Bifurcation Tracking Research Articles

Bifurcation tracking on moving meshes and with consideration of azimuthal symmetry breaking instabilities

We present a black-box method to numerically investigate the linear stability of arbitrary multi-physics problems. While the user just has to enter the system's residual in weak formulation, e.g. by a finite element method, all required discretized matrices are automatically assembled based on just-in-time generated and compiled C codes. Based on this method, entire phase diagrams in the parameter space can be obtained by bifurcation tracking and continuation at low computational costs. Particular focus is put on problems with moving domains, e.g. free surface problems in fluid dynamics, since a moving mesh introduces a plethora of complicated nonlinearities to the system. By symbolic differentiation before the code generation, however, these moving mesh problems are made accessible to bifurcation tracking methods. In a second step, our method is generalized to investigate symmetry-breaking instabilities of axisymmetric stationary solutions by effectively utilizing the symmetry of the base state. Each bifurcation type is validated on the basis of results reported in the literature on versatile fluid dynamics problems, for which we subsequently present novel results as well.

Application of Numerical Bifurcation Tracking Strategy to Blade-Tip/Casing Interactions in Aircraft Engines

Abstract Building on the regularized-Lanczos harmonic balance method (RL-HBM), a previously developed frequency method, this paper presents a numerical bifurcation tracking strategy dedicated to high-dimensional nonlinear mechanical systems. In order to demonstrate its applicability to industrial applications, it is here used to obtain original results in the context of blade-tip/casing interactions in aircraft engines. The emphasis is put specifically on the tracking of predicted limit point (LP) bifurcations as key parameters—such as the amplitude of the aerodynamic forcing applied on the blade, the friction coefficient or the operating clearances—vary. Overall, presented results underline that the employed frequency method is well-suited to tackle the numerical challenges inherent to such computations on high-dimensional systems. For the mechanical system of interest, the industrial fan blade National Aeronautics and Space Administration (NASA) rotor 67, it is shown that the application of the presented strategy yields an efficient way to identify isolated branches of solutions, which may be of critical importance from a design standpoint.

Control of isolated response curves through optimization of codimension-1 singularities

We introduce a computational framework for controlling the location of isolated response curves, i.e. responses that are not connected to the main solution branch and form a closed curve in parameter space. The methodology relies on bifurcation tracking to follow the evolution of fold bifurcations in a codimension-2 parameter space. Singularity theory is used to distinguish points of isola formation and merger from codimension-2 bifurcations and an optimization problem is formulated to delay or advance the onset or merger of isolated response curves or control their position in the state/parameter space. We illustrate the methodology on three examples: a finite element model of a cantilever beam with cubic nonlinearity at its tip, a two-degree-of-freedom oscillator with asymmetry and a two-degree-of-freedom base-excited oscillator exhibiting multiple isolas. Our results show that the location of points of isola formation and mergers can effectively be controlled through structural optimization.

Multi-parametric optimization for controlling bifurcation structures

Bifurcations organize the dynamics of many natural and engineered systems. They induce qualitative and quantitative changes to a system’s dynamics, which can have catastrophic consequences if ignored during design. In this paper, we propose a general computational method to control the local bifurcations of dynamical systems by optimizing design parameters. We define an objective functional that enforces the appearance of local bifurcation points at targeted locations or even encourages their disappearance. The methodology is an efficient alternative to bifurcation tracking techniques capable of handling many design parameters ( > 10 2 ). The method is demonstrated on a Duffing oscillator featuring a hardening cubic nonlinearity and an autonomous van der Pol-Duffing oscillator coupled to a nonlinear tuned vibration absorber. The finite-element model of a clamped-free Euler–Bernoulli beam, coupled with a reduced-order modelling technique, is also used to show the extension to the shape optimization of more complicated structures. Results demonstrate that several local bifurcations of various types can be handled simultaneously by the bifurcation control framework, with both parameter and state target values.

