Return Map Research Articles

The concentric Sitnikov problem: Circular case

A new configuration in the sense of concentric Sitnikov problem is introduced in this paper. This configuration consists of four primaries Pi(i=1,2,3,4) of masses mi, respectively. Here, all the primaries are placed on a straight line and moving in concentric circular orbits around their common center of mass under the restrictions of masses as m1=m2=m, m3=m4=ḿ and m>ḿ. The objective of the present manuscript is to study the existence of equilibrium points and their linear stability, first return map, periodic orbits, and N-R BoA in the proposed model. Under the restriction of mass parameter 0<μ́<1/4, it is found that there exist three equilibrium points and all the equilibrium points are linearly unstable for all values of mass parameter 0<μ́<1/4. Graphically, the nature of orbits have been examined with the help of first return map. The families of periodic orbits of infinitesimal mass m5 around the primaries and equilibrium points have been obtained for different values of mass parameter μ́. Finally, we explore the Newton–Raphson Basins of Attraction (N-R BoA), related to the equilibrium points in the concentric problem of Sitnikov.

Nonlinear Pulse Wave Dynamics in Prediction of Coronary Heart Disease and Myocardial Infarction

Atherosclerosis may induce coronary heart diseases (CHD) that progress to myocardial infarction (MCI), if untreated. The pulse plethysmogram (PPG) plot demonstrated the change in peripheral blood volume influence induced by the sympathetic branch of the autonomic nervous system that causes vasoconstriction. Thus, the present work has been aimed at proposing a noninvasive, low-cost, and wearable technology for the forecasting of CHD and MCI subjects using PPG-derived features with a support vector machine (SVM). A total of 70 volunteers (50-55 years) including MCI (n = 10), CHD (n = 30), and Control (n = 30) has been participated. Digital PPG was recorded for 10 min from the index finger of the left hand in the supine position. Ten samples from each subject were selected from the PPG signal for deriving nonlinear pulse rate variability (PRV) features using Kubios 2.0. The sensitivity, specificity, and accuracy were calculated from the obtained confusion matrix in the classification of control, CHD, and MCI classes. The findings suggested that the reduced value of PRV features in the Poincare plot, detrended fluctuation analysis recurrence plot, and correlation dimension while higher entropy measures in MCI than the CHD subjects. The best accuracy of 98.2% and 98.3% were observed with quadratic functions in the prediction of CHD and MCI subjects from the control group, respectively. The obtained results suggested the applicability of nonlinear PRV features in designing the low-cost real-time wearable technique in the prediction of CHD and MCI subjects.

A constitutive model for amorphous thermoplastics from low to high strain rates: Formulation and computational aspects

In this paper, a recently proposed finite strain visco-elastic visco-plastic (three-dimensional) constitutive model is extended to predict the nonlinear response of amorphous polymers from low to high strain rates. The model accounts for the influence of distinct molecular mechanisms, which become active at different deformation rates. Therefore, the constitutive equations include two relaxation phenomena to describe the strain rate sensitivity of amorphous polymers. Well-established rheological elements are adopted to define visco-elasticity (generalized Maxwell elements) and visco-plasticity (Eyring dashpots). In addition, strain hardening is modeled with a plasticity-induced (nonlinear) hardening element which is extended to distinguish between the contribution of the two transitions. From a computational viewpoint, a fully implicit integration algorithm is derived, and a highly efficient implementation is obtained. It is shown that it is possible to reduce the return mapping system of equations to only two independent (scalar) nonlinear equations. A four-stage optimization-based calibration procedure is proposed to identify the model’s material parameters in a completely unsupervised way. The predictive capability of the constitutive model is validated against literature results for polycarbonate and poly(methyl methacrylate), accounting for temperature and strain rate dependencies under different loading conditions. The results show that the model can capture the transition in the yield behavior and predict the post-yield large strain behavior over a wide range of strain rates. The efficiency of the calibration procedure and the overall numerical strategy is also demonstrated. Despite the adiabatic conditions observed under high strain rates, the model replicates the associated effect of temperature through strain rate dependency.

