Stable Relaxation Research Articles

A general numerical unsteady non-linear lifting line model for engineering aerodynamics studies

ABSTRACTThe lifting-line theory is widely used for obtaining aerodynamic performance results in various engineering fields, from aircraft conceptual design to wind-power generation. Many different models were proposed, each tailored for a specific purpose, thus having a rather narrow applicability range. This paper presents a general lifting-line model capable of accurately analysing a wide range of engineering problems involving lifting surfaces, both steady-state and unsteady cases. It can be used for lifting surface with sweep, dihedral, twisting and winglets and includes features such as non-linear viscous corrections, unsteady and quasi-steady force calculation, stable wake relaxation through fictitious time marching and wake stretching and dissipation. Possible applications include wing design for low-speed aircraft and unmanned aerial vehicles, the study of high-frequency avian flapping flight or wind-turbine blade design and analysis. Several validation studies are performed, both steady-state and unsteady, the method showing good agreement with experimental data or numerical results obtained with more computationally expensive methods.

Relaxation periodic solutions of one singular perturbed system with delay

In this paper, we consider a singularly perturbed system of two differential equations with delay, simulating two coupled oscillators with a nonlinear compactly supported feedback. We reduce studying nonlocal dynamics of initial system to studying dynamics of special finite-dimensional mappings: rough stable (unstable) cycles of these mappings correspond to exponentially orbitally stable (unstable) relaxation solutions of initial problem. We show that dynamics of initial model depends on coupling coefficient crucially. Multistability is proved.

Dynamics of a System of Two Simplest Oscillators with Compactly Supported Nonlinear Feedbacks

In this paper, we consider a singularly perturbed system of two differential equations with delay, simulating two coupled oscillators with a nonlinear feedback. The feedback function is assumed to be compactly supported and piecewise-continuous and it is assumed that its sign is constant. In this paper, we prove the existence of relaxation periodic solutions and make conclusions about their stability. Using a special large-parameter method, we construct asymptotics of all solutions of the considered system under the assumption that the initial-value conditions belong to a certain class. Using this asymptotics, we construct a special mapping principally describing the dynamics of the original model. It is shown that the dynamics changes fundamentally as the coupling coefficient decreases: we have a stable homogeneous periodic solution if the coupling coefficient is on the order of unity and the dynamics become more complex as the coupling coefficient decreases (it is described by a special map). For small values of the coupling, we show that there are values of the parameters such that several different stable relaxation periodic regimes coexist in the original problem.

Effect of freeze-thaw cycles in mechanical behaviors of frozen loess

A series of triaxial compression, creep and stress relaxation tests at temperature of −6°C were conducted on frozen loess, experienced different freeze-thaw numbers (0–12times), in order to study the thermal cycling effect in mechanical behaviors. The freeze-thaw process has a strong influence in the mechanical behaviors of frozen loess. The strength, stiffness and viscosity properties of frozen loess weaken gradually with the increase of the freeze-thaw procedure until the cycle number gets to the critical value for steady state of soil sample. The freezing low temperature ranged from −6°C to −12°C has no evident effect in the testing results of same freeze-thaw numbers. The strength distribution loci of frozen loess in the equivalent stress-mean stress space exhibits a unidirectional shrinkage over the range of the freeze-thaw process studied. Triaxial strength characteristics of frozen loess, experienced different freeze-thaw numbers, are more in agreement with the Ma's failure criterion for frozen soils (Ma et al., 1993). The stiffness properties (elastic modulus, elastoplastic coupled degree and damage degree) and the viscosity properties (relaxing stress, stable relaxation time, stable creep rate and viscoelastic modulus) of frozen loess both depend on the freeze-thaw process. The microstructure change of frozen loess induced by the thermal cycling effect is analyzed detailly so as to explore the micro-mechanism of property degeneration. Finally, a developed constitutive model, considering the freeze-thaw and confining pressure effect, is proposed to research and predict the triaxial strength and deformation characteristics of frozen loess experienced different thermal cycling process.

A stable and accurate relaxation technique using multiple penalty terms in space and time

A new method for data relaxation based on weak imposition of external data is introduced. The technique is simple, easy to implement, and the resulting numerical scheme is unconditionally stable. Numerical experiments show that the error growth naturally present in long term simulations can be prevented by using the new technique.

Increased beat-to-beat variation in diastolic phase percentages in patients with congestive heart failure.

