Additional Integral Term Research Articles

Updated formulation of magnetic body force in ferrofluids

The magnetic body force is critical for modelling of convection in ferrofluids. Despite a long history in the development of the theories for ferro-hydrodynamics, literature from the last five years shows that a universal consensus has not been reached concerning the formulation of this term for ferrofluids. We present an updated derivation of the body force directly from the Lorentz force and Maxwell's equations. The derivation requires that the differential volume experiencing the body force only contains complete dipole current loops. This has the effect that an additional surface integral term to account for bound surface current is not needed when modelling situations where the ferrofluid has interfaces with other materials. We compare results from our derived body force with five other formulations from the literature for the case of a single conductor in ferrofluid under static and convection conditions. Most formulations become similar in the limit of small magnetic susceptibility. For a susceptibility of the order of 1, as is typical for ferrofluids, the calculated body force from the formulations differed by a factor of about four, greatly affecting thermomagnetic convection predictions for a heated microwire.

Longterm existence of solutions of a reaction diffusion system with non-local terms modeling an immune response—An interpretation-orientated proof

This paper shows the global existence and boundedness of solutions of a reaction diffusion system modeling liver infections. The existence proof is presented step by step and the focus lies on the interpretation of intermediate results in the context of liver infections which is modeled. Non-local effects in the dynamics between the virus and the immune system cells coming from the immune response in the lymphs lead to an integro-partial differential equation. While existence theorems for parabolic partial differential equations are textbook examples in the field, the additional integral term requires new approaches to proving the global existence of a solution. This allows to set up an existence proof with a focus on interpretation leading to more insight in the system and in the modeling perspective at the same time.We show the boundedness of the solution in the L1(Ω)- and the L2(Ω)-norms, and use these results to prove the global existence and boundedness of the solution. A core element of the proof is the handling of oppositely acting mechanisms in the reaction term, which occur in all population dynamics models and which results in reaction terms with opposite monotonicity behavior. In the context of modeling liver infections, the boundedness in the L∞(Ω)-norm has practical relevance: Large immune responses lead to strong inflammations of the liver tissue. Strong inflammations negatively impact the health of an infected person and lead to grave secondary diseases. The gained rough estimates are compared with numerical tests.

Weakly Singular Symmetric Galerkin Boundary Element Method for Fracture Analysis of Three-Dimensional Structures Considering Rotational Inertia and Gravitational Forces

The Symmetric Galerkin Boundary Element Method is advantageous for the linear elastic fracture and crackgrowth analysis of solid structures, because only boundary and crack-surface elements are needed. However, for engineering structures subjected to body forces such as rotational inertia and gravitational loads, additional domain integral terms in the Galerkin boundary integral equation will necessitate meshing of the interior of the domain. In this study, weakly-singular SGBEM for fracture analysis of three-dimensional structures considering rotational inertia and gravitational forces are developed. By using divergence theorem or alternatively the radial integration method, the domain integral terms caused by body forces are transformed into boundary integrals. And due to the weak singularity of the formulated boundary integral equations, a simple Gauss-Legendre quadrature with a few integral points is suffcient for numerically evaluating the SGBEM equations. Some numerical examples are presented to verify this approach and results are compared with benchmark solutions.

The Immersion and Invariance Wind Speed Estimator Revisited and New Results

The Immersion and Invariance (I&I) wind speed estimator is a powerful and widely-used technique to estimate the rotor effective wind speed on horizontal axis wind turbines. Anyway, its global convergence proof is rather cumbersome, which hinders the extension of the method and proof to time-delayed and/or uncertain systems. In this letter, we illustrate that the circle criterion can be used as an alternative method to prove the global convergence of the I&I estimator. This also opens up the inclusion of time-delays and uncertainties. First, we demonstrate that the I&I wind speed estimator is equivalent to a torque balance estimator with a proportional correction term. As the nonlinearity in the estimator is sector bounded, the well-known circle criterion is applied to the estimator to guarantee its global convergence for time-delayed systems. By looking at the theoretical framework from this new perspective, this letter further proposes the addition of an integrator to the correction term to improve the estimator performance. Case studies show that the proposed estimator with an additional integral correction term is effective at wind speed estimation. Furthermore, its global convergence can be guaranteed by the circle criterion for time-delayed systems.

