Dynamics of Large Structures and Inverse Problems

By Abdelkhalak El Hami and Bouchaib Radi

This book deals with the various aspects of stochastic dynamics, the resolution of large mechanical systems, and inverse problems. It integrates the most recent ideas from research and industry in the field of stochastic dynamics and optimization in structural mechanics over 11 chapters. These chapters provide an update on the various tools for dealing with uncertainties, stochastic dynamics, reliability and optimization of systems. The optimization–reliability coupling in structures dynamics is approached in order to take into account the uncertainties in the modeling and the resolution of the problems encountered.

Accompanied by detailed examples of uncertainties, optimization, reliability, and model reduction, this book presents the newest design tools. It is intended for students and engineers and is a valuable support for practicing engineers and teacher-researchers.

• Language: English
• Publisher: Wiley
• Release date: Jul 17, 2017
• ISBN: 9781119427315

Abdelkhalak El Hami

Abdelkhalak El Hami is Professeur des universités at the Institut National des Sciences Appliquées (INSA-Rouen) in France and is in charge of the Normandy Conservatoire National des Arts et Metiers (CNAM) Chair of Mechanics and Head of the department of mechanical engineering of INSA Normandy, as well as several European pedagogical projects. He is an expert in fluid–structure interaction studies, reliability and optimization.

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Preface

In recent years, engineers, scientists and decision makers have expressed a strong interest in the dynamics of large structures and inverse problems. These two fields have attracted growing interest due to their industrial applications. The problems in structure dynamics are very important, notably the trend of building structures that are more and more supple and subject to frenzies that fluctuate more and more quickly in time. However, a dynamic analysis of large industrial structures is often based on model reduction techniques. With this aim, we will present some solution methods for large systems.

Within the sphere of calculating structures, the finite elements method enables one to determine a structure’s physical response to an applied force. This technique not only enables one to determine the stress states on a mechanical structure’s interior, but also to model the complete manufacturing processes, for example. Nowadays, the significantly reduced calculation time allows us to address so-called inverse problems. By repeating the calculations by finite elements while modifying the material’s parameters or the structure’s geometry, one comes to identify an optimal solution for the problem in question. The procedure, which couples optimization and calculations by finite elements, is of utmost importance for the manufacturing industry, for example, as this virtual development reduces the time and costs involved in developing new products. Those who understand the difference use the terminology of inverse problem, as opposed to that of direct problem, to refer to solving a differential equation based on the known parameters in order to calculate the system’s response. In the instance of an inverse problem, the system’s response is assumed to be known. Therefore, we aim to determine the physical or geometrical parameters which, when used in direct problems, allow one to find the prescribed system’s response. Inverse problems also involve an objective function, to be constructed according to the application, measuring a gap between the known response and the responses obtained from the sets of different parameters, by solving the direct problem. There are two large categories of techniques for solving an inverse problem:

1) Gradient-type techniques: They consist of identifying the minimum of the objective function as a point where the gradient of this function is cancelled.

2) Stochastic methods [ELH 16].

This book includes the most recent ideas resulting from research and from the industry in the field of large structure dynamics and inverse problems. It consists of 11 chapters. These chapters take stock of the different tools used to handle condensation methods, linear and nonlinear model synthesis, identification, resetting, sensitivity, optimization, reliability and some inverse problems.

Each chapter has clear explanations of the techniques used and developed, and are accompanied by fully illustrated examples.

Chapter 1 introduces problems related to inverse problems.

Chapter 2 encompasses analyzing and solving first-order linear differential equations with constant coefficients. The chapter introduces a way in which it is applied to mechanical engineering for dynamic systems.

Chapter 3 presents an introduction to linear dynamics of structures. In each of the various industrial sectors (automobile, aeronautics, civil engineering, nuclear engineering, defense, aerospace, oceanic and marine engineering, etc.), it is important to determine the structure’s response to different applied forces for designing and dimensioning it. To assess this response (displacements, stresses, speed and acceleration) to a dynamic load (variable in time), there are two approaches: the determinist approach and the stochastic or non-determinist approach [ELH 16]. In this chapter, we present the general principles of linear determinist structure dynamics. This study enables one to establish the essential relations when calculating dynamic responses, when calculating frequencies, normal modes and response functions in frequencies. Finally, a few simple examples are introduced.

Chapter 4 introduces the dynamics of nonlinear structures. The objective of the chapter is to raise awareness about nonlinear specific characteristic in basic cases. Returning to the linear structure, a few basic avenues for analysis are presented, which may be sufficient for certain industrial applications.

In Chapter 5, some condensation methods are introduced. Currently, discrete models for forward calculations of structural behavior tend to be the finite elements type. Given the complexity of industrial structures, these knowledge models often involve a significant amount of degrees of freedom (DOFs). When making a dynamic analysis of such models, the size may exceed the capacity of the computers available. The discrete mechanical models considered are conservative linear models of the second order.

