43 research outputs found

    Fast Computation of Fourier Integral Operators

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    We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation and general hyperbolic equations. The problem is to evaluate numerically a so-called Fourier integral operator (FIO) of the form e2πiΦ(x,ξ)a(x,ξ)f^(ξ)dξ\int e^{2\pi i \Phi(x,\xi)} a(x,\xi) \hat{f}(\xi) \mathrm{d}\xi at points given on a Cartesian grid. Here, ξ\xi is a frequency variable, f^(ξ)\hat f(\xi) is the Fourier transform of the input ff, a(x,ξ)a(x,\xi) is an amplitude and Φ(x,ξ)\Phi(x,\xi) is a phase function, which is typically as large as ξ|\xi|; hence the integral is highly oscillatory at high frequencies. Because an FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size NN by NN would require O(N4)O(N^4) operations. This paper develops a new numerical algorithm which requires O(N2.5logN)O(N^{2.5} \log N) operations, and as low as O(N)O(\sqrt{N}) in storage space. It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel e2πiΦ(x,ξ)a(x,ξ)e^{2 \pi i \Phi(x,\xi)} a(x,\xi) into two components: 1) a diffeomorphism which is handled by means of a nonuniform FFT and 2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is that the separation rank of the residual kernel is \emph{provably independent of the problem size}. Several numerical examples demonstrate the efficiency and accuracy of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology.Comment: 31 pages, 3 figure

    Spherical Harmonics on constitutive equations for biological cells

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    Tese (doutorado)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Civil e Ambiental, 2019.Desenvolvem-se e avaliam-se neste trabalho modelos constitutivos não-lineares incluindo o estudo de grandes deformações com o objetivo de modelar células biológicas representadas por elementos de cascas finas. É utilizada como ponto de partida a formulação clássica de elementos de cascas finas, considerando as hipóteses de Kirchhoff que apresentam como mais importante característica a redução dimensional. Esta é atingida derivando tensões 2D como médias das tensões 3D pela integração direta sob a espessura da casca. Para a definição da deformação do continuo é utilizada uma descrição Lagrangiana. As células biológicas não podem ser modeladas de forma correta utilizando modelos constitutivos lineares. Especificamente no estudo dos glóbulos vermelhos devem ser considerados: o comportamento elástico não linear e o aporte da viscosidade da parede da célula. Consequentemente, neste trabalho, modelos hiperelasticos são implementados junto ao modelo de Kelvin-Voigth para obter um modelo viscoelástico. Na implementação computacional Funções de Esféricos Harmônicos são utilizadas para sintetizar as principais variáveis, esforços e deslocamentos. Isto se deve a que a geometria dos glóbulos vermelhos pode ser descrita de forma simples utilizando coordenadas esféricas. Resultando numa implementação de baixo custo computacional que consegue lidar com altas não linearidades. Este trabalho apresenta uma formulação de um método indireto pois consiste no cálculo de coeficientes da expansão de Esféricos Harmônicos, sendo que estes coeficientes não têm sentido físico. É importante mencionar que o projeto se encontra num estágio inicial e não foi encontrado na literatura uma aplicação utilizando teoria de cascas, Harmônicos Esféricos junto com modelos constitutivos lidando com grandes deformações. Finalmente o método é validado e estudado suas possíveis aplicações.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) e Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).In this work, constitutive models are developed and evaluated with the aim of modeling biological cells represented by thin shell elements in a second-order analysis. The classical formulation of thin shell elements is used while considering dimensional reduction, which is the main feature of the Kirchhoff hypotheses. This reduction is achieved by deriving two-dimensional stresses as averages of the true three-dimensional stresses by means of direct integration through the shell thickness. A Lagrangian description is used to define the deformation of the continuum. Biological cells cannot be correctly modeled using linear constitutive relations. Specifically, in the study of red blood cells, one should consider both their nonlinear elastic behavior and the contribution of the cell wall viscosity. Consequently, hyperelastic constitutive equations are implemented using the Kelvin-Voigt approach to obtain a viscoelastic model. In the computational implementation, spherical harmonic functions are used to synthesize the main variables, resultant forces and displacements since the geometry of red blood cells can be simply described using spherical coordinates. As a result, a low-cost computational implementation for highly nonlinear analyses is obtained. This work presents a formulation of an indirect method since consists on the calculation of the expansion coefficients of a Spherical Harmonic Analysis, these coefficients have no physical meaning. It is important to mention that this work is part of a project that is at an early stage. In the literature no application was found using shell theory, Spherical Harmonics with constitutive models dealing with large deformations. Finally, the method is validated and its possible applications are discussed

