Galerkin Procedure Research Articles

Far Field Splitting for the Helmholtz Equation

Given far field data of a time-harmonic wave radiated by an ensemble of well separated acoustic or electromagnetic sources as well as a priori information on the locations of these sources, we discuss an algorithm to approximate the far field data radiated by each of these sources separately. The method is based on a Galerkin procedure considering subspaces spanned by the singular vectors of “restricted” far field operators that map local source distributions to the corresponding radiated far field patterns. We provide an error analysis for this algorithm and consider its stability. Furthermore, we exemplify a means to extract the required a priori knowledge directly from the far field data, and we show how to utilize the split far fields to recover information on the supports of the individual sources beyond that a priori information. Numerical results for the most important example of inverse obstacle scattering illustrate our theoretical findings.

Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the non-linear internal feedback

We consider the viscoelastic wave equation in a bounded domain with a delay term in the non–linear internal feedback utt(x,t)−Δxu(x,t)+∫t0h(t−s)Δxu(x,s)ds+μ1g1(ut(x,t))+μ2g2(ut(x,t−τ))=0 and prove a global existence result using the energy method combined with the Faedo–Galerkin procedure under assumption of a relation between the weight of the delay term in the feedback and the weight of the term without delay. Furthermore, we study the asymptotic behaviour of solutions using a perturbed energy method.

The response of a micro-electro-mechanical system (MEMS) cantilever-paddle gas sensor to mechanical shock loads

This work presents an investigation into the effect of nonlinearities on the response of a micro-electro-mechanical system gas sensor under mechanical shock and electrostatic loading. The gas sensor consists of a cantilever microbeam with a rigid microplate (micro-paddle) attached to its tip. A nonlinear Euler-Bernoulli beam theory is used to model the system, accounting for both geometric and inertia nonlinearities, in addition to the nonlinear electrostatic force used to actuate the system. The system of integro-partial-differential equations is discretized using a Galerkin procedure to extract a reduced-order model, which is then used for dynamic simulations of the system responses. The influences of the different components of nonlinearity such as geometric and inertia nonlinearities are examined. The results of the nonlinear model are compared to results obtained from linear beam theory and finite element simulations. For mechanical shock loading, both quasi-static and dynamic responses of a microbeam are considered. The effect of nonlinearity is found to be significant when the deflection of the microbeam exceeds around 30% of its length. The consequence of the large deflection is that the geometric nonlinearity has a much stronger influence on the response in comparison to the inertia nonlinearity. It is also apparent that the effect of the paddle is to enhance the dynamic, as opposed to the quasi-static, response of the microbeam to mechanical shock. For electrostatic actuation, it is found that using a nonlinear beam model to predict the pull-in and the deflection produces a slight improvement over using a linear beam model.

Large Amplitude Free and Forced Oscillations of Functionally Graded Beams

The free and forced vibration of a functionally graded beam is studied in this article within the framework of Euler-Bernoulli beam theory and von Kármán geometric nonlinearity. It is assumed that material properties follow either exponential or power law distributions through thickness direction. It is assumed that the beam may be hinged-hinged or clamped-hinged at its ends. The Galerkin procedure is used to obtain a second-order nonlinear ordinary equation with cubic and quadratic nonlinear terms. The natural frequencies are obtained for a nonlinear problem by using the Multiple Time Scales method. Also, forced vibration of a system in primary resonance has been studied and the effects of different parameters and end supports on the frequency-response have been investigated.

On the accuracy and efficiency of discontinuous Galerkin, spectral difference and correction procedure via reconstruction methods

Numerical accuracy and efficiency of several discontinuous high-order methods, including the quadrature-based discontinuous Galerkin (QDG), nodal discontinuous Galerkin (NDG), spectral difference (SD) and flux reconstruction/correction procedure via reconstruction (FR/CPR), for the conservation laws are analyzed and compared on both linear and curved quadrilateral elements. On linear elements, all the above schemes are one-dimensional in each natural coordinate direction. However, on curved elements, not all schemes can be reduced to a one-dimensional form, although the SD and CPR formulations remain one-dimensional by design. The efficiency and accuracy of various formulations are compared on highly skewed curved elements. Several benchmark problems are simulated to further evaluate the performance of these schemes.