Diastolic time (DT) may reflect the functioning status of cardiac relaxation/diastolic filling. Previous studies shown that beat-to-beat variation in RR interval (i.e., HRV) is preferentially expressed in the variation in DT series, raising up a question that whether DT possesses intrinsic variation, or whether the DT variation is simply a surrogate of HRV without additional information. In this study, we defined the diastolic phase percentages (DP) by correcting DT by the corresponding RR interval and proposed to analyze DP variation (DPV) in order to eliminate the masking effect of HRV. We studied DPV of 60 patients with congestive heart failure (CHF) and 60 healthy control subjects. Radial artery pressure (RAP) was monitored in synchrony with electrocardiography (ECG) for 5 min in order to extract the proposed DPV. Additionally, DPV was corrected by the individual mean DP level which resulted in the coefficient of DPV (cDPV). The differences in DPV and cDPV between the two groups were determined by linear regression models controlled for age and sex. Results showed that both DPV and cDPV increased significantly in CHF patients compared with healthy control subjects (both p <; 0.05). Those results, highly endorsing the existence of intrinsic physiological variation in DT, may suggest that CHF patients have less stable cardiac relaxation (that is not due to HRV) or/and less stable contractility (since the diastolic and systolic phases are mutually complementary). This study may provide helpful reference for the noninvasive evaluation of cardiac functionality and for the further understanding of multiple physiological variabilities.

Efficient simulation of a slow-fast dynamical system using multirate finite difference schemes

We consider a system of ordinary differential equations describing a slow-fast dynamical system, in particular, a predator-prey system that is highly susceptible to local time variations. This model exhibits coexistence of predatorprey dynamics in the case when the prey population grows much faster than that of the predators with a quite diversified time response. For particular parametric values their interactions show a stable relaxation oscillation in the positive octant. Such characteristics are difficult to mimic using conventional time integrators that are used to solve systems of ordinary differential equations. To resolve this, we design and analyze multirate time integration methods to solve a mathematical model for a slow-fast dynamical system. Proposed methods are based on using extrapolation multirate discretisation algorithms. Through these methods, we reduce the integration time by integrating the slow sub-system with a larger step length than the fast sub-system. This allows us to efficiently solve multiscale ordinary differential equations. Besides theoretical results, we provide thorough numerical experiments which confirm that these multirate schemes outperform corresponding single-rate schemes substantially both in terms of computational work and CPU times.

Integrated experimental measurement and computational analysis of relaxation behavior of cement and asphalt mortar

The stress relaxation behavior of cement and asphalt mortar (CA mortar) under different initial strain levels was investigated. Results show that the stress relaxation of CA mortar can be divided into two parts: the attenuation stress relaxation stage and stable stress relaxation stage. With the increasing strain levels, the relaxed stress, stress relaxation rate, stress relaxation modulus and specific stress relaxation of CA mortar increase gradually. Under the same initial strain level, the relaxed stress of type I CA mortar and type II CA mortar are little different, whereas the specific stress relaxation of type I CA mortar is much greater than that of type II CA mortar. In the strain range of 0.32–0.80%, the isochronous stress-strain relation of type I CA mortar is linear. When the strain is greater than 0.80%, the isochronous stress-strain relationship concaves to the strain axis gradually. The isochronous stress-strain relationship of type II CA mortar show linear increase in the strain range of 0.47–1.42%. A stress relaxation model based on thermodynamic theory was established, which agrees very well with the experimental results.

Stable Grip Control on Soft Objects With Time-Varying Stiffness

Humans can hold a live animal like a hamster without overly squeezing despite the fact that its soft body undergoes impedance and size variations due to breathing and wiggling. Although the exact nature of such biological motor controllers is not known, existing literature suggests that they maintain metastable interactions with dynamic objects based on prediction rather than reaction. Most robotic gripper controllers find such tasks very challenging mainly due to hard constraints imposed on the stability of closed-loop control and inadequate rates of convergence of adaptive controller parameters. This paper presents experimental and numerical simulation results of a control law based on a relaxed stability criterion of reducing the probability of failure to maintain a stable grip on a soft object that undergoes temporal variations in its internal impedance. The proposed controller uses only three parameters to interpret the probability of failure estimated using a history of grip forces to adjust the grip on the dynamic object. Here, we demonstrate that the proposed controller can maintain smooth and stable grip tightening and relaxing when the object undergoes random impedance variations, compared with a reactive controller that involves a similar number of controller parameters.

Dynamics of a System of Two Simplest Oscillators with Finite Non-linear Feedbacks

In this paper, we consider a singularly perturbed system of two differential equations with delay which simulates two coupled oscillators with nonlinear feedback. Feedback function is assumed to be finite, piecewise continuous, and with a constant sign. In this paper, we prove the existence of relaxation periodic solutions and make conclusion about their stability. With the help of the special method of a large parameter we construct asymptotics of the solutions with the initial conditions of a certain class. On this asymptotics we build a special mapping, which in the main describes the dynamics of the original model. It is shown that the dynamics changes significantly with the decreasing of coupling coefficient: we have a stable homogeneous periodic solution if the coupling coefficient is of unity order, and with decreasing the coupling coefficient the dynamics become more complex, and it is described by a special mapping. It was shown that for small values of the coupling under certain values of the parameters several different stable relaxation periodic regimes coexist in the original problem.

