Frequency-domain Formulation Research Articles

Numerical algorithms for the FDiTD and FDFD simulation of slowly varying electromagnetic fields

The simulation of slowly varying electromagnetic fields is possible for very large, realistic problems with finite-difference implicit time-domain (FDiTD) and frequency-domain (FDFD) formulations on the basis of the consistent Finite-Integration Technique (FIT). Magneto-quasistatic time-domain formulations combined with implicit time marching schemes require the repeated solution of real-valued symmetric systems. The solution of driven frequency domain problems usually consists in the solution of one non-Hermitean system. Preconditioned conjugate gradient-type methods are well-suited for this task. They allow the efficient solution even for consistent singular or near-singular systems, which typically arise from formulations for slowly varying electromagnetic fields using the Maxwell-Grid-Equations of the FI-Method. Numerical results for TEAM workshop 11 benchmark problem and for a large practical problem, a shading ring sensor, show that the presented algorithms are capable of solving realistic problems for large numbers of unknowns in acceptable calculation times on contemporary medium sized workstations. Copyright © 1999 John Wiley & Sons, Ltd.

Numerical algorithms for the FDiTD and FDFD simulation of slowly varying electromagnetic fields

The simulation of slowly varying electromagnetic fields is possible for very large, realistic problems with finite-difference implicit time-domain (FDiTD) and frequency-domain (FDFD) formulations on the basis of the consistent Finite-Integration Technique (FIT). Magneto-quasistatic time-domain formulations combined with implicit time marching schemes require the repeated solution of real-valued symmetric systems. The solution of driven frequency domain problems usually consists in the solution of one non-Hermitean system. Preconditioned conjugate gradient-type methods are well-suited for this task. They allow the efficient solution even for consistent singular or near-singular systems, which typically arise from formulations for slowly varying electromagnetic fields using the Maxwell-Grid-Equations of the FI-Method. Numerical results for TEAM workshop 11 benchmark problem and for a large practical problem, a shading ring sensor, show that the presented algorithms are capable of solving realistic problems for large numbers of unknowns in acceptable calculation times on contemporary medium sized workstations. Copyright © 1999 John Wiley & Sons, Ltd.

H∞ optimization with plant uncertainty and semidefinite programming

SUMMARY The fundamental H= problem of control is that of finding the stable frequency response function that best fits worst case frequency domain specifications. This is a non-smooth optimization problem that underlies the frequency domain formulation of the H= problem of control; it is the main optimization problem in qualitative feedback theory for example. It is shown in this article how the fundamental H= optimization problem of control can be naturally treated with modern primal—dual interior point (PDIP) methods. The theory introduced here generalizes and unifies approaches to solving large classes of optimization problems involving matrix-valued functions, a subclass of which are commonly treated with linear matrix inequalities techniques. Also, in this article new optimality conditions for H= optimization problems over matrix-valued functions are proved, and numerical experience on natural (PDIP) algorithms for these problems is reported. In experiments we find the algorithms exhibit (local) quadratic convergence rate in many instances. Finally, H= optimization problems with an uncertainty parameter are considered. It is shown how to apply the theory developed here to obtain optimality conditions and derive algorithms. Numerical tests on simple examples are reported. ( 1998 John Wiley & Sons, Ltd.

A hybrid time frequency domain formulation for non-linear soil-structure interaction

Methods that combine frequency and time domain techniques offer an attractive alternative for solving Soil-Structure-interaction problems where the structure exhibits non-linear behaviour. In the hybrid-frequency-time-domain procedure a reference linear system is solved in the frequency domain and the difference between the actual restoring forces and those in the linear model are treated as pseudo-forces. In the solution scheme explored in this paper, designated as the hybrid-time-frequency-domain (HTFD) procedure, the equations of motion are solved in the time domain with due consideration for non-linearities and with the unbounded medium represented by frequency-independent springs and dampers. The frequency dependency of the impedance coefficients is introduced by means of pseudo-forces evaluated in the frequency domain at the end of each iteration. A criterion of stability for the HTFD approach is derived analytically and its validity is sustained numerically. As is often the case, the criterion takes the form of a limit of unity on the spectral radius of an appropriately defined matrix. Inspection of the terms in this matrix shows that convergence can be guaranteed by suitable selection of the reference impedance. The CPU times required to obtain converged solutions with the HTFD are found, in a number of numerical simulations, to be up to one order of magnitude less than those required by the alternative hybrid-frequency-time-domain approach.

