Eigenvalue Technique Research Articles

A NEW LOOK AT NUMERICAL ANALYSIS OF UNIFORM FIBER BRAGG GRATINGS USING COUPLED MODE THEORY

The coupled mode theory (CMT) is used to analyze uniform Fiber Bragg gratings. The multi-mode CMT is expressed as the flrst-order vector ordinary difierential equations (ODEs) with coe-cients depending on the propagation distance. We show in this paper that by changing variables, the original couple mode equations (CMEs) can be re-casted as constant coe-cient ODEs. The eigenvalue and eigenvector technique (EVVT), the analytic method for solving constant coe-cient ODEs, is then applied to solve the coupled mode equations. Furthermore, we also investigate the application of Runge- Kutta method (RKM) to the calculation of the global transfer-function matrix for CMEs. We compare the transmission and the re∞ection

A predictive tool to evaluate disk brake squeal using a fully coupled thermo-mechanical finite element model

This paper presents a new integrated approach to the analysis of brake squeal modelling. It focuses on developing a validated thermo-mechanical finite element model by taking full account of the effect of thermal loading on the structural response of the brake. An integrated study involving time-dependent non-linear contact and a fully coupled transient thermal analysis are carried out to provide the contact and temperature distribution within the brake before executing an instability study using the complex eigenvalue technique. The results, in turn, demonstrate the fugitive nature of brake squeal through the system eigenvalues that are extracted throughout the braking period.

Experimental Testing of a Moving Force Identification Bridge Weigh-in-Motion Algorithm

Bridge weigh-in-motion systems are based on the measurement of strain on a bridge and the use of the measurements to estimate the static weights of passing traffic loads. Traditionally, commercial systems employ a static algorithm and use the bridge influence line to infer static axle weights. This paper describes the experimental testing of an algorithm based on moving force identification theory. In this approach the bridge is dynamically modeled using the finite element method and an eigenvalue reduction technique is employed to reduce the dimension of the system. The inverse problem of finding the applied forces from measured responses is then formulated as a least squares problem with Tikhonov regularization. The optimal regularization parameter is solved using the L-curve method. Finally, the static axle loads, impact factors and truck frequencies are obtained from a complete time history of the identified moving forces.

FREE VIBRATION ANALYSIS OF CROSS-PLY LAYERED COMPOSITE BEAMS WITH FINITE LENGTH ON ELASTIC FOUNDATION

In this paper, free vibration analysis of cross-ply layered composite beams (LCB) with finite length and rectangular cross-section rested on an elastic foundation is investigated by finite element method. Based on the Timoshenko beam theory which includes the shear deformation and rotary inertia, the stiffness and mass matrices of a LCB are obtained using the energy method. Then, the natural frequencies are calculated by employing eigenvalue technique. The obtained results are verified against existing data in the literatures for a LCB with no foundation and uniform cross-section. Good agreements are observed between these cases. In the same way, the natural frequencies of a specific case, i.e. the stepped beam are calculated and finally, free vibrations of a symmetric and non-symmetric LCB are compared with each others.

Conditionality of inverse solutions to discretised integral equations in geoid modelling from local gravity data

The eigenvalue decomposition technique is used for analysis of conditionality of two alternative solutions for a determination of the geoid from local gravity data. The first solution is based on the standard two-step approach utilising the inverse of the Abel-Poisson integral equation (downward continuation) and consequently the Stokes/Hotine integration (gravity inversion). The second solution is based on a single integral that combines the downward continuation and the gravity inversion in one integral equation. Extreme eigenvalues and corresponding condition numbers of matrix operators are investigated to compare the stability of inverse problems of the above-mentioned computational models. To preserve a dominantly diagonal structure of the matrices for inverse solutions, the horizontal positions of the parameterised solution on the geoid and of data points are identical. The numerical experiments using real data reveal that the direct gravity inversion is numerically more stable than the downward continuation procedure in the two-step approach.

An efficient conceptual model to simulate surface water body‐aquifer interaction in conjunctive use management models

