Renormalization Analysis of a Composite Ultrasonic Transducer With a Fractal Architecture

Algehyne, Ebrahem A.; Mulholland, Anthony J.

doi:10.1142/s0218348x17500153

Cited by 4 publications

(2 citation statements)

References 38 publications

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“…9,[12][13][14]30,31 A renormalization adaptation was used to investigate the potential performance of a fractal-like transducer. The fractal used as inspiration for the transducer design was the Sierpinski carpet.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

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The Effectiveness of a Sierpinski Carpet-Inspired Transducer

Walker

Roach

2017

Fractals

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Piezoelectric ultrasonic transducers have the ability to act both as a receiver and a transmitter of ultrasound. Standard designs have a regular structure and therefore operate effectively over narrow bandwidths due to their single length scale. Naturally occurring transducers benefit from a wide range of length scales giving rise to increased bandwidths. It is therefore of interest to investigate structures which incorporate a range of length scales, such as fractals. This paper applies an adaptation of the Green function renormalization method to analyze the propagation of an ultrasonic wave in a series of pre-fractal structures. The structure being investigated here is the Sierpinski carpet. Novel expressions for the non-dimensionalized electrical impedance and the transmission and reception sensitivities as a function of the operating frequency are presented. Comparisons of metrics between three new designs alongside the standard design (Euclidean structure) and the previously investigated Sierpinski gasket device are performed. § Corresponding author. The results indicate a significant improvement in the reception sensitivity of the device, and improved bandwidth in both the receiving and transmitting responses. 1750050-1

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Section: Conclusion and Discussionmentioning

confidence: 99%

“…Research on transducer performance designed on a modified Sierpinski gasket has already begun. 31 The purpose of adjusting such fractal designs would be to yield wider length scaled devices.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

The Effectiveness of a Sierpinski Carpet-Inspired Transducer

Walker

Roach

2017

Fractals

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Analysis of a fractal ultrasonic transducer with a range of piezoelectric length scales

Algehyne

Mulholland²

2019

IMA Journal of Applied Mathematics

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The transmission and reception sensitivities of most piezoelectric ultrasonic transducers are enhanced by their geometrical structures. This structure is normally a regular, periodic one with one principal length scale, which, due to the resonant nature of the devices, determines the central operating frequency. There is engineering interest in building wide-bandwidth devices, and so it follows that, in their design, resonators that have a range of length scales should be used. This paper describes a mathematical model of a fractal ultrasound transducer whose piezoelectric components span a range of length scales. There have been many previous studies of wave propagation in the Sierpinski gasket but this paper is the first to study its complement. This is a critically important mathematical development as the complement is formed from a broad distribution of triangle sizes, whereas the Sierpinski gasket is formed from triangles of equal size. Within this structure, the electrical and mechanical fields fluctuate in tune with the time-dependent displacement of these substructures. A new set of basis functions is developed that allow us to express this displacement as part of a finite element methodology. A renormalization approach is then used to develop a recursion scheme that analytically describes the key components from the discrete matrices that arise. Expressions for the transducer’s operational characteristics are then derived and analysed as a function of the driving frequency. It transpires that the fractal device has a significantly higher reception sensitivity (18 dB) and a significantly wider bandwidth (3 MHz) than an equivalent Euclidean (standard) device.

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Dynamics of Sir Mathematical Model for Covid-19 Outbreak in Pakistan Under Fractal-Fractional Derivative

et al. 2021

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There are still mathematical predictions in the fight against epidemics. Speedy expansion, ways and procedures for the pandemic control require early understanding when solutions with better computer-based mathematical modeling and prognosis are developed. Despite high uncertainty in each of these models, one of the important tools for public health management system is epidemiology models. The fractional order is shown to be more effective in modeling epidemic diseases, in relation to the memory effects. Notably, recently founded calculus tools, called fractal-fractional calculus, having a fractional order and fractal dimension, enable us to study the behavior of a real-world problem under both fractal and fractional tools. This paper is about the dynamical behavior of a new mathematical model of novel corona disease (COVID-19) under the fractal-fractional Atangana–Baleanu derivative. The considered model has three compartments, namely, susceptible, infected and recovered or removed [Formula: see text]. The existence and uniqueness of the model’s solution will be proved via Krasnoselskii’s and Banach’s fixed point theorems, respectively. The stability of the solution in the sense of Hyers–Ulam [Formula: see text] will be built up by nonlinear functional analysis. Moreover, the numerical simulations for different values of isolation parameters corresponding to various fractal-fractional orders are analyzed using fractional Adams–Bashforth [Formula: see text] method with two-step Lagrange polynomial. Finally, the obtained simulation results are applied to the real data of disease spread from Pakistan. The graphical interpretations demonstrate that increasing the isolation parameters which is caused by strict precautionary measures will reduce the disease infection transmission in society.

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Renormalization Analysis of a Composite Ultrasonic Transducer With a Fractal Architecture

Cited by 4 publications

References 38 publications

The Effectiveness of a Sierpinski Carpet-Inspired Transducer

The Effectiveness of a Sierpinski Carpet-Inspired Transducer

Analysis of a fractal ultrasonic transducer with a range of piezoelectric length scales

Dynamics of Sir Mathematical Model for Covid-19 Outbreak in Pakistan Under Fractal-Fractional Derivative

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