scholarly journals Sensitivity analysis of waveguide eigenvalue problems

2011 ◽  
Vol 9 ◽  
pp. 85-89 ◽  
Author(s):  
N. Burschäpers ◽  
S. Fiege ◽  
R. Schuhmann ◽  
A. Walther
Keyword(s):  
Sensitivity Analysis ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Wave Number ◽  
Design Parameters ◽  
Dielectric Waveguides ◽  
Integration Technique ◽  
Numerical Examples ◽  
Numerical Effort ◽  
Complex Solutions

Abstract. We analyze the sensitivity of dielectric waveguides with respect to design parameters such as permittivity values or simple geometric dependencies. Based on a discretization using the Finite Integration Technique the eigenvalue problem for the wave number is shown to be non-Hermitian with possibly complex solutions even in the lossless case. Nevertheless, the sensitivity can be obtained with negligible numerical effort. Numerical examples demonstrate the validity of the approach.

scholarly journals Higher-order sensitivity analysis of periodic 3-D eigenvalue problems for electromagnetic field calculations

2017 ◽  
Vol 15 ◽  
pp. 215-221 ◽  
Author(s):  
Philipp Jorkowski ◽  
Rolf Schuhmann
Keyword(s):  
Sensitivity Analysis ◽  
Discrete Model ◽  
Eigenvalue Problems ◽  
Periodic Structures ◽  
Higher Order ◽  
Integration Technique ◽  
Periodic Boundaries ◽  
Eigenvalue Derivatives ◽  
Eigenvector Derivatives ◽  
Eigenvalue And Eigenvector

Abstract. An algorithm to perform a higher-order sensitivity analysis for electromagnetic eigenvalue problems is presented. By computing the eigenvalue and eigenvector derivatives, the Brillouin Diagram for periodic structures can be calculated. The discrete model is described using the Finite Integration Technique (FIT) with periodic boundaries, and the sensitivity analysis is performed with respect to the phase shift φ between the periodic boundaries. For validation, a reference solution is calculated by solving multiple eigenvalue problems (EVP). Furthermore, the eigenvalue derivatives are compared to reference values using finite difference (FD) formulas.

Uncertainty Propagation in the Band Gap Structure of a 1D Array of Magnetically Coupled Oscillators

2013 ◽  
Vol 135 (4) ◽  
Author(s):  
Brian P. Bernard ◽  
Benjamin A. M. Owens ◽  
Brian P. Mann
Keyword(s):  
Eigenvalue Problem ◽  
Band Gap ◽  
Propagation Constant ◽  
Uncertainty Propagation ◽  
Nonlinear Effects ◽  
Excitation Amplitude ◽  
Standard Uncertainty ◽  
Wave Number ◽  
Design Parameters ◽  
Band Gap Structure

The propagation constant technique has previously been used to predict band gap regions in linear oscillator chains by solving an eigenvalue problem for frequency in terms of a wave number. This paper describes a method by which selected design parameters can be separated from the eigenvalue problem, allowing standard uncertainty propagation techniques to provide closed form solutions for the uncertainty in frequency. Examples are provided for different types of measurement or environmental uncertainty showing the varying robustness of a band gap region to changes in parameters of the same or different order. The system studied in this paper is comprised of repelling magnetic oscillators using a dipole model. Numerical simulation has been performed to confirm the accuracy of analytical solutions up to a certain level of base excitation amplitude after which nonlinear effects change the predicted band gap regions to low energy chaos.

Frequency Spectra of Laminated Piezoelectric Cylinders

1991 ◽  
Author(s):  
J. C.-T. Siao ◽  
S. B. Dong ◽  
J. Song
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Wave Number ◽  
Complex Conjugate ◽  
Element Formulation ◽  
Frequency Spectra ◽  
Quadratic Polynomials ◽  
Propagating Waves ◽  
Long Cylinder

Abstract A method is given for determining the entire frequency spectra of a cylinder with cylindrical piezoelectric properties for both propagating modes and edge vibrations. Finite element modeling occurs in the radial direction so that layered profiles can be accommodated. In the vector of displacements and electric potential q(r)exp{iξz + nθ + ωt)}, quadratic polynomials are taken for q(r). Two algebraic eigenvalue problems can be deduced from this finite element formulation depending on whether the axial wave number ξ or the frequency ω acts as the eigenvalue. When real wave number ξ are assigned, an eigenvalue problem results for the natural frequency ω. When ω is assigned, then an eigenvalue problem for ξ emerges, where the values for ξ may be real (for propagating waves) or complex conjugate pairs (for edge vibrations in a semi-infinitely long cylinder). Two examples are given to illustrate the method of analysis, viz., a solid piezoelectric cylinder of PZT-4 ceramic material and a two-layer cylinder of PZT-4 covering an isotropic material.

scholarly journals Identifying linear absolute instabilities from differential eigenvalue problems using sensitivity analysis

