTECHNOLOGY PROGRESS, Issue 2 2010Bart J. Kemps Abstract During this work, it was investigated whether spectral measurements can be used to monitor embryonic growth. An experiment was conducted in which both the transmission spectra and embryonic weight were determined on 240 eggs (Cobb, 37 weeks) between Day 5 and Day 10 of incubation. The spectral data were linked to embryonic weight by means of a partial least squares analysis. Different preprocessing procedures were compared during this work, that is, smoothing, multiplicative scatter correction (MSC), and first- and second-order derivative. Compared to the remainder of the preprocessing procedures, MSC leads to a considerable improvement of the prediction capability of the embryonic weight. The ratio of performance to deviation obtained for the MSC spectra equaled 4.5 indicating that a very accurate prediction of embryonic weight is feasible based on the VIS/NIR transmission measurements. Important regions for the prediction are situated around 685,740 nm. It is suggested that the spectral changes in these spectral regions result from the displacement of carotenoids from the yolk into the blood circuitry. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2010 [source] Comparison of methods to model the gravitational gradients from topographic data basesGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2006Christopher Jekeli SUMMARY A number of methods have been developed over the last few decades to model the gravitational gradients using digital elevation data. All methods are based on second-order derivatives of the Newtonian mass integral for the gravitational potential. Foremost are algorithms that divide the topographic masses into prisms or more general polyhedra and sum the corresponding gradient contributions. Other methods are designed for computational speed and make use of the fast Fourier transform (FFT), require a regular rectangular grid of data, and yield gradients on the entire grid, but only at constant altitude. We add to these the ordinary numerical integration (in horizontal coordinates) of the gradient integrals. In total we compare two prism, two FFT and two ordinary numerical integration methods using 1, elevation data in two topographic regimes (rough and moderate terrain). Prism methods depend on the type of finite elements that are generated with the elevation data; in particular, alternative triangulations can yield significant differences in the gradients (up to tens of Eötvös). The FFT methods depend on a series development of the topographic heights, requiring terms up to 14th order in rough terrain; and, one popular method has significant bias errors (e.g. 13 Eötvös in the vertical,vertical gradient) embedded in its practical realization. The straightforward numerical integrations, whether on a rectangular or triangulated grid, yield sub-Eötvös differences in the gradients when compared to the other methods (except near the edges of the integration area) and they are as efficient computationally as the finite element methods. [source] An improved meshless collocation method for elastostatic and elastodynamic problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2008P. H. Wen Abstract Meshless methods for solving differential equations have become a promising alternative to the finite element and boundary element methods. In this paper, an improved meshless collocation method is presented for use with either moving least square (MLS) or compactly supported radial basis functions (RBFs). A new technique referred to as an indirect derivative method is developed and compared with the direct derivative technique used for evaluation of second-order derivatives and higher-order derivatives of the MLS and RBF shape functions at the field point. As the derivatives are obtained from a local approximation (MLS or compact support RBFs), the new method is computationally economical and efficient. Neither the connectivity of mesh in the domain/boundary nor integrations with fundamental/particular solutions is required in this approach. The accuracy of the two techniques to determine the second-order derivative of shape function is assessed. The applications of meshless method to two-dimensional elastostatic and elastodynamic problems have been presented and comparisons have been made with benchmark analytical solutions. Copyright © 2007 John Wiley & Sons, Ltd. [source] Sensitivity analyses of FORM-based and DRM-based performance measure approach (PMA) for reliability-based design optimization (RBDO)INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2010Ikjin Lee Abstract In gradient-based design optimization, the sensitivities of the constraint with respect to the design variables are required. In reliability-based design optimization (RBDO), the probabilistic constraint is evaluated at the most probable point (MPP), and thus the sensitivities of the probabilistic constraints at MPP are required. This paper presents the rigorous analytic derivation of the sensitivities of the probabilistic constraint at MPP for both first-order reliability method (FORM)-based performance measure approach (PMA) and dimension reduction method (DRM)-based PMA. Numerical examples are used to demonstrate that the analytic sensitivities agree very well with the sensitivities obtained from the finite difference method (FDM). However, as the sensitivity calculation at the true DRM-based MPP requires the second-order derivatives and additional MPP search, the sensitivity derivation at the approximated DRM-based MPP, which does not require the second-order derivatives and additional MPP search to find the DRM-based MPP, is proposed in this paper. A convergence study illustrates that the sensitivity at the approximated DRM-based MPP converges to the sensitivity at the true DRM-based MPP as the design approaches the optimum design. Hence, the sensitivity at the approximated DRM-based MPP is proposed to be used for the DRM-based RBDO to enhance the efficiency of the optimization. Copyright © 2009 John Wiley & Sons, Ltd. [source] | ||||||||||