A stable and accurate Davies-like relaxation procedure using multiple penalty terms for lateral boundary conditions

A lateral boundary treatment using summation-by-parts operators and simultaneous approximation terms is introduced. The method is similar to Davies relaxation technique used in the weather prediction community and have similar areas of application, but is also provably stable. In this paper, it is shown how this technique can be applied to the shallow water equations, and that it reduces the errors in the computational domain.

Blue sky catastrophe in systems with nonclassical relaxation oscillations

The feasibility of the well-known blue sky bifurcation in a class of three-dimensional singularly perturbed systems of ordinary differential equations with one fast and two slow variables is studied. A characteristic property of the considered systems is that so-called nonclassical relaxation oscillations occur in them. The same name is used for oscillations with slow components, which are asymptotically close to some time-discontinuous functions and a δ-like fast component. Cases when the blue sky catastrophe results in a stable relaxation cycle or a stable two-dimensional invariant torus are analyzed. The problem of the appearance of homoclinic structures is also considered.

Truncation effect on Taylor–Aris dispersion in lattice Boltzmann schemes: Accuracy towards stability

The Taylor dispersion in parabolic velocity field provides a well-known benchmark for advection–diffusion (ADE) schemes and serves as a first step towards accurate modeling of the high-order non-Gaussian effects in heterogeneous flow. While applying the Lattice Boltzmann ADE two-relaxation-times (TRT) scheme for a transport with given Péclet number (Pe) one should select six free-tunable parameters, namely, (i) molecular-diffusion-scale, equilibrium parameter; (ii) three families of equilibrium weights, assigned to the terms of mass, velocity and numerical-diffusion-correction, and (iii) two relaxation rates. We analytically and numerically investigate the respective roles of all these degrees of freedom in the accuracy and stability in the evolution of a Gaussian plume. For this purpose, the third- and fourth-order transient multi-dimensional analysis of the recurrence equations of the TRT ADE scheme is extended for a spatially-variable velocity field. The key point is in the coupling of the truncation and Taylor dispersion analysis which allows us to identify the second-order numerical correction δkT to Taylor dispersivity coefficient kT. The procedure is exemplified for a straight Poiseuille flow where δkT is given in a closed analytical form in equilibrium and relaxation parameter spaces. The predicted longitudinal dispersivity is in excellent agreement with the numerical experiments over a wide parameter range. In relatively small Pe-range, the relative dispersion error increases with Péclet number. This deficiency reduces in the intermediate and high Pe-range where it becomes Pe-independent and velocity-amplitude independent. Eliminating δkT by a proper parameter choice and employing specular reflection for zero flux condition on solid boundaries, the d2Q9 TRT ADE scheme may reproduce the Taylor–Aris result quasi-exactly, from very coarse to fine grids, and from very small to arbitrarily high Péclet numbers. Since free-tunable product of two eigenfunctions also controls stability of the model, the validity of the analytically established von Neumann stability diagram is examined in Poiseuille profile. The simplest coordinate-stencil subclass, which is the d2Q5 TRT bounce-back scheme, demonstrates the best performance and achieves the maximum accuracy for most stable relaxation parameters.

Vertical and adiabatic excitations in anthracene from quantum Monte Carlo: Constrained energy minimization for structural and electronic excited-state properties in the JAGP ansatz.

We study the ionization energy, electron affinity, and the π → π(∗) ((1)La) excitation energy of the anthracene molecule, by means of variational quantum Monte Carlo (QMC) methods based on a Jastrow correlated antisymmetrized geminal power (JAGP) wave function, developed on molecular orbitals (MOs). The MO-based JAGP ansatz allows one to rigorously treat electron transitions, such as the HOMO → LUMO one, which underlies the (1)La excited state. We present a QMC optimization scheme able to preserve the rank of the antisymmetrized geminal power matrix, thanks to a constrained minimization with projectors built upon symmetry selected MOs. We show that this approach leads to stable energy minimization and geometry relaxation of both ground and excited states, performed consistently within the correlated QMC framework. Geometry optimization of excited states is needed to make a reliable and direct comparison with experimental adiabatic excitation energies. This is particularly important in π-conjugated and polycyclic aromatic hydrocarbons, where there is a strong interplay between low-lying energy excitations and structural modifications, playing a functional role in many photochemical processes. Anthracene is an ideal benchmark to test these effects. Its geometry relaxation energies upon electron excitation are of up to 0.3 eV in the neutral (1)La excited state, while they are of the order of 0.1 eV in electron addition and removal processes. Significant modifications of the ground state bond length alternation are revealed in the QMC excited state geometry optimizations. Our QMC study yields benchmark results for both geometries and energies, with values below chemical accuracy if compared to experiments, once zero point energy effects are taken into account.