Tracking of bifurcations and hysteresis in electrostatically actuated resonators by motion-induced current

In this work, we present an experimental technique to detect the onset of bifurcation and measure the extent of hysteresis in electrostatic MEMS. The device-under-test comprises a microcantilever beam actuated via a side electrode. The beam vibrations result in varying the resonator’s capacitance, therefore inducing a current that can be measured and used for characterization and performance analysis. The motion-induced current is measured, and its harmonics are extracted using a lock-in amplifier. Locking on the third harmonic of the induced current enables us to bypass parasitic capacitance and extract a signal directly related to the resonator motions. We show that this signal can be used to investigate the nonlinear dynamic behavior of MEMS resonators undergoing large motions. Specifically, our experiments demonstrate our technique's ability to detect various bifurcations, to track the onset of hysteresis, and to measure the hysteresis bandwidth. We also use our technique to analyze the quantitative relationships between the actuation voltage, the location of the bifurcation point, and the hysteretic bandwidth. Finally, we show that our technique can be used to generate the calibration curves for inertial MEMS sensors.

Robust design of vibro-impacting geared systems with uncertain tooth profile modifications via bifurcation tracking

The present work investigates the influence of uncertain tooth profile modifications on the nonlinear dynamic response of a spur gear pair induced by a backlash nonlinearity. To this end, an original approach based on bifurcation tracking is developed. The equations of motion are solved in the frequency domain with the harmonic balance method (HBM) coupled to an arc-length continuation algorithm and a bordering technique. The evolution of the bifurcation points with respect to the uncertain parameter is computed in a deterministic way. The study focuses on minimizing the amplitude-jump instabilities induced by the backlash nonlinearity around the primary resonance peak. The proposed methodology allows for a fast and reliable estimation of the tooth profile modification that minimizes the amplitude-jump instability by defining two criteria using the results of the bifurcation tracking algorithm. Probability density functions (PDF) of various indicators of the severity of vibro-impacts can be computed with Monte-Carlo (MC) simulation with minimal computational burden. Results show that the tooth profile modification that minimizes the amplitude-jump instabilities differs from the optimum obtained with static computations.

Dynamics of a non-linear Jeffcott rotor in supercritical regime

This work is concerned with the prediction and simulation of limit cycles of a nonlinear Jeffcott rotor. Geometrical nonlinearity (related to large flexural displacements) in both stiffness and internal damping is considered. Classical instabilities due to linear internal damping occurring in the supercritical range are modified by nonlinear effects and lead to oscillating limit cycles at a radius whose value is found analytically as a function of the rotating speed. If the rotor is additionally excited by unbalance, the response spectrum also contains the excitation frequency, and the bifurcation leads to quasi-periodic limit cycles. The dynamics of this system is studied numerically via the harmonic balance method, path continuation and bifurcation tracking. In particular, an original procedure for switching to a branch of quasi-periodic solutions is proposed. To preserve the integrity of the machine post bifurcation, one then considers the possibility of constraining the transverse displacement of the shaft. Such possible rotor–stator interaction generates three-frequency quasi-periodic oscillations. Frequency response curves show that the quasi-periodic solutions created by the bouncing rotor are mainly driven by the friction coefficient.

Model reduction of a cyclic symmetric structure exhibiting geometric nonlinearity with a normal form approach

This papers deals with the computationally costly simulation of the frequency response function of geometrically nonlinear structures exhibiting cyclic symmetry. Its objective is to propose a reduction method that extends the model of a single sector reduced via the direct normal form to a full cyclically symmetric system. After validation on simple structures, the methodology is applied on a 3D meshed blade, representative component of the new generation of high bypass ratio turbofans. Based on both the proposed reduction methodology and theoretical conditions that predict the possible appearance of internal resonances as a function of the number of sectors and the engine order of the excitation, the complex nonlinear dynamics of a finely meshed full fan cyclic system is then investigated. This could not have been achieved without an appropriate reduction method. Branch switching algorithm, stability analysis, and bifurcation tracking compose the numerical tools employed along the harmonic balance method to simulate the forced response of the structure.