A new approach for nonlinear transient dynamic analysis of micro SMA beam considering size effect and instantaneous phase transformation

Shape Memory Alloys (SMAs) have attracted lots of attraction in the recent years due to their unique characteristics. However, studies in the field of behavior of SMAs at micro scale is not enough and more research is required. The present paper presents a nonlinear dynamic solution for a micro made of NiFeGa SMA, located on a viscoelastic foundation. The recommended approach takes into account instantaneous variations of martensite volume fraction and size effect which are the main novelties of the present study. In this regard, the formulation proposed by Hernandez and Lagoudas is used to capture and model the size effect of the SMAs. The governing differential equations obtained for Euler-Bernoulli micro beam using the Hamilton’s principle, considering the Von-Karman nonlinear strain. The nonlinear equation of motion is solved applying the semi-exact method. In this study, the return mapping convex cutting plane is used, in order to overcome the material nonlinearity of the problem. The obtained results are compared with the reference papers which shows the accuracy of the present model. Results show that with considering the size effect, the loss factor of vibration will increase by increasing the thickness of the micro SMA beam. While regardless of the size effect, this factor remains almost constant. The reason for this phenomenon is that when size effect is taken into account, the thermodynamic properties and critical stresses of phase transformation will change with resizing the SMA. Outputs also show that by increasing the damping coefficient and decreasing the stiffness of foundation, more energy is wasted in the vibration of the micro SMA beam. In addition, it was observed that micro beams with lower thicknesses are more sensitive to foundation characteristics meanwhile the loss factor of vibration with lesser thicknesses changes more by changing the foundation stiffness and damping coefficients.

Heart-Rate Variability Correlates to Choroidal Thickness in Central Serous Chorioretinopathy.

Patients with central serous chorioretinopathy (CSC) have previously been shown to have a lower heart rate variability (HRV), implying a lower vagal tone. Vagal tone alters mineralocorticoids, which in turn affect the thickness of the choroid. Since increased choroidal thickness is characteristic of CSC, we wanted to investigate its correlation with HRV. In this case-control study, 21 acute CSC patients and 31 healthy controls were included. Diagnosis was confirmed by optical coherence tomography (OCT) and retinal examination. HRV was evaluated following accepted standards. Outcome measures were chosen beforehand as follows: Standard deviation of N-N intervals (SDNN), root mean square of successive differences (RMSDD), low frequency/high frequency ratio (LF/HF ratio), and standard deviation ratio from commonly used Poincare plot (SD2/SD1 ratio). Choroidal thickness was measured using OCT directly under the foveola. Patients and healthy controls did not differ in health and medical characteristics in addition to CSC disease. Choroidal thickness was greatest in patients with CSC (mean±SD: 342±80 μm) compared to controls (235±60 μm, p<0.0001). A correlation was observed between LF/HF ratio and choroidal thickness in patients with CSC (Pearson correlation 0.63, p=0.02), where the CSC group had a lower LF/HF ratio (Median 2.39 ms2, IQR: 1.2-4.34 ms2) compared to controls (Median 1.2 ms2, IQR: 0.9-2.4 ms2, p=0.06) and SD2/SD1 ratio (CSC 0.59±0.2 vs controls 0.74±0.3, p=0.06). We found a correlation between the thickness of the choroid and the HRV-measured LF/HF ratio in patients with CSC and showed a borderline significant reduction in HRV measurements in patients with CSC. The data imply that vagal alterations exist in patients with CSC. Due to the low n, this should be considered as a pilot study. Further studies are warranted to elucidate mechanisms and validate findings.

Impact of coupling on neuronal extreme events: Mitigation and enhancement

We focus on the emergence of extreme events in a collection of aperiodic neuronal maps, under local diffusive coupling, as well as global mean-field coupling. Our central finding is that local diffusive coupling enhances the probability of occurrence of both temporal and spatial extreme events, while in marked contrast, global mean-field coupling suppresses extreme events. So the nature of the coupling crucially determines whether the extreme events are enhanced or mitigated by coupling. Further, in globally coupled systems, there exist initial states in a window of coupling strength that exhibit spatial extreme events, but not temporal extreme events, suggesting that spatial extreme events do not imply temporal extreme events. We also explored the existence of discernible patterns in the return maps of successive inter-event intervals in order to gauge short-term risk-assessment. We find that single neuronal maps, as well as systems under strong diffusive coupling, display broad noisy patterns in these return maps, with clusters around characteristic intervals, allowing some short-term predictability in the extreme event sequence. In contrast, under weak diffusive coupling and global coupling, inter-event intervals lose all perceptible correlations, and the distribution extends to very large inter-event intervals. Lastly, we investigated a non-local diffusive coupling form. Interestingly, this coupling yielded a large window where temporal extreme events occurred, but the spatial profile was synchronized, namely, we found synchronized temporal extreme events. Such synchronized extreme spiking is reminiscent of the neuronal activity leading to epileptic seizures and is of potential relevance to extreme events in brain activity.

