Volterra Theory Research Articles

Changing dynamic of bridge aerodynamics

The assurance of safety and reliability of long-span bridges under winds requires the accurate modelling of wind-induced effects. In this study, the existing conventional linear and non-linear models of bridge aerodynamics are first systematically reviewed with a focus on the study of the relationships among them. After highlighting the shortcomings of conventional models, three advanced non-linear models, based on artificial neural networks, non-linear moving average (within the framework of Volterra theory) and Volterra series, are introduced. Example applications of linear and non-linear analysis schemes are presented, underscoring the changing dynamic of bridge aerodynamics under smooth and turbulent wind conditions.

A nonlinear analysis framework for bluff-body aerodynamics: A Volterra representation of the solution of Navier-Stokes equations

A nonlinear analysis framework for bluff-body aerodynamics based on Volterra theory is introduced to capture the linear and nonlinear aerodynamic effects. The Volterra kernels based on the impulse function concept are identified by way of the simulation of Navier-Stokes equations using computational fluid dynamics (CFD). The computational schemes used here are validated through theoretical consideration, i.e., Blasius solution for the steady-state and Theodorsen solution for the system dynamic-state simulation. The source of nonlinearities in the aerodynamics of bluff bodies is systematically investigated. The simulation of bluff-body aerodynamics based on the Volterra reduced-order modeling scheme is obtained by the convolution of the identified kernels with the external inputs, e.g., turbulent inflow or body motion for aerodynamic or aeroelastic response, respectively. It is demonstrated that the Volterra theory-based nonlinear analysis framework for bluff-body aerodynamics combined with the identification of kernels using CFD promises to capture the salient features of bluff-body aerodynamics and offers an accurate reduced-order approximation of the Navier-Stokes equations with reduced level of computational effort.

Modeling of Unsteady Aerodynamics for a Flapping wing

Insect-like flapping flight offers a power-efficient and highly manoeuvrable basis for a micro air vehicle. In this paper, investigation was made into the aerodynamic analysis of a flapping wing applicable to micro aerial vehicles. First, the three dimensional aerodynamic calculation and flow field simulation of a flapping wing were studied. Then, the generation of reduced order models for computing the unsteady and nonlinear aerodynamic loads on the flapping wing from flapping motions is described. The models considered are based on Volterra theory using nonlinear kernels. One of the biggest challenges in creating these reduced order modeling techniques is accurate identification of the Volterra kernels. The validity of models studied was assessed by comparison of the model outputs with unsteady CFD result. It was proved that the percent error between CFD and reduced order model is 8.55.

Fractional rheological models for thermomechanical systems. Dissipation and free energies

Abstract Within the fractional derivative framework, we study thermomechanical models with memory and compare them with the classical Volterra theory. The fractional models involve significant differences in the type of kernels and predicts important changes in the behavior of fluids and solids. Moreover, an analysis of the thermodynamic restrictions provides compatibility conditions on the kernels and allows us to determine certain free energies, which in turn enables the definition of a topology on the history space. Finally, an analogous analysis is carried out for the phenomenon of heat propagation with memory.

Transonic Aerodynamic Load Modeling of X-31 Aircraft Pitching Motions

The generation of reduced-order models for computing the unsteady and nonlinear aerodynamic loads on an aircraft from pitching motions in the transonic speed range is described. The models considered are based on Duhamel’s superposition integral using indicial (step) response functions, Volterra theory using nonlinear kernels, radial basis functions, and a surrogate-based recurrence framework, both using time-history simulations of a training maneuver(s). Results are reported for the X-31 configuration with a sharp leading-edge cranked delta wing geometry, including canard/wing vortex interactions. The validity of the various models studied was assessed by comparison of the model outputs with time-accurate computational-fluid-dynamics simulations of new maneuvers. Overall, the reduced-order models were found to produce accurate results, although a nonlinear model based on indicial functions yielded the best accuracy among all models. This model, along with a time-dependent surrogate approach, helped to produce accurate predictions for a wide range of motions in the transonic speed range.

A multi-harmonic model taking into account coupling effects of long- and short-term memory in SSPAs

This paper presents a new macro modeling methodology for solid-state power amplifiers (SSPAs) and packaged transistors used in communication systems. The model topology is based on the principle of harmonic superposition recently introduced by Agilent Technologies' X-parametersTMcombined with dynamic Volterra theory. The resulting multi-harmonic bilateral model takes into account the coupling effects of both short- and long-term memory in SSPAs. In this work, the behavioral model was developed from time-domain load pull and used to simulate the amplifier's response to a 16-QAM signal with specific regards to ACPR and IM3.

