Numerical and Experimental Analysis of the Nonlinear Dynamics due to Impacts of a Continuous Overhung Rotor

1997 ◽  
Author(s):  
Mohammed F. Abdul Azeez ◽  
Alexander F. Vakakis
Keyword(s):  
Nonlinear Dynamics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Experiments ◽  
The Other ◽  
Experimental Setup ◽  
Partial Differential ◽  
Qualitative Comparison ◽  
Set Up ◽  
Theory And Experiments

Abstract This work is aimed at obtaining the transient response of an overhung rotor when there are impacts occurring in the system. An overhung rotor clamped on one end, with a flywheel on the other and impacts occurring in between, due to a bearing with clearance, is considered. The system is modeled as a continuous rotor system and the governing partial differential equations are set up and solved. The method of assumed modes is used to discretize the system in order to solve the partial differential equations. Using this method numerical experiments are run and a few of the results are presented. The different numerical issues involved are also discussed. An experimental setup was built to run experiments and validate the results. Preliminary experimental observations are presented to show qualitative comparison of theory and experiments.

scholarly journals Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach

2015 ◽  
Vol 65 (5) ◽  
Author(s):  
Nathan Bush ◽  
Baasansuren Jadamba ◽  
Akhtar A. Khan ◽  
Fabio Raciti
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Optimization Problem ◽  
Short Note ◽  
Variable Parameter ◽  
Partial Differential ◽  
Convergence Results ◽  
Equation Error

AbstractThe objective of this short note is to employ an equation error approach to identify a variable parameter in fourth-order partial differential equations. Existence and convergence results are given for the optimization problem emerging from the equation error formulation. Finite element based numerical experiments show the effectiveness of the proposed framework.

scholarly journals ANNs-Based Method for Solving Partial Differential Equations : A Survey

2021 ◽  
Author(s):  
Danang Adi Pratama ◽  
Maharani Abu Bakar ◽  
Mustafa Man ◽  
M. Mashuri
Keyword(s):  
Machine Learning ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Review Paper ◽  
Partial Differential ◽  
Poisson Equations ◽  
Finite Difference Approximations ◽  
Higher Dimensional ◽  
Set Up ◽  
Discretization Process

Conventionally, partial differential equations (PDE) problems are solved numerically through discretization process by using finite difference approximations. The algebraic systems generated by this process are then finalized by using an iterative method. Recently, scientists invented a short cut approach, without discretization process, to solve the PDE problems, namely by using machine learning (ML). This is potential to make scientific machine learning as a new sub-field of research. Thus, given the interest in developing ML for solving PDEs, it makes an abundance of an easy-to-use methods that allows researchers to quickly set up and solve problems. In this review paper, we discussed at least three methods for solving high dimensional of PDEs, namely PyDEns, NeuroDiffEq, and Nangs, which are all based on artificial neural networks (ANNs). ANN is one of the methods under ML which proven to be a universal estimator function. Comparison of numerical results presented in solving the classical PDEs such as heat, wave, and Poisson equations, to look at the accuracy and efficiency of the methods. The results showed that the NeuroDiffEq and Nangs algorithms performed better to solve higher dimensional of PDEs than the PyDEns.

A Multiobjective Mesh Optimization Algorithm for Improving the Solution Accuracy of PDE Computations

2016 ◽  
Vol 13 (01) ◽  
pp. 1650002 ◽  
Author(s):  
Jibum Kim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimization Algorithm ◽  
Numerical Experiments ◽  
Optimization Algorithms ◽  
Mesh Optimization ◽  
Partial Differential ◽  
Solution Accuracy ◽  
Single Objective

Mesh qualities affect both the efficiency and accuracy for solving partial differential equations (PDEs). In this paper, we present a multiobjective mesh optimization algorithm, which improves the accuracy for solving PDEs. Our algorithm is designed to simultaneously improve more than two aspects of the mesh, while being able to successfully decrease errors for solving various PDEs. Numerical experiments show that our algorithm is able to significantly decrease errors compared with existing single objective mesh optimization algorithms.