Well-posedness and Blowup of the Geophysical Boundary Layer Problem

The proposal of this paper is to study the local existence of analytic solutions, and blowup of solutions in a finite time for the geophysical boundary layer problem. In contrast with the classical Prandtl boundary layer equation, the geophysical boundary layer equation has an additional integral term arising from the Coriolis force. Under the assumption that the initial velocity and outer flow velocity are analytic in the horizontal variable, we obtain the local well-posedness of the geophysical boundary layer problem by using energy method in the weighted Chemin-Lerner spaces. Moreover, when the initial velocity and outer flow velocity satisfy certain condition on a transversal plane, for any smooth solution decaying exponentially in the normal variable to the geophysical boundary layer problem, it is proved that its $$W^{1,\infty }-$$ norm blows up in a finite time. Comparing with the blowup result obtained in Kukavica et al. (Adv Math 307:288–311, 2017) for the classical Prandtl equation, we find that the integral term in the geophysical boundary layer equation triggers the formulation of singularities earlier.

Static Kinks in Chains of Interacting Atoms

We theoretically analyse the equation of topological solitons in a chain of particles interacting via a repulsive power-law potential and confined by a periodic lattice. Starting from the discrete model, we perform a gradient expansion and obtain the kink equation in the continuum limit for a power-law exponent n ≥ 1 . The power-law interaction modifies the sine-Gordon equation, giving rise to a rescaling of the coefficient multiplying the second derivative (the kink width) and to an additional integral term. We argue that the integral term does not affect the local properties of the kink, but it governs the behaviour at the asymptotics. The kink behaviour at the center is dominated by a sine-Gordon equation and its width tends to increase with the power law exponent. When the interaction is the Coulomb repulsion, in particular, the kink width depends logarithmically on the chain size. We define an appropriate thermodynamic limit and compare our results with existing studies performed for infinite chains. Our formalism allows one to systematically take into account the finite-size effects and also slowly varying external potentials, such as for instance the curvature in an ion trap.

An improved optimal integral sliding mode control for uncertain robotic manipulators with reduced tracking error, chattering, and energy consumption

This paper proposes an improved optimal integral sliding mode control (IOISMC) for a robust and optimal tracking control of robotic manipulators under the presence of coupling effects, parameter variations, and external disturbances. By representing the system dynamics in the error coordinate and introducing an additional optimal integral term in the nominal control part of IOISMC, the effects of unmeasured forces, reaching phase issue, chattering issue, and position offset associated with conventional optimal integral sliding mode control are significantly addressed. Hence, the proposed approach has several notable merits, namely fast recovery response, low chattering, high tracking performance, high energy efficiency, low computational complexity, and robustness against uncertainties. The idea has been proven in two ways: a strong theoretical framework using Lyapunov stability theory and extensive simulation verification. The superiority of the proposed method has been confirmed through significant reductions of chattering and energy consumption without significant performance loss. Furthermore, the proposed method is capable of sustaining high tracking performance despite the presence of the aforementioned uncertainties and eliminates the reaching phase. It is envisaged that the proposed method can be useful in designing a robust and optimal tracking control for all types of uncertain multivariate systems without compromising on the computational cost.

Homogenization of an elastic material reinforced with thin rigid von Kármán ribbons

We study the homogenization of an elastic material reinforced with periodic parallel elastic ribbons of higher rigidity. We assume that the ribbons are isotropic von Kármán plates of highly contrasting rigidity with respect to that of the matrix. We assume perfect adhesion along the common interface between the matrix and the ribbons. We derive effective energy functionals involving additional degrees of freedom implying nonlocal effects associated with thin boundary layer phenomena and additional integral terms, including macroscopic fields of extension, flexion and torsion.