Bifurcation tracking of geared systems with parameter-dependent internal excitation

We herein propose an algorithm for tracking smooth bifurcations of nonlinear systems with interdependent parameters. The approach is based on a complex formulation of the well-known harmonic balance method (HBM). Hill’s method is used to assess the stability of the computed forced response curves, and a minimally extended system is built to allow for the parametric continuation of the detected bifurcation points. The feasibility of coupling HBM-based minimally extended systems and arc-length continuation algorithms is established and demonstrated. The method offers an efficient way of determining the stability regions of the system. The methodology is applied on a spur gear pair model including the backlash nonlinearity and subjected to transmission error and mesh stiffness fluctuation whose harmonic contents depend on several parameters that do not appear explicitly in the equations of motion.

Multiscale physics of rotating detonation waves: Autosolitons and modulational instabilities.

Proposed is a phenomenological modeling framework that is capable of reproducing the diverse experimental observations of the nonlinear, combustion wave propagation in a rotating detonation engine (RDE), specifically the nucleation and formation of combustion pulses, the soliton-like interactions between these combustion fronts, and the fundamental, underlying Hopf bifurcation to time-periodic modulation of the waves. In this framework, the mode-locked structures are classified as autosolitons or stably propagating nonlinear waves where the local physics of nonlinearity, gain, and dissipation exactly balance. We find that the global dominant balance physics in the RDE combustion chamber are dissipative and multiscale in nature: The fast combustion physics provide the energy input to form the fundamental mode-locked autosoliton state, while the slow physics of exhaust and propellant recovery shape the waveform and dictate the number of autosolitons. In this manner, the global multiscale balance physics give rise to the stable structures-not exclusively the frontal dynamics prescribed by classical detonation theory. Experimental observations and numerical models of the RDE combustion chamber are in strong qualitative agreement. Moreover, numerical continuation (computational bifurcation tracking) of the RDE analog system indicates that a Hopf bifurcation of the steadily propagating pulse train leads to the fundamental instability of the RDE, or time-periodic modulation of the waves. Along branches of Hopf orbits in parameter space exist a continuum of wave-pair interactions that exhibit solitonic interactions of varying strength.

A nonlinear optimization bifurcation tracking method for periodic solution of nonlinear systems

A continuation strategy which exploits the proposed optimization formulation to search periodic solution is presented for bifurcation tracking. The proposed optimization correction scheme integrated with the prediction-correction strategy is characterized by the minimization of the pseudo arclength equation with the imposition of optimization constraints on the harmonic balance equations and bifurcation conditions. Stability and sensitivity analysis are carried out in the frequency domain by making use of the Floquet theory. The derivatives of the stability factor with respect to the Fourier coefficients, vibration frequency, and bifurcation parameters are given in the explicit form. Finally, the effectiveness of the proposed methodology is illustrated by three numerical examples which include a Duffing oscillator, a nonlinear energy sink, and a Jeffcott rotor, respectively. Numerical results have demonstrated that the proposed method offers a convenient scheme to trace bifurcation solution.

A nonlinear optimization shooting method for bifurcation tracking of nonlinear systems

A continuation method for bifurcation tracking is presented based on the proposed optimization problem formulation which is designed to locate the bifurcation periodic solution. The bifurcation detection problem is formulated as a constrained optimization problem. The nonlinear constraints of the optimization problem are imposed on the shooting function and bifurcation conditions derived from the Floquet theory whereas the objective function associated with the pseudo-arclength correlation equation is devised to solution continuation. The proposed optimization formulation is integrated with the prediction–correction strategy to achieve bifurcation tracking. Two numerical examples about the Jeffcott rotor and the nonlinear tuned vibration absorber are illustrated to validate the effectiveness of the proposed methodology. Numerical results have demonstrated that the proposed method offers a convenient scheme to follow bifurcation periodic solution.

