Dissipative Equations Research Articles

Numerical and Analytical Study of the Cyclic Behavior of ADAS Damper and the Effect of Axial Force on its Behavior

ABSTRACT The objective of this research was to study ADAS dampers and develop equations to calculate their seismic parameters. A model was used to analyze the dampers with various variables. The study proposed equations for stiffness, energy dissipation, and strength of the dampers. It also identified the variables with the most and least impact on the results. The effect of axial force on damper performance was investigated, including the calculation of critical load and examining its influence on stiffness, strength, and ductility. Reducing the opening area and height and increasing the width of the damper increased energy dissipation and effective stiffness.

Binary robustness of random attractors for 2D-Ginzburg–Landau equations with Wong–Zakai noise

Consider a non-autonomous 2D-Ginzburg–Landau equation driven by Wong–Zakai noise or white noise, respectively, we first show the existence of pullback random attractors, which are random compact attracting sets indexed by two parameters: the size of Wong–Zakai noise and the current time. We then establish the robustness of the attractors when both parameters are simultaneously convergent. An essential difficulty arises from the possible loss of the convergence of solutions and only part convergence of solutions is available, which is a new phenomenon for 2D-GL equation distinguishing with the 1D case. So, by using part joint-convergence, regularity, eventual local-compactness and recurrence, we establish a binary robustness theorem of pullback random attractors and apply it to the weakly dissipative stochastic equation.

Wave-breaking phenomena for a new weakly dissipative quasilinear shallow-water waves equation

Wave-breaking phenomena for a new weakly dissipative quasilinear shallow-water waves equation

Existence of a unique global solution, and its decay at infinity, for the modified supercritical dissipative quasi-geostrophic equation

Existence of a unique global solution, and its decay at infinity, for the modified supercritical dissipative quasi-geostrophic equation

Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era

This study presents a collection of purely data-driven workflows for constructing reduced-order models (ROMs) for distributed dynamical systems. The ROMs we focus on, are data-assisted models inspired by, and templated upon, the theory of Approximate Inertial Manifolds (AIMs); the particular motivation is the so-called post-processing Galerkin method of Garcia-Archilla, Novo and Titi. Its applicability can be extended: the need for accurate truncated Galerkin projections and for deriving closed-formed corrections can be circumvented using machine learning tools. When the right latent variables are not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a manifold learning scheme) can be used to discover good sets of latent variables and test their explainability. The proposed methodology can express the ROMs in terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both Black-Box and (theoretically-informed and data-corrected) Gray-Box models are described; the necessity for the latter arises when truncated Galerkin projections are so inaccurate as to not be amenable to post-processing. We use the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative partial differential equations to illustrate and successfully test the overall framework.

Semigroup theory and finite element method applied to a non-linear dissipative wave equation

We study the wave equation with a non-linear dissipative term associated to a bidimensional membrane with fixed boundary. We use the semigroup theory to consider the existence and uniqueness of solutions to the problem and we implement the finite element method to analyse the vibrating evolutionary equation. In particular we use Comsol Multiphysics software with a rectangular mesh to analyze the corresponding evolutionary system. We mention that this system can appear, for example, in the diaphragm of a centrifugal pump in mining processes.

Towards provably efficient quantum algorithms for large-scale machine-learning models

Large machine learning models are revolutionary technologies of artificial intelligence whose bottlenecks include huge computational expenses, power, and time used both in the pre-training and fine-tuning process. In this work, we show that fault-tolerant quantum computing could possibly provide provably efficient resolutions for generic (stochastic) gradient descent algorithms, scaling as O(T2×polylog(n))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{O}}}}}}}}({T}^{2}\ imes {{{{{{{\\rm{polylog}}}}}}}}(n))$$\\end{document}, where n is the size of the models and T is the number of iterations in the training, as long as the models are both sufficiently dissipative and sparse, with small learning rates. Based on earlier efficient quantum algorithms for dissipative differential equations, we find and prove that similar algorithms work for (stochastic) gradient descent, the primary algorithm for machine learning. In practice, we benchmark instances of large machine learning models from 7 million to 103 million parameters. We find that, in the context of sparse training, a quantum enhancement is possible at the early stage of learning after model pruning, motivating a sparse parameter download and re-upload scheme. Our work shows solidly that fault-tolerant quantum algorithms could potentially contribute to most state-of-the-art, large-scale machine-learning problems.

