Deslauriers–Dubuc interpolating wavelet beam finite element

Burgos, Rodrigo; Santos, Marco Antônio Cetale; Silva, Raul Rosas e

doi:10.1016/j.finel.2013.07.004

Cited by 16 publications

(8 citation statements)

References 16 publications

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“…Our proposed numerical method utilizes the biorthogonal interpolating wavelet family of basis functions, defined in Donoho (1992). These bases are sometimes referred to as the Deslauriers-Dubuc wavelets (Burgos et al, 2013;Fujii and Hoefer, 2003), or the autocorrelation of the Daubechies wavelets (Bertoluzza and Naldi, 1996). Since modified bases are required for wavelet representation on an interval (Alpert et al, 2002;Sweldens, 1998), we use the secondgeneration wavelets defined in Villiers et al (2003) near spatial boundaries.…”

Section: Wavelet Representationmentioning

confidence: 99%

Adaptive Wavelet Algorithm for Solving Nonlinear Initial–boundary Value Problems With Error Control

Harnish

Matouš

Livescu

2018

Int J Mult Comp Eng

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We present a numerical method which exploits the biorthogonal interpolating wavelet family, and second-generation wavelets, to solve initial-boundary value problems on finite domains. Our predictor-corrector algorithm constructs a dynamically adaptive computational grid with significant data compression, and provides explicit error control. Error estimates are provided for the wavelet representation of functions, their derivatives, and the nonlinear product of functions. The method is verified on traditional nonlinear problems such as Burgers' equation and the Sod shock tube. Numerical analysis shows polynomial convergence with negligible global energy dissipation.

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Section: Wavelet Representationmentioning

confidence: 99%

Adaptive Wavelet Algorithm for Solving Nonlinear Initial–boundary Value Problems With Error Control

Harnish

Matouš

Livescu

2018

Int J Mult Comp Eng

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“…To evaluate the integral of the weak form the unit-integral property of the Deslauriers-Dubuc interpolating wavelets is used [37]. The integral of the mother scaling function φ(θ) constructed on a dyadic wavelet grid with sufficient grid points (≥ 2m − 1), coordinate θ and a grid spacing Δθ = 1 is given by [4].…”

Section: Integration Of the Wavelet Representationmentioning

confidence: 99%

Wavelet based reduced order models for microstructural analyses

et al. 2018

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This paper proposes a novel method to accurately and efficiently reduce a microstructural mechanical model using a wavelet based discretisation. The model enriches a standard reduced order modelling (ROM) approach with a wavelet representation. Although the ROM approach reduces the dimensionality of the system of equations, the computational complexity of the integration of the weak form remains problematic. Using a sparse wavelet representation of the required integrands, the computational cost of the assembly of the system of equations is reduced significantly. This wavelet-reduced order model (W-ROM) is applied to the mechanical equilibrium of a microstructural volume as used in a computational homogenisation framework. The reduction technique however is not limited to micro-scale models and can also be applied to macroscopic problems to reduce the computational costs of the integration. For the sake of clarity, the W-ROM will be demonstrated using a one-dimensional example, providing full insight in the underlying steps taken.

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“…Since the expression above leads to an infinite number of solutions, there is the need for a normalization rule that provides a unique eigenvector. This unique solution comes with the inclusion of the so-called moment equation, derived from the wavelet property of exact polynomial representation, which is given originally in [15] and adapted for Interpolets by [8]: The dimensionless expressions for the stiffness k and mass m matrices are in wavelet space and need to be transformed to physical space, using a transformation matrix T obtained by evaluating the wavelet basis at the element node coordinates using Equation (16). It can be noticed that some terms related to the length (L) of the element emerge from coordinate changes.…”

Section: Finite Element Formulationmentioning

confidence: 99%

“…A complete basis of wavelets can be generated through dilation and translation of a mother scaling function. Although many applications use only the wavelet filter coefficients of the multiresolution analysis, there are some which explicitly require the values of the basis functions and their derivatives, such as the Wavelet Finite Element Method (WFEM) [4]- [8]. Compactly supported wavelets have a finite number of derivatives which can be highly oscillatory.…”

Section: Introductionmentioning

confidence: 99%

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

Burgos¹,

Santos²

2016

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This paper presents the formulation of finite elements based on Deslauriers-Dubuc interpolating scaling functions, also known as Interpolets, for their use in wave propagation modeling. Unlike other wavelet families like Daubechies, Interpolets possess rational filter coefficients, are smooth, symmetric and therefore more suitable for use in numerical methods. Expressions for stiffness and mass matrices are developed based on connection coefficients, which are inner products of basis functions and their derivatives. An example in 1-D was formulated using Central Difference and Newmark schemes for time differentiation. Encouraging results were obtained even for large time steps. Results obtained in 2-D are compared with the standard Finite Difference Method for validation.

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Deslauriers–Dubuc interpolating wavelet beam finite element

Cited by 16 publications

References 16 publications

Adaptive Wavelet Algorithm for Solving Nonlinear Initial–boundary Value Problems With Error Control

Adaptive Wavelet Algorithm for Solving Nonlinear Initial–boundary Value Problems With Error Control

Wavelet based reduced order models for microstructural analyses

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

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