Global detection of detached periodic solution branches of friction-damped mechanical systems

Two novel methods to determine detached periodic solution branches of low-dimensional and large-scale friction-damped mechanical systems have been developed. The approach for low-dimensional systems is an extension of the global terrain method and consists of three steps: First, the non-smooth elastic Coulomb slider is temporarily modified to pose a $$C^2$$ continuous energy surface in the frequency domain. Second, the global terrain method is extended to rescale the search space consecutively along multiple eigendirections of the Hessian at the solution points, facilitating the determination of disconnected solutions. Finally, found solutions are used as the starting points for a homotopy strategy to detect the solutions of the original non-smooth problem. The method for large-scale systems based on an invariant manifold approach consists of three steps: First, the system’s dimension is decreased following a nonlinear modal reduction leading to a two-dimensional surrogate problem. Second, various subspaces for direct extrema, direct bifurcation, constant frequency, and constant amplitude solution detection are defined. Finally, a deterministic line search, a stochastic global solution method, and a newly developed deterministic line deflation are applied to the problem. The proposed methods are applied to low-dimensional and large-scale friction-damped mechanical systems simultaneously subjected to external and self-excitation as well as a low-dimensional mechanical system with shape-memory alloy nonlinearity subjected to external excitation. Their ability to find all solutions is discussed and compared with existing solution procedures such as bifurcation tracking, deflation, homotopy, and the global terrain method. Depending on the chosen subspace for the global search, the proposed methods are capable of providing additional detached solution branches (isola).

Bifurcation and mixed tracking of the discrete fractional LPA model

The memory effect in fractional calculus has made it a more realistic approach for the practical modeling of real life phenomena ranging from population, electric circuits, etc. In this paper, a discrete fractional order model for the Larva–Pupa–Adult (LPA) beetle population dynamics is proposed. The bifurcation of the proposed model showed novel dynamics. The mixed tracking control of the discrete fractional order LPA model was also considered to adjust individually and independently the population of each of larva, pupa or adult.

Coupled Multiple Timescale Dynamics in Populations of Endocrine Neurons: Pulsatile and Surge Patterns of GnRH Secretion

The gonadotropin-releasing hormone (GnRH) is secreted by hypothalamic neurons into the pituitary portal blood in a pulsatile manner. The alternation between a frequency-modulated pulsatile regime and the ovulatory surge is the hallmark of the GnRH secretion pattern in ovarian cycles of female mammals. In this work, we aim at modeling additional features of the GnRH secretion pattern: the possible occurrence of a two-bump surge (``camel surge'') and an episode of partial desynchronization before the surge. We propose a six-dimensional extension of a former four-dimensional model with three timescales and introduce two mutually coupled, slightly heterogenous GnRH subpopulations (secretors) regulated by the same slow oscillator (regulator). We consider two types of coupling functions between the secretors, including dynamic state-dependent coupling, and we use numerical and analytic tools to characterize the coupling parameter values leading to the generation of a two-bump surge in both coupling cases. We reveal the impact of the slowly varying control exerted by the regulator onto the pulsatile dynamics of the secretors, which leads to dynamic bifurcations and gives rise to desynchronization. To assess the occurrence time of desynchronization during the pulsatile phase, we introduce asymptotic tools based on quasi-static and geometric approaches, as well as analytic tools based on the H-function derived from the phase equation and numerical tracking of period-doubling bifurcations. We discuss the role of coupling parameters in the two-bump surge generation and the speed of desynchronization.

On the buckling of an elastic holey column.

We report the results of a numerical and theoretical study of buckling in elastic columns containing a line of holes. Buckling is a common failure mode of elastic columns under compression, found over scales ranging from metres in buildings and aircraft to tens of nanometers in DNA. This failure usually occurs through lateral buckling, described for slender columns by Euler’s theory. When the column is perforated with a regular line of holes, a new buckling mode arises, in which adjacent holes collapse in orthogonal directions. In this paper, we firstly elucidate how this alternate hole buckling mode coexists and interacts with classical Euler buckling modes, using finite-element numerical calculations with bifurcation tracking. We show how the preferred buckling mode is selected by the geometry, and discuss the roles of localized (hole-scale) and global (column-scale) buckling. Secondly, we develop a novel predictive model for the buckling of columns perforated with large holes. This model is derived without arbitrary fitting parameters, and quantitatively predicts the critical strain for buckling. We extend the model to sheets perforated with a regular array of circular holes and use it to provide quantitative predictions of their buckling.