In conjunctive use management models, the surface and subsurface components must be simultaneously simulated because of the hydraulic interactions and to the combined operating rules inherent to such schemes. When many management alternatives have to be simulated over long time horizon or when optimizing complex large‐scale systems, an efficient tool for surface water body‐aquifer interaction simulation is needed. In this paper, we present a parsimonious conceptual model as an alternative approach to simulate aquifer storage and surface water body‐aquifer interaction for linear aquifers (linear response of head to stresses). The formulation is deduced from a finite difference model preserving its approach of the geometry, boundary conditions, hydrodynamic parameters (heterogeneity), and spatial distribution of the stresses. It does not require any additional assumption. The solution, obtained using eigenvalue techniques, is formulated with an explicit state equation that is a function of time and does not need to be calculated in a sequential way (that is, time is not discretized). The structure of the solution finally deduced allows a simple conceptual interpretation of surface water body‐aquifer interaction phenomenon as the drainage of a number of independent linear reservoirs. The parameters and initial conditions of each linear reservoir are mathematically defined in a univocal way from the original calibrated finite difference model preserving its characteristics. In most practical cases, an accurate solution can be obtained with a reduced number of linear reservoirs. Therefore this multireservoir conceptual definition enables an important reduction in the dimension of the model matrices and provides a straightforward means of incorporating the accuracy of more complex calibrated distributed‐parameter numerical models for simulating surface water body‐aquifer interaction within conjunctive use management models and with a minimum computational cost. Several case studies (synthetic and real world cases) are used to illustrate the applicability of the methodology.

Eigenvalue optimization-based formulations for nonlinear dynamics and control problems

Eigenvalues play an important role in many fields of applied mathematics to engineering. For some applications it may be desirable to calculate the variables of a model in order to optimize an objective function and/or to verify constraints that involve the eigenvalues of a certain matrix. In general the elements of such a matrix depend nonlinearly on the optimization variables. Despite its potential to address diverse chemical engineering problems, eigenvalue optimization techniques have not been extensively used in the Process Systems Engineering discipline. The objectives of this contribution are to review most relevant topics on eigenvalue optimization and to present formulations and solution strategies to practically address eigenvalue optimization problems in the field of chemical engineering. In order to illustrate the ideas, several small size applications, which have to do with the analysis and control of nonlinear dynamic systems, are developed. Other potential applications and future lines of research are also suggested.

A general methodology to simulate groundwater flow of unconfined aquifers with a reduced computational cost

The computational cost of groundwater flow simulation can be crucial when analyzing complex conjunctive use water resources systems that need to simulate simultaneously surface and groundwater components. A general methodology for accurate simulation of unconfined groundwater flow with low computational cost is presented. It requires linearizing the unconfined groundwater flow problem governed by the Boussinesq equation. The technique is based on a change of variable and depends on the reference level adopted. Some recommendations have been provided to set the reference level to estimate the spatially variant parameters required to define the linearized problem. Using this linearization, more accurate results can be obtained than those derived with the classical assumption of invariant transmissivity. Solving the problem with eigenvalue techniques, the solution can be defined with a semi-explicit state equation with low computational cost. Some case studies have been analyzed in order to demonstrate that the methodology can be applied to any aquifer geometry (including non-horizontal bottoms), hydrodynamic properties and boundary conditions (even different prescribed head values). The results have been compared with those obtained with other linearization methods and MODFLOW [McDonald, M.G., Harbaugh, A.W., 1988. A modular three dimensional finite difference ground water flow model. Open – File Report 83-875, US Geological Survey, Washington DC] for unconfined aquifers. A case study defined from a previously calibrated finite-difference model of the “Delta Adra” aquifer, located in southern Spain, has been also analyzed.

Semantic mediation modelling in the service component design

Semantic mediation is one of the big challenges in this century from scale and complexity. The main objective of mediation is to minimise the architecture gap, and reduce expenses. Achieving this objective requires multi-dimensional approaches including service strategy and integration. Therefore, architecture alignment and coherence are extremely important. We further extend our investigation by using AHP and eigenvalue techniques in component evaluation. Our proposal mainly argues that semantic mediation should not be initiated from the macro architecture; instead we propose that mediation should impact components first, to push semantics strategy directly into the micro process to be synthesised into the service layer.

An Eigenvector-Based Approach for Multidimensional Frequency Estimation With Improved Identifiability

This paper presents an algebraic method for two-dimensional (2-D) and multidimensional frequency estimation by exploiting eigenvector structure. The algorithm is based on multidimensional smoothing and data folding, and offers significantly improved identifiability (ID) over existing algebraic approaches, thus is termed the improved multidimensional folding (IMDF) algorithm. The ID, performance, and computational complexity of the proposed algorithm are analyzed in detail. In the 2-D case, it is shown that with the IMDF algorithm, up to approximately 0.34M1(M2+1) 2-D frequencies can be uniquely resolved with probability one from an M1 by M2 data mixture (assuming M1gesM2), while the most relaxed ID bound offered by existing algebraic approaches is approximately M1M2/4. Unlike most eigenvalue techniques that usually require an extra frequency association step, the IMDF algorithm achieves automatic frequency pairing once an eigenvalue decomposition problem is solved because frequencies are estimated from the eigenvectors instead of the eigenvalues. Theoretical analysis and simulation results demonstrate its competitive performance compared to the Crameacuter-Rao bound (CRB)

5-chromatic strongly regular graphs

In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5. Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph. For each parameter set, we must decide which strongly regular graphs, if any, possessing the set are 5-chromatic. In this way, we deal completely with 34 of these parameter sets using eigenvalue techniques and computer enumerations.

