43 research outputs found
Fast Computation of Fourier Integral Operators
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 at points given on
a Cartesian grid. Here, is a frequency variable, is the
Fourier transform of the input , is an amplitude and
is a phase function, which is typically as large as ;
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 by would require operations.
This paper develops a new numerical algorithm which requires operations, and as low as 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 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
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
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
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