Interior Estimates for Elliptic Partial Differential Equations in the ℒ(q,λ) Spaces of Strong Type

1984 ◽  
Vol 36 (3) ◽  
pp. 385-404
Author(s):  
Akira Ono
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Partial Differential Equations ◽  
The Other ◽  
Usual Sense ◽  
Elliptic Type ◽  
Strong Type ◽  
Partial Differential ◽  
Interior Estimates ◽  
Isomorphism Theorems

Recently the ℒ(q,λ) spaces have been investigated by many authors and the theory of these spaces has proved to be particularly important for research in partial differential equations (see for example [15], [16] and [18]).The equations of elliptic type in these spaces were first studied by C. B. Morrey [8], [9], who applied his well-known imbedding theorems, and afterwards by S. Campanato [3], [4] with the aid of isomorphism theorems and the so-called fundamental inequalities due to him.On the other hand, G. Stampacchia introduced the ℒ(q,λ) spaces of strong type [17], the structures of which are more general and complicated than those of ℒ(q,λ) Spaces in the usual sense, and greater part of them were characterized by him, L. C. Piccinini, Y. Furusho, the author and others (see [5], [11]-[14], [16] and [17]).

Adding police to a mathematical model of burglary

2010 ◽  
Vol 21 (4-5) ◽  
pp. 401-419 ◽  
Author(s):  
ASHLEY B. PITCHER
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Continuum Limit ◽  
The Other ◽  
Residential Burglary ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
The Continuum

We review the Short model of urban residential burglary derived from taking the continuum limit of two difference equations – one of which models the attractiveness of individual houses to burglary, and the other of which models burglar movement. This leads to a system of non-linear partial differential equations. We propose a change to the Short model and also add deterrence caused by the presence of uniformed officers to the model. We solve the resulting system of non-linear partial differential equations numerically and present results both with and without deterrence.

Explicit Solutions for Nonlinear Partial Differential Equations Using Bezier Functions

2007 ◽  
Author(s):  
P. Venkataraman ◽  
J. G. Michopoulos
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Metal Composite ◽  
Partial Differential ◽  
Ionic Polymer ◽  
Linear Partial Differential Equations ◽  
Polymer Metal ◽  
Set Up ◽  
Domain Discretization

This paper presents a methodology for generating solutions of non linear partial differential equations through Bezier functions. These functions define corresponding Bezier surfaces using a bipolynomial Bernstein basis function. The solution, or essentially the coefficients, is identified through design optimization. The set up is direct, elegantly simple, and involves minimizing the error in the residuals of the differential equations over the domain. No domain discretization is necessary. The procedure is not problem dependent and is adaptive through the selection of the order of the Bezier functions. Two examples: (1) the laminar flow over a flat plate; and (2) displacement of an ionic polymer-metal composite membrane are solved. Alternate solution to these problems is referenced in the paper.

Pricing futures by deterministic methods

Acta Numerica ◽  
2012 ◽  
Vol 21 ◽  
pp. 577-671
Author(s):  
Olivier Pironneau
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Narrow Range ◽  
Numerical Algorithms ◽  
The Other ◽  
Financial Mathematics ◽  
Pragmatic Approach ◽  
Partial Differential ◽  
Deterministic Methods ◽  
Better Than

In this article we will focus on only a small part of financial mathematics, namely the use of partial differential equations for pricing futures. Even within this narrow range it is hard to be systematic and complete, or even to do better than existing books such as Wilmott, Howison and Dewynne (1995), Achdou and Pironneau (2005), or software manuals such as Lapeyre, Martini and Sulem (2010). So this article may be valuable only to the extent that it reflects ten years of teaching, conferences and interaction with the protagonists of financial mathematics.Also, because the theory of partial differential equations is not always well known, we have chosen a pragmatic approach and left out the details of the theory or the proofs of some results, and refer the reader to other books. The numerical algorithms, on the other hand, are given in detail.

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