Postbuckling of functionally graded cylindrical shells with tangential edge restraints and temperature-dependent properties

This paper presents an analytical approach to investigate the buckling and postbuckling behavior of functionally graded cylindrical shells subjected to thermal and axial compressive loads. Material properties are assumed to be temperature dependent and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The governing equations are established within the framework of classical thin shallow shell theory taking both geometrical nonlinearity in von Karman–Donnell sense and initial imperfection into consideration. Thermal stability analysis also incorporates the effects of tangential edge constraints. A Galerkin procedure is applied to derive expressions of load-deflection relations from which the thermal buckling loads and postbuckling curves of the shells are obtained by an iteration. Effects played by material and geometrical properties, tangential stiffness, imperfection and buckling modes are discussed.

A Waveguide Shunt Slot-Fed Microstrip Patch Antenna—Analysis Using the Method-of-Moments

This paper presents a new waveguide shunt-slot feed for a microstrip patch antenna that is analyzed using a formulation based on the method-of-moments (MoM). Starting from a generalized formulation, expressions are obtained for a rectangular patch radiator excited by a rectangular slot using entire-domain basis functions with a Galerkin procedure to obtain the moment matrix. Expressions for the dielectric region are derived using Green's function in Fourier transform domain. The developed analysis is used on a prototype C-band radiator presenting details. Subsequently, parametric studies are performed on the key design variables of the configuration, i.e., slot dimensions, axial offset of slot and patch position w.r.t. slot center. A slot length of 0.08λ is found a good initial design estimate to obtain resonance. The variation of coupled power with axial slot offset is reported while showing little effect on resonant frequency. This can be a useful design aid for linear arrays to control amplitude excitations. Patch offset from slot center is found to be noncritical in regard to alignment tolerances. However, changes in coupling factor and resonant frequency are reported. The proposed technique is validated through a commercial e.m. simulator, prototype hardware measurements and comparison to an earlier reported antenna. The paper thus establishes the proposed feeding technique using a developed MoM based analysis and presents useful parametric studies to serve as design aids.

On the dynamic buckling of truncated conical shells with functionally graded coatings subject to a time dependent axial load in the large deformation

A study has been made to determine the critical time parameters of truncated conical shells with functionally graded coatings (FGCs) and subjected to a time dependent axial load in the large deformation. The method of solution utilizes Superposition principle and Galerkin procedure. Donnell–Karman type non-linear differential equations for the truncated conical shell with FGCs are derived and reduced to ordinary differential equation with the time dependent coefficient. The Runge–Kutta method and modified Budiansky–Roth criterion are then used to solve this non-linear differential equation with the time dependent coefficient. Finally, effects of compositional profiles of coatings, variation of truncated conical shell parameters and loading speed on the dimensionless linear and non-linear critical time parameters have been studied. Comparing the results of this study with those in the literature validates the present analysis.

Vortex dynamics effects on microwave propagation in high temperature superconducting coplanar waveguides

An analysis of the microwave signal propagation through a High Temperature Superconducting (HTS) transmission line should take into account the vortex dynamics effects of the HTS material used for the study. Taking Coplanar Waveguide (CPW) as a model transmission line, we simulated the transmission characteristics using a field computation method based on Galerkin's procedure. The vortex effects were incorporated into the study using the surface impedance derived from the modified two-fluid model proposed by Coffey and Clem which takes into account the field and thermal effects in a self consistent manner. The increase of temperature and magnetic field significantly affected the attenuation due to the enhanced vortex motion. The observed dip in the value of attenuation of the transmission line at low field values in high temperature range is explained using the vortex effects. It is found that the dispersion is considerably low for the proposed geometrical structures. Simulations were performed for varying strip-to-slot ratios. The impact of vortex motion on the signal propagation is made out for a wide range of temperature, magnetic field, and line geometry.

Study nonlinear vibration of cross‐ply laminated plates using scale models

This article describes the establishment of necessary similarity conditions for geometrically nonlinear vibrations of thin cross‐ply laminated plates. The Von‐karman's strain‐displacement relations have been employed to model structural nonlinearity of the system. The Galerkin procedure is used to reduce the nonlinear partial differential equations to a nonlinearly second‐order ordinary differential equation. An analytical investigation based on the direct use of the governing equations of the systems is undertaken to derive the necessary scaling laws and similarity conditions. In this study, a set of scaling laws are introduced that can predict the nonlinear free vibration frequency of the prototypes by projecting the frequency of the model, accurately. The effects of distorted models with relaxations in the number of plies, stacking sequence, and different vibration amplitudes are studied. The results presented herein indicate that models with different relaxations can predict the nonlinear vibration frequency of prototypes with good accuracy. POLYM. COMPOS., 35:752–758, 2014. © 2013 Society of Plastics Engineers

Dynamics of an axially functionally graded beam under axial load

This paper considers the dynamics of a simply supported beam under axial time–dependent load. The beam is made of an axially functionally graded material. The motion equations are deduced from the equilibrium in deformed configuration and no restriction is made on the amplitude of the transversal displacement, but that naturally imposed by the inextensibility assumption that is adopted in the present study. The transversal motion equation, that is a partial differential equation, is approximated by its Taylor expansion until third order and then discretized through the Galerkin procedure.