Flicker Noise in a Locally Nonequilibrium Medium

A model implying that particles of a medium are subjected to external forces with a flicker noise spectrum has been proposed to describe fluctuations in locally nonequilibrium physical media. The Langevin equation with additional integral terms describing the action of the locally nonequilibrium medium has been derived. The spectral density of fluctuations of an electric current flowing in the locally nonequilibrium medium has been calculated. It has been found that this density in the low-frequency spectral range is flicker noise.

The Interface Between Fresh and Salt Groundwater in Horizontal Aquifers: The Dupuit\u2013Forchheimer Approximation Revisited

We analyze the motion of a sharp interface between fresh and salt groundwater in horizontal, confined aquifers of infinite extend. The analysis is based on earlier results of De Josselin de Jong (Proc Euromech 143:75–82, 1981). Parameterizing the height of the interface along the horizontal base of the aquifer and assuming the validity of the Dupuit–Forchheimer approximation in both the fresh and saltwater, he derived an approximate interface motion equation. This equation is a nonlinear doubly degenerate diffusion equation in terms of the height of the interface. In that paper, he also developed a stream function-based formulation for the dynamics of a two-fluid interface. By replacing the two fluids by one hypothetical fluid, with a distribution of vortices along the interface, the exact discharge field throughout the flow domain can be determined. Starting point for our analysis is the stream function formulation. We derive an exact integro-differential equation for the movement of the interface. We show that the pointwise differential terms are identical to the approximate Dupuit–Forchheimer interface motion equation as derived by De Josselin de Jong. We analyze (mathematical) properties of the additional integral term in the exact interface motion formulation to validate the approximate Dupuit–Forchheimer interface motion equation. We also consider the case of flat interfaces, and we study the behavior of the toe of the interface. In particular, we give a criterion for finite or infinite speed of propagation.

Stokes–Cahn–Hilliard formulations and simulations of two-phase flows with suspended rigid particles

Three formulations of the Stokes-Cahn–Hilliard (SCH) system are investigated for the simulation of rigid particles in two-phase flows. Using the SCH framework, we assume that the interface between the two fluids is diffuse, whereas the interface between the fluids and the particle is assumed to be sharp. To describe the sharp boundary of the particle, a moving, boundary-fitted mesh is used which is refined near the fluid–fluid interface.The three formulations, the “stress form” and two “potential forms”, are first investigated in the absence of traction boundary conditions, by simulating a retracting droplet in a closed, cylindrical container. We show that the three formulations perform similar in terms of accuracy, although the velocity is slightly more accurate for the potential forms. When investigating mesh-convergence, superconvergence of the velocity and chemical potential is observed in the three forms. In equilibrium, the stress form shows higher parasitic currents near the interface. When comparing the pressure as it is defined in the stress form, the potential forms show higher peaks in the pressure near the interface. The three methods are stable when simulating a stationary droplet for a long period of time.We proceed by simulating a suspended rigid, spherical particle in the Cahn–Hilliard fluid, where a traction boundary conditions is applied to the particle boundary. For the potential forms, an additional integral term arises on the particle boundary. When investigating mesh-convergence, we observe superconvergence of the location and velocity of the particle, the velocity field and the chemical potential if the stress form is used. However, subconvergence is observed for these variables when using the potential forms. The three methods are stable when simulating a particle that is captured at a fluid–fluid interface for a long period of time.

The Swift–Hohenberg equation with a nonlocal nonlinearity

It is well known that aspects of the formation of localised states in a one-dimensional Swift–Hohenberg equation can be described by Ginzburg–Landau-type envelope equations. This paper extends these multiple scales analyses to cases where an additional nonlinear integral term, in the form of a convolution, is present. The presence of a kernel function introduces a new lengthscale into the problem, and this results in additional complexity in both the derivation of envelope equations and in the bifurcation structure.When the kernel is short-range, weakly nonlinear analysis results in envelope equations of standard type but whose coefficients are modified in complicated ways by the nonlinear nonlocal term. Nevertheless, these computations can be formulated quite generally in terms of properties of the Fourier transform of the kernel function. When the lengthscale associated with the kernel is longer, our method leads naturally to the derivation of two different, novel, envelope equations that describe aspects of the dynamics in these new regimes. The first of these contains additional bifurcations, and unexpected loops in the bifurcation diagram. The second of these captures the stretched-out nature of the homoclinic snaking curves that arises due to the nonlocal term.

