scholarly journals An FFT-based approach for Bloch wave analysis: application to polycrystals

2021 ◽  
Author(s):  
Javier Segurado ◽  
Ricardo A. Lebensohn
Keyword(s):  
Acoustic Waves ◽  
Differential Operators ◽  
Bloch Wave ◽  
Linear Operators ◽  
Fourier Space ◽  
Subspace Iteration ◽  
Analysis Application ◽  
Iteration Methods ◽  
Bloch Wave Analysis ◽  
Fourier Series Analysis

AbstractA method based on the Fast Fourier Transform is proposed to obtain the dispersion relation of acoustic waves in heterogeneous periodic media with arbitrary microstructures. The microstructure is explicitly considered using a voxelized Representative Volume Element (RVE). The dispersion diagram is obtained solving an eigenvalue problem for Bloch waves in Fourier space. To this aim, two linear operators representing stiffness and mass are defined through the use of differential operators in Fourier space. The smallest eigenvalues are obtained using the implicitly restarted Lanczos and the subspace iteration methods, and the required inverse of the stiffness operator is done using the conjugate gradient with a preconditioner. The method is used to study the propagation of acoustic waves in elastic polycrystals, showing the strong effect of crystal anistropy and polycrystaline texture on the propagation. It is shown that the method combines the simplicity of classical Fourier series analysis with the versatility of Finite Elements to account for complex geometries proving an efficient and general approach which allows the use of large RVEs in 3D.

scholarly journals DIFFUSIVE REPRESENTATIONS FOR THE ANALYSIS AND SIMULATION OF FLARED ACOUSTIC PIPES WITH VISCO-THERMAL LOSSES

2006 ◽  
Vol 16 (04) ◽  
pp. 503-536 ◽  
Author(s):  
TH. HÉLIE ◽  
D. MATIGNON
Keyword(s):  
Finite Order ◽  
Acoustic Waves ◽  
Complex Analysis ◽  
Differential Operators ◽  
Transfer Functions ◽  
Representation Formula ◽  
Input Output ◽  
Numerical Approximations ◽  
Pseudo Differential Operators ◽  
Scattering Matrices

Acoustic waves travelling in axisymmetric pipes with visco-thermal losses at the wall obey a Webster–Lokshin model. Their simulation may be achieved by concatenating scattering matrices of elementary transfer functions associated with nearly constant parameters (e.g. curvature). These functions are computed analytically and involve diffusive pseudo-differential operators, for which we have representation formula and input-output realizations, yielding direct numerical approximations of finite order. The method is based on some involved complex analysis.

Non-Local Elliptic Boundary-Value Problems

1968 ◽  
Vol 20 ◽  
pp. 1365-1382 ◽  
Author(s):  
Bui An Ton
Keyword(s):  
Differential Operators ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Linear Operators ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Open Set ◽  
Non Local ◽  
Image Position

Let G be a bounded open set of Rn with a smooth boundary ∂G. We consider the following elliptic boundary-value problem:where A and Bj are, respectively singular integro-differential operators on G and on ∂G, of orders 2m and rj with rj < 2m; Ck are boundary differential operators, and Ljk are linear operators, bounded in a sense to be specified.

scholarly journals Coupled Bloch-Wave Analysis of Active PhC Waveguides and Cavities

2018 ◽  
Author(s):  
Marco Saldutti ◽  
Jesper Mork ◽  
Paolo Bardella ◽  
Ivo Montrosset ◽  
Mariangela Gioannini
Keyword(s):  
Bloch Wave ◽  
Wave Analysis ◽  
Bloch Wave Analysis

scholarly journals Subspace Iteration Methods in terms of Matrix Product States

PAMM ◽  
2012 ◽  
Vol 12 (1) ◽  
pp. 641-642 ◽  
Author(s):  
Thomas Huckle ◽  
Konrad Waldherr
Keyword(s):  
Matrix Product ◽  
Matrix Product States ◽  
Subspace Iteration ◽  
Iteration Methods

Bloch Wave Analysis of Long Leaky Coaxial Cables

2008 ◽  
Vol 56 (6) ◽  
pp. 1548-1554 ◽  
Author(s):  
Giuseppe Addamo ◽  
Renato Orta ◽  
Riccardo Tascone
Keyword(s):  
Bloch Wave ◽  
Wave Analysis ◽  
Bloch Wave Analysis

Regularly solvable extensions of non-self-adjoint ordinary differential operators

1984 ◽  
Vol 97 ◽  
pp. 79-95 ◽  
Author(s):  
W. D. Evans
Keyword(s):  
Hilbert Space ◽  
Weighted Space ◽  
Differential Operators ◽  
Linear Operators ◽  
Order Linear ◽  
Adjoint Pair ◽  
Closed Operators ◽  
Ordinary Differential Operators ◽  
Linear Differential ◽  
Densely Defined

