Contenido

El MEF fue al principio desarrollado en 1943 por Richard Courant, quien utilizó el método de Ritz de análisis numérico y minimización de las variables de cálculo para obtener soluciones aproximadas a un sistema de vibración. Poco después, un documento publicado en 1956 por M. J. Turner, R. W. Clough, H. C. Martin, y L. J. Topp estableció una definición más amplia del análisis numérico.[7]​ El documento se centró en «la rigidez y deformación de estructuras complejas».

Con la llegada de los primeros ordenadores instaura el cálculo matricial de estructuras. Éste parte de la discretización de la estructura en elementos lineales tipo barra de los que se conoce su rigidez frente a los desplazamientos de sus nodos. Se plantea entonces un sistema de ecuaciones resultado de aplicar las ecuaciones de equilibrio a los nodos de la estructura. Este sistema de ecuaciones se esquematiza de la siguiente manera:.

Donde las incógnitas son los desplazamientos en los nodos (vector u) que se hallan a partir de las "fuerzas" o "solicitaciones" en los nodos (vector ) y de la rigidez de las barras (matriz de rigidez ). Conocidos dichos desplazamientos es posible determinar los esfuerzos en las barras. La solución obtenida es bastante precisa si se cumplen las hipótesis implícitas en la formulación.

Practical use of the method around 1950

When the first computer equipment arrived in the 1950s, the calculation of structures was at a point where the predominant calculation methods consisted of iterative methods (Cross and Kani methods) that were carried out manually and, therefore, were quite tedious. The calculation of a multi-story building structure, for example, could take several weeks, which represented a substantial cost of time to the detriment of the possibility of investing time in optimizing the structure.

The arrival of the computer allowed the resurgence of the method of displacements already known in previous centuries (Navier, Lagrange, Cauchy), but which were difficult to apply since in the end they led to the resolution of enormous systems of equations that were unapproachable from a manual point of view.

From 1960 to 1970

As practical applications of finite elements grew in size, the computing time and memory requirements of computers grew. At that point the development of more efficient algorithms became important. To solve the systems of equations, the study of the adaptability of already known algorithms (Gauss, Cholesky, Crout, Conjugate Gradient, etc.) is promoted. Saving time is unthinkable and with it the use of the matrix method spreads. This development is especially notable in building structures where the discretization of the frames into bars is practically immediate from the beams and columns.

However, and despite developing modeling of surface elements using bars (slabs with grids, curved elements using approximations of straight elements, etc.), great difficulties arise in the face of continuous structures (surfaces and volumes) and with complex geometries. Hence, it is precisely within the aerospace field where new FEM techniques begin to be developed.

Given its generality, the method was extended to other non-structural fields such as heat conduction, fluid mechanics, etc., where it competed with other numerical methods such as the finite difference method, which, although more intuitive, once again had difficulties in planning for complex geometries.

With the arrival of computer centers and the first commercial programs in the 60s, the MEF, while becoming popular in the industry, reinforced its theoretical bases in university centers.

In the 70s there was a great growth in the bibliography as well as the extension of the method to other problems such as non-linear ones. In this decade, the MEF was limited to expensive mainframe computers generally owned by the aeronautical, automotive, defense and nuclear industries. New types of elements are studied and the rigorous mathematical foundations of the method are laid, which had previously appeared more as an engineering technique than as a numerical method of mathematics.

Since 1980

Finally, starting in the 1980s, with the widespread use of personal computers, the use of commercial programs that specialize in various fields spread, establishing the use of graphic pre- and post-processors that perform meshing and graphical representation of the results. The study of the application of the method to new behavioral models (plasticity, fracture, continuous damage, etc.) and the analysis of errors continue.

Currently, within the structural field, the FEM shares prominence with the matrix method, with many programs that mix analysis by both methods, mainly due to the greater need for memory required by finite element analysis. Thus, the application of the FEM has been left to the analysis of continuous slab or screen type elements, while the frames are still being discretized into bars and using the matrix method. And since the rapid decline in the cost of computers and the phenomenal increase in computing power, the MEF has developed incredible precision. Today, supercomputers are capable of giving quite precise results for all types of parameters.

El desarrollo de un algoritmo de elementos finitos para resolver un problema definido mediante ecuaciones diferenciales y condiciones de contorno requiere en general cuatro etapas:.

1. • El problema debe reformularse en forma variacional.

2. • El dominio de variables independientes (usualmente un dominio espacial) debe dividirse mediante una partición "Partición (matemática)") en subdominios, llamados elementos finitos. Asociada a la partición anterior se construye un espacio vectorial de dimensión finita, llamado espacio de elementos finitos. Siendo la solución numérica aproximada obtenida por elementos finitos una combinación lineal en dicho espacio vectorial.

3. • Se obtiene la proyección del problema variacional original sobre el espacio de elementos finitos obtenido de la partición. Esto da lugar a un sistema con un número de ecuaciones finito, aunque en general con un número elevado de ecuaciones incógnitas. El número de incógnitas será igual a la dimensión del espacio vectorial de elementos finitos obtenido y, en general, cuanto mayor sea dicha dimensión tanto mejor será la aproximación numérica obtenida.

4. • El último paso es el cálculo numérico de la solución del sistema de ecuaciones.

Los pasos anteriores permiten construir un problema de cálculo diferencial en un problema de álgebra lineal. Dicho problema en general se plantea sobre un espacio vectorial de dimensión no-finita, pero que puede resolverse aproximadamente encontrando una proyección sobre un subespacio de dimensión finita, y