On numerical methods for highly oscillatory problems in circuit simulation

We propose in this paper a novel technique for an efficient numerical approximation of systems of highly oscillatory ordinary differential equations. In particular, we consider electronic systems subject to modulated signals. A Filon-type method is proposed for use and compared with traditional trapezoidal rule and Runge–Kutta methods. The Filontype method is combined with the waveform relaxation technique for nonlinear systems. Preliminary numerical examples highlight the efficacy of this approach

Application Of Nonlinear Models For A Well-defined Description Of The Dynamics Of Rotors In Magnetic Bearings

A research report has been submitted. It deals with implementing a method for a mathematical description of the nonlinear dynamics of rotors in magnetic bearings of different types (passive and active). The method is based on Lagrange-Maxwell differential equations in a form similar to that of Routh equations in mechanics. The mathematical models account for such nonlinearities as the nonlinear dependencies of magnetic forces on gaps in passive and active magnetic bearings and on currents in the windings of electromagnets; nonlinearities related to the inductances in coils; the geometric link between the electromagnets in one AMB and the link between all AMBs in one rotor, which results in relatedness of processes in orthogonal directions, and other factors. The suggested approach made it possible to detect and investigate different phenomena in nonlinear rotor dynamics. The method adequacy has been confirmed experimentally on a laboratory setup, which is a prototype of a complete combined magnetic-electromagnetic suspension in small-size rotor machinery. Different variants of linearizing the equations of motion have been considered. They provide for both linearization of restoring magnetic or electromagnetic forces in passive and active magnetic bearings, and exclusion of nonlinear motion equation terms. Calculation results for several linearization variants have been obtained. An appraisal of results identified the drawbacks of linearized mathematical models and allowed drawing a conclusion on the necessity of applying nonlinear models for a well-defined description of the dynamics of rotor systems with magnetic bearings

Computational strategies for the solution of large nonlinear problems via quasi-newton methods

The usefulness of quasi-Newton methods for the solution of nonlinear systems of equations is demonstrated. After a review of the Newton iterative method, several quasi-Newton updates are presented and tested. Special attention is devoted to the solution of large sparse systems of equations such as those issued from spatial discretization of continua by finite elements. The numerical examples presented comprise static and dynamic analyses of geometrical, material and combined nonlinear structural problems and a model fluid flow problem with different levels of nonlinearity. All the results are assorted with a complete discussion of the different methods used, of the convergence rates and of the associated computer costs. From the present studies, it can be concluded that computational costs for the solution of large nonlinear systems of equations can be reduced drastically by using convenient quasi-Newton updates or by adequate combined Newton/quasi-Newton strategies

Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations

AbstractBy establishing a comparison result and using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence of solutions for systems of nonlinear fractional differential equations

Response statistics and failure probability determination of nonlinear stochastic structural dynamical systems

Novel approximation techniques are proposed for the analysis and evaluation of nonlinear dynamical systems in the field of stochastic dynamics. Efficient determination of response statistics and reliability estimates for nonlinear systems remains challenging, especially those with singular matrices or endowed with fractional derivative elements. This thesis addresses the challenges of three main topics. The first topic relates to the determination of response statistics of multi-degree-of-freedom nonlinear systems with singular matrices subject to combined deterministic and stochastic excitations. Notably, singular matrices can appear in the governing equations of motion of engineering systems for various reasons, such as due to a redundant coordinates modeling or due to modeling with additional constraint equations. Moreover, it is common for nonlinear systems to experience both stochastic and deterministic excitations simultaneously. In this context, first, a novel solution framework is developed for determining the response of such systems subject to combined deterministic and stochastic excitation of the stationary kind. This is achieved by using the harmonic balance method and the generalized statistical linearization method. An over-determined system of equations is generated and solved by resorting to generalized matrix inverse theory. Subsequently, the developed framework is appropriately extended to systems subject to a mixture of deterministic and stochastic excitations of the non-stationary kind. The generalized statistical linearization method is used to handle the nonlinear subsystem subject to non-stationary stochastic excitation, which, in conjunction with a state space formulation, forms a matrix differential equation governing the stochastic response. Then, the developed equations are solved by numerical methods. The accuracy for the proposed techniques has been demonstrated by considering nonlinear structural systems with redundant coordinates modeling, as well as a piezoelectric vibration energy harvesting device have been employed in the relevant application part. The second topic relates to code-compliant stochastic dynamic analysis of nonlinear structural systems with fractional derivative elements. First, a novel approximation method is proposed to efficiently determine the peak response of nonlinear structural systems with fractional derivative elements subject to excitation compatible with a given seismic design spectrum. The proposed methods involve deriving an excitation evolutionary power spectrum that matches the design spectrum in a stochastic sense. The peak response is approximated by utilizing equivalent linear elements, in conjunction with code-compliant design spectra, hopefully rendering it favorable to engineers of practice. Nonlinear structural systems endowed with fractional derivative terms in the governing equations of motion have been considered. A particular attribute pertains to utilizing localized time-dependent equivalent linear elements, which is superior to classical approaches utilizing standard time-invariant statistical linearization method. Then, the approximation method is extended to perform stochastic incremental dynamical analysis for nonlinear structural systems with fractional derivative elements exposed to stochastic excitations aligned with contemporary aseismic codes. The proposed method is achieved by resorting to the combination of stochastic averaging and statistical linearization methods, resulting in an efficient and comprehensive way to obtain the response displacement probability density function. A stochastic incremental dynamical analysis surface is generated instead of the traditional curves, leading to a reliable higher order statistics of the system response. Lastly, the problem of the first excursion probability of nonlinear dynamic systems subject to imprecisely defined stochastic Gaussian loads is considered. This involves solving a nested double-loop problem, generally intractable without resorting to surrogate modeling schemes. To overcome these challenges, this thesis first proposes a generalized operator norm framework based on statistical linearization method. Its efficiency is achieved by breaking the double loop and determining the values of the epistemic uncertain parameters that produce bounds on the probability of failure a priori. The proposed framework can significantly reduce the computational burden and provide a reliable estimate of the probability of failure