SynopsisLetL0,M0be closed densely defined linear operators in a Hilbert spaceHwhich form an adjoint pair, i.e.. In this paper, we study closed operatorsSwhich satisfyand are regularly solvable in the sense of Višik. The abstract results obtained are applied to operators generated by second-order linear differential expressions in a weighted spaceL2(a, b; w).

scholarly journals Some analytic properties of the Weyl function of a closed linear relation

2018 ◽  
Vol 10 (1) ◽  
pp. 197-205
Author(s):  
O.G. Storozh
Keyword(s):  
Operator Function ◽  
Differential Operators ◽  
Weyl Function ◽  
Linear Operators ◽  
Bounded Linear ◽  
Multivalued Operators ◽  
Boundary Operators ◽  
Gamma Field ◽  
Abstract Boundary ◽  
Closed Linear Relation

Let $L$ and $L_{0}$, where $L$ is an expansion of $L_{0}$, be closed linear relations (multivalued operators) in a Hilbert space $H$. In terms of abstract boundary operators (i.e. in the form which in the case of differential operators leads immediately to boundary conditions) some analytic properties of the Weyl function $M(\lambda)$ corresponding to a certain boundary pair of the couple $(L, L_{0}),$ are studied. In particular, applying Hilbert resolvent identity for relations, the criterion of invertibility in the algebra of bounded linear operators in $H$ for transformation $M(\lambda) - M(\lambda_{0})$ in certain small punctured neighbourhood of $\lambda_{0} $ is established. It is proved that in this case $\lambda _{0}$ is a first-order pole for the operator-function $\left(M(\lambda )- M(\lambda_{0} )\right)^{-1} $. The corresponding residue and Laurent series expansion are found. Under some additional assumptions, the behaviour of so called $\gamma$-field $Z_{\lambda}$ (being an operator-function closely connected to $M(\lambda)$) as $\lambda \to - \infty $ is investigated.

scholarly journals Implicit Subspace Iteration to Improve the Stability Analysis in Grinding Processes

2020 ◽  
Vol 10 (22) ◽  
pp. 8203 ◽  
Author(s):  
Jorge Alvarez ◽  
Mikel Zatarain ◽  
David Barrenetxea ◽  
Jose Ignacio Marquinez ◽  
Borja Izquierdo
Keyword(s):  
Stability Analysis ◽  
Dynamic Stability ◽  
Time Domain ◽  
Grinding Processes ◽  
Subspace Iteration ◽  
Efficient Calculation ◽  
Chatter Vibrations ◽  
Stability Maps ◽  
Iteration Methods ◽  
The Stability

An alternative method is devised for calculating dynamic stability maps in cylindrical and centerless infeed grinding processes. The method is based on the application of the Floquet theorem by repeated time integrations. Without the need of building the transition matrix, this is the most efficient calculation in terms of computation effort compared to previously presented time-domain stability analysis methods (semi-discretization or time-domain simulations). In the analyzed cases, subspace iteration has been up to 130 times faster. One of the advantages of these time-domain methods to the detriment of frequency domain ones is that they can analyze the stability of regenerative chatter with the application of variable workpiece speed, a well-known technique to avoid chatter vibrations in grinding processes so the optimal combination of amplitude and frequency can be selected. Subspace iteration methods also deal with this analysis, providing an efficient solution between 27 and 47 times faster than the abovementioned methods. Validation of this method has been carried out by comparing its accuracy with previous published methods such as semi-discretization, frequency and time-domain simulations, obtaining good correlation in the results of the dynamic stability maps and the instability reduction ratio maps due to the application of variable speed.

scholarly journals A Sharp Rearrangement Principle in Fourier Space and Symmetry Results for PDEs with Arbitrary Order

2020 ◽  
Author(s):  
Enno Lenzmann ◽  
Jérémy Sok
Keyword(s):  
Fourier Transform ◽  
Phase Retrieval ◽  
Differential Operators ◽  
Arbitrary Order ◽  
Radial Symmetry ◽  
Fourier Space ◽  
Polyharmonic Operator ◽  
Pseudo Differential Operators ◽  
Retrieval Problem ◽  
The Fourier Transform

Abstract We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in $\mathbb{R}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in $\mathbb{R}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-\Delta )^s$ with $s&gt; 0$ and, in particular, any polyharmonic operator $(-\Delta )^m$ with integer $m \geqslant 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for (1) Gagliardo–Nirenberg inequalities with derivatives of arbitrary order, (2) ground states for bi- and polyharmonic nonlinear Schrödinger equations (NLS), and (3) Adams–Moser–Trudinger type inequalities for $H^{d/2}(\mathbb{R}^d)$ in any dimension $d \geqslant 1$. As a technical key result, we solve a phase retrieval problem for the Fourier transform in $\mathbb{R}^d$. To achieve this, we classify the case of equality in the corresponding Hardy–Littlewood majorant problem for the Fourier transform in $\mathbb{R}^d$.

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