Non-affine Model Research Articles

Affine and non-affine microsphere models for chain scission in polydisperse elastomer networks

Under mechanical loading, elastomers undergo discrete chain rupture events in their network architecture which collectively contribute to macroscale failure. This phenomenon is expected to have a significant influence due to the polydispersity of elastomer networks, which occurs naturally due to the probabilistic nature of the chain polymerization process. To study the deterioration of the network due to polydispersity under loading, a set of micromechanically motivated constitutive models are developed. At the chain level, an extensible inverse Langevin model is utilized, accounting for the internal energy due to stretching of Kuhn segments. Further, chain scission is introduced following an energetic criterion for bond rupture. To obtain the macroscale response, affine and non-affine microsphere models are developed incorporating the effects of chain scission through a microscopic chain damage variable. Utilizing the microsphere for homogenization purposes allows for the connection of anisotropic microscale damage to the effective macroscopic response. The theory allowing for rupture of chains is first applied to an affine microsphere model where two variants are considered, assuming equal force and equal strain load sharing. A non-affine microsphere model is then subsequently developed that takes chain rupture into account. Interestingly, the resulting non-affine formulation, specialized for the monodisperse case in the absence of damage, is identical to the maximal advance path theory by Tkachuk and Linder (2012) despite the different assumptions in the development of each model. Through numerical simulations of uniaxial tension tests, the stress-stretch response and concurrent damage evolution are studied. The impact of microsphere quadrature order and chain damage evolution on stability is also studied. Stereographic projections of the microsphere visually display chain deformation and damage behavior as a function of load level and constitutive model assumptions.

Effect of polymer chains entanglements, crosslinks and finite extensibility on the nonlinear dynamic oscillations of dielectric viscoelastomer actuators

Soft materials exhibiting large deformation under external stimuli have gained an increasing attention in the recent past because of their potential applications in soft transducers aimed at achieving biomimetic actuation. This paper theoretically analyzes the effect of internal properties, including entanglements, crosslinks and finite extensibility of polymer chains along with inherent viscoelastic properties of polymers on the performance of Dielectric Elastomer actuator (DEA) in dynamic modes of actuation. A physics based nonaffine material model proposed by Davidson and Goulbourne is used to model the polymer chains entanglements, crosslinks and finite extensibility. To incorprate the viscoelastic properties, a rheological material model based on the additive decomposition of the isotropic strain energy density into equilibrium and viscous parts is implemented. A computationally efficient method, which relies on the principle of least action is used for extracting the governing equation representing the dynamic motion of the DE actuator. The results demonstrate that DEAs with strong entanglements and crosslinks along with small finite extensibility of polymer chains exhibits lower deformation level in DC dynamic modes of actuation. It is inferred that the strong entanglements and crosslinks in polymer chains enhances the resonant frequency, but debilitate the intensity of vibration of viscoelastic DEAs. Further, the periodicity and stability of the nonlinear oscillations exhibited by the viscoelastic DEAs are assessed by employing the Poincare maps and phase portraits. The results of the present investigation can be applied effectively in bridging the mechanism between the microcosmic polymer chains and macroscopic dynamic behavior of viscoelastic DE actuators.

Novel Prescribed Performance Control Scheme for Flexible Hypersonic Flight Vehicles with Nonaffine Dynamics and Neural Approximation

This study develops a novel neural-approximation-based prescribed performance controller for flexible hypersonic flight vehicles (HFVs). Firstly, a new prescribed performance mechanism is exploited, which develops new performance functions guaranteeing velocity and altitude tracking errors with small overshoots. Compared with the existing prescribed performance mechanism, it has better preselected transient and steady-state performance. Then, the nonaffine model of HFV is decomposed into a velocity subsystem and an altitude subsystem. A prescribed performance-based proportional-integral controller is designed in the velocity subsystem. In the altitude subsystem, the model is expressed as a nonaffine pure feedback form, and control inputs are derived from neural approximations. In order to reduce the amount of computation, only one neural network approximator is used to approximate the subsystem uncertainties, and an advanced regulation algorithm is applied to the devise adaptive law for neural estimation. At the same time, the complex design process of back-stepping can be avoided. Finally, numerical simulation results are presented to verify the efficiency of the design.

