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The finite element method in engineering.

engineering calculation (Finite Element Analysis)
One of the fundamental tasks of the engineer is the analysis and calculation, this is the quantitative prediction of system behavior (system) technology or process to proceed with design efficient or to meet production specifications.
Examples thereof may be found in areas of heat flow, fluid mechanics, electromagnetics, chemical reactions and others.
This should make use of concepts of physics, chemistry and mathematics, to develop a mathematical model of the system or process under consideration. This model is only a system of equations whose unknowns represent magnitudes of technological interest for describing the behavior of the object under analysis. Consequently, to perform the prediction itself, the engineer has to solve the above equations quantitatively to engage, then to the technical interpretation and analysis of Results. many situations, the relevant models involve boundary value problems governed by partial differential equations. To mention some of these cases may include the structural study of automobiles, airplanes, bridges, or field analysis of heat flow in machine components, fluid flow, filtration in earth dams, etc.Debido to the great difficulty obtain analytical solutions to the equations cited, engineering has been used historically to use simplified models based on experimental results, experience and the best in a few particular mathematical solutions for a more accurate model. This general engineering methodology has been very successful and is still doing. However, it is important to note that this is a methodology that presents strong limitations on the scope of analysis, a fact that becomes more serious when one considers the growing needs of technology has changed moderna.Este box with the advent of electronic computer and the associated development of computational methods. In the context referred important numerical techniques have appeared among the most important are the methods of finite differences, boundary elements and elements finitos.En particular the latter is the more powerful and therefore more utilizado.A the following is a brief description of it and then comment on some of its many applications in engineering practice.



engineering problems.

engineering problems are studied with mathematical models that represent physical situations, these models are differential equations with boundary and initial conditions determinadas.Las differential equations are derived, which are derived by applying fundamental laws and principles of nature to systems, these represent the mass balance, strength or energy.
When possible, the exact solution These equations show the behavior of a system under study under certain conditions, analytical solutions are composed of two parts (1) a homogeneous part and (2) a particular part.
In any engineering problem, there are two kinds of parameters that influence the way the system behaved, first are the parameters that give information on the natural behavior of a given system, these include properties such as modulus of elasticity, thermal conductivity and viscosity (see picture).
Then there are the parameters that produce a disturbance or alteration in the system as external forces, moments, temperature difference the middle and pressure differences in fluid flow.
natural behavior of a system that follows is the homogeneous part of the solution of differential equations, in contrast, the parameters that cause disturbances appear in the particular solution.
is important to understand the role of these parameters in modeling with finite element techniques, in terms of their respective appearances of the so-called matrix hardness or rigidity (stiffness) and conductance matrices and load or force.
characteristic systems always have the stiffness matrix, the matrix of conductance or resistance matrix, while the parameters that produced riots in the matrix load.

numerical analysis.

There are many practical problems in engineering which can not obtain the exact solution, this can be attributed to the natural complexity of differential equations or difficulties that may occur with the boundary conditions or initial.
To address these problems we numericas.En contrasting approaches to the analytical solution, which shows the exact behavior of a system at any point on it, the numerical solutions approximate the exact solution only at discrete points called nodes.
The first step in any numerical procedure is the discretization, this process splits the middle interest in a small number of subregions and nodes.
There are two kinds of numerical methods: (1) finite difference method and (2) finite element method.
With the finite difference method, the differential equation is written for each node and the derivatives are replaced by difference equations, this is achieved with a set of simultaneous linear equations, although this method is easy to understand and use simple problems, presents difficulties when applied to complex geometries and complex boundary conditions, this situation is true for problems with non-isotropic material properties (which have the same properties in all directions).
In contrast, the finite element method uses integral formulations rather than a difference equation for a system of algebraic equations, moreover approximate a continuous function is assumed to represent the solution for each element, the complete solution is generated by connecting or assembling the individual solutions, allowing the continuity of the boundaries ínterelementales.
then drawn some engineering systems and their parameters,





Basics finite element method

is a general method for solving boundary value problems governed by ordinary or partial differential equations. In essence it is a technique which replaces the algebraic differential problem on the other, roughly equivalent to the general techniques of known resolution. To do this makes use of "discretizing" or subdivision of a region over which the equations are defined in simple geometric shapes called finite elements. The material properties and relationships governing these elements are expressed in terms of unknown values \u200b\u200bin the "corners" of the elements or nodes (see Figure 1).
One advantage of this method is its ease of implementation in a computer program, which in turn is a condition base for use as the treatment of a particular problem must be a very large number of operations to solve algebraic systems of the order of hundreds or thousands of equations. However, this amount is not a limitation with today's standard computers.
The basic ideas of this method originated in progress in the structural analysis of the aviation industry in the 50s. In the '60s the method was generalized to the approximate solution of problems of stress analysis, fluid flow and heat transfer. The first book on finite elements was published in 1967 by Zienkiewicz and Cheung. In the 70's The method was extended to the analysis of nonlinear problems of continuum mechanics. Today, the method can solve almost any physical situation can be formulated by a system of differential equations.
In the beginning the finite element method failed massively to the practice of engineering because of the unavailability of computers in engineering education and on the other the requirement of expert knowledge not only of technical and mathematical models but also relevant computer programming. Currently, the situation is completely different, and that modern personal computers seamlessly support powerful general-purpose programs easy to use.



Figure 1

The process of analyzing a physical problem using finite elements shown in Figure 1. The geometry can be defined by the analyst or created from a CAD program. The second step is to define the mathematical model to solve. This is the fundamental step which specifies the type of equations to determine the boundary conditions, material properties, and other details about the method itself. Once completed this definition the program automatically solves the relevant equations and provides results in a form appropriate to the analyst. Applications



elastohydrodynamic model of a bearing.

Figure 2 shows the pressure distribution developed in a bearing. Such a solution was reached by considering the two-dimensional elastohydrodynamic coupled problem, ie trying together to obtain the pressures of the flow of lubricant within the bearing (using the modified Reynolds equation) and modeling it as an elastic plate simply supported at its ends. This treatment allows an approximation better than conventional analytic models of pressure distribution inside the bearing when considering strain simultaneously its effects. It is also possible to consider the effects on the edges of the bearings to work with a two-dimensional model. Figure 2



Dynamic study of a dam - reservoir

This example has developed a dynamic analysis of a gravity dam with the reservoir of liquid that interacts with it. It has been considered for this horizontal sinusoidal excitation (representative, eg an earthquake) on the solid part of the domain (prey). From the classical analytical standpoint, this problem must be addressed by obtaining, first, the hydrodynamic pressure in the reservoir, considering the dam as a rigid body, at a later stage can determine the elastic problem in the structure. Using a finite element computational model can be analyzed simultaneously the dynamics in the reservoir and dam. Can easily be other complexities such as interaction with stratified foundations, inspection of galleries in the dam, the dam complex geometries, etc. Figure 3 shows the finite element mesh of the model, while Figure 4 shows the pressure at the interface between liquid and solid media.


Figure 3

Figure 4


Concluding remarks

This article has sought to spread briefly one of the most important tools with which the engineer has to analyze complex problems a few years ago were intractable.
Paradoxically, the curriculum of the engineering still gives this method the proper place, thus reducing the possibilities of analysis of future engineers. This situation must be remedied in the short term in order to train engineers to be able to use powerful computer technology that are at hand for modern and more efficient designs. Link to

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English words for searching: Finite Element Procedures, engineering analysis, stress analyzes, Vibration Analysis, computer simulation.


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