New closed-form expressions for the estimation of arterial windkessel compliance

New closed-form mathematical expressions, in the time- and frequency-domain, are derived for estimating the arterial windkessel compliance. The proposed expressions assume the three-element windkessel to model the arterial system and require the measurements of the entire waveforms of arterial pressure and flow. The resistance parameters are estimated using the recently proposed energy-balance method, then compliance is analytically calculated in order to minimize the pressure error in the compliant element. The derived expressions remain valid even when the windkessel compliance is assumed to be pressure-dependent. Also, it is shown that the method, either time- or frequency-domain formulation, provides parameter estimates, which minimize the arterial pressure square error. The method has been applied to simulated data as well as to pressure and flow data measured in the ascending aorta of three anaesthetized dogs under different circulatory conditions.

Wavelets in electromagnetics: the impedance matrix compression (IMC) method

The use of wavelet expansions in numerical solutions of electromagnetic frequency-domain integral equation formulations is steadily growing. In this paper we review the recently suggested impedance matrix compression (IMC) method for a more effective integration of wavelet-based transforms into existing numerical solvers. The difference between the IMC method and the previous approaches to applying wavelets in computational electromagnetics is twofold. Firstly, the transformation is effected by means of a digital filtering approach. This approach renders the transform algorithm adaptive and facilitates the derivation of a basis which best suits the problem at hand. Secondly, the conventional thresholding procedure applied to the impedance matrix is substituted for by a compression process in which only the significant terms in the expansion of the (yet-unknown) current are retained and hence a substantially smaller number of coefficients has to be determined. A few numerical results are included to demonstrate the advantages of the presented method over the currently used ones. The feasibility of ensuring a slow growth in the number of unknowns even when there is a rapid increase in the problem complexity is shown by an illustrative example. © 1998 John Wiley & Sons, Ltd.

Accelerated FD analysis of dielectric resonators

Finite-difference frequency-domain (FDFD) formulation, based on Yee's mesh, is used for determination of resonant frequencies of rotationally symmetric shielded dielectric resonators. Resulting eigenproblems are solved by means of the Arnoldi method with Chebyshev preconditioning.

Wavelets in electromagnetics: the impedance matrix compression (IMC) method

The use of wavelet expansions in numerical solutions of electromagnetic frequency-domain integral equation formulations is steadily growing. In this paper we review the recently suggested impedance matrix compression (IMC) method for a more effective integration of wavelet-based transforms into existing numerical solvers. The difference between the IMC method and the previous approaches to applying wavelets in computational electromagnetics is twofold. Firstly, the transformation is effected by means of a digital filtering approach. This approach renders the transform algorithm adaptive and facilitates the derivation of a basis which best suits the problem at hand. Secondly, the conventional thresholding procedure applied to the impedance matrix is substituted for by a compression process in which only the significant terms in the expansion of the (yet-unknown) current are retained and hence a substantially smaller number of coefficients has to be determined. A few numerical results are included to demonstrate the advantages of the presented method over the currently used ones. The feasibility of ensuring a slow growth in the number of unknowns even when there is a rapid increase in the problem complexity is shown by an illustrative example. © 1998 John Wiley & Sons, Ltd.

Inner loop control of supersonic aircraft in the presence of aeroelastic modes

Longitudinal control system design is considered for a linearized dynamic model of a supersonic transport aircraft concept characterized by relaxed static stability and significant aeroelastic interactions. Two LQG-type controllers are designed using the frequency-domain additive uncertainty formulation to ensure robustness to unmodeled flexible modes. The first controller is based on a fourth-order model containing only the rigid-body modes, while the second controller is based on an eighth-order model that additionally includes the two most prominent flexible modes. The performance obtainable from the fourth-order controller is not adequate, while the eighth-order controller is found to provide better performance. Frequency domain and time-domain (Lyapunov) methods are subsequently used to assess the robustness of the eighth-order controller to parametric uncertainties in the design model.

On earthquake excitation of polar motion : Gu Zhen-nian Shanghai Observatory, Chinese Academy of Sciences, Shanghai 200030

The free-core nutation (FCN) is a rotational normal mode of the Earth's outer core. We derive the equations of motion for FCN w.r.t. both the inertia space F0 and the uniformly rotating frame FΩ, and show that the two sets of equations are invariant in form under the reference frame transformation, as required by physics. The frequency-domain formulation describes the FCN resonance (to nearby tidal signals), which has been exploited to estimate the complex eigenfrequency of FCN, or its eigenperiod P and quality factor Q. On the other hand, our time-domain formulation in terms of temporal convolution describes the response of the free FCN under a (continual) excitation. The convolution well explains the dynamic behaviors of FCN in the observed very-long-baseline interferometry (VLBI) data (in F0), including the undulation of the FCN amplitude and the apparent fluctuations in the period and phase over time, as well as the temporal concurrence of a large phase jump with the near-zero amplitude during ∼1998–2000, in complete analogy to the observed behavior of the Chandler wobble (in FΩ). The reverse, deconvolution process is further exploited to yield optimal estimates for FCN's eigenfrequency using the VLBI data, following the approach of Furuya and Chao (1996) of locating minimum excitation power. While this method is found to be insensitive to Q owing to the short timespan of the data, we obtain the estimate of P=441±4.5 sidereal days (sd) where the 1-sigma uncertainty is assessed via extensive Monte Carlo simulations. This value is closer to the theoretical value of ∼460 sd predicted by Earth models assuming hydrostatic equilibrium than do the prior estimates (425–435 sd) by the resonance method. The deconvolution process also yields the excitation function as a by-product, the physical sources of which await further studies.

