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原文：A finite element is an analysis method that uses an infinitely refined mesh instead of an entity and computes it.Engineers use a wide range of tools and techniques to ensure that the designs they create are safe. However, accidents sometimes happen and when they do, companies need to know if a product failed because the design was inadequate or if there is another cause, such as an user error. But they have to ensure that the product works well under a wide range of conditions, and try to avoid to the maximum a failure produced by any cause. One important tool to achieve this is the finite element method.“The finite element method is one of the most powerful numerical techniques ever devised for solving differential (and integral) equations of initial and boundary-value problems in geometrically complicated regions.” (Reddy, 1988). There is some data that can not be ignored when analyzing an element by the finite element method. This input data is to define the domain, the boundary and initial conditions and also the physical properties. After knowing this data, if the analysis is done carefully, it will give satisfactory results. It can be said that the process to do this analysis is very methodical, and that it is why it is so popular, because that makes it easier to apply. “The finite element analysis of a problem is so systematic that it can be divided into a set of logical steps that can be implemented on a digital computer and can be utilized to solve a wide range of problems by merely changing the data input to the computer program.” (Reddy,1988).The finite element analysis can be done for one, two and three-dimensional problems. But generally, the easier problems are those including one and two dimensions, and those can be solved without the aid of a computer, because even if they give a lot of equations, if they are handled with care, an exact result can be achieved. But if the analysis requires three-dimensional tools, then it would be a lot more complicated, because it will involve a lot of equations that are very difficult to solve without having an error. That is why engineers have developed softwares that can perform these analyses by computer, making everything easier. These softwares can make analysis of one, two and three dimensional problems with a very good accuracy.A basic thing to understand how finite element works is to know that it divides the whole element into a finite number of small elements. “The domain of the problem is viewed as a collection of nonintersecting simple subdomains, called finite elements The subdivision of a domain into elements is termed finite element discretization. The collection of the elements is called the finite element mesh of the domain.” (Reddy,1988). The advantage of dividing a big element into small ones is that it allows that every small element has a simpler shape, which leads to a good approximation for the analysis. Another advantage is that at every node (the intersection of the boundaries) arises an interpolant polynomial, which allows an accurate result at a specific point. Before the finite element method, engineers and physicians used a method that involved the use of differential equations, which is known as the finite difference method.The method of the finite element is a numerical technique that solves or at least approximates enough to a solution of a system of differential equations related with a physics or engineering problem. As explained before, this method requires a completely defined geometrical space, and then it would be subdivided into small portions, which together will form a mesh. The difference between the method of the finite element and the method of the finite difference is that in the second one, the mesh consists of lines and rows of orthogonal lines, while in the method of the finite element the division does not necessarily involves orthogonal lines, and this results in a more accurate analysis The equations used for the finite element method are a lot, but they have the basison some single equations that describe a particular phenomenon. For example :The elliptic equation is described by220dxyThe parabolic equation is described by 2tThe hyperbolic equation is described by120txNo matter which is the cause of the internal forces and the deformation that they cause, there are three basic conditions that allow the finite element analysis: the equilibrium of forces, the compatibility of displacements and the laws of material behavior. “The first condition merely requires that the internal forces balance the external applied loads.” (Rocket et. al., 1983). That is the most important condition, but the other two assure that the system will be a statically determinate problem. Another condition that must be taken into account is that there exists a relationship between the load applied and the deformation, and this is given by Hookes law, as explained in past chapters, but only in the elastic range.Boundary conditions are the limitations set for the problem. These limitations are necessary in order to solve it, because otherwise, the system would be taken as a rigid body. The limitations stated by these boundary conditions are like where does the element is likely to move, and were it is restricted. If there were not boundary conditions, the body would be floating in the space, and under the action of any load, it would not suffer any deformation, but it would move around the space as a rigid body. So, when assuming boundary conditions, it has to be assured that the element has enough of them in order to prevent moving as a rigid body. Once this is done, the values of the displacements are obtained and can be substituted in the last equation seen, which will give that the displacement is equal to zero, because the element can not move in any direction. Then, algebra is applied, and the values of the forces or of the stiffness are known. 