Lattice-Based Hedging Schemes Under GARCH Models

This paper proposes an efficient way to implement quadratic hedging schemes for European options when the asset return process follows an asymmetric non-affine GARCH model driven by Gaussian innovations. More specifically, using a lattice approximation for the underlying, we construct locally risk-minimizing (LRM) hedge ratios under both physical and risk-neutral measures, as well as standard delta strategies. We investigate the convergence of option prices and hedges resulting from the LRM strategies relative to the number of intra-daily periods used in the lattice. Several numerical experiments are conducted to assess the sensitivity of the hedge ratios to the equity risk premium and leverage effect parameters, and to compare their performance by computing the corresponding one-period and terminal hedging errors. Our results suggest that the LRM scheme under the physical measure consistently outperforms competing hedging strategies.

Micromechanical modeling of the multi-axial deformation behavior in double network hydrogels

The Mullins-type damage behaviors in double network hydrogels have attracted a broad research interest in recent years. However, most of current works focus on characterizing and modeling the uniaxial deformation behaviors of these materials. In this work, we combine experimental and theoretical approaches to investigate the anisotropic damage behaviors of double network hydrogels revealed in multi-axial deformation conditions. We demonstrate that an isotropic damage model based on the eight-chain model and the network alteration theory fails to capture the stress response in multi-axial loading tests. An anisotropic damage theory based on the microsphere model has also been developed, while both the affine and non-affine approaches are adopted to obtain the micro-macro mapping. The results show that the affine microsphere model cannot describe the experimental results in pure shear and unequal biaxial tests. Remarkably, the non-affine microsphere model with three parameters captures all the important features of the experimental observations. This is because the non-affine model accurately predicts the directional damage of the primary cross-linked network. The non-affine microsphere model is also able to describe the damage cross-effect in double network hydrogels. The developed theoretical framework can promote the fundamental understanding of the anisotropic damage behaviors in various types of tough gels.

Robust control for a quadrotor aircraft with small overshoot and high-precision position tracking performance

A tracking control problem for a quadrotor aircraft with external disturbances and model uncertainties is considered. An error manifold involving the velocity and position tracking error is used to handle the dynamics of the translational subsystem. Novel performance functions combined with primary conditions are designed to drive the error manifold into a prescribed envelope, which ensures a small overshoot and avoids control singularity. A control-oriented non-affine model for the angular velocity subsystem is utilized to ensure a better applicability of the controller. By means of backstepping technology, prescribed performance controller is constructed for the rotational subsystem. Different from the existing flight control methods, no estimator or conservative assumption is adopted to solve the coupling term between the rotational and translational subsystems in this paper. The proposed methodology is estimation-free and independent of exact knowledge of the model parameters. The effectiveness of the developed controller is verified by two simulation examples.

Key Technique of Almost Exact Simulation for Non-affine Heston Model

In order to sample asset price more accurately under the non-affine Heston model in the situation where the Feller condition was unsatisfied, we proposed the key technique of almost exact simulation for non-affine Heston model, by fusing the approximate analytic solution of conditional variance characteristic function, the second-order Newton’s method and interpolation method. Numerical results on four representative benchmarks show that our newly proposed simulation scheme has both good convergence and accuracy, which are much better than those of the Itô-Taylor schemes. Especially, our scheme performs well in the case where the Feller condition is extremely unsatisfied, while the alternatives don’t work.

Approximate Analytical Pricing of European Options under the Non-affine Stochastic Volatility Model

The non-affine stochastic volatility model has attracted increasing attention in recent years, due to its excellent performance in describing the nonlinear characteristics of asset price path. However, the fact that there is no close-form of option pricing formula under this model, restricts its use badly. To remove this restriction, two approximate analytical option pricing approaches are proposed in this paper, that is, the piecewise first-order Taylor expansion method and the perturbation-based asymptotic expansion method of implied volatility. The first method is used to derive an approximate characteristic function, then based on which the Fourier-Cosine expansion method calculates the European option price. In the second way, implied volatility of the European option price is asymptotically expanded around non-affine term of volatility. Compared with the existing methods in literature, numerical experiments show that the first method has higher accuracy, while the second method is of more practical significance. In addition, the accuracy of the first method is generally higher than that of the second method in most cases, while the second method performs better when the volatility is extremely small.

