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Basic Tools for Reliability Analysis

1.1. Introduction

The modeling of a mechanical system can be defined as the mathematical idealization of physical phenomena that control it. This obviously requires us to define input variables (geometric parameters of the system, loading conditions, etc.) and output variables (displacements, stresses, etc.) that will help to understand the evolution of the mechanical system. The models used are more complex and accurate and the current issue is the identification of the parameters constituting them. In fact, we can no longer afford, in dealing with certain types of problems, to use deterministic models where the mean values interpose only because that generally leads to a very erroneous representation of reality. This chapter presents the basic tools for the development of a model of reliability in biomechanics especially in prosthesis design. This model is considered implicit and requires a numerical simulation to identify the parameters of structural response. To estimate this response, with good accuracy, the finite element method (FEM) appears as a preferential numerical simulation tool. This method consists of discretizing a structure, such as a prosthesis, into a set of subdomains, called finite elements or mesh, linked to each other by nodes. The calculation of a structure is to establish an equation system for the displacements of the set of all meshing nodes and deduct, pursuant to their resolution, the approximations of the deformation and stress fields. Then, the reliability itself is performed by a process of optimization. To perform the optimization and reliability, a sensitivity analysis is required to determine or identify the role of each parameter with respect to the constraints and objective functions.

1.2. Advantages of numerical simulation and optimization

In general, biomechanical models are very complicated. It is necessary to have tools and methods to design these models for analyzing satisfaction levels. With regard to the mechanical behavior of the structure, the engineer or designer has a wide range of methods: methods based on knowledge (empirical laws, databases, etc.), simplified calculation methods (strength of materials), the FEM which is the most widely used and methods of optimization. The implementation and relevant use of the FEM require a certain experience. However, in a highly competitive industrial context, this method allows us to:

• - reduce costs (optimization of shapes and material volumes, choice of materials, reduction of the number of prototypes, etc.);
• - reduce the time (reduce the iteration number in the design process, directly propose viable solutions from the behavioral perspective, focus testing, etc.);
• - improve the quality (ensure the respect of the various functions and constraints in terms of reliability, comfort, ergonomics, etc.).

The scope of the FEM is very large. They have proved their effectiveness in the case of problems simple such as in that of great complexity calculations. This field covers all applications of the structural mechanics (statics, plasticity, composites, dynamics, shock, friction, etc.) and also the mechanics of fluids, rheology, heat exchange, electromagnetic calculations, etc. [POU 99].

The use of the FEM in the medical field has an additional interest. In fact, contrary to the fields of automotive or aeronautics, the designed products are not intended to equip a car or an airplane but a human being. This induced on the first hand that it is often impossible to test a device on humans as we perform a crash test on a car to test the operation of an airbag, for example. On the second hand, for medical devices, in particular the implantable devices, the reliability is essential because maintenance is not possible. For the design of these devices, the FEM brings not only the advantages mentioned previously (cost, time and quality) but also helps provide the anatomical models (part of the body, biological tissues, etc.). The use of such models is common and can simulate the behavior of devices in location [AOU 10, RAM 11].

1.3. Numerical simulation by finite elements

1.3.1. Use

The design process causes a succession of choices and decisions that lead to the final definition of the product. Numerical simulation has an important role to play in the realization of these choices and can therefore be used at various stages of this process. Depending on the stage of the design, we can differentiate two types of numerical calculation: the calculation for assistance choice and the calculation for validation [POU 99]. The first consists of comparing technical solutions in order to choose the one that meets the criteria set. Contrary to the calculation for validation, it is not the absolute character of the result which is interesting but rather the comparison of the result with that of one or several competing solutions. This type of use allows us to realize simplified models simulations (geometry, behavior laws of materials, etc.). A complete knowledge of the product being not necessary for the simplified model calculation, it may be undertaken in the early stages of the design and prove its interest. In the case of the calculation for validation, the implementation of a numerical simulation requires as a starting point a semi-complete definition of the studied device (forms, dimensions and materials) and its surroundings (boundary conditions, loads, etc.). It comes at a relatively advanced stage of the project to validate a product definition, and therefore a first set of design choices relative to the specifications. The concept of validation implies a good reliability level of the obtained results. It is therefore appropriate to build a suitable calculation model to reproduce a sufficiently precise mechanical behavior of the piece(s). One step of the model validation is then unavoidable. The use of the FEM for the validation of a product is reflected generally (except some very simple cases) by a fine representation of the geometry and thus a model which is complex and costly in terms of computing resources.

1.3.2. Principle

To treat a problem, FE calculation software requires a certain number of input data. These data consist of a complete description of the mechanical problem to be treated (under a unique formalism to each software) as well as knowledge of the parameters related to the treatment method (for example, the type and distribution of finite elements, the steps of resolution, etc.). The software returns results on the physical quantities of interest in the problem (displacement, temperatures, stresses, etc.). The use of this type of tool therefore requires that the user should be able not only to provide relevant input data, but also to assess the reliability of the obtained results. All that requires a good knowledge of the concerned domain as well as an experience of the theoretical aspects of the method. The problem is considered in the form of a model that first defines the geometry of the structure and the boundary conditions (efforts and displacement). In this case, the FE is more widespread and exploited and it is called the "displacement method". It consists of determining the "displacement field", which means the displacement at each point of the structure. In order to represent the displacement field of a structure, the studied field must be discretized using elements of simple geometric shapes (straight line, triangles, quadrangles, tetrahedra, etc.) and of finite dimensions, called "finite elements". The set of discretization constitutes the "mesh" of the structure. The displacement approximation is then performed independently on each FE. The accuracy of the results of the calculation is directly related, on the one hand to the choice of elements and on the other hand to the quality of the realized mesh (number of elements, distribution in the structure, shape of elements, etc.).

1.3.3. General approach

In general, the development of a FE model can be described by a succession of steps (Figure 1.1). Some of these steps may differ slightly or be reversed depending on the modeling software used [AOU 11]. The first step is to create the model geometry either by drawing or by importing when there are already files of the STEP, IGES, etc. type. Next, it should be to assign material characteristics to each part of the geometry and then to perform the mesh. The mesh step includes the choice of the element type, their number and distribution depending on the geometry. The next step defines the boundary conditions (loads, displacements, fixations, contacts, etc.). It allows us to recreate utilization conditions of the product or the modeled part. When these four steps are performed, the model is constituted and the calculation can be started. In the case where the calculation is not converged (divergence case), it is necessary to return to the previous steps to refine the different parameters: geometry, behavior laws of materials, mesh, or boundary conditions and loadings.

When the calculation is successful, the numerical results (output data) can be analyzed. Following the development of a FE model, a validation step is essential to prove its credibility. It is therefore essential, as a first step, to simulate a known situation to compare the simulation results with proven results (for example, experimental results) and then to make modifications that will harden the model. Only the correlation of these results allows the validation of the...