Analytical Volterra-based models for nonlinear low order flight dynamics approximation systems

AbstractAnalytical methodology is presented to conduct dynamical assembly of simple low order nonlinear responses for system synthesis and prediction using Volterra theory. The procedure is set forth generically and then applied to several atmospheric flight examples. A two-term truncated Volterra series, which is enough to capture the quadratic and bilinear nonlinearities, is developed for first and second order generalised nonlinear single degree of freedom systems. The resultant models are given in the form of first and second kernels. A parametric study of the influence of each linear and nonlinear term on kernel structures is investigated. A step input is then employed to quantify and qualify the nonlinear response characteristics. Uniaxial surge and pitch motions are presented as examples of the low order flight dynamic systems. These examples show the ability of the proposed analytical Volterra-based models to predict, understand, and analyse the nonlinear aircraft behaviour beyond that attainable by linear-based models. The proposed analytical Volterra-based model offers an efficient nonlinear preliminary design tool in qualifying the aircraft responses before computer simulation is available or invoked.

Multiharmonic Volterra Model Dedicated to the Design of Wideband and Highly Efficient GaN Power Amplifiers

This paper presents a complete validation of the new behavioral model called the multiharmonic Volterra (MHV) model for designing wideband and highly efficient power amplifiers with packaged transistors in computer-aided design (CAD) software. The proposed model topology is based on the principle of the harmonic superposition introduced by the Agilent X-parameters, which is combined with the dynamic Volterra theory to give an MHV model that can handle short-term memory effects. The MHV models of 10- and 100-W packaged GaN transistors have been extracted from time-domain load-pull measurements under continuous wave and pulsed modes, respectively. Both MHV models have been implemented into CAD software to design 10- and 85-W power amplifiers in L- and S-bands. Finally, the first power amplifier exhibited mean measured values of 10-W output power and 65% power-added efficiency over 36% bandwidth centered at 2.2 GHz, while the second one exhibited 85-W output power and 65% drain efficiency over 50% bandwidth centered at 1.6 GHz.

Equations for the Quantum Mind Evolution in the Struggle for Life

The purpose of this paper is to indicate and demonstrate how to include the involvement and participation of the quantum mind into the mathematical theory of the living species evolution. For this purpose a generalized system of equations in the spirit of Vito Volterra theory of the struggle for existence was obtained. NeuroQuantology | June 2012 | Volume 10 | Issue 2 | Page 305-310

Global stability analysis of epidemiological models based on Volterra–Lyapunov stable matrices

In this paper, we study the global dynamics of a class of mathematical epidemiological models formulated by systems of differential equations. These models involve both human population and environmental component(s) and constitute high-dimensional nonlinear autonomous systems, for which the global asymptotic stability of the endemic equilibria has been a major challenge in analyzing the dynamics. By incorporating the theory of Volterra–Lyapunov stable matrices into the classical method of Lyapunov functions, we present an approach for global stability analysis and obtain new results on some three- and four-dimensional model systems. In addition, we conduct numerical simulation to verify the analytical results.

Stochastic analysis of tsunami runup due to heterogeneous coseismic slip and dispersion

Most tsunami models apply dislocation models that assume uniform slip over the entire fault plane, followed by standard analytical models based on Volterra's theory of elastic dislocations for the seabed deformation. In contrast, we quantify tsunami runup variability for an earthquake with fixed magnitude but with heterogeneous rupture distribution assuming plane wave propagation (i.e., an infinitely long rupture). A simple stochastic analysis of 500 slip realizations illustrates the expected variability in coseismic slip along a fault plane and the subsequent runup that occurs along a coastline in the near field. Because of the need for systematically analyzing different fault geometries, grid resolutions, and hydrodynamic models, several hundred thousand model runs are required. Thus, simple but efficient linear models for the tsunami generation, propagation, and runup estimation are used. The mean value and variability of the maximum runup is identified for a given coastal slope configuration and is analyzed for different dip angles. On the basis of the ensemble runs, nonhydrostatic effects are discussed with respect to their impact on generation, nearshore propagation, and runup. We conclude that for the geometry and magnitude investigated, nonhydrostatic effects reduce the variability of the runup; that is, hydrostatic models will produce an artificially high variability.

Existence of solutions for a class of nonlinear Volterra singular integral equations

In this paper we prove some results concerning the existence of solutions for a large class of nonlinear Volterra singular integral equations in the space C [ 0 , 1 ] consisting of real functions defined and continuous on the interval [ 0 , 1 ] . The main tool used in the proof is the concept of a measure of noncompactness. We also present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results. Moreover, we compare our theory with the approach depending on the use of the theory of Volterra–Stieltjes integral equations. We also show that the results of the paper are applicable in the study of the so-called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems.