Direct adaptive fuzzy position controller for an electropneumatic actuator: Design and experimental evaluation

This paper proposes a direct adaptive fuzzy position controller for an electropneumatic actuator. This control approach is based on the estimation of an ideal inverse dynamic control, which is capable of satisfying the control objectives in terms of tracking desired trajectories and stability. For this purpose, a fuzzy system is used to approximate as best as possible this ideal inverse dynamic control. The adjustable parameters of the used fuzzy system are updated online with a stable adaptation law. This latter is developed in order to minimize the error between the unknown ideal inverse dynamic control and the used fuzzy controller. The general non-affine model of the controlled actuator is used to develop the control law and its adaptation mechanism, meaning that the presented adaptive fuzzy position controller does not rely on a physical model of the pneumatic actuator. The stability of the closed-loop system is performed using a Lyapunov approach. Finally, real-time experimental results are presented to show the performances of the proposed position controller.

Micromechanics of Stress-Softening and Hysteresis of Filler Reinforced Elastomers with Applications to Thermo-Oxidative Aging.

A micromechanical concept of filler-induced stress-softening and hysteresis is established that describes the complex quasi-static deformation behavior of filler reinforced rubbers upon repeated stretching with increasing amplitude. It is based on a non-affine tube model of rubber elasticity and a distinct deformation and fracture mechanics of filler clusters in the stress field of the rubber matrix. For the description of the clusters we refer to a three-dimensional generalization of the Kantor–Webman model of flexible chain aggregates with distinct bending–twisting and tension deformation of bonds. The bending–twisting deformation dominates the elasticity of filler clusters in elastomers while the tension deformation is assumed to be mainly responsible for fracture. The cluster mechanics is described in detail in the theoretical section, whereby two different fracture criteria of filler–filler bonds are considered, denoted “monodisperse” and “hierarchical” bond fracture mechanism. Both concepts are compared in the experimental section, where stress–strain cycles of a series of ethylene–propylene–diene rubber (EPDM) composites with various thermo-oxidative aging histories are evaluated. It is found that the “hierarchical” bond fracture mechanism delivers better fits and more stable fitting parameters, though the evolution of fitting parameters with aging time is similar for both models. From the adaptations it is concluded that the crosslinking density remains almost constant, indicating that the sulfur bridges in EPDM networks are mono-sulfidic, and hence, quite stable—even at 130 °C aging temperature. The hardening of the composites with increasing aging time is mainly attributed to the relaxation of filler–filler bonds, which results in an increased stiffness and strength of the bonds. Finally, a frame-independent simplified version of the stress-softening model is proposed that allows for an easy implementation into numerical codes for fast FEM simulations

Observer-Based Sampled-Data Model-Free Adaptive Control for Continuous-Time Nonlinear Nonaffine Systems With Input Rate Constraints

A sampled-data model-free adaptive control (SMFAC) strategy is proposed for continuous-time nonlinear nonaffine systems with input rate constraints. By using differential and integral mean value theorems as two basic mathematic tools, a sampled-data local dynamic linearization method is proposed at first to transform the continuous-time nonlinear nonaffine model into a sampled-data nonlinear affine I/O model, including a linear parametric term affined to the control input and a nonlinear uncertainty term. On this basis, we consequently propose an observer-based SMFAC (ObSMFAC) scheme, including a sampled-data parameter estimator to estimate the unknown partial derivatives and a sampled-data observer to estimate the residual nonlinear uncertainty, respectively. Note that the sampling period is incorporated explicitly in the proposed ObSMFAC which enhances the control performance by reducing its negative influence on the system stability. The constraint on the input rate is also considered in the control law as the transition condition of the input updating algorithms. The convergence of the proposed ObSMFAC is proved by using the contraction mapping principle. The simulation study demonstrates the theoretical results.

A Neural Network-Based Adaptive Backstepping Control Law With Covariance Resetting for Asymptotic Output Tracking of a CSTR Plant

A robust nonlinear adaptive controller merging a backstepping approach with neural networks is proposed for a nonlinear non-affine model. The work presented here is evaluated on a complex uncertain model of a continuous stirred tank reactor plant including an unknown varying parameter that enters the complexity model. By exploiting NN and adaptive backstepping approximation methods, an equivalent adaptive NN controller is constructed to achieve robust asymptotic output tracking control. The robustness to uncertainties as well as the lack of informative process data is the main enhancement of this work. This is attained through the implementation of the covariance resetting algorithm in the least square estimation of the NN weight tuning algorithm. The proposed novel control algorithm has been analyzed using Lyapunov analysis. In addition to excellent output trajectory tracking performance, the proposed approach has a profound benefit in terms of substantially lower control effort in comparison to the established work in the literature. In terms of applications in the petrochemical industry, lower control effort can translate to a more energy-efficient actuator, leading to lower costs over a long-run operation. The proposed method's feasibility for chemical process control was shown via numerical simulation.