Genetic Algorithm Optimization of a Finned-Tube Heat Exchanger Modeled With Volume-Averaging Theory

This paper proposes and implements a new methodology for optimizing finned-tube heat exchangers (FTHEs) using a volume-averaging theory (VAT) hierarchical physical model and a genetic algorithm (GA) numerical optimizer. This method allows for multiple-parameter constrained optimization of FTHEs by design of their basic morphological structures. A consistent model is used to describe transport phenomena in a FTHE based on VAT, which allows for the volume-averaged conservation of mass, momentum, and energy equations to be solved point by point, with the morphology of the structure directly incorporated into the field equations and full conjugate effects included. The equations differ from those often presented in porous media modeling and are developed using a rigorous averaging technique, hierarchical modeling methodology, and fully turbulent models with Reynolds stresses and fluxes in every pore space. These averaged equations have additional integral and differential terms that must be dealt with in order for the equation set to be closed, and recent work has provided this closure for FTHEs. The resulting governing equation set is relatively simple and is discretized and quickly solved numerically. Such a computational solution algorithm is fast running, but still able to present a detailed picture of the temperature fields in both of the fluid flows as well as in the solid structure of the heat exchanger. A GA is integrated with the VAT-based solver to carry out the FTHE numerical optimization, which is a ten parameter problem, and the FTHE is optimized subject to imposed constraints. This method of using the VAT-based solver fully integrated with a GA optimizer results in a new all-in-one tool for performing multiple-parameter constrained optimization on FTHEs.

A priorierror estimates for reduced order models in finance

Mathematical models for option pricing often result in partial differential equations. Recent enhancements are models driven by Lévy processes, which lead to a partial differential equation with an additional integral term. In the context of model calibration, these partial integro differential equations need to be solved quite frequently. To reduce the computational cost the implementation of a reduced order model has shown to be very successful numerically. In this paper we give a priori error estimates for the use of the proper orthogonal decomposition technique in the context of option pricing models.

Boundary element method for thermoelastic analysis of three-dimensional transversely isotropic solids

Thermal effects are well known to manifest themselves as additional volume integral terms in the direct formulation of the boundary integral equation (BIE) for linear elastic solids when using the boundary element method (BEM). This domain integral has been successfully transformed in an exact manner to surface ones only in isotropy and in 2D anisotropy, thereby restoring the BEM as a truly boundary solution technique. The difficulties with extending it to 3D general anisotropic solids lie in the mathematical complexity of the Green’s function and its derivatives for such materials. These quantities are required items in the BEM formulation. In this paper, the exact, analytical transformation of the volume integral associated with thermal effects to surface ones is achieved for a transversely isotropic material using a similar approach which the authors have previously employed for the same task in BEM for 2D general anisotropy. A numerical scheme, however, needs to be employed to evaluate some of the new terms introduced in the surface integrals that arise from this process here. The mathematical soundness of the formulation is demonstrated by a few examples; the numerical results obtained are checked by alternative means, including those obtained from the commercial FEM code, ANSYS.

Further improvement of the Lyapunov functional and the delay-dependent stability criterion for a neural network with a constant delay

This paper investigates the asymptotical stability problem of a neural system with a constant delay. A new delay-dependent stability condition is derived by using the novel augmented Lyapunov—Krasovskii function with triple integral terms, and the additional triple integral terms play a key role in the further reduction of conservativeness. Finally, a numerical example is given to demonstrate the effectiveness and lower conservativeness of the proposed method.

