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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
- Veröffentlichungsjahr: 2017

This book is dedicated to the general study of fluid structure interaction with consideration of uncertainties. The fluid-structure interaction is the study of the behavior of a solid in contact with a fluid, the response can be strongly affected by the action of the fluid. These phenomena are common and are sometimes the cause of the operation of certain systems, or otherwise manifest malfunction. The vibrations affect the integrity of structures and must be predicted to prevent accelerated wear of the system by material fatigue or even its destruction when the vibrations exceed a certain threshold.

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Cover

Title

Copyright

Preface

1 Fluid–Structure Interaction

1.1. Introduction

1.2. Fluid–structure interaction problem

1.3. Vibroacoustics

1.4. Aerodynamics

2 Fluid–Structure Interaction with Ansys/Fluent

2.1. Presentation of Ansys

2.2. Coupling with Ansys

2.3. Example of fluid–structure interaction using the “physics” environment

2.4. Example of interaction using Fluent

3 Vibroacoustics

3.1. Introduction

3.2. Equations of the acoustic and structure problems

3.3. Vibroacoustic problem

3.4. Study of an elastic plate coupled with a fluid cavity

3.5. Study of the propeller of a boat

4 Aerodynamics

4.1. Introduction

4.2. Computational method

4.3. Aerodynamic problem’s resolution

4.4. Finite element method for the solid

4.5. Finite volumes for the fluid

4.6. Coupling procedures

4.7. Numerical results

4.8. Study of a 3D airplane wing

4.9. Transient analysis

5 Modal Reduction for FSI

5.1. Introduction

5.2. Dynamic substructuring methods

5.3. Nonlinear substructuring method

5.4. Proper orthogonal decomposition for flows

5.5. Dynamic substructure/acoustic subdomain coupling

5.6. Numerical simulation

6 Reliability-based Optimization for FSI

6.1. Introduction

6.2. Reliability in mechanics

6.3. Failure in mechanics

6.4. Reliability index

6.5. Mechanoreliability coupling

6.6. Reliability-based optimization in mechanics

6.7. SP method

6.8. Numerical results

Bibliography

Index

End User License Agreement

3 Vibroacoustics

Table 3.1. The first five natural frequencies of the structure, both dry and immersed, based on a formulation in terms of (u, p)

Table 3.2. The first five natural frequencies of the structure, both dry and immersed, based on a formulation in terms of (u, p, Φ)

Table 3.3. The first five natural frequencies of the dry structure

Table 3.4. The first five natural frequencies of the immersed structure

Table 3.5. The first three natural frequencies of the full propeller in air

Table 3.6. The first three natural frequencies of the full propeller in water

Table 3.7. The first three natural frequencies of the blade, both in air and water, calculated with Ansys code

4 Aerodynamics

Table 4.1. Geometric and material properties of the problem

Table 4.2. Total displacement

Table 4.3. Total displacement

Table 4.4. Natural frequencies of the wing

Table 4.5. Geometric and material properties of the problem

5 Modal Reduction for FSI

Table 5.1 Geometric and physical properties

Table 5.2. Characterization of the inertial effects

Table 5.3. Analytical and numerical calculations of the natural frequencies for the immersed elastic ring

Table 5.4. Natural frequencies of the circular cavity

Table 5.5. Natural frequencies of the elastic ring

Table 5.6. Natural frequencies of the immersed ring

Table 5.7. Material properties

Table 5.8. Natural frequencies of the propeller

6 Reliability-based Optimization for FSI

Table 6.1. Random variables and distributions

Table 6.2. Natural frequencies of the wing

Table 6.3. Random variables and their statistical moments

Table 6.4. Comparison of the drag coefficient (Cd) and lift coefficient (Cl)

Table 6.5. Parametrization of the wing

Table 6.6. Parameters of the three models

Table 6.7. Results for the normal distribution

Table 6.8. Results for the log-normal distribution

Cover

Table of Contents

Begin Reading

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Reliability of Multiphysical Systems Set

coordinated byAbdelkhalak El Hami

Volume 6

Abdelkhalak El Hami

Bouchaib Radi

First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK

www.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.wiley.com

© ISTE Ltd 2017

The rights of Abdelkhalak El Hami and Bouchaib Radi to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2016960066

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-939-7

The progress achieved by digital and software tools in the past 40 years has allowed scientists to dramatically improve their understanding of the world. The development of mathematical models has allowed us to work on increasingly sophisticated problems in a wide range of fields: predicting the behavior of production tools, transportation, the environment, etc. Managing these complex problems has been facilitated within each discipline separately but also from a cross-disciplinary perspective, allowing more general phenomena to be tackled.