Novel Quadrature-Element Analysis of Plane Stress Elastoplasticity

A novel quadrature element for performing plane stress elastoplastic analysis is introduced in this paper. Differing from the popular finite-element method, the quadrature-element method (QEM) first evaluates the integration in the weak-form statement of the problem by an integral quadrature scheme, and then approximates the differentiation at the discrete integration points by the differential quadrature analog. As a result, the QEM avoids construction of shape functions and obtains higher-order elements easily by just increasing the order of integration. The usage of higher-order elements leads to more accurate solutions and coarser geometric meshes without detriment to the computational scale because the number of degrees of freedom is maintained. In addition, the element nodes in the QEM are the same as the integration points that possess physical meanings such as strains and stresses, which is crucial in the elastoplastic analysis for determining the elastic/plastic state of the nodes. A straightforward elastoplastic quadrature-element formulation is developed. Incremental-iterative and return mapping solution schemes are adopted to implement the quadrature element for the elastoplastic analysis. Numerical examples are presented to demonstrate the effectiveness and high accuracy of the proposed approach.

Return mapping scheme for the Hoek-Brown model with tension cut-off

The present paper revisits a stress update procedure for the Hoek-Brown plasticity model enhanced by a Rankine type of tension cut-off failure criterion. Limiting tensile stresses not only supports the behavior of weak rock masses with a very low tensile strength but in combination with the original Hoek-Brown model improves robustness of the stress update procedure, i.e. the stress return mapping algorithm. Herein, the primary focus is on the stress return from a space of inadmissible trial stresses for which neither the Hoek-Brown yield surface nor its derivative can be evaluated. All potential scenarios are thoroughly discussed both in the framework of single- and multi-surface plasticity. The presented procedures were implemented and verified with the help of the Geo5 FEM software.

Analysis of the fractal characteristics for combustion instability in a premixed natural gas engine

To investigate the influence of gas injection timing (GIT) on the combustion instability of a premixed natural gas engine, experiments were conducted under low load conditions using various GITs. Multifractal and multiscale entropy analyses were employed to examine the fractal characteristics and complexity of the experimental time series for indicated mean effective pressure (IMEP) and heat release (Q) at different scales. Statistical analysis and return maps of the IMEP and Q time series were utilized to verify the results. The findings revealed that the combustion process of the natural gas engine demonstrates clear fractal characteristics at different scales. A strong correlation is found between the combustion instability and the fractal characteristics. Furthermore, the probability densities of the IMEP and Q time series exhibit super-Gaussian distributions. Retarding the GIT results in an initial increase, followed by a decrease in the difference value of the Hurst index and singular spectrum width. The mapping point distributions of the IMEP and Q time series initially disperse and subsequently concentrate. The fractal complexity and chaotic characteristics of combustion instability initially strengthen and then gradually diminish. Moreover, under lower load conditions, the anti-persistent correlation becomes more pronounced, and the intermittence and complexity of the fractal characteristics also intensify, signifying a more significant impact of GIT on the combustion instability of the natural gas engine. Notably, when the GIT is approximately 60°CA after top dead center, the combustion process exhibits stronger fractal characteristics, accompanied by a greater dispersion degree of the mapping points. This study provides a theoretical basis for enhancing the lean-burn stability of natural gas engines.