An FD-FD formulation for the analysis of the optical axis misalignment effect on propagation characteristics of anisotropic dielectric waveguides

A finite-difference frequency-domain (FD-FD) formulation is developed to study the dispersion characteristics of anisotropic dielectric waveguides with their optical axes not aligned with the coordinate-system axes. In this analysis, the optical axes are initially assumed to be aligned with the coordinate-system axes such that the electric-permittivity and magnetic-permeability tensors are diagonal. The optical axes of the anisotropic dielectric are then rotated an angle /spl theta/ (or /spl phi/) with respect to the coordinate-system axes. While the FD-FD formulation developed is general, it is applied here only to waveguides containing uniaxial anisotropic dielectrics. The results show that accurate optical-axis orientation is important in the design of dielectric waveguides.

Discrete irregular sampling with larger gaps

Discrete irregular sampling is the problem of recovering a band-limited discrete signal from its nonuniformly sampled values. We introduce a new method for discrete irregular sampling. It features a new sampling geometry where the sampling points are chosen to come in pairs. A frequency domain formulation reduces the problem to a system of linear equations for which the matrix is of Toeplitz type and is positive definite as soon as there are enough samples. An explicit estimate of the condition number of that matrix is given under some condition on the maximum sampling gap. This condition allows gaps larger than the Nyquist gap. We recombine the new method and the conjugate method to deduce an efficient iterative reconstruction scheme for band-limited discrete signals. Numerical results are provided to illustrate the efficiency of the method.

RESPONSE OF EARTH DAMS TO P AND SV WAVES USING COUPLED FE-BE FORMULATION

A study of the effects of dam-foundation interaction on the response of earth dams to obliquely incident P and SV waves is presented. Emphasis is placed on the effects of the foundation flexibility and the spatial variability of the ground motion. The study is based on a rigorous hybrid numerical formulation that combines the efficiency and versatility of the Finite Element Method (FEM) and the ability of Boundary Element Method (BEM) to account for the radiation conditions at the far field. The developed hybrid method is very powerful and can be used efficiently to obtain accurate solutions of problems of complex geometry, material heterogeneity and, for time-domain analyses, material nonlinearity. The 2-D frequency-domain formulation is used here to investigate the response of infinitely long earth dams to obliquely incident P and SV waves. By accounting rigorously for the energy radiated back into the half-space, the study demonstrates the dramatic effect of the flexibility of the foundation rock in reducing the overall response of the dam. The effects of the spatial variability of the ground motion for P and SV waves travelling across the width of the dam are also important, but somewhat less pronounced than those of the foundation flexibility.

Geometric Model Predictive Control: A Practical Approach to Uncertain Industrial Processes

This paper presents a set-theoretic approach to dealing explicitly with uncertainty in Model Predictive Control (MPC). Specifically, all control objectives, disturbances, model uncertainties, nonlinear effects, and constraints are described by sets of admissible values. The corresponding sets of control sequences consistent with this problem formulation may then be determined from the mathematical theory of convex sets and geometric inequalities. This formulation provides an alternative to both the time-domain stochastic formulations and robust frequency domain formulations without having to treat the important phenomena of disturbance modeling, model uncertainty, nonlinearity, and constraints in an ad hoc fashion. The explicit geometric nature of the problem formulation and the practical implementation of this technique are illustrated with a two-state process example for which all aspects of the control problem may be viewed explicitly in terms of sets in the plane, and the resulting solutions interpreted in practical terms.

Constrained least squares design of 2-D FIR filters

We consider the design of 2-D linear phase finite impulse response (FIR) filters according to the least squares (LS) error criterion subject to equality and/or inequality constraints. Since we use a frequency domain formulation, these constraints can be used to explicitly prescribe (frequency-dependent) error tolerances, the maximum, minimum, or fixed values of the frequency response at certain points and/or regions. Our method combines Lagrange multiplier and Kuhn-Tucker theory to solve a much wider class of problems than do standard methods. It allows arbitrary compromises between the LS and the equiripple design.