7.1 Applications of the finite element method in engineeringEvery designed made has to fulfill certain specifications, and among them is working under a variety of conditions: temperature, humidity, vibrations, etc. The job of the designer is to achieve this, and to assure that the product will work effectively, taking care of the user and of the element. An engineer has to follow certain steps in order to create a good product with high quality. First, the steps of the design flow chart must be followed:1. recognize a need2. specifications and requirements3. feasibility sturdy4. creative design and development5. detailed drawings6. prototype building and testing7. design for manufactureThese are the basic concepts of the design, but it involves a lot of other things to be able to assure a good performance of the product. After the designer has the calculations of the dimensions, tolerances, manufacturing parameters, etc, some other tests should be done. For example, an engineer has to know whether the product is going to support certain loads, or how it is going to behave with temperature variations, or what could happen if vibrations are present. This is where the finite element method enters in action in engineering. In the last pages, it has been explained how the finite element works and which are the basis of it. Now it is time to explain what is it good for, the applications, the benefits, etc. For example, the first application of the method was introduced by Richard Courant to solve torsion on a cylinder. Then, in the middle 50s, the method started to be applied for airframe and structural analysis, and then used in civil engineering. In general, finite element methods are used in a wide variety of engineering applications, like in computer graphics, heat transfer, electrical and magnetic fields, among others. Use of finite element method in mechanical engineering is very wide. For example, it is used in mechanics of materials, for structures and trusses. It is used to understand and to prevent how some structures are going to behave under the action of some loads. For example, for a bridge, how is it going to behave with the vibrations, or with the effect of the air, or with the variations of temperature. The aircraft industry uses this method to determine the static and dynamic answer of planes and space crafts to the great variety of environments and conditions that can be found during their operation.In the case of mechanics of fluids, the method is used to know how a wing of a plane is going to behave with the air flux passing through it, if it is going to resist, how much vibration is this going to cause, in order to avoid resonance (because of the vibrations caused by the turbines too). The finite element method allows calculating the drag and the flotation forces caused during operation. For heat transfer, it allows to know how a turbine is going to behave, and how the material is going to be affected for the effect of the heat. It is known that heat also creates stresses, and this is a fundamental concept when talking about design of turbines. Another important thing to mention here is that when turbines are working, they reach very high temperatures, and they use a coolant to maintain the temperature under some point. This coolant, when touching the hot turbine, creates a thermal shock, which also produces stresses. These stresses can be measured through the finite element method. Something else that it is important is that some components of the turbine (like blades) have holes to let the air in to act as a coolant to avoid overheating. These holes can act as stress concentrators, and by this method, it can be analyzed which is the way they affect less the stress distribution.7.2 AdvantagesThe use of the finite element method has a lot of advantages, and most of them have already been commented in this thesis, but here it is going to be made a short condensation of them.First, it is a very important tool for stress and strain analysis, not only because it provides accurate information, but because it also saves a lot of money and time by simulating the events in computer and not in real life. It is a really easy to use tool. Once some tutorials are followed, the only thing left to do is to explore a little bit of the software, and then to apply all the knowledge acquired. The principles of the method are easy to understand, even if when the model is complex, the analyses are also complex. For every engineer it is a very reliable tool, because it is very specific for each occasion, and it is able to perform different analyses of the same model under different circumstances with simple changes in the boundary conditions, in the loads, in the material, or whatever the problem demands.The method can help to modify each design in order to increase the service life of it as much as possible. That is the case of study of this thesis, where some analyses are run in the computer to see how the presence of a stress concentrator affects the behavior of the element. If the stress concentration is high, the element can be modified with ease and then subjected to analysis again, and depending on the results, a decision has to be taken to see if it needs more changes or if it has reached or if it is even close to its maximum service life condition.The combination of this as a software with other types of softwares is a very useful tool, because the program of finite element analysis allows the designer to import models from other CAD softwares, and that way it does not forces the engineer to make the solid in the FEA CAD. That way, complex models can be created and analyzed with the combination of those two powerful tools.
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