A thermodynamics-based micro-macro elastoplastic micropolar continuum model for granular materials

A micro–macro elastoplastic micropolar continuum model for granular materials in the dry and pendular states is proposed on the basis of thermodynamics. The independent rotational degrees of freedom are considered besides the translational degrees of freedom for particles among the discrete particle system, which is described by the equivalent micropolar continuum. In the framework of the micropolar continuum, the elastic part and yield locus at macroscale are firstly rigorously deduced from the general elastic constitutive relation and Coulomb-type yield criterion at inter-particle contacts, respectively. Furthermore, the micro–macro elastoplastic micropolar continuum model is proposed, in which the model parameters have clear physical meanings. Subsequently, a return mapping algorithm for the integration of the micropolar elastoplastic model is presented. Finally, the proposed model, coded into ABAQUS software, is validated by comparisons of test and discrete element method results in terms of the stress–strain relation and strain localization phenomenon. The stress–strain relation and shear band modes are well reproduced, and the effects of the confining pressure and the partially saturated state on granular materials can also be correctly predicted by the proposed model.

A non-autonomous mega-extreme multistable chaotic system

Megastable and extreme multistable systems comprise two major new branches of multistable systems. So far, they have been studied separately in various chaotic systems. Nevertheless, to the best of our knowledge, no chaotic system has so far been reported that possesses both types of multistability. This paper introduces the first three-dimensional non-autonomous chaotic system that displays megastability and extreme multistability, jointly called mega-extreme multistability. Our model shows extreme multistability for a variation of an initial condition associated with one system variable and megastability concerning another variable. The different types of coexisting attractors are characterized by the corresponding phase portraits and first return maps, as well as by constructing the appropriate bifurcation diagrams, calculating the Lyapunov spectra, the Kaplan-Yorke dimension and the connecting curves, and by determining the corresponding basins of attraction. The system is explicitly shown to be dissipative, with the dissipation being state-dependent. We demonstrate the feasibility and applicability of our model by designing and simulating an appropriate analog circuit.

Bifurcation diagrams of global connections in Filippov systems

In this paper, we are concerned about the qualitative behavior of planar Filippov systems around some typical invariant sets, namely, polycycles. In the smooth context, a polycycle is a simple closed curve composed by a collection of singularities and regular orbits, inducing a first return map. Here, this concept is extended to Filippov systems by allowing typical Filippov singularities lying on the switching manifold. Our main goal consists in developing a method to investigate the unfolding of polycycles in Filippov systems. In addition, we apply this method to describe bifurcation diagrams of Filippov systems around certain polycycles.

Variational approach for nonsmooth elasto-plastic dynamics with contact and impacts

The objective of this article is the modeling and the numerical simulation of the response of elastoplastic structures to impacts. To this end, a numerical method is proposed that takes into account one-sided contact (Signorini condition) and impact phenomena together with plasticity in a monolithic solver, while accounting for the non-smooth character of the dynamics. The formulation of the plasticity and the contact laws are based on inclusions into normal cones of convex sets, or equivalently, variational inequalities following the pioneering work of Moreau (1974) and Halphen and Nguyen (1975), who introduced the assumptions of normal dissipation and of generalized standard materials (GSM) in the framework of associated plasticity with strain hardening. The proposed time–stepping method is an extension of the Jean and Moreau (1987) scheme for nonsmooth dynamics. The discrete energy balance shows that spurious numerical damping can be suppressed and that the scheme is in practice unconditionally stable. Furthermore, the finite-dimensional variational inequality at each time–step is well–posed and can be solved by optimization methods for convex quadratic programs, providing an interesting alternative to the return mapping algorithm coupled with a dedicated frictional contact method. The paper is completed by illustrative numerical examples of impacts on steel structures consisting of beams.

Investigation on the inhomogeneous deformation of magnesium alloy during bending using an advanced plasticity model

In the present work, an advanced plasticity model is established to character the anisotropic and asymmetric yielding and hardening behaviors of magnesium alloy. This model is comprised of the stress invariant-based Yoon2014 yield function, non-associated flow rule and yield surface-interpolation method in the description of distortional hardening. For effectively numerical implementation of the plasticity model, a semi-explicit stress integration scheme based on the return mapping method is provided. The implemented model is then applied to simulate the inhomogeneous deformations during three-point bending of an extruded magnesium alloy. Simulation results show that there are three distinct zones in the bent plate according to its deformation patterns, i.e., the compressive inner zone, the tensile outer zone, and the central zone which experiences tensile, then compressive and again tensile deformations. Compared to the counterpart Hill48 function, Yoon2014 function has better performance in simulating the inhomogeneous deformations of magnesium alloy.