Robust integral feedback control based on interval observer for stabilising parameter‐varying systems

This study develops a novel method for designing a controller based on an interval observer to stabilise a class of non-linear parameter-varying systems. The interval observer, that provides estimates of lower and upper bounds on main states, is described in a non-affine piecewise form with states and inputs-dependent switching mechanisms. Non-affine piecewise model of interval observer is converted to an affine piecewise model by applying a state variable change. Then, by choosing an appropriate piecewise quadratic Lyapunov function, the control input is developed to stabilise the interval observer in all operating regions regarding input-to-state stability that leads to stabilisation of the system. The Lyapunov analysis is based on the subspace-partitions information and preserves the continuity of Lyapunov function in the boundaries of the regions. Compared with the existing interval observer-based control methods, the proposed method considers less conservative assumptions and can be applied to a much broader class of parameter-varying systems. The simulation results for several examples indicate the effectiveness of the proposed method.

Pricing generalized variance swaps under the Heston model with stochastic interest rates

Unlike vanilla variance swaps, generalized variance swaps such as gamma, corridor variance and conditional variance swaps are expected to be not free from interest rates because of their weight processes. To examine the impact of stochastic interest rates on the generalized variance swaps, this paper considers discrete sampling times and the Heston stochastic volatility model incorporated by stochastic interest rates driven by the Cox–Ingersoll–Ross process. Based on the explicit calculation of the discounted characteristic function of Duffie et al. (2000), we obtain exact solutions for the fair strike prices of the generalized variance swaps for an affine version of the hybrid model. The solutions are given in closed form expression for the vanilla variance and gamma swaps and in Fourier integral expression for the corridor and conditional variance swaps. We apply the projection techniques of Grzelak and Oosterlee (2011) to the original non-affine model with a generalized correlation structure and obtain affine approximate solutions. We show the effects of stochastic interest rates on the strike prices of the generalized variance swaps.

Robust tracking control of hypersonic flight vehicles: A continuous model-free control approach

It is extremely difficult to establish precise motion model for hypersonic flight vehicles (HFVs). However, the majority of the existing control techniques focus on model-based methodologies. This paper investigates a novel model-free controller for a continuous non-affine model of HFV. Firstly, the nonaffine vehicle dynamics is transformed into a pure feedback formulation consisting of velocity subsystem and altitude subsystem. Then, a model-free tracking control law is exploited for velocity subsystem. Furthermore, the developed model-independent control technique is expanded into the altitude subsystem that is a high-order dynamic system. The proposed model-free controller contains a Hurwitz stable term and a filtering term. In addition, compensated filters are devised to further strength robustness. The special novelty is that the addressed controller doesn't require accurate affine model, exhibiting fine universal property and strong robustness. Finally, theoretical conclusions are validated and the effectiveness of the design is verified by simulation results.

Polarized Raman Scattering Predictions for a Molecular Constitutive Model of a Topologically Entangled and Permanently Cross-Linked Rubber

A new molecular constitutive model for rubber-like elasticity has been developed based on the same theory structure and terminology as Doi–Edwards type constitutive models for topologically entangled polymeric fluids. The new model using the independent alignment approximation (IAA) successfully predicts the effective modulus (Mooney ratio) in uniaxial extension and compression with accuracy nearly equal to the nonaffine slip tube model of Rubinstein and Panyukov. However, unlike the nonaffine slip tube model of Rubinstein and Panyukov, our Doi–Edwards IAA rubber model can be simply extended to arbitrary deformation types as well as make theoretical predictions for the fourth moment of the segmental orientation distribution function in general uniaxial deformations which is relevant to polarized Raman scattering experiments. We demonstrate this capability explicitly. The new Doi–Edwards rubber model demonstrates that there are apparently other, simpler, essentially ad hoc means to address the effects and character of the trapped topological entanglements in elastomers at moderate and high strains than the much more complex nonaffine slip tube model.