2019 ◽  
Vol 870 ◽  
pp. 941-969 ◽  
Author(s):  
L. S. de B. Alves ◽  
S. C. Hirata ◽  
M. Schuabb ◽  
A. Barletta
Keyword(s):  
Sensitivity Analysis ◽  
Dispersion Relation ◽  
Eigenvalue Problem ◽  
Group Velocity ◽  
Algebraic System ◽  
Eigenvalue Problems ◽  
Computational Cost ◽  
Root Finding ◽  
Alternative Methodology ◽  
Zero Group

Identifying the convective/absolute instability nature of a local base flow requires an analysis of its linear impulse response. One must find the appropriate singularity in the eigenvalue problem with complex frequencies and wavenumbers and prove causality. One way to do so is to show that the appropriate integration contour of this response, a steepest decent path through the relevant singularity, exists. Due to the inherent difficulties of such a proof, one often verifies instead whether this singularity satisfies the collision criterion. In other words, one must show that the branches involved in the formation of this singularity come from distinct halves of the complex wavenumber plane. However, this graphical search is computationally intensive in a single plane and essentially prohibitive in two planes. A significant computational cost reduction can be achieved when root finding procedures are applied instead of graphical ones to search for singularities. They focus on locating these points, with causality being verified graphically a posteriori for a small parametric sample size. The use of root-finding procedures require auxiliary equations, often derived by applying the zero group velocity conditions to the dispersion relation. This relation, in turn, is derived by applying matrix forming to the differential eigenvalue problem and taking the determinant of the resulting system of algebraic equations. Taking the derivative of the dispersion relation with respect to the wavenumbers generates the auxiliary equations. If the algebraic system is decoupled, this derivation is straightforward. However, its computational cost is often prohibitive when the algebraic system is coupled. Other methods exist, but often they can also be too costly and/or not reliable for two wavenumber plane searches. This paper describes an alternative methodology based on sensitivity analysis and adjoints that allow the zero group velocity conditions to be applied directly to the differential eigenvalue problem. In doing so, the direct and auxiliary differential eigenvalue problems can be solved simultaneously using standard shooting methods to directly locate singularities. Auxiliary dispersion relations no longer have to be derived, although it is shown that they are the algebraic equivalent of the auxiliary differential eigenvalue problems obtained by this alternative methodology. Using the latter dramatically reduces computational costs. The search for arbitrary singularities is then not only accelerated in single wavenumber planes but it also becomes viable in two wavenumber planes. Finally, the new method also allows group velocity calculations, greatly facilitating the verification of causality. Several test cases are presented to illustrate the capabilities of this new method.

scholarly journals Super-Exponentially Convergent Parallel Algorithm for a Fractional Eigenvalue Problem of Jacobi-Type

2018 ◽  
Vol 18 (1) ◽  
pp. 21-32 ◽  
Author(s):  
Ivan Gavrilyuk ◽  
Volodymyr Makarov ◽  
Nataliia Romaniuk
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Exponential Convergence ◽  
Linear Boundary ◽  
Recursive Procedure ◽  
Content Type ◽  
Numerical Examples ◽  
Discrete Scheme ◽  
Symbolic Algorithm

AbstractA new algorithm for eigenvalue problems for the fractional Jacobi-type ODE is proposed. The algorithm is based on piecewise approximation of the coefficients of the differential equation with subsequent recursive procedure adapted from some homotopy considerations. As a result, the eigenvalue problem (which is in fact nonlinear) is replaced by a sequence of linear boundary value problems (besides the first one) with a singular linear operator called the exact functional discrete scheme (EFDS). A finite subsequence of m terms, called truncated functional discrete scheme (TFDS), is the basis for our algorithm. The approach provides super-exponential convergence rate as {m\to\infty}. The eigenpairs can be computed in parallel for all given indexes. The algorithm is based on some recurrence procedures including the basic arithmetical operations with the coefficients of some expansions only. This is an exact symbolic algorithm (ESA) for {m=\infty} and a truncated symbolic algorithm (TSA) for a finite m. Numerical examples are presented to support the theory.

scholarly journals CONCEPTUAL DESIGN FOR ASSEMBLY IN AEROSPACE INDUSTRY: SENSITIVITY ANALYSIS OF MATHEMATICAL FRAMEWORK AND DESIGN PARAMETERS

2021 ◽  
Vol 1 ◽  
pp. 731-740
Author(s):  
Giovanni Formentini ◽  
Claudio Favi ◽  
Claude Cuiller ◽  
Pierre-Eric Dereux ◽  
Francois Bouissiere ◽  
...  
Keyword(s):  
Sensitivity Analysis ◽  
Conceptual Design ◽  
Design Parameters ◽  
Design For Assembly ◽  
Mathematical Framework ◽  
Commercial Aircraft ◽  
System Architectures ◽  
Complex Engineering ◽  
Aircraft System ◽  
Definition Of

