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. 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. 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:

• - Behavioral laws:

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

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 s (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. 1) a solution u(z, t) exists ? z? O, ?t ? [0, T] for and given;
2. 2) the solution u(z, t) is unique;
3. 3) the solution permanently depends on and .

In particular, this posture requires ?O = ?Ou ? ?fO and ?uO n ?fO = 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 ?fuO 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 ?0O can lead to non-unicity. This is particularly the case when ?fuO = Ø. In this instance, prescribing boundary data on force or displacement on ?0O makes the problem well-posed.

In our case, we are aiming to find the set of model parameters ? 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 O 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...