Stationary response of nonlinear magneto-piezoelectric energy harvester systems under stochastic excitation

Recent years have shown increasing interest of researchers in energy harvesting systems designed to generate electrical energy from ambient energy sources, such as mechanical excitations. In a lot of cases excitation patterns of such systems exhibit random rather than deterministic behaviour with broad-band frequency spectra. In this paper, we study the efficiency of vibration energy harvesting systems with stochastic ambient excitations by solving corresponding Fokker-Planck equations. In the system under consideration, mechanical energy is transformed by a piezoelectric transducer in the presence of mechanical potential functions which are governed by magnetic fields applied to the device. Depending on the magnet positions and orientations the vibrating piezo beam system is subject to characteristic potential functions, including single and double well shapes. Considering random excitation, the probability density function (pdf) of the state variables can be calculated by solving the corresponding Fokker-Planck equation. For this purpose, the pdf is expanded into orthogonal polynomials specially adapted to the problem and the residual is minimized by a Galerkin procedure. The power output has been estimated as a function of basic potential function parameters determining the characteristic pdf shape.

Coifman Wavelets in 3-D Scattering From a Calibration Target Consisting of Doubly Periodic Sharp Metal Cones

The magnetic field integral equation (MFIE) is solved by Coiflet-based Galerkin's procedure in conjunction with the Floquet theorem for plane wave scattering from a doubly periodic array of sharp conducting circular cones. To avoid using intervallic wavelets to handle the boundary truncations, we split the unknown surface current into four components enforcing periodic boundary conditions, so that the standard Coiflets can be employed. Owing to the orthogonality, compact support, continuity and smoothness of the Coiflets, well-conditioned impedance matrices are obtained at the resolution level 5. Majority of the matrix entries are obtained in the spectral domain by one-point quadrature with high precision in O(h5), imposing the Dirac- δ property of the Coiflets. For the self elements and elements associated with the same elevation of the field and source points, z=z', spatial domain computation is applied, bypassing the slow convergence of the spectral summation of the non-damping propagating modes. The simulation results of induced current and far-zone scattering pattern are compared with the solutions from an RWG-MLFMA based commercial software, FEKO, and excellent agreement is observed.

An analytical approach for nonlinear vibration and thermal stability of shape memory alloy hybrid laminated composite beams

Abstract In this article, large amplitude vibration and thermal post-buckling of shape memory alloy (SMA) fiber reinforced hybrid composite beams with symmetric and asymmetric lay-up are analytically investigated. To predict the behavior of the smart laminated beam, the Euler–Bernoulli beam theory and the nonlinear von-Karman strain field are employed. Also, one-dimensional Brinson SMA model is utilized to calculate the recovery stress of SMA fibers in the case of restrained strain. Nonlinear governing equations of motion are derived via the Hamilton principle. Using an analytical approach based on the Galerkin procedure together with the simple harmonic motion assumption, a closed-form solution is obtained for the thermal post-buckling and nonlinear free vibration analysis of SMA fiber reinforced hybrid composite beams. Due to lack of any results on the free vibration and thermal stability of SMA fiber reinforced composite beams, the results obtained from the present solution for laminated composite beams without SMA fiber are compared with counterpart data in the open literature, which validate the present solution. Then, a set of parametric study is carried out to show the influence of SMA volume fraction, amount of prestrain in the SMA fiber, orientation of composite fiber, SMA-reinforced layer thickness to total thickness ratio, location of SMA layer, vibration amplitude, boundary conditions and temperature on the vibration characteristic of the laminated beam reinforced with SMA in the pre- and post-buckled domains.

An efficient Karhunen–Loève Galerkin method for sequential solution of inverse problems

A practical method is developed to solve inverse natural convection problems sequentially. The sequential algorithm is based on the Kalman filtering technique. By reducing the Boussinesq equation to a reduced order model through the Karhunen–Loève Galerkin procedure, the computational difficulties associated with the covariance equation are overcome. Although this method is found to yield accurate results with efficiency, its accuracy depends on that of the reduced order model employed. In the present investigation, it is demonstrated that this method can be further developed to be reasonably accurate and computationally feasible even with a relatively inaccurate reduced order model by employing the Boussinesq equation as the state equation while adopting the Kalman filter constructed from less accurate reduced order model.

