Qualitative Analysis of an Additional Food Provided Predator–Prey Model in the Presence of Allee Effect

Singh, Manoj Kumar; Bhadauria, B. S.

doi:10.1007/s40819-017-0409-2

Cited by 5 publications

(3 citation statements)

References 36 publications

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“…The first-order three-point numerical derivatives of y(x) at nodes x i−1 , x i and x i+1 for nonuniform grids are approximated by (e.g., Singh & Bhadauria 2009)…”

Section: Smoothness Indicators For Nonuniform Gridsmentioning

confidence: 99%

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A novel fourth-order WENO interpolation technique

Janett

Steiner

Ballester

et al. 2019

A&A

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Context. Several numerical problems require the interpolation of discrete data that present at the same time (i) complex smooth structures and (ii) various types of discontinuities. The radiative transfer in solar and stellar atmospheres is a typical example of such a problem. This calls for high-order well-behaved techniques that are able to interpolate both smooth and discontinuous data. Aims. This article expands on different nonlinear interpolation techniques capable of guaranteeing high-order accuracy and handling discontinuities in an accurate and non-oscillatory fashion. The final aim is to propose new techniques which could be suitable for applications in the context of numerical radiative transfer. Methods. We have proposed and tested two different techniques. Essentially non-oscillatory (ENO) techniques generate several candidate interpolations based on different substencils. The smoothest candidate interpolation is determined from a measure for the local smoothness, thereby enabling the essentially non-oscillatory property. Weighted ENO (WENO) techniques use a convex combination of all candidate substencils to obtain high-order accuracy in smooth regions while keeping the essentially non-oscillatory property. In particular, we have outlined and tested a novel well-performing fourth-order WENO interpolation technique for both uniform and nonuniform grids. Results. Numerical tests prove that the fourth-order WENO interpolation guarantees fourth-order accuracy in smooth regions of the interpolated functions. In the presence of discontinuities, the fourth-order WENO interpolation enables the non-oscillatory property, avoiding oscillations. Unlike Bézier and monotonic high-order Hermite interpolations, it does not degenerate to a linear interpolation near smooth extrema of the interpolated function. Conclusion. The novel fourth-order WENO interpolation guarantees high accuracy in smooth regions, while effectively handling discontinuities. This interpolation technique might be particularly suitable for several problems, including a number of radiative transfer applications such as multidimensional problems, multigrid methods, and formal solutions.

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“…The first-order three-point numerical derivatives of y(x) at nodes x i−1 , x i and x i+1 for nonuniform grids are approximated by (e.g., Singh & Bhadauria 2009)…”

Section: Smoothness Indicators For Nonuniform Gridsmentioning

confidence: 99%

“…The formulae inSingh & Bhadauria (2009) contain some typos, which have been corrected in the present manuscript.…”

mentioning

confidence: 99%

A novel fourth-order WENO interpolation technique

Janett

Steiner

Ballester

et al. 2019

A&A

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“…Although the spatial resolution of monitoring displacement can be enhanced by utilizing advanced measurement instrumentation, Gautschi found that placing sensors too close to each other tends to cause more noise effect when using the central difference approximation to calculate the curvature. Because the central difference approximation is only applicable to uniformly spaced measurement data, Singh and Bhadauria derived a second‐order finite difference approximation for unequally spaced data, that is,

v_{i}^{″} = \frac{2 ((),, v_{i + 1} h_{2} - v_{i} ((),, h_{1} + h_{2}) + v_{i - 1} h_{1})}{h_{1} h_{2} ((),, h_{1} + h_{2})},

where h 1 and h 2 denote distances between two successive measured locations i and i + 1, and i and i − 1, respectively. However, Gautschi indicated that the finite difference approximation may possibly generate noticeable truncation errors when the sampling data are not uniformly spaced.…”

Section: Introductionmentioning

confidence: 99%

Damage detection in shear buildings using different estimated curvature

Shi

Spencer

Chen

2017

Struct Control Health Monit

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SummaryThis study investigates the damage localization employing the curvature of the lateral displacement envelope in shear building structures. Both the finite difference method and the proposed interpolation method are applied to evaluate the curvature of mode shapes and frequency response functions (FRFs) in a 12-story shear building. The interpolated displacement function used in the proposed method considers appropriate continuity and boundary conditions for the shear buildings.Numerical studies show that using the curvature by the proposed method could reduce the occurrences of false damage localization when the vibration responses include small simulated noise. Moreover, the existing FRF curvature method could perform worse than random guessing to find correct damage locations at the cost of the considerable number of false alarms. However, the poor detection performance of the existing method may be enhanced significantly by using the proposed method to evaluate the curvature of the FRFs if the simulated noise is under a small level.The proposed method is shown to perform better than the finite difference method to improve the effectiveness of the curvature-based methods for damage detection. | INTRODUCTIONNumerous vibration-based damage detection (VBDD) methods have been developed to identify and localize damage in structures during the last few decades. VBDD methods detect and quantify the structural damage by analyzing the changes of the dynamic responses for structures. The development of VBDD methods has been reviewed in recent studies by Farrar and Worden [1] and Fan and Qiao. [2] Modal response has been frequently employed for damage detection. Several studies [3][4][5][6][7][8][9][10][11][12] investigated the changes of mode shapes between the intact and damaged states of the structure to localize and quantify the structural damage. Pandey et al. [3] first presented a method to detect damage from changes in curvature mode shapes and also indicated that the difference in a curvature mode shape between an intact and a damaged beam showed a significant peak at a damaged location. Because the curvature of an elastic beam is inversely proportional to the flexural rigidity of the section, a loss of flexural rigidity will cause an increase of curvature. However, for the higher modes, there were also some small peaks observed at the undamaged locations. If the small peaks are observed in practical applications, the engineer may be misled regarding the damage locations. Wahab and Roeck [4] further proposed a curvature damage factor (CDF) that summarizes in one number the differences in curvature mode shapes for all modes for each measured point. They found that the spatial distribution of CDF can show a clear peak at the position of damage in a continuous beam. Chandrashekhar and Ganguli [5] proposed a fuzzy logic system for structural damage detection using modal curvatures. Their parametric studies

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