The field of fluid–structure interactions unites the study of all phenomena involving a coupling between the motion of a structure and the motion of a fluid. The range of studied phenomena is very broad, from vibrating cylinders in flows, such as in the nuclear industry, to vibrating structures in turbulent flows and free-surface phenomena in reservoirs. One well-known example of fluid–structure interaction and the complexity of the couplings involved is the collapse of the Tacoma bridge in 1940, which began vibrating to the point of resonance frequency under the effect of violent winds, causing it to be completely destroyed. This shows just how important it is to prepare reliable models in advance of any project so that this kind of behavior may be predicted.

Wind tunnels, such as in aeronautics, allow us to inspect the behavior of the structure on the ground without needing to perform tests in flight. The Euler and Navier–Stokes equations have made it possible to rigorously define a physical framework for characterizing the behavior of the aircraft in terms of a set of different parameters such as the velocity or Mach number. Finite element models have greatly simplified the process of representing an aircraft model and its structure, as well as the way that the aircraft responds to stress.

However, the complexity of the studied phenomena is reflected in the prohibitive computational costs, which motivates us to search for reduced models with more realistic computation times. By a reduced model, we mean a description as a low-dimensional system obtained by analyzing classical numerical formulations. Acheiving this reduction incurs an initial cost, but this cost is largely offset if the reduced model is later found to be applicable for configurations of parameters other than those of the initial formulation.

Thus, just like in other areas of the industry, optimization research is extremely active within the aviation sector. One significant development since the late 1980s has been the introduction of uncertainty parameters into numerical models. Optimization techniques in the presence of uncertainty in aerodynamics have only been studied more recently, beginning in the early 2000s. Their introduction was motivated by the need to account for specific types of situations that make it too difficult to precisely evaluate the aircraft’s behavior. For example, during the aircraft design phase, in order to meet the various different criteria or eliminate certain problems encountered by the model, the model is able to adjust itself to more effectively meet the requirements and needs that it is designed to satisfy. The initial drafts of the model are not fixed, but for safety reasons it is necessary to ensure throughout the development process that the structure is capable of withstanding the stresses that it is likely to encounter in operating conditions. One way of accounting for these potential changes is to introduce uncertainty into the model.

Furthermore, when designing aircraft, manufacturers are naturally interested in maximizing the performance of each vehicle: reducing pollution, noise, drag, increasing the range, maximizing stability, etc. Minimizing the structural mass is an important objective for manufacturers as it allows other optimization criteria such as reducing pollution or extending the range to be satisfied. But less mass will also have negative repercussions on other criteria, including the stability of the aircraft in flight, for example by rendering it susceptible to the phenomenon of “fluttering”.

Manufacturers must therefore perform constrained optimization: minimizing the weight of the wing while ensuring that fluttering cannot affect the airplane within its flight envelope. In such a case, optimization problems have a cross-disciplinary character, since they exhibit behaviors that include both structural and aerodynamic aspects.

The goal is now to integrate the aspect of uncertainty mentioned above into the optimization process. However, to do this, we must first identify the nature of these uncertainties, and decide how we should represent them. Several types of uncertainty have been identified and classified according to their nature.

Accounting for uncertainty has been studied in a number of research areas, but, until recently, in aeronautics research it was not possible to account for or quantify structural uncertainties within the optimization procedures due to the limitations of numerical tools and a lack of theoretical understanding of their impact within reliability studies. Engineers have therefore been forced to implement alternative procedures to simplify the integration of structural uncertainties into model development. The first studies on this topic in the aviation sector were only conducted in the 1990s, at which point this field of research began to produce tangible results.

In the case of optimization problems with probabilistic constraints, reliability-based optimization, which is extremely common in industrial contexts, replaces these probabilistic constraints with another deterministic optimization problem derived by techniques of approximation. The primary difficulty lies in evaluating the reliability of the structure, which is itself the result of another given optimization procedure. Reliability analysis is performed at the optimal point in order to determine the reliability index of the limiting state that is being considered.

This book presents the different aspects of fluid–structure interaction: vibroacoustics and aerodynamics, and the various numerical methods used to simulate them numerically.