Simplified Phenomenological Model for Ferroelectric Micro-Actuator.

As smart structures are becoming increasingly ubiquitous in our daily life, the need for efficient modeling electromechanical coupling devices is also rapidly advancing. Smart structures are often made of piezoelectric materials such as lead zirconate titanate (PZT), which exhibits strong nonlinear behavior known as hysteresis effect under a large applied electric field. There have been numerous modeling techniques that are able to capture such an effect; some techniques are suitable for obtaining physical insights into the micro-structure of the material, while other techniques are better-suited to practical structural analyses. In this paper, we aim to achieve the latter. We propose a simplified phenomenological macroscopic model of a nonlinear ferroelectric actuator. The assumption is based on the direct relation between the irreversible strain and irreversible electric field, and the consequently irreversible polarization. The proposed model is then implemented in a finite element framework, in which the main features such as local return mapping and the tangent moduli are derived. The outcomes of the model are compared and validated with experimental data. Therefore, the development presented in this paper can be a useful tool for the modeling of nonlinear ferroelectric actuators.

A post‐processing procedure for predicting high‐ and low‐cycle fatigue life of welded structures based on the master E–N curve

AbstractThis work provides a post‐processing procedure to predict the high‐ and low‐cycle fatigue life of welded structures. The post‐processor calculates the equivalent structural strain range, given the traction stress state at the weld notch, by enforcing the equilibrium condition and Navier's hypothesis. The return mapping algorithm is applied to deal with the low‐cycle fatigue condition in which through‐thickness plastic deformation develops. The proposed method predicts the fatigue life of 793 welded joints of different configurations made from steel, magnesium, titanium, and aluminum alloy. The predicted high‐ and low‐cycle fatigue lives agree equally well with the experimental result.

Variational formulation of Cahn–Hilliard-type diffusion coupled with crystal plasticity

This article presents a relatively general framework for the description of the Cahn–Hilliard-type diffusion in solids undergoing infinitesimal elastic and plastic deformations. The coupled chemo-mechanical problem, characterised by phenomena such as phase segregation, microstructure coarsening and swelling, is treated using the variational framework which is governed by continuous-time, discrete-time and discrete-space–time incremental variational principles. It is shown that the governing equations of the coupled problem can be derived as Euler equations of minimisation and saddle point principles. A point of departure from the existing works is the coupling of crystal plasticity to the problem of diffusion and optimising the potential with respect to the plastic variables such that they are solved locally at the integration points. This is done using a return map algorithm which results in a reduced global problem. The variational framework results in a system of symmetric non-linear algebraic equations that are solved by Newton–Raphson-type iterative methods. This is a novel and attractive feature with respect to numerical implementation, as models resulting from the proposed variational framework are computationally less expensive in comparison with non-symmetric formulations. The numerical simulations presented at the end predict the applicability of models resulting from the proposed variational framework for multiple scenarios.

Re-interpretation of the Homogenized Constrained Mixture Theory within the plasticity framework and application to soft tissue growth and remodeling

The homogenized constrained mixture theory (H-CMT) is an attractive and efficient computational framework to simulate growth and remodeling (G&R) of soft tissues within finite deformations. It considers several prestressed constituents within a mixture and it enables their continuous individual mass removal and production to be taken into account. However, the referred theory was developed for specific mixtures, whose remodeling occurred on unidimensional constituents (fibers) only, while being embedded in an isotropic matrix. As the microstructure of soft tissues is generally more complex, we propose an extension of the H-CMT, which enables remodeling to occur on tridimensional constituents. This was achieved by manipulating the remodeling stress rate equation of the H-CMT. By rearranging the tensorial expression, it was possible to re-interpret its terms as variables of the classical plasticity theory and the resulting equation is a particular case of kinematic hardening. This interpretation, in turn, enables standard return mapping algorithms, which are classical in plasticity, to be quickly adapted to G&R problems. Therefore, not only we explore the intersection of both the H-CMT and the plasticity frameworks, but we also propose new algorithmic implementations of G&R that closely resemble those used in standard elasto-plastic problems. Applications to the simulation of G&R in mixtures composed of anisotropic constituents are eventually shown to demonstrate the capabilities of the new algorithms.