A two‐step explicit solution of the Boussinesq equation for efficient simulation of unconfined aquifers in conjunctive‐use models

Mathematical models that simultaneously simulate surface and groundwater components and their interaction are required to evaluate management alternatives in conjunctive‐use systems. Complex systems and scenarios defined over long time periods often force the use of computationally efficient aquifer models. Thus the groundwater flow equation needs to be solved using explicit techniques such as influence functions or the eigenvalue technique. The eigenvalue technique provides a continuous‐in‐time solution using a state equation with important computational advantages. These methods are strictly applicable only to confined aquifers, which are modeled with a linear groundwater flow equation. However, many commonly exploited aquifers connected with the surface system are unconfined and should be simulated using the nonlinear Boussinesq equation. A two‐step explicit solution of the unconfined groundwater flow problem is presented for aquifers with nearly horizontal bottom. It is based on a new approach to linearize the Boussinesq equation. Using a change of variable, it is possible to define an equation with a structure similar to the linear groundwater flow equation. The only difference is found in a term that depends on the solution. Approaching this term by means of a fictitious stress, we obtain a linear equation analogous to the confined groundwater flow equation. The boundary conditions can also be formulated as a function of this new variable with linear expressions. Therefore the unconfined groundwater flow problem can be solved by applying the superposition principle, and the solution can be obtained with a reduced computational cost using the eigenvalue technique.

능동 동조질량감쇠의 고층빌딩 해석을 위한 동적압축법

자유도 수가 많은 고층 구조물의 해석하기 위해 모든 층에 sensors를 설치하는 것은 비 실용적이다. 따라서 이러한 문제를 해결하기 위해 "reduced-order control" 방법이 소개되었다. 본 논문은 동적압축법(dynamic condensation method)이 제안되었다. 이 압축법은 반복법으로 "Guyan condensation"의 initial approximation을 적용하였다. 본 논문에서 제시된 동적압축법(dynamic condensation)은 원하는 값을 얻을 때까지 지속적으로 updated가 되며, 결과는 기존의 "Guyan condensation"보다 정확한 결과를 나타내었다. 또한 "eigenvalue shifting technique"을 적용하여 iteration으로 계산되는 시간을 크게 단축하였다. "Reduced-order system"을 도입하기 위한 두가지 schemes이 토의되었다. 제시된 동적압축법 효과의 증명을 위해 능동 동조질량감쇠 고층빌딩의 수치 해석이 토의되었고, 단지 두 번의 반복(iterations)을 통한 결과는 매우 정확한 것으로 나타났다. It is impractical to install sensors on every floor of a tall building to measure the full state vector because of the large number of degrees of freedom. This makes it necessary to introduce reduced order control. A kind of system reduction scheme (dynamic condensation method) is proposed in this paper. This method is iterative and Guyan condensation is looked upon as an initial approximation of the iteration. Since the reduced order system is updated repeatedly until a desired one is obtained, the accuracy of the reduced order system resulting from the proposed method is much higher than that obtained from the Guyan condensation method. An eigenvalue shilling technique is applied to accelerate the convergence of Iteration. Two schemes to establish the reduced order system by using the proposed method are also presented and discussed in this paper. The results for a tail building with active tuned mass damper show that the proposed method is efficient for the reduced order modelling and the accuracy is very close to exact only after two iterations.

A hybrid regularization method for inverse heat conduction problems

AbstractThis paper presents a hybrid regularization method for solving inverse heat conduction problems. The method uses future temperatures and past fluxes to reduce the sensitivity to temperature noise. A straightforward comparison technique is suggested to find the optimal number of the future temperatures. Also, an eigenvalue reduction technique is used to further improve the accuracy of the inverse solution. The method provides a physical insight into the inverse problems under study. The insight indicates that the inverse algorithm is a general purpose algorithm and applicable to various numerical methods (although our development was based on FEM), and that the inverse solutions can be obtained by directly extending Stolz's equation in the least‐squares error (LSE) sense. Direct extension of the present method to the inverse internal heat generation problems is made. Four numerical examples are given to validate the method. The effects of the future temperatures, the past fluxes, the eigenvalue reduction, the varying number of future temperatures and local iterations for non‐linear problems are studied. Copyright © 2005 John Wiley & Sons, Ltd.