Blue Sky Catastrophe in Systems with Non-classical Relaxation Oscillations

The feasibility of the well-known blue sky bifurcation in a class of three-dimensional singularly perturbed systems of ordinary differential equations with one fast and two slow variables is studied. A characteristic property of the considered systems is that so-called nonclassical relaxation oscillations occur in them. The same name is used for oscillations with slow components, which are asymptotically close to some time-discontinuous functions and a δ-like fast component. Cases when the blue sky catastrophe results in a stable relaxation cycle or a stable two-dimensional invariant torus are analyzed. The problem of the appearance of homoclinic structures is also considered.

Spatially distributed control of the dynamics of the logistic delay equation

The influence exerted by a small spatially inhomogeneous control on the dynamics of the logistic delay equation is studied. This paper consists of two parts. The first deals with the case where the logistic delay equation has a stable relaxation cycle. It is shown that a small control function can give rise to complex relaxation objects, namely, to a large number of different attractors. In the second part, the local dynamics of the stability problem is analyzed in a neighborhood of equilibrium in a close-to-critical case of “infinite” dimension. Special quasi-normal forms are constructed whose nonlocal dynamics determine the local behavior of solutions to the original equation. Some results of a numerical analysis are presented.

Relaxation and square-wave oscillations in a semiconductor laser with polarization rotated optical feedback

The rate equations for a laser with a polarization rotated optical feedback are investigated both numerically and analytically. The frequency detuning between the polarization modes is now taken into account and we review all earlier studies in order to motivate the range of values of the fixed parameters. We find that two basic Hopf bifurcations leading to either stable sustained relaxation or square-wave oscillations appear in the detuning versus feedback rate diagram. We also identify two key parameters describing the differences between the total gains of the two polarization modes and discuss their effects on the periodic square-waves.

Chaos disappearance in a piecewise linear Bonhoeffer–van der Pol dynamics with a bistability of stable focus and stable relaxation oscillation under weak periodic perturbation

This study elucidates the bifurcation structure causing chaos disappearance in a four-segment piecewise linear Bonhoeffer–van der Pol oscillator with a diode under a weak periodic perturbation. The parameter values of this oscillator are chosen such that stable focus and stable relaxation oscillation can coexist in close proximity in the phase plane if no perturbation is applied. Chaos disappearance occurs through a previously unreported novel and unconventional bifurcation mechanism. To rigorously analyze these phenomena, the diode in this oscillator is assumed to operate as a switch. In this case, the governing equation is represented as a constraint equation, and the Poincare map is constructed as an one-dimensional map. By analyzing the Poincare map, we clearly demonstrate why the stable relaxation oscillation that exists when no perturbation is applied disappears via chaotic oscillation when an extremely weak perturbation is applied.

Sudden change from chaos to oscillation death in the Bonhoeffer–van der Pol oscillator under weak periodic perturbation

In this paper, we analyze the sudden change from chaos to oscillation death generated by the Bonhoeffer-van der Pol (BVP) oscillator under weak periodic perturbation. The parameter values of the BVP oscillator are chosen such that a stable focus and a stable relaxation oscillation coexist if no perturbation is applied. In such a system, complicated bifurcation structure is expected to emerge when weak periodic perturbation is applied because the stable focus and the stable relaxation oscillation coexist in close proximity in the phase plane. We draw a bifurcation diagram of the fundamental harmonic entrainment. The bifurcation structure is complex because there coexist two bifurcation sets. One is the bifurcation set generated in the vicinity of the stable focus, and the other is that generated in the vicinity of the stable relaxation oscillation. By analyzing the bifurcation diagram in detail, we can explain the sudden change from chaos with complicated waveforms to oscillation death. We make it clear that this phenomenon is caused by a saddle-node bifurcation.

The theory of relaxation oscillations for Hutchinson's equation

Hutchinson's equation is a scalar equation with time delay which is well known in ecology. In this paper a complete asymptotic representation is constructed for a stable relaxation cycle of this equation, in the form of series in integer powers of a certain small parameter. The techniques of asymptotic integration developed on the way are then applied to analyse the question of attractors for a system of circularly interrelated Hutchinson equations.Bibliography: 8 titles.