One chapter is devoted to the question of model reduction in fluid–structure interaction problems. We begin by presenting dynamic substructuring methods in linear and nonlinear cases. We then give a description of the method of proper orthogonal decomposition for fluid flows. Finally, we present a modal synthesis method for solving large-scale coupled fluid-structure problems. This method couples a dynamic substructuring method of the type proposed by Craig and Bampton with an acoustic subdomain method based on an acoustic formulation of the velocity potential.

To account for uncertainty, one chapter presents concepts associated with reliability and its objectives and benefits in mechanics, methods for calculating the probability of failure, simulation methods such as the Monte Carlo and response surface methods, and approximate methods for analyzing the reliability and calculating the reliability index by the first-order reliability method (FORM) and the second-order reliability method (SORM). We then give a detailed presentation of the implementation of the latter approach in the context of a certain set of reliability-based optimization problems encountered when designing structures that interact with flowing fluids, with the goal of detecting the critical frequency bands that might cause the structure to experience damage or destruction.

We would like to thank everyone who has contributed in some way to this book, and in particular the engineering and PhD students at INSA in Rouen whom we have supervised over the past few years.

Abdelkhalak EL HAMIBouchaïb RADIDecember 2016

Recently, several new problems have been formulated in the area of fluid–structure coupling, for example in the automotive industry with the dynamics of airbag inflation and fluid sloshing inside tanks; in aeronautics with the fluttering phenomenon affecting airplane wings, which involves a coupling between the vibrational dynamics of a structure and the flow of a fluid; and in the transportation industry with studies on noise reduction inside vehicles based on vibroacoustic analysis.

Each and every structure in contact with a fluid is subject to phenomena involving mechanical fluid–structure couplings to some extent. This kind of multiphysics coupling often significantly affects the dynamic behavior of mechanical systems. Taking it into account is one of the major challenges in calculating the dimensions of structures, especially when the objective is to ensure that their design meets the necessary safety requirements.

In this chapter, we will examine problems relating to the interaction of a structure with fluids both at rest and in flow. We will give a description of the motion of the fluid based on vibration theory, considering small vibrations in the structure and fluctuations in the pressure of the fluid around a stable equilibrium state, and we will present the relevant equations in the case of flowing fluids and the corresponding numerical methods for calculating couplings with dynamic structures.

The mechanical coupling between the two media acts in both directions at their surface of contact: deformations in the structure resulting from the forces applied by the fluid flow modify the state of the fluid–structure interface; this affects the flow conditions of the fluid, which induces a change in the forces exerted on the structure at the interface, thus bringing the interaction cycle to a close.

Figure 1.1.Fluid–structure coupling mechanism. For a color version of this figure, see www.iste.co.uk/elhami/interactions.zip

The fluid–structure interaction is described as the exchange of mechanical energy between a fluid and structure. This definition encompasses a wide range of problems. We can classify these problems using two criteria according to the physics of the problem at hand. The first criterion, proposed by Axisa [AXI 01], is based on the nature of the fluid flow. If the flow is negligible or non-existent, we say that the fluid is stagnant. Otherwise, we say that the fluid is flowing. In the first case, the objective is to describe small movements of the fluid and the structure around an equilibrium rest state. In these conditions, we choose to describe the dynamics of the interaction as a function of frequency; the equations describing the behavior of the structure and the fluid are written in terms of the reference (rest) state and generally lead to linear problems. In the second case, the objective is to establish a description of larger scale motion in the fluid and/or the structure. In these conditions, we choose to describe the dynamics of the interaction as a function of time; the equations describing the behavior of the structure and the fluid are written in terms of the current state of the system and generally lead to nonlinear problems.

The second criterion considers the coupling strength, which may be defined as the magnitude of the interactions or exchanges between the two media.

A coupling is said to be strong if there are high levels of exchange between the two media, i.e. the fluid has a significant impact on the structure, and vice versa. A coupling is said to be weak if the effect of one of the media dominates that of the other (Figure 1.2).

Figure 1.2.Examples of fluid–structure interaction problems [GAU 11]. For a color version of this figure, see www.iste.co.uk/elhami/interactions.zip

Three dimensionless numbers have been suggested to classify these problems [DEL 01]:

– The

mass number M

A

is defined as the ratio between the density of the fluid

ρ

f

and that of the structure

ρ

s

:

[1.1]

This describes the significance of the inertial effects of the fluid and the structure. If its value is close to one, the inertial effects of the fluid are comparable to those of the structure, and so must be taken into account.