Adaptive interconnected observer-based backstepping control design for sensorless induction motor

In this paper, an observer-based controller scheme is designed to robustly drive a sensorless Induction Motor (IM) even for the case of low frequencies with unknown load torque. Combining the field oriented control strategy with the backstepping control method, we introduce additional integral terms to improve the robustness properties of the controller in spite of uncertainties and perturbations. The estimation of the fluxes, the speed, the load torque and moreover the stator resistance is given by an adaptive interconnected observer. The practical stability of the proposed observer–controller scheme is proved. Finally, some experimental test results are given on the framework of a specific sensorless induction motor benchmark.

Conjugate Heat Transfer Within a Heterogeneous Hierarchical Structure

Optimization of heat exchangers (HE), compact heat exchangers (CHE) and microheat exchangers, by design of their basic structures is the focus of this work. Consistant models are developed to describe transport phenomena in a porous medium that take into account the scales and other characteristics of the medium morphology. Equation sets allowing for turbulence and two temperature or two concentration diffusion are obtained for nonisotropic porous media with interface exchange. The equations differ from known equations and were developed using a rigorous averaging technique, hierarchical modeling methodology, and fully turbulent models with Reynolds stresses and fluxes in the space of every pore. The transport equations are shown to have additional integral and differential terms. The description of the structural morphology determines the importance of these terms and the range of application of the closure schemes. A natural way to transfer from transport equations in a porous media with integral terms to differential equations with coefficients that could be experimentally or numerically evaluated and determined is described. The relationship between computational fluid dynamics, experiment and closure needed for the volume averaged equations is discussed. Mathematical models for modeling momentum and heat transport based on well established averaging theorems are developed. Use of a “porous media” length scale is shown to be very beneficial in collapsing complex data onto a single curve yielding simple heat transfer and friction factor correlations. The general transport equations developed for a single phase fluid in a heat exchange medium have many more integral and differential terms than the homogenized or classical continuum mechanics equations. Once these terms are dealt with by closure, the resulting equation set is relatively simple and their solution is obtained using simple numerical methods quickly enough for multiple parameter optimization using design of experiment or genetic algorithms. Current efforts to significantly improve the performance of an HE for electronic cooling, a two temperature problem, and of a finned tube heat exchanger, a three temperature problem, are described.

HOPF‐Bifurcations, Global Coupling and Hysteresis in dc‐Driven Oxygen Discharges

AbstractStarting from a hydrodynamic model to describe the positive column of a glow discharge in oxygen, we investigate the stability of the homogeneous state. Near the critical points of the instability curve the wave dynamics are approximated by an amplitude equation of the Ginzburg‐Landau type with complex coefficients and an additional integral term. The nonlocal coupling term describes the influence of the external circuit on the plasma properties. The complex coefficients are calculated for selected values of the plasma parameters. For sufficiently large values of the external resistor a subcritical Hopf‐bifurcation is found. This is in agreement with the observation that in oxygen discharges a strong hysteresis occurs at the transition from the H ‐mode to the T ‐mode. Moreover, a numerical approach is used to study the hysteresis as a transition phenomenon (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Integral sliding-mode-based current-control strategy for single-phase current-source inverters

This paper proposes two current-control strategies adopting the sliding-mode concept for single-phase current-source inverters (CSI) feeding a resistive load. The first strategy is based on forming a general sliding surface using the error variables of the inductor current and capacitor voltage. However, it is observed that with this type of control there exists a steady-state error in the capacitor voltage. Therefore, an integral sliding-mode-based current control using an additional integral term of the error variables in the sliding surface is proposed. It is observed that the additional integral term alleviates the steady-state error in the capacitor voltage. The existence conditions for stability are determined for both control strategies. It is shown that the compliance of the existence conditions for full operating range of the CSI can be assured if the complete ranges of operating conditions are taken into consideration. In order to reduce the switching losses and improve the quality of the output voltage, the three-level hysteresis function is implemented as two separate two-level functions for the switching devices. Computer simulations are performed to verify the theoretical results and for comparison with existing methods. Experimental results are also presented to support the theoretical considerations.