Experimental demonstration of intra-channel nonlinearity mitigation in coherent QPSK systems with nonlinear electrical equaliser

The mitigation of intra-channel nonlinearities in 10 Gbaud coherent NRZ-QPSK systems via a nonlinear electrical equaliser based on Volterra theory is experimentally demonstrated. Without the requirement of prior knowledge of the fibre link, the equaliser significantly improves the system performance compared to digital dispersion compensation only.

Application of Multi-Input Volterra Theory to Nonlinear Multi-Degree-of-Freedom Aerodynamic Systems

This paper presents a reduced-order-modeling approach for nonlinear, multi-degree-of-freedom aerodynamic systems using multi-input Volterra theory. The method is applied to a two-dimensional, 2 degree-of-freedom transonic airfoil undergoing simultaneous forced pitch and heave harmonic oscillations. The so-called Volterra cross kernels are identified and shown to successfully model the aerodynamic nonlinearities associated with the simultaneous pitch and heave motions. The improvements in accuracy over previous approaches that effectively ignored the cross kernels by using superposition are demonstrated.

Aeroservoelastic Analysis for Transonic Missile Based on Computational Fluid Dynamics

Aeroservoelastic Analysis for Transonic Missile Based on Computational Fluid Dynamics

Piecewise Global Volterra Nonlinear Modeling and Characterization for Aircraft Dynamics

Evaluating global aircraft dynamic behavior by Volterra theory as an efficient nonlinear approximation method is investigated. In previous research, Volterra theory was implemented to evaluate the system behavior in a selected local region, which restricts the ability to capture the aircraft behavior while moving from one flight region to another. The current research presents a global piecewise Volterra kernel approach, which uses submodel Volterra kernels to build global kernels. A piecewise interpolation is employed to switch between the submodels. An approximate nonlinear pitch-plunge model for a high-performance aircraft was used to assess the proposed approach. A set of experimental simulations is conducted to verify the capability of the proposed model. The approach is compared with a global linear approach while employing the nonlinear simulation as the benchmark.

Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer

A novel nonlinear electrical equalizer (NLEE) based on Volterra theory is proposed for mitigating intra-channel nonlinearities in pseudo-linear coherent optical communication systems. According to the temporal matching condition of intra-channel nonlinear pulse interactions, we just select specific nonlinear terms and reduce the computation complexity considerably. Numerical simulations of 10GBaud coherent quadrature phase-shift keying (QPSK) transmission show that the simplified NLEE can efficiently mitigate intra-channel nonlinearities.

Fault Detection for Shipboard Monitoring – Volterra Kernel and Hammerstein Model Approaches

In this paper nonlinear fault detection for in-service monitoring and decision support systems for ships will be presented. The ship is described as a nonlinear system, and the stochastic wave elevation and the associated ship responses are conveniently modelled in frequency domain. The transformation from time domain to frequency domain has been conducted by use of Volterra theory. The paper takes as an example fault detection of a containership on which a decision support system has been installed.

Nonlinear Electrical Equalization for Different Modulation Formats With Optical Filtering

Based on bit-error-ratio simulations, we investigate electrical-dispersion-compensation performance by using a nonlinear electrical equalizer based on nonlinear Volterra theory for different modulation formats. This nonlinear equalizer is compared to conventional decision-feedback equalizer as well as to maximum-likelihood sequence estimation. Especially, we show that nonlinear equalizers, in conjunction with narrowband optical filtering of the light signal, result in improved performance. First of all, the system can benefit from the noise reduction due to narrowband filtering. Second, interacting with chromatic dispersion, nonlinear equalizers benefit from the improved dispersion tolerance through spectrum reshaping by narrowband optical filtering. Finally, nonlinear equalizers can efficiently mitigate the distortion resulting from strong optical filtering

Measuring Volterra Kernels of Analog-to-Digital Converters Using a Stepped Three-Tone Scan

The Volterra theory can be used to mathematically model nonlinear dynamic components such as analog-to-digital converters (ADCs). This paper describes how frequency-domain Volterra kernels of an ADC are determined from measurements. The elements of the Volterra theory are given, and practical issues are considered, such as methods for signal conditioning and finding the appropriate test signals scenario and suitable sampling frequency. The results show that, for the used pipeline ADC, the frequency dependence is significantly stronger for second-order difference products than for sum products and the linear frequency dependence was not as pronounced as that of the second-order Volterra kernel. It is suggested that the Volterra kernels have the symmetry properties of a specific box model, namely, the parallel Hammerstein system.