AbstractOne of the most challenging activity in the engineering design process is the definition of a framework (model and parameters) for the characterization of specific processes such as installation and assembly. Aircraft system architectures are complex structures used to understand relation among elements (modules) inside an aircraft and its evaluation is one of the first activity since the conceptual design. The assessment of aircraft architectures, from the assembly perspective, requires parameter identification as well as the definition of the overall analysis framework (i.e., mathematical models, equations).The paper aims at the analysis of a mathematical framework (structure, equations and parameters) developed to assess the fit for assembly performances of aircraft system architectures by the mean of sensitivity analysis (One-Factor-At-Time method). The sensitivity analysis was performed on a complex engineering framework, i.e. the Conceptual Design for Assembly (CDfA) methodology, which is characterized by level, domains and attributes (parameters). A commercial aircraft cabin system was used as a case study to understand the use of different mathematical operators as well as the way to cluster attributes.

Direct Sensitivity Analysis of Spatial Multibody Systems With Joint Friction Using Index-1 Formulation

2021 ◽  
Author(s):  
Adwait Verulkar ◽  
Corina Sandu ◽  
Daniel Dopico ◽  
Adrian Sandu
Keyword(s):  
Sensitivity Analysis ◽  
Dynamic Systems ◽  
Multibody Systems ◽  
Friction Model ◽  
Design Parameters ◽  
Adjoint Sensitivity ◽  
Joint Friction ◽  
Multibody Dynamic ◽  
Model Dynamics ◽  
Direct Sensitivity

Abstract Sensitivity analysis is one of the most prominent gradient based optimization techniques for mechanical systems. Model sensitivities are the derivatives of the generalized coordinates defining the motion of the system in time with respect to the system design parameters. These sensitivities can be calculated using finite differences, but the accuracy and computational inefficiency of this method limits its use. Hence, the methodologies of direct and adjoint sensitivity analysis have gained prominence. Recent research has presented computationally efficient methodologies for both direct and adjoint sensitivity analysis of complex multibody dynamic systems. The contribution of this article is in the development of the mathematical framework for conducting the direct sensitivity analysis of multibody dynamic systems with joint friction using the index-1 formulation. For modeling friction in multibody systems, the Brown and McPhee friction model has been used. This model incorporates the effects of both static and dynamic friction on the model dynamics. A case study has been conducted on a spatial slider-crank mechanism to illustrate the application of this methodology to real-world systems. Using computer models, with and without joint friction, effect of friction on the dynamics and model sensitivities has been demonstrated. The sensitivities of slider velocity have been computed with respect to the design parameters of crank length, rod length, and the parameters defining the friction model. Due to the highly non-linear nature of friction, the model dynamics are more sensitive during the transition phases, where the friction coefficient changes from static to dynamic and vice versa.

Transparent boundary conditions for discretized non-Hermitian eigenvalue problems

2021 ◽  
pp. 2150138
Author(s):  
Jonathan Heinz ◽  
Miroslav Kolesik
Keyword(s):  
Quantum Mechanics ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Complex Scaling ◽  
Transparent Boundary Conditions ◽  
Drastic Reduction ◽  
Optical Cavities ◽  
Energy Dependent ◽  
Lossy Waveguides

A method is presented for transparent, energy-dependent boundary conditions for open, non-Hermitian systems, and is illustrated on an example of Stark resonances in a single-particle quantum system. The approach provides an alternative to external complex scaling, and is applicable when asymptotic solutions can be characterized at large distances from the origin. Its main benefit consists in a drastic reduction of the dimesnionality of the underlying eigenvalue problem. Besides application to quantum mechanics, the method can be used in other contexts such as in systems involving unstable optical cavities and lossy waveguides.

A Study on Optimum Design Using Fuzzy Numbers As Design Variables

1995 ◽  
Author(s):  
Masao Arakawa ◽  
Hiroshi Yamakawa
Keyword(s):  
Fuzzy Sets ◽  
Optimum Design ◽  
Fuzzy Numbers ◽  
Fuzzy Optimization ◽  
Design Parameters ◽  
Numerical Examples ◽  
Design Variables ◽  
Conventional Methods

Abstract In this study, we summerize the method of fuzzy optimization using fuzzy numbers as design variables. In order to detect flaw in fuzzy calculation, we use LR-fuzzy numbers, which is known as its simplicity in calculation. We also use simple fuzzy numbers’ operations, which was proposed in the previous papers. The proposed method has unique characteristics that we can obtain fuzzy sets in design variables (results of the design) directly from single numerical optimizing process. Which takes a large number of numerical optimizing processes when we try to obtain similar results in the conventional methods. In the numerical examples, we compare the proposed method with several other methods taking imprecision in design parameters into account, and demonstrate the effectiveness of the proposed method.

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