Chapter 6 is dedicated to introducing linear modal synthesis methods. The reader is reminded of the substructuration strategy, which was initially formulated for static problems. It consists of processing structures such as assembling substructures that are interconnected with each other. The modal synthesis methods differ in the choice of modes for representing the dynamics of each substructure and in assembly procedures. We then propose a strategy for reducing junction DOFs, after assembly. This strategy is based on the use of interface modes. These modes are obtained from condensation on the complete structure’s Guyan interfaces.

Chapter 7 introduces different reduction methods for models in nonlinear dynamics.

Chapter 8 is dedicated to analyzing a model’s sensitivity. It studies the variations of the output variations compared to the input parameters. It enables one to have improved understanding of the model’s behavior and to quantify the influence of different input parameters on the variability of the system’s output. We are often led to assess the dynamic behavioral variations due to given modifications of design variables. These are the direct problems. The design modifications variables leading to a given variation of dynamic behavior are the inverse problems. In this chapter, direct and inverse sensitivity methods are introduced.

Chapter 10 presents the robustness function in structure dynamics for inverse problems. In the probabilistic approach, the parameters are described by the probability densities and we aim to propagate this probabilistic characteristic through the mechanical model. The approach by convex models of uncertainty problems in mechanics has mainly been approached by Ben-Haim [BEN 90]. The info-gap convex models of uncertainty are defined as the gap between what is known, the nominal values of parameters, and what we want to determine, the uncertainties, to satisfy a given design criterion. We present two methods for solving inverse problems. The first one is based on interval arithmetic. The second one is a minimization problem under stress. Finally, we introduce some digital applications. A structure’s own pulse is chosen as a performance function. We use different model synthesis methods to calculate this function. We compare the results obtained with the complete model.

The objective of Chapter 11 is to introduce a methodology that couples modal synthesis techniques with optimizing the design’s reliability. We introduce an algorithm that enables modal synthesis methods to be integrated into the reliability optimization process. Finally, we assess this algorithm when used on different applications to show the effectiveness and robustness of the method presented.

Finally, this book constitutes an invaluable support for teachers and researchers. It is also aimed at engineering students, practicing engineers and master’s and PhD engineering students.

Acknowledgments

We would like to thank every person who has contributed in both big and small ways to the development of this publication, our families and particularly, the Rouen INSA PhD students for whom we have been responsible for the last few years.

Abdelkhalak EL HAMI

HAMI Bouchaïb RADI

May 2017

1

Introduction to Inverse Methods

1.1. Introduction

In the field of structural calculations, the finite elements method allows for determining a structure’s physical response to an applied force. This technique not only enables us to determine the stress states on a mechanical structure’s interior, but also to model the complete manufacturing processes, for example. Nowadays, the significantly reduced calculation time allows us to address so-called inverse problems. By repeating the calculations by finite elements while modifying the material’s parameters or the structure’s geometry, we can identify an optimal solution for the problem in question. The procedure, which couples optimization and calculations by finite elements, is of utmost importance for the manufacturing industry, for example, as this virtual development reduces the time and costs involved in developing new products.

For those who understand the difference, the terminology of inverse problem is used, as opposed to that of direct problem, to refer to solving a differential equation based on the known parameters in order to calculate the system’s response. In the instance of an inverse problem, the system’s response is assumed to be known. Therefore, we aim to determine the physical or geometrical parameters that, when used in direct problems, allow us to find the prescribed system’s response. Inverse problems also involve an objective function to be constructed according to the application, measuring a gap between the known response and the responses obtained from the sets of different parameters by solving the direct problem. Various inverse problems can be distinguished: for example, restoring a system to its past state by knowing its current state (if this system is invariable) or determining the system’s parameters by knowing (one part of) its evolution. This last problem is that of identifying parameters, which will be dealt with in section 1.2 (see Figure 1.2).

Figure 1.1. Illustration of a direct problem and its inverse problem. For a color version of this figure see, www.iste.co.uk/elhami/dynamics.zip

There are two main categories of techniques for solving an inverse problem:

1) Gradient-type techniques have often been considered in applications for which the necessary time to assess a direct problem is significant. They consist of identifying the minimum of the objective function as a point where this function’s gradient cancels itself out. This approach does not guarantee that the global minimum will be identified, but it has the benefit of quickly converging toward a minimum. This minimum will be global if the initial one is close enough to the desired solution, which is quite often the case in engineering problems.

2) Stochastic [BAR 01] or progressive methods have major significance in non-differentiable optimization and are a recourse for problems, which have local minima. Gradient methods are used when the function to be optimized is differentiable. They use the information given by the partial derivatives. In the instance of differentiable functions whose convexity cannot be guaranteed, hybrid or mixed algorithms are often used to combine the advantages of stochastic algorithms and gradient algorithms. The method chosen depends on the nature of the inverse problem (differentiable, non-differentiable, etc.) and above all on the calculation time necessary for assessing the system’s response.