Null-controllability for weakly dissipative heat-like equations

Null-controllability for weakly dissipative heat-like equations

Pullback Exponential Attractors with Explicit Fractal Dimensions for Non-Autonomous Partial Functional Differential Equations

The aim of this paper is to propose a new method to construct pullback exponential attractors with explicit fractal dimensions for non-autonomous infinite-dimensional dynamical systems in Banach spaces. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalize the method established for autonomous systems in Hilbert spaces (Eden A, Foias C, Nicolaenko B, and Temam R Exponential attractors for dissipative evolution equations, Wiley, New York, 1994). The method is especially effective for non-autonomous partial functional differential equations for which phase space decomposition based on the exponential dichotomy of the linear part or variation techniques are available for proving squeezing property. The theoretical results are illustrated by applications to several specific non-autonomous partial functional differential equations, including a retarded reaction–diffusion equation, a retarded 2D Navier–Stokes equation and a retarded semilinear wave equation. The constructed exponential attractors possess explicit fractal dimensions which do not depend on the entropy number but only on some inner characteristics of the studied equations including the spectra of the linear part and the Lipschitz constants of the nonlinear terms and hence do not require the smooth embedding between two spaces in the previous work.

Analysing of different wave structures to the dissipative NLS equation and modulation instability

Analysing of different wave structures to the dissipative NLS equation and modulation instability

Long time dynamics of stochastic fractionally dissipative quasi-geostrophic equations with stochastic damping

Long time dynamics of stochastic fractionally dissipative quasi-geostrophic equations with stochastic damping

Energy Transfer and Dissipation in Combined-Stiffness Nonlinear Energy Sink Systems

Abstract Nonlinear energy sinks (NES) are highly efficient vibration energy absorption and dissipation devices, and play an important vibration-suppression role in many types of structures. In this study, the influence of parameters on the combined stiffness nonlinear energy sink system is revealed from the perspective of energy, in which combined-stiffness terms are composed of piecewise linear stiffness and cubic stiffness. First, the slow-varying derivative of the combined-stiffness nonlinear energy sink system is calculated based on the complexification-averaging and multiscale methods. Second, an approximate expression for the extreme points on the slow-invariant manifold (SIM) of the system is derived by polynomial approximation, and the energy dissipation equation of the combined-stiffness nonlinear energy sink system is derived. The impacts of the stiffness gap, piecewise linear stiffness coefficient, and cubic stiffness coefficient on the system are analyzed by studying the energy transfer efficiency equation. Additionally, the relationship between the damping ratio of the primary structure and the dissipation time is analyzed.

Stochastic Quasi-Geostrophic Equation with Jump Noise in Lp Spaces

In this paper, we consider a 2D stochastic quasi-geostrophic equation driven by jump noise in a smooth bounded domain. We prove the local existence and uniqueness of mild Lp(D)-solutions for the dissipative quasi-geostrophic equation with a full range of subcritical powers α∈(12,1] by using the semigroup theory and fixed point theorem. Our approach, based on the Yosida approximation argument and Itô formula for the Banach space valued processes, allows for establishing some uniform bounds for the mild solutions and we prove the global existence of mild solutions in L∞(0,T;Lp(D)) space for all p>22α−1, which is consistent with the deterministic case.

Performance of affine-splitting pseudo-spectral methods for fractional complex Ginzburg-Landau equations

We evaluate the performance of novel numerical methods for solving one-dimensional nonlinear fractional dispersive and dissipative evolution equations. The methods are based on affine combinations of time-splitting integrators and pseudo-spectral discretizations using Hermite and Fourier expansions. We show the effectiveness of the proposed methods by numerically computing the dynamics of soliton solutions of the standard and fractional variants of the nonlinear Schrödinger equation (NLSE) and the complex Ginzburg-Landau equation (CGLE), and by comparing the results with those obtained by standard splitting integrators. An exhaustive numerical investigation shows that the new technique is competitive when compared to traditional composition-splitting schemes for the case of Hamiltonian problems both in terms accuracy and computational cost. Moreover, it is applicable straightforwardly to irreversible models, outperforming high-order symplectic integrators which could become unstable due to their need of negative time steps. Finally, we discuss potential improvements of the numerical methods aimed to increase their efficiency, and possible applications to the investigation of dissipative solitons that arise in nonlinear optical systems of contemporary interest. Overall, the method offers a promising alternative for solving a wide range of evolutionary partial differential equations.