Non-affine parameter dependent LPV model and LMI based adaptive control for turbofan engines

The precise control of turbofan engines thrust is an important guarantee for an aircraft to obtain good flight performance and a challenge due to complex nonlinear dynamics of engines and time-varying parameters. The main difficulties lie in the following two aspects. Firstly, it is hard to obtain an accurate kinetic model for the turbofan engine. Secondly, some model parameters often change in different flight conditions and states and even fluctuate sharply in some cases. These variable parameters bring huge challenge for the turbofan engine control. To solve the turbofan engine control problem, this paper presents a non-affine parameter-dependent Linear Parameter Varying (LPV) model-based adaptive control approach. In this approach, polynomial-based LPV modeling method is firstly employed to obtain the basis matrices, and then the Radial Basis Function Neural Networks (RBFNN) is introduced for the online estimation of the non-affine model parameters to improve the simulation performance. LPV model-based Linear Matrix Inequality (LMI) control method is applied to derive the control law. A robust control term is introduced to fix the estimation error of the nonlinear time-varying model parameters for better control performance. Finally, the Lyapunov stability analysis is performed to ensure the asymptotical convergence of the closed loop system. The simulation results show that the states of the engine can change smoothly and the thrust of the engine can accurately follow the desired trajectory, indicating that the proposed control approach is effective. The contribution of this work lies in the combination of linear system control and nonlinear system control methods to design an effective controller for the turbofan engine and to provide a new way for turbofan engine control research.

A fully equilibrated microsphere model with damage for rubberlike materials

A non-affine microsphere model for rubberlike materials is proposed, based on a local minimization of the network free energy under a maximal advance path constraint. It accounts for any chain weight distribution and for damage such as Mullins softening observed in filled rubber materials. The non-affine equal-force model is compared to the common affine model and a hybrid equal-force model from the literature, when considering the isotropic hyperelastic behavior without damage of rubber materials presenting chains of various lengths. The non-affine model shows an improved deformability compared to the affine model limited by the maximal extension of the shorter chains and a significantly softer behavior. Possible damage is introduced by increasing the chain lengths according to the submitted maximal chain traction force. Each chain is impacted independently resulting in a directional softening that introduces the evolution of the stress-free configuration that needs to be assessed over the loadings. The model was successfully tested on the cyclic uniaxial tension stretch-stress responses of carbon-black filled styrene butadiene rubbers that were well fitted with three parameters only.

Bending a gel beam with consideration of entanglements and finite extensibility

A hybrid beam containing a gel layer attached on the top of an elastic non-swellable substrate has been normally used to manufacture diverse sensors and actuators. Generally, when the system uses different gels, it is necessary to develop differentiated models for the hybrid beam. Recently, on the basis of the generalized ideal elastomeric gel model, a unified relationship between the gels' swelling and the elastic beam's bending curvature is formulated by Cai, which is separated from some particular swelling mechanisms of gels. He plugged neo-Hookean model and Gent model of the stretching polymer network to the generalized equations. However, the polymer networks usually have many entanglements due to the uncrossability of the network chains as well as a finite extensibility resulting from the full stretching of the network chains. The effect of entanglements cannot be captured by these two models. In this letter, the nonaffine model proposed by Davidson and Goulbourne is adopted to the generalized equations derived by Cai. The results show that entanglements as well as chain extensibility have important influences on the mechanical bending behaviors of the gels' beam.

A three‐scale compressible microsphere model for hyperelastic materials

SummaryThis paper presents a seminumerical homogenization framework for porous hyperelastic materials that is open for any hyperelastic microresponse. The conventional analytical homogenization schemes do apply to a limited number of elementary hyperelastic constitutive models. Within this context, we propose a general numerical scheme based on the homogenization of a spherical cavity in an incompressible unit hyperelastic solid sphere, which is denoted as the mesoscopic representative volume element (mRVE). The approach is applicable to any hyperelastic micromechanical response. The deformation field in the sphere is approximated via nonaffine kinematics proposed by Hou and Abeyaratne (JMPS 40:571‐592,1992). Symmetric displacement boundary conditions driven by the principal stretches of the deformation gradient are applied on the outer boundary of the mRVE. The macroscopic quantities, eg, stress and moduli expressions, are obtained by analytically derived pointwise geometric transformations. The macroscopic expressions are then computed numerically through quadrature rules applied in the radial and surface directions of the sphere. A three‐scale compressible microsphere model is derived from the developed seminumerical homogenization framework where the micro‐meso transition is based on the nonaffine microsphere model at every point of the mRVE. The numerical scheme developed for the derivation of macroscopic homogenized stresses and moduli terms as well as the modeling capability of the three‐scale microsphere model is investigated through representative boundary value problems.