Nonlinear static and dynamic buckling analysis of imperfect eccentrically stiffened functionally graded circular cylindrical thin shells under axial compression

An analytical approach is presented to investigate the nonlinear static and dynamic buckling of imperfect eccentrically stiffened functionally graded thin circular cylindrical shells subjected to axial compression. Based on the classical thin shell theory with the geometrical nonlinearity in von Karman–Donnell sense, initial geometrical imperfection and the smeared stiffeners technique, the governing equations of motion of eccentrically stiffened functionally graded circular cylindrical shells are derived. The functionally graded cylindrical shells with simply supported edges are reinforced by ring and stringer stiffeners system on internal and (or) external surface. The resulting equations are solved by the Galerkin procedure to obtain the explicit expression of static critical buckling load, post-buckling load–deflection curve and nonlinear dynamic motion equation. The nonlinear dynamic responses are found by using fourth-order Runge–Kutta method. The dynamic critical buckling loads of shells under step loading of infinite duration are found corresponding to the load value of sudden jump in the average deflection and those of shells under linear-time compression are investigated according to Budiansky–Roth criterion. The obtained results show the effects of stiffeners and input factors on the static and dynamic buckling behavior of these structures.

Time discretization effect on the nonlinear vibration of embedded SWBNNT conveying viscous fluid

In this study, the effect of time discretization on the nonlinear transverse vibration and instability of single-walled boron nitride nanotube (SWBNNT) conveying viscous fluid is investigated based on the nonlocal piezoelasticity theory. SWBNNT is considered as an Euler–Bernoulli beam and is subjected to combined mechanical loading, thermal changes and electrical field. The elastic medium is simulated as Winkler and Pasternak foundation. The interaction between the inner viscous fluid and SWBNNT is obtained using Navier–Stokes equation. The axial inertia is neglected and a new approach is introduced to decouple the mechanical and electrical fields considering charge equation. Motion equations are derived by Hamilton’s principle using the Von-Kármán nonlinearity theory. In the first approach, time and space domains are discretized using the method of multiple scale (MMS) and Galerkin procedure respectively, and in the second one differential quadrature method (DQM) is utilized to space discretization. Good agreement is shown between the results of first and second approach. Numerical results indicate the significant effects of aspect ratio, elastic medium and nonlocality on the frequency and instability of the SWBNNT.

A spectral method for calculation of rectangular shielded waveguides with an arbitrary anisotropic filling

A method for calculation of shielded guiding structures with an arbitrary anisotropic filling is proposed. The method is based on a modified Galerkin procedure. Field components are expanded into the eigenfunction spectrum of the comparison waveguide. The components are related via expansion coefficients, which are determined in the course of solution of a dispersion problem.

Minimal residual methods for large scale Lyapunov equations

Projection methods have emerged as competitive techniques for solving large scale matrix Lyapunov equations. We explore the numerical solution of this class of linear matrix equations when a Minimal Residual (MR) condition is used during the projection step. We derive both a new direct method, and a preconditioned operator-oriented iterative solver based on CGLS, for solving the projected reduced least squares problem. Numerical experiments with benchmark problems show the effectiveness of an MR approach over a Galerkin procedure using the same approximation space.

A New Dynamical Model of Flexible Cracked Wind Turbines for Health Monitoring

We develop a mathematical model of a large-scale cracked horizontal axis wind turbine (HAWT) describing the flapping flexure of the flexible tower and blades. The proposed model has enough fidelity to be used in health monitoring applications. The equations of motion account for the effect of the applied aerodynamic forces, modeled using the blade element momentum (BEM) theory, and the location and shape of a crack introduced into one of the blades. We first examine the static response of the HAWT in presence of the crack, and then we formulate the eigenvalue problem and determine the natural frequencies and associated mode shapes. We show that both shape and location of the crack influence the first four natural frequencies. The dynamic response of the HAWT subjected to wind and gravity is obtained using a Galerkin procedure. We conduct a parametric analysis to investigate the influence of the crack on the eigenstructure and overall dynamics. The simulations depict that the first four natural frequencies are reduced as the crack size become more important. We also conclude that the tower root moment may be considered as potential indicators for health monitoring purposes.