    A new mixed model based on the enhanced-Refined Zigzag Theory for the analysis of thick multilayered composite plates

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    The Refined Zigzag Theory (RZT) has been widely used in the numerical analysis of multilayered and sandwich plates in the last decay. It has been demonstrated its high accuracy in predicting global quantities, such as maximum displacement, frequencies and buckling loads, and local quantities such as through-the-thickness distribution of displacements and in-plane stresses [1,2]. Moreover, the C0 continuity conditions make this theory appealing to finite element formulations [3]. The standard RZT, due to the derivation of the zigzag functions, cannot be used to investigate the structural behaviour of angle-ply laminated plates. This drawback has been recently solved by introducing a new set of generalized zigzag functions that allow the coupling effect between the local contribution of the zigzag displacements [4]. The newly developed theory has been named enhanced Refined Zigzag Theory (en- RZT) and has been demonstrated to be very accurate in the prediction of displacements, frequencies, buckling loads and stresses. The predictive capabilities of standard RZT for transverse shear stress distributions can be improved using the Reissner’s Mixed Variational Theorem (RMVT). In the mixed RZT, named RZT(m) [5], the assumed transverse shear stresses are derived from the integration of local three-dimensional equilibrium equations. Following the variational statement described by Auricchio and Sacco [6], the purpose of this work is to implement a mixed variational formulation for the en-RZT, in order to improve the accuracy of the predicted transverse stress distributions. The assumed kinematic field is cubic for the in-plane displacements and parabolic for the transverse one. Using an appropriate procedure enforcing the transverse shear stresses null on both the top and bottom surface, a new set of enhanced piecewise cubic zigzag functions are obtained. The transverse normal stress is assumed as a smeared cubic function along the laminate thickness. The assumed transverse shear stresses profile is derived from the integration of local three-dimensional equilibrium equations. The variational functional is the sum of three contributions: (1) one related to the membrane-bending deformation with a full displacement formulation, (2) the Hellinger-Reissner functional for the transverse normal and shear terms and (3) a penalty functional adopted to enforce the compatibility between the strains coming from the displacement field and new “strain” independent variables. The entire formulation is developed and the governing equations are derived for cases with existing analytical solutions. Finally, to assess the proposed model’s predictive capabilities, results are compared with an exact three-dimensional solution, when available, or high-fidelity finite elements 3D models. References: [1] Tessler A, Di Sciuva M, Gherlone M. Refined Zigzag Theory for Laminated Composite and Sandwich Plates. NASA/TP- 2009-215561 2009:1–53. [2] Iurlaro L, Gherlone M, Di Sciuva M, Tessler A. Assessment of the Refined Zigzag Theory for bending, vibration, and buckling of sandwich plates: a comparative study of different theories. Composite Structures 2013;106:777–92. https://doi.org/10.1016/j.compstruct.2013.07.019. [3] Di Sciuva M, Gherlone M, Iurlaro L, Tessler A. A class of higher-order C0 composite and sandwich beam elements based on the Refined Zigzag Theory. Composite Structures 2015;132:784–803. https://doi.org/10.1016/j.compstruct.2015.06.071. [4] Sorrenti M, Di Sciuva M. An enhancement of the warping shear functions of Refined Zigzag Theory. Journal of Applied Mechanics 2021;88:7. https://doi.org/10.1115/1.4050908. [5] Iurlaro L, Gherlone M, Di Sciuva M, Tessler A. A Multi-scale Refined Zigzag Theory for Multilayered Composite and Sandwich Plates with Improved Transverse Shear Stresses, Ibiza, Spain: 2013. [6] Auricchio F, Sacco E. Refined First-Order Shear Deformation Theory Models for Composite Laminates. J Appl Mech 2003;70:381–90. https://doi.org/10.1115/1.1572901

    1-D broadside-radiating leaky-wave antenna based on a numerically synthesized impedance surface

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    A newly-developed deterministic numerical technique for the automated design of metasurface antennas is applied here for the first time to the design of a 1-D printed Leaky-Wave Antenna (LWA) for broadside radiation. The surface impedance synthesis process does not require any a priori knowledge on the impedance pattern, and starts from a mask constraint on the desired far-field and practical bounds on the unit cell impedance values. The designed reactance surface for broadside radiation exhibits a non conventional patterning; this highlights the merit of using an automated design process for a design well known to be challenging for analytical methods. The antenna is physically implemented with an array of metal strips with varying gap widths and simulation results show very good agreement with the predicted performance
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