Frequency-domain formulation for nonlinear method of moments

In this paper a frequency domain formulation of the method of moments taking into account the presence of ferromagnetic materials is presented. By combining the electric field definition, Ohm's law and Lorentz gauge inside every volume element into which the system is subdivided, a linear algebraic system of integral equations is obtained. Considering the nonlinear constitutive equation H=H(B) and the magnetization M a nonlinear algebraic system of equations is obtained. The unknowns of these systems are the time Fourier transforms of magnetizations and conduction currents. The solution of these systems for a finite number of harmonics gives the frequency domain solution of the problem. The use of pulse functions as subsection bases allows a quick matrix set up especially when regular volume shapes are selected. Calculated results are compared with results obtained with other methods.

Coupled aeroelastic and aerodynamic response analysis of long-span bridges

Abstract Multi-mode responses of suspended-span bridges to turbulent wind excitation have been estimated in the past by considering the response of each mode and then taking the root-sum-square for the total response. Other procedures, mainly in the time domain, also exist but they have certain simplifying assumptions that limit the complete analysis of the system as well as the estimation of certain important criteria, e.g., flutter speed. A more comprehensive procedure, in the frequency domain, has been developed to capture the full bridge aerodynamics by considering simultaneously the effect of more than one mode. Frequency-domain functions (flutter derivatives) are preferred by most bridge dynamists because they are easier to obtain experimentally. The determination of multi-mode response is especially important for longer-span bridges since their increased flexibility is more likely to elicit a coupled response. In this paper, the fundamental single-mode frequency-domain formulation is extended into the multi-modal domain and a detailed example is presented to highlight the importance of using a coupled analysis procedure for suspended bridges with longer spans.

Augmented Hooke's law in frequency domain. A three dimensional, material damping formulation

An isothermal, fully three-dimensional material damping modelling technique, to some extent alternative to classic viscoelasticity, is proposed. The method is formulated in the frequency domain as an augmented Hooke's law (AHL) with a constitutive matrix in which material damping is introduced by adding frequency dependent, complex valued terms to the classical material modulus matrix of Hooke's generalized law. The derivations are based on linear, irreversible thermodynamics and the concept of hidden coordinates as introduced by Biot [(1955) Phys. Rev.97, 1463–1469]. The one-dimensional concept of multiple augmenting thermodynamic fields by Lesieutre [(1992) Int. J. Solids Structures29, 1567–1579] is generalized to a suitable three-dimensional continuum form through the introduction of a special type of hidden coordinate vectors with linear, first order, time domain relaxation equations. Consistent with the hidden coordinates, a free energy density function, assuming isothermal conditions, is introduced as the time domain basis of the augmented Hooke's law. Through a time-domain model of the coupled evolution of the mechanical displacements and the thermodynamical variables, issues of causality are avoided completely in the final frequency domain formulation. The general time-domain model used is shown to be equivalent to a three-dimensional, multiple anelastic displacement field model. An isotropic augmented Hooke's law with both dilatational and shearing damping has been implemented and tested using a 20-node volume element in the finite element code ASKA Acoustics. A close agreement between finite element calculations and the corresponding analytically exact results for the studied rod and beam cases is obtained.

Transient currents on lightning protection systems due to the indirect lightning effect

A general efficient method for analysing transient currents on lightning-control installations for protecting various structures (buildings) against lightning electromagnetic pulse (LEMP) effects is presented. The method is based upon the frequency-domain mixed-potential electric field integral equation formulation combined with the method of moments and the Fourier transform technique. Attention is focused on predicting the indirect effects of LEMP. The transmission-line representation of lightning return stroke, and the double exponential pulse model for the current at the channel base are employed in LEMP calculation. Several example problems are considered to illustrate the capability of the method. Some theoretical results are compared with experimental data obtained from measurements on the reduced-size models of lightning protection systems.

The buffeting wind loading of structural members at an arbitrary attitude in the flow

Abstract The fluctuating buffeting load (per unit length) is established for a structural member at an arbitrary attitude in the flow, i.e. the angle of incidence and the yaw angle may be substantially different from zero. The formulation is based on a regular “quasi-steady ” linearized theory for a structure in motion, and the relevant loading components (projections), conveniently split into mean, turbulence induced and motion-dependent parts, comprise all the well-known buffeting terms as well as the basic terms regarding galloping and divergence instability. The loading is expressed both in body axes and in flow axes. Auto-spectral density functions for the fluctuating loads from wind turbulence and a proposition for spatially general calculation of co-coherence are included. The solution strategy in a general frequency domain formulation is briefly touched upon. Possible further improvements of the theory are also indicated.