1.2. Identification methods

In the general context of physics and particularly in solid mechanics, it is often necessary to assess or identify the physical quantities governing the system studied. In many cases, the quantities being searched for (Young’s modulus, coefficient damping, etc.) may not be directly measurable and one must use other measurable quantities (accelerations, strains, speeds, etc.) to obtain more information. The principle of the identification methods consists of establishing a mathematical relation based on physical laws, also called models, so that the quantities searched for (sometimes called parameters) are found from the measurements available. Thus, from a mathematical point of view, the solution to such a problem may encounter problems relating to solutions’ existence, unicity and continuity. Consequently, the identification methods can be considered to fall into the category of inverse problems where, unlike the solutions to direct problems, one must overcome the difficulty of the problem being ill-posed.

From a mechanical point of view, the reference problem that we are aiming to solve consists of studying the evolution of a structure occupying a volume in an interval of time t ∈ [0, T] (see Figure 1.2).

Figure 1.2. Area studied and its limits using the data available. For a color version of this figure see, www.iste.co.uk/elhami/dynamics.zip

The structural behavior is given by the solution to the reference problem defined by:

Find the displacement and the stresses

– Behavior equation:

[1.1]

– Behavioral laws:

[1.2]

where ε is the strain tensor and θ represents a given set of model parameters defining structural parameters (material, geometry, etc.). Moreover, the space of admissible displacements U(ū) and admissible stresses is defined by:

[1.3]

where s.r. designates functions that are sufficiently regular, defined on the confined stress and the kinetic energy for u (z, t) and integrable squared for σ (z, t), and n is the normal vector on the surface ∂f.

The problem is said to be well-posed in the sense of Hadamard [BUI 93] if, and only if, the three following conditions are verified:

1) a solution u(z, t) exists ∀ z∊ Ω, ∀t ∊ [0, T] for and given;

2) the solution u(z, t) is unique;

3) the solution permanently depends on and .

In particular, this posture requires ∂Ω = ∂Ωu ∪ ∂fΩ and ∂uΩ ∩ ∂fΩ = Ø (see Figure 1.2). In this description, the direct problem will generally be ill-posed for at least two reasons:

– the presence of overdetermined data and in ∂fuΩ generally leads to the inexistence of the solution, with the exception of the instance where and are compatible with the constitutive relation [1.2];

– the lack of data in a certain area of the boundary ∂0Ω can lead to non-unicity. This is particularly the case when ∂fuΩ = Ø. In this instance, prescribing boundary data on force or displacement on ∂0Ω makes the problem well-posed.

In our case, we are aiming to find the set of model parameters θ and the solution field u satisfying the equations of a model [1.1] and [1.2] above, which better represent the available data. Because the available data and can be noisy and overdetermined, just as the equations of the inexact model in comparison to the real physics (discretization of the area, material, etc.), the solution of this inverse problem could often be ill-posed in Hadamard’s sense as it cannot comply with one or many of the conditions listed above.

In the field of solid mechanics, various authors have studied the identification of the model’s properties based on observed data. To give an example, it has been shown in [BON 05] that, in the elastic example, the problem of finding a field of properties distributed E(z) in the entire space Ω is an ill-posed problem in Hadamard’s sense and it becomes necessary to introduce a priori knowledge, which draws near the solution.

There are various methods that exist for solving problems related to identifying a model’s properties, depending on the nature of the problem (static, dynamic, available data, etc.). The identification problem generally ends up being formulated as an optimization problem, namely researching the minimum of a cost function that quantifies the difference between a model forecast and the available data to some extent.

Among the different approaches that exist for building a suitable cost, the following families can be distinguished:

– the least squares approach [TAR 82] where the difference between the data and the solution of the direct model projected on the observation space is measured with an L2 regulation;

– an approach based on auxiliary fields. In linear mechanics, the Maxwell–Betti reciprocity theorem and the cost functions are generally constructed on the overdetermined data on the boundary area. An interesting example of using this approach can be found in [AND 97] for detecting fissures on the inside of an elastic body;

– an approach consisting of these functional functions with an energy base, and in particular those based on the error in the constitutive relation for which a detailed description is given further on.

On the other hand, if the identification problem is ill posed, it will generally lead to the solution becoming sensitive or unstable against the noisy data. In order to overcome this problem, we will distinguish two classical approaches:

– Tikhonov’s regularization techniques [TIK 77], which are largely used and where an additional term is introduced into the aforesaid cost functions. This term represents an a priori knowledge of the solution being searched for and the property of stabilizing the results with regard to the noise in the data;

– the probabilistic approaches [TAR 05, ARN 07] where the uncertainties of the data and the model are quantified with the help of a stochastic framework, and a probability density function for the unknown parameters is generally searched for.

1.3. Identification of the strain hardening law

Some authors [GAV 96, MAH 97, GHO 98, YOS 98, YOS 03, DIO 03] have instigated a new approach within the context of identifying plastic behavior. In one research paper, the authors proposed to extend the sphere of classical testing analysis by carrying out a shift of measured response