The valuations of the near hexagons related to the Witt designs S(5,6,12) and S(5,8,24)

AbstractValuations of dense near polygons were introduced in 16. In the present paper, we classify all valuations of the near hexagons 𝔼1 and 𝔼2, which are related to the respective Witt designs S(5,6,12) and S(5,8,24). Using these classifications, we prove that if a dense near polygon S contains a hex H isomorphic to 𝔼1 or 𝔼2, then H is classical in S. We will use this result to determine all dense near octagons that contain a hex isomorphic to 𝔼1 or 𝔼2. As a by‐product, we obtain a purely geometrical proof for the nonexistence of regular near 2d‐gons, d ≥ 4, whose parameters s, t, ti (0 ≤ i ≤ d) satisfy (s, t2, t3) = (2, 1, 11) or (2, 2, 14). The nonexistence of these regular near polygons can also be shown with the aid of eigenvalue techniques. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 214–228, 2006

Generic boundary conditions for lattice Boltzmann models and their application to advection and anisotropic dispersion equations

We address a “multi-reflection” approach to model Dirichlet and Neumann time-dependent boundary conditions in lattice Boltzmann methods for arbitrarily shaped surfaces. The multi-reflection condition for an incoming population represents a linear combination of the known population solutions. The closure relations are first established for symmetric and anti-symmetric parts of the equilibrium functions, independently of the nature of the problem. The symmetric part is tuned to build second- and third-order accurate Dirichlet boundary conditions for the scalar function specified by the equilibrium distribution. The focus is on two approaches to advection and anisotropic-dispersion equations (AADE): the equilibrium technique when the coefficients of the expanded equilibrium functions match the coefficients of the transformed dispersion tensor, and the eigenvalue technique when the coefficients of the dispersion tensor are built as linear combinations of the eigenvalue functions associated with the link-type collision operator. As a particular local boundary technique, the “anti-bounce-back” condition is analyzed. The anti-symmetric part of the generic closure relation allows to specify normal flux conditions without inversion of the diffusion tensor. Normal and tangential constraints are derived for bounce-back and specular reflections. The bounce-back closure relation is released from the non-physical tangential flux restriction at leading orders. Solutions for the Poisson equation and for convection–diffusion equations are presented for isotropic/anisotropic configurations with specified Dirichlet and Neumann boundary conditions.

Optimal dimensionality reduction of sensor data in multisensor estimation fusion

When there exists the limitation of communication bandwidth between sensors and a fusion center, one needs to optimally precompress sensor outputs-sensor observations or estimates before the sensors' transmission in order to obtain a constrained optimal estimation at the fusion center in terms of the linear minimum error variance criterion, or when an allowed performance loss constraint exists, one needs to design the minimum dimension of sensor data. This paper will answer the above questions by using the matrix decomposition, pseudo-inverse, and eigenvalue techniques.

Design of Vibration Absorbers for Step Motions and Step Disturbances

Mechanical systems with flexible dynamics often suffer from vibration induced by changes in the reference command or from external disturbances. The technique of adding a vibration absorber has proven useful at eliminating vibrations from external disturbances and rotational imbalances. Traditionally, vibration absorbers have been designed for systems subject to sinusoidal or random excitations. Here the applicability of vibration absorbers to systems with steplike changes in the reference command or similar disturbances is studied. This type of motion is more common in robotic applications. Here absorbers are designed using two methods; the first technique uses a weighting on peak overshoot and settling time to allow tradeoffs between the two performance criteria. The second simpler method utilizes an eigenvalue technique to predict the time constant. Both of these techniques provide the possibility of significant improvement in settling time. The performance of this absorber design strategies is compared with previously proposed vibration absorbers and experimental results verify its utility.

Stabilization of nonlinear systems by bifurcation placement

The stabilization problem for multi-input nonlinear systems exhibiting bifurcation phenomena is considered. The objective is to stabilize a part of the equilibrium manifold, while preserving all system equilibria. The problem is addressed by means of bifurcation placement (also known as anti-control of bifurcations). Using eigenvalue assignment techniques a washout-filter-aided dynamic state feedback control law is designed that introduces a Hopf bifurcation at a certain equilibrium and achieves the objective described above. The proposed method is applied to a simple model of a continuous stirred tank reactor.

Analysis of shaft torsional oscillation in a power system using a Lure Lyapunov function

A method based on a Lure Lyapunov function is developed to analyse subsynchronous resonance (SSR) phenomena in power systems. An improved Lure Lyapunov function used to analyse the stability of a power system including the turbine-generator (T-G) shaft is presented first. Then, a standard Lure control model for a power system with a T-G shaft is obtained. The stability of such nonlinear systems at the equilibrium points is discussed in detail. The stable operation region of the system is discussed with the proposed method. The effectiveness of the proposed method is verified by a case study. This study has shown potential benefits and the validity of the Lyapunov direct method in the subsynchronous resonance analysis. Compared with the conventional analysis method, such as eigenvalue and frequency scanning techniques, the proposed method is based on the original nonlinear system. Error, that may be introduced by linearisation, is avoided completely. Moreover, the stable operation region of power systems can be obtained by the proposed method.