– The

Cauchy number C

y

is the ratio between the dynamic pressure and the elasticity of the structure, which is quantified by Young’s modulus

E

.

[1.2]

This indicates the significance of the deformations induced by the flow. If this number is small, i.e. if the structure is rigid or the fluid velocity is small, structural deformations are negligible.

– The reduced velocity

V

r

is the ratio between the characteristic flow velocity and the velocity of wave propagation inside the structure:

[1.3]

If this number is large, the fluid dominates the problem from the perspective of time, and the dynamics of the structure are not important. By contrast, the dynamics of the structure increasingly dominate as this number tends to zero. If the number is close to 1, both dynamics carry similar weight in the problem.

These numbers are highly convenient for checking the importance of each phenomenon within the context of a given problem. However, as is the case for most dimensionless numbers, it is still difficult to define a priori threshold values applicable to all problems. In each problem, the large or small terms in the above will correspond to very different numerical values.

Using numerical simulations allows us to understand and predict the dynamic behavior of structures coupled with fluids, which is valuable in a number of industrial sectors. The numerical methods that we will use require us to solve the mathematical equations that model the behavior of the coupled fluid–structure system.

In general, the formulation of a coupled problem is based on the following description:

– the structure problem is formulated in terms of the displacement; the goal is to describe the behavior of the structure as a function of the displacement

u,

strain

ε

(

u

), stress

σ

(

u

), and to solve the equations of this dynamic to find the

u

,

ε

(

u

) and

σ

(

u

) fields in the structure domain;

– the fluid problem is formulated in terms of the pressure/velocity; the goal is to describe the behavior of the fluid as a function of the pressure

p

and the velocity

v,

to solve the equations of conservation of mass and momentum and to find the

p

and

v

fields in the fluid domain;

– at the fluid–structure interface, the mechanical exchanges are represented, on the one hand, by considering the force

φ

exerted by the fluid as a boundary condition for the structure problem and, on the other and, by considering the velocity imposed by the structure as a boundary condition for the fluid problem.

The energy exchanges between the fluid and the structure occur simultaneously. This needs to be taken into account by the numerical simulation. Coupled simulations can implement a single computational program to simultaneously solve the equations of the fluid and structure problems or alternatively can have two separate programs, one dedicated to the fluid problem and the other to the structure problem. The degree of complexity of the numerical simulation depends on the problem and methods of spatial and temporal discretization used to solve the equations of the problem.

Figure 1.3 proposes an overview of the most suitable general methods for simulating fluid–structure interaction problems:

Figure 1.3.General methods for numerically simulating fluid–structure interactions

There are several suitable coupling methods for the kinds of problems that we typically encounter. The following methods are used for stagnant fluids:

– The

decoupled method

finds the load or hydrostatic pressure on the structure, and then uses the results as an input to solve the deformation in the structure problem.

–

Acoustic fluid formulations (in terms of frequency)

allow small displacements around the equilibrium position of a structure to be determined. If the fluid is heavy, the vibrations of the structure and the fluid are strongly coupled. This coupling is reflected in the distinct natural frequencies of the modes, and the shapes of these modes. These methods use formulations that can be either non-symmetric (

u

,

p

) or symmetric (

u

,

p

,

φ

). They were proposed by Morand and Ohayon [MOR 95] and illustrated by Sigrist [SIG 11].

Frequency-based formulations of the fluid potential are applicable to problems with stagnant fluids, but can also be used to describe the elevation of a free surface subject to sloshing. The goal is to determine the motion of the free surface in order to find the pressure variations along the walls. These methods are similar to acoustic fluid formulations, which use either symmetric or non-symmetric expressions for the coupling equations, written as (u, p0) and (u, h, φ) [SIG 11].

In this book, we will consider fluids that are flowing. It is important to note that the solutions of flow problems are based on Eulerian formulations. This kind of formulation is particularly well suited to the study of flows and greatly simplifies the process of solving the equations of the fluid problem.

Solving the deformation of a structure more naturally leads to a Lagrangian description. Expressing the interaction problem between a flowing fluid and a structure introduces an additional complication into the choice of formulation for the problem, as this formulation must be compatible with the models of both the fluid and the structure. Existing methods tackle this issue in different ways, which allows different levels of interaction to be taken into account.

–

Monolithic approaches