Development of an efficient bifurcation tracking method

The buffet onset boundary is associated with a change in stability of the flow solution. The buffet onset boundary has been computed for a NACA 0012 aerofoil using a modified bifurcation tracking method. The algorithm combines the projection aspect of the Recursive Projection Method with two different direct bifurcation tracking methods. This considerably reduces the cost of the bifurcation tracking methods by solving for the bifurcation point on a small subspace of the system containing only the least stable dynamics. The method has been extended to allow the boundary to be computed as geometrical parameters, camber and thickness, are changed. This enables rapid evaluation of the effect of different aerofoil designs on the boundary. Results are presented comparing the full bifurcation tracking methods to their projected equivalent and show that although the projected methods lack the numerical accuracy of the full counterpart, the trends of the boundaries agree well.

Bifurcation Tracking for High Reynolds Number Flow Around an Airfoil

High Reynolds number flows are typical for many applications including those found in aerospace. In these conditions nonlinearities arise which can, under certain conditions, result in instabilities of the flow. The accurate prediction of these instabilities is vital to enhance understanding and aid in the design process. The stability boundary can be traced by following the path of a bifurcation as two parameters are varied using a direct bifurcation tracking method. Historically, these methods have been applied to small-scale systems and only more recently have been used for large systems as found in Computational Fluid Dynamics. However, these have all been concerned with flows that are inviscid. We show how direct bifurcation tracking methods can be applied efficiently to high Reynolds number flows around an airfoil. This has been demonstrated through the presentation of a number of test cases using both flow and geometrical parameters.

Experimental Tracking of Limit-Point Bifurcations and Backbone Curves Using Control-Based Continuation

Control-based continuation (CBC) is a means of applying numerical continuation directly to a physical experiment for bifurcation analysis without the use of a mathematical model. CBC enables the detection and tracking of bifurcations directly, without the need for a post-processing stage as is often the case for more traditional experimental approaches. In this paper, we use CBC to directly locate limit-point bifurcations of a periodically forced oscillator and track them as forcing parameters are varied. Backbone curves, which capture the overall frequency-amplitude dependence of the system’s forced response, are also traced out directly. The proposed method is demonstrated on a single-degree-of-freedom mechanical system with a nonlinear stiffness characteristic. Results are presented for two configurations of the nonlinearity — one where it exhibits a hardening stiffness characteristic and one where it exhibits softening-hardening.

A finite element method for a weakly nonlinear dynamic analysis and bifurcation tracking of thermo-acoustic instability in longitudinal and annular combustors

The aim of this paper is to investigate how nonlinear flame models influence the bifurcation process that characterize the transition to self-sustained thermo-acoustic pressure oscillations in gas turbine combustors. The analysis is carried out by means of a Finite Element Method solver able to treat complex combustor systems with multiple burners. The heat release fluctuations are coupled to the velocity fluctuations in the burner by means of nonlinear dependence. Two polynomial expressions of the third and of the fifth order are respectively considered. At first the proposed numerical procedure is validated in a longitudinal configuration against analytical results obtained in a low-order framework. Then, the ability of the proposed numerical approach to treat combustion systems with multiple independent flames is verified on an annular configuration equipped with twelve burners. In both configurations, in order to track bifurcation diagrams, the amplitudes of velocity fluctuations at limit cycles are plotted against the acoustic-combustion interaction index n considered as a control parameter. Regardless of the configuration, supercritical and subcritical bifurcations are obtained depending of the chosen flame model. The influence of time delay and acoustic damping is also investigated.