Improved Endothelial and Autonomic Function after Transcatheter Aortic Valve Implantation.

Degenerative aortic stenosis is an atherosclerotic-like process associated with impaired endothelial and autonomic function. Transcatheter aortic valve implantation (TAVI) has become a treatment of choice for patient with severe degenerative aortic stenosis at high surgical risk. The effect of this procedure on endothelial function measured with flow mediated dilatation (FMD) and autonomic function measured with heart rate variability (HRV) at different time-points of disease management (early and late follow-up) remains unknown. We prospectively included 50 patients with severe aortic stenosis who were deemed suitable for TAVI by the Heart Team. FMD and HRV parameters were collected at baseline ( 24 h pre-TAVI), at early follow-up (up to 48 h post-TAVI) and at late follow-up (3-6 months post-TAVI). 43 patients (mean age 81 (75-85); 60% women) completed the study. FMD significantly improved from 2.8 1.5% before TAVI to 4.7 2.7% early after TAVI (p 0.001) and was later maintained on late follow-up (4.8 2.7%, p = 0.936). Conversely, high-resolution ECG parameters remained preserved at early and improved at late follow-up after TAVI. Significant improvement was detected in a high frequency-domain parameter-HF (from 5231 1783 to 6507 1789 ; p = 0.029) and in two Poincare plot parameters: ratio of the short- and long-term R-R variability in the Poincare plot-SD1/SD2 (from 0.682 to 0.884 ; p = 0.003) and short-term R-R variability in the Poincare plot-SDRR (from 9.6 to 23.9 ms; p = 0.001). Echocardiographic parameters comprising baseline maximal aortic valve velocity (R = 0.415; p = 0.011), mean aortic gradient (R = 0.373; p = 0.018), indexed stroke volume (R = 0.503; p = 0.006), change in aortic valve maximal velocity (R = 0.365; p = 0.031), change in mean aortic gradient (R = 0.394; p = 0.019) and NT-proBNP (R = 0.491; p = 0.001) were found as significant predictors of change in FMD. Endothelial function measured with FMD and autonomic function obtained with HRV parameters significantly improve after TAVI. While endothelial function improves early and is maintained later after TAVI, autonomic function remains stable and improves on late follow-up. This is most likely caused by early hemodynamic changes after resolution of aortic valve obstruction and gradual left ventricular remodeling. www.clinicaltrials.gov, identifier NCT04286893.

Two novel alternative integration schemes for multi-invariants dependent isotropic finite deformation plasticity

Multi-invariants dependent isotropic yield function in terms of first invariant I1 of stress tensor, second J2 and third J3 invariants of deviatoric stress tensor is required for accurate description of plasticity of polycrystalline solids. In the context of multi-invariants dependent multiplicative hyperelasto-plasticity, two integration schemes (IS-1 and IS-2) are formulated and their computational costs are compared for the first time in this paper. IS-1 is fully-implicit in which backward Euler difference approximation of equivalent plastic strain rate is coupled with the exponential approximation of return mapping in principal Kirchhoff stress space and accordingly principal Kirchhoff stresses, equivalent plastic strain and plastic Lagrange multiplier need to be updated iteratively. IS-2 is semi-implicit in which plastic Lagrange multiplier evaluated at current time step and semi-implicit forward Euler difference approximation of equivalent plastic strain rate is coupled with logarithmic approximation of return mapping in Kirchhoff stress space and accordingly only plastic Lagrange multiplier needs to be updated iteratively. For J2 dependent plasticity, computation time involved in numerical simulations for both the integration schemes is nearly the same for same accuracy. For IS-1, computation time involved in simulation for multi-invariants dependent plasticity is significantly greater than that for J2 dependent plasticity due to the computation of the Jacobian matrix in return mapping involving lengthy terms of second order partial derivatives of multi-invariants dependent yield function. For IS-2, computation time involved in simulation for multi-invariants dependent plasticity is nearly the same as that for J2 dependent plasticity. This is due to the fact that the computation of the Jacobian matrix in IS-2 is not required and return mapping involves only first partial derivatives of yield function. Thus, IS-2 is computationally cost effective than IS-1.