An approximate inertial manifold for computing Burgers' equation

1992 ◽  
Vol 60 (1-4) ◽  
pp. 175-184 ◽  
Author(s):  
L.G. Margolin ◽  
D.A. Jones
1999 ◽  
Vol 60 (2) ◽  
pp. 319-330
Author(s):  
Anibal Rodriguez-Bernal ◽  
Bixiang Wang

In this paper, we study approximate inertial manifolds for nonlinear evolution partial differential equations which possess symmetry. The relationship between symmetry and dimensions of approximate inertial manifolds is established. We demonstrate that symmetry can reduce the dimensions of an approximate inertial manifold. Applications for concrete evolution equations are given.


Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Lisha Xu ◽  
Hua Deng ◽  
Chong Lin ◽  
Yi Zhang

The dynamic characteristics of the mechanical arm with a rigid-flexible structure are very complex. The reason is that it is a complex DPS (distributed parameter system) with infinite dimension and nonlinearity in essence due to the rigid-flexible coupling. So, accurately positioning and controlling the rigid-flexible mechanical arms could be difficult. Therefore, a model reduction method of rigid-flexible mechanical arms based on the approximate inertial manifold is put forward. To repress the residual vibration of the end of the mechanical arm, a feedforward control strategy is designed. The high-dimensional solution of the vibration equation of the rigid-flexible mechanical arms is projected into the complete space composed of orthogonal decomposition modes. By using Galerkin’s method, the system is simplified and the approximate solution is obtained through the interaction between high-order and low-order modes. The truncated finite mode is also used to construct a lowest-order dynamic model on the basis of approximate inertia manifold. Given the reduced-order rigid-flexible mechanical arms dynamic model, dynamic response analysis is conducted to optimize the target position error and end residual vibration. A limited number of sinusoidal signals approximately combine the input signal, by using the particle swarm optimization algorithm to optimize the input signal, and the amplitude of the sinusoidal signal is corrected. The simulation results depict the superiority of the proposed method, which greatly suppresses the end residual vibration of the mechanical arm and realizes the accurate positioning of the end of the mechanical arm. In addition, the hardware experimental device of the rigid-flexible mechanical arms is constructed, and the experimental verification of the above method is put into effect. The simulation results of angular displacement and end vibration of the reduced model are accordant which is shown by the experimental results of the hardware platform.


2000 ◽  
Vol 13 (3) ◽  
pp. 269-285 ◽  
Author(s):  
Nejib Smaoui

We study numerically the long-time dynamics of a system of reaction-diffusion equations that arise from the viscous forced Burgers equation (ut+uux−vuxx=F). A nonlinear transformation introduced by Kwak is used to embed the scalar Burgers equation into a system of reaction diffusion equations. The Kwak transformation is used to determine the existence of an inertial manifold for the 2-D Navier-Stokes equation. We show analytically as well as numerically that the two systems have a similar, long-time dynamical, behavior for large viscosity v.


2002 ◽  
Vol 15 (1) ◽  
pp. 53-69 ◽  
Author(s):  
Nejib Smaoui ◽  
Fethi Belgacem

The convective diffusion equation with drift b(x) and indefinite weight r(x), ∂ϕ∂t=∂∂x[a∂ϕ∂x−b(x)ϕ]+λr(x)ϕ,   (1) is introduced as a model for population dispersal. Strong connections between Equation (1) and the forced Burgers equation with positive frequency (m≥0), ∂u∂t=∂2u∂x2−u∂u∂x+mu+k(x),   (2) are established through the Hopf-Cole transformation. Equation (2) is a prime prototype of the large class of quasilinear parabolic equations given by ∂u∂t=∂2u∂x2+∂(f(v))∂x+g(v)+h(x).  (3) A compact attractor and an inertial manifold for the forced Burgers equation are shown to exist via the Kwak transformation. Consequently, existence of an inertial manifold for the convective diffusion equation is guaranteed. Equation (2) can be interpreted as the velocity field precursed by Equation (1). Therefore, the dynamics exhibited by the population density in Equation (1) show their effects on the velocity expressed in Equation (2). Numerical results illustrating some aspects of the previous connections are obtained by using a pseudospectral method.


2021 ◽  
Vol 260 ◽  
pp. 03014
Author(s):  
Lisha Xu ◽  
Xiaoshan Qian ◽  
Chong Lin

An order reduction method for the flexible deformation response analysis of rigid flexible manipulators is proposed based on the approximate inertial manifold theory. This method allows a lower dimensional simplified model to be constructed from a subspace smaller than the entire state space. In this paper, truncated three-order modes are used to construct a first-order system of AIM. Compared with the traditional Galerkin method, the results show that the proposed method can reduce the degree of freedom of the system and improve the computational efficiency without obviously losing the precision of the solution, which is convenient for the subsequent vibration analysis and controller design of the system.


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