A cVEM-DG space-time method for the dissipative wave equation

A novel space-time discretization for the (linear) scalar-valued dissipative wave equation is presented. It is a structured approach, namely, the discretization space is obtained tensorizing the Virtual Element (VE) discretization in space with the Discontinuous Galerkin (DG) method in time. As such, it combines the advantages of both the VE and the DG methods. The proposed scheme is implicit and it is proved to be unconditionally stable and accurate in space and time.

Uniform sparse domination and quantitative weighted boundedness for singular integrals and application to the dissipative quasi-geostrophic equation

In this paper, we consider a kind of singular integralsTλf(x)= p.v. ∫RnΩ(y)|y|n−λf(x−y)dy for any f∈Lq(Rn),1<q<∞ and 0<λ<n, which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation(0.1)∂tθ+u⋅∇θ+κΛ2βθ=0,(x,t)∈R2×R+,κ>0, where u=−∇⊥Λ−2+2αθ, α∈[0,12] and β∈(0,1]. Firstly, we give a uniform sparse domination for this kind of singular integral operators. Secondly, we obtain the uniform quantitative weighted bounds for the operator Tλ with rough kernel. As an application, we obtain the uniform quantitative weighted bounds for the commutator [b,Tλ] with rough kernel and study solutions to the generalized 2D dissipative quasi-geostrophic (QG) equation.

Routes to chaos and generation of cnoidal wave envelopes in exciton-polariton microcavities

We show the presence of intermittent chaos as a function of the variation of the chemical potential during the condensation of exciton-polaritons using the bifurcation diagram and the Lyapunov Exponent. We then show the possibility of controlling the chaos in the homogeneous solution by the variation of the chemical potential when the pump increases. The influence of the chemical potential is demonstrated in stationary state by studying stability using the Routh-Hurwitz criterion. We also prove the existence of cnoidal waves and through phase portraits, their stabilities in propagation are established using the notion of symmetry. For this, we use a dissipative Gross-Pitaevskii equation for the polariton field, coupled to a reservoir dynamics equation, commonly used to describe polariton condensates under non-resonant pumping.

ω-Limit Sets of Impulsive Semigroups for Hyperbolic Equations

In this paper, we investigate the qualitative behavior of an evolutionary problem consisting of a hyperbolic dissipative equation whose trajectories undergo instantaneous impulsive discontinuities at the moments when the energy functional reaches a certain threshold value. The novelty of the current study is that we consider the case in which the entire infinite-dimensional phase vector undergoes an impulsive disturbance. This substantially broadens the existing results, which admit discontinuities for only a finite subset of phase coordinates. Under fairly general conditions on the system parameters, we prove that such a problem generates an impulsive dynamical system in the natural phase space, and its trajectories have nonempty compact ω-limit sets.

Generalized Pitchfork Bifurcations in D-Concave Nonautonomous Scalar Ordinary Differential Equations

AbstractThe global bifurcation diagrams for two different one-parametric perturbations ($$+\lambda x$$ + λ x and $$+\lambda x^2$$ + λ x 2 ) of a dissipative scalar nonautonomous ordinary differential equation $$x'=f(t,x)$$ x ′ = f ( t , x ) are described assuming that 0 is a constant solution, that f is recurrent in t, and that its first derivative with respect to x is a strictly concave function. The use of the skewproduct formalism allows us to identify bifurcations with changes in the number of minimal sets and in the shape of the global attractor. In the case of perturbation $$+\lambda x$$ + λ x , a so-called generalized pitchfork bifurcation may arise, with the particularity of lack of an analogue in autonomous dynamics. This new bifurcation pattern is extensively investigated in this work.

Topology Optimization of Turbulent Flow Cooling Structures Based on the k-ε Model.

Topology optimization (TO) is an effective approach to designing novel and efficient heat transfer devices. However, the TO of conjugate heat transfer has been essentially limited to laminar flow conditions only. The present study proposes a framework for TO involving turbulent conjugate heat transfer based on the variable density method. Different from the commonly used and oversimplified Darcy model, this approach is based on the more accurate and widely accepted k-ε model to optimize turbulent flow channels. We add penalty terms to the Navier-Stokes equation, turbulent kinetic energy equation, and turbulent energy dissipation equation, and use interpolation models for the thermal properties of materials. A multi-objective optimization function, aiming to minimize the pressure drop and the average temperature, is set up to balance the thermal and hydraulic performance. A case study is conducted to compare various optimization methods in the turbulent regime, and the results show that the present method has substantially higher optimization effectiveness while remaining computationally inexpensive.