Finite Elements Based on Deslauriers-Dubuc Wavelets for Wave Propagation Problems

Burgos, Rodrigo; Santos, Marco Antônio Cetale

doi:10.4236/am.2016.714128

Cited by 6 publications

(3 citation statements)

References 12 publications

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“…Since M k m the moment of the wavelet scales concerning the x k monomial, where k is the degree of the polynomial, m and j are the translation and resolution of the ϕ wavelet. The justification for the construction of the equation is found in the work of [8][9][10], in which the author concludes that the c j m coefficients for approximating a monomial of the x k form, using a Daubechies wavelet base ϕ, looks like this:…”

Section: Generating An Analytical Function Of the Type X K Using Wavementioning

confidence: 99%

Use of Daubechies Wavelets in the Representation of Analytical Functions

Silva¹

2021

Wavelet Theory

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This chapter aims to use Daubechies' wavelets as basis functions to generate analytical functions, thus being able to rewrite the Taylor series using these wavelets. This makes it possible to analyze functions with a high degree of complexity, in problems that require a high degree of precision in their solution. Wavelet analysis can be applied to practical problems that require a high degree of precision, for example, in the study and analysis of electromagnetic propagation in optical fibers, solutions of differential equations involving engineering problems, in the transmission of WiFi signals, in the treatment and analysis of biomedical images, detection of oil sources through the study of seismic signals.

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Section: Generating An Analytical Function Of the Type X K Using Wavementioning

confidence: 99%

Use of Daubechies Wavelets in the Representation of Analytical Functions

Silva¹

2021

Wavelet Theory

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“…Diaz et al [4] have also developed a Daubechies wavelet-based Euler-Bernoulli beam element and a Mindlin-Reisner plate element for static problems. Except from Daubechies wavelets, Deslauries-Dubuc interpolating wavelets, known as interpolets, have been employed for the construction of beam elements for static [5] and wave propagation analysis [6]. Ko, Kurdila and Pilant [7] have developed a class of finite elements based on orthonormal compactly supported wavelets, and used Daubechies wavelets to solve a second order Neumann problem.…”

Section: Introductionmentioning

confidence: 99%

Multi-Resolution Finite Wavelet Domain Method for Fast Transient Dynamic Analysis in Homogeneous and Heterogeneous Rods and Beams

Dimitriou¹,

Nastos²,

Saravanos³

2021

Proceedings of the 8th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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A Multi-resolution wavelet-based numerical method is developed for the fast prediction of transient response in elastic homogeneous and heterogeneous rods and beams. The method takes advantage of the remarkable mathematical properties of Daubechies wavelet and scaling functions as basis functions for the spatial approximation of state variables. The Multiresolution capability of the Daubechies wavelet family, provides the hierarchical computational framework that incorporates both scaling and wavelet functions. An uncoupled solution system between each resolution is formulated, using an explicit time integration scheme. The first level of analysis provides the coarse solution, while finer approximations are sequentially calculated and superimposed on the coarse solution, until the desired level of accuracy is achieved, without discarding the previous results obtained at coarser resolutions. Additionally, due to the orthogonality and compact support of Daubechies wavelet family, the decoupled mass matrices of each resolution are diagonal, or block diagonal and the stiffness matrices are banded. The proposed method uses a uniform grid which remains practically unchanged when increasing the order of interpolation (p-method), owing to its meshless character. Numerical results for the simulation of high-frequency wave propagation in isotropic and orthotropic rods and beams are presented and compared against confirmed models, demonstrating substantial reduction in computational effort. Furthermore, additional benefits of the proposed method are shown, such as the localization capabilities of the fine solutions, which exhibit high sensitivity in detecting discontinuities resulted from material inhomogeneity.

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“…Smith (1974) proposed a method which was claimed to work perfectly for all incident angles [14]. In this method, the simulation is ex- [16].…”

Section: Introductionmentioning

confidence: 99%

Time Dependent Wave Propagation Modeling Using Finite Difference Scheme of 2D Wave Equation Based on Absorbing and Reflecting Boundaries

Alam¹,

Loskor²,

Mohiuddin³

2021

JAMP

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Boundary procedure is an important phenomenon in numerical simulation. To reduce or eliminate the spurious reflections significantly which is occurred in boundary is a challenging and vital approach. The appropriate artificial numerical boundaries can be applied to eliminate the effect of unnecessary spurious reflections in case of the numerical simulations of wave propagation phenomena problems. Typically, to reduce the artificial reflections, the absorbing boundary conditions are necessary. In this paper, we overview and investigate the appropriate typical absorbing boundary conditions and analyzed the boundary effect of two dimensional wave equation numerically. Reflections over the wide-ranging incident angles are complicated to eliminate, but the absorbing boundary conditions that we have applied are computationally cost efficient, easy to apply and able to reduce reflections significantly. For numerical solution, finite difference method is applied to develop numerical scheme using 2D wave equation. Using the developed numerical scheme, we obtain the numerical solution of the governing equation as an initial boundary value problem and realize the qualitative behavior of the solution in infinite space. The finite difference numerical scheme has been investigated by developing MATLAB programming language code. Numerical results have been discussed and analyzed with presenting different qualitative behavior of the numerical scheme. The accuracy and efficiency of the numerical scheme has been illustrated. The stability analysis was discussed and verified stability condition. Using the numerical scheme and absorbing boundary conditions, the boundary effects and absorption of spurious reflection of boundary have been demonstrated.

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Finite Elements Based on Deslauriers-Dubuc Wavelets for Wave Propagation Problems

Cited by 6 publications

References 12 publications

Use of Daubechies Wavelets in the Representation of Analytical Functions

Use of Daubechies Wavelets in the Representation of Analytical Functions

Multi-Resolution Finite Wavelet Domain Method for Fast Transient Dynamic Analysis in Homogeneous and Heterogeneous Rods and Beams

Time Dependent Wave Propagation Modeling Using Finite Difference Scheme of 2D Wave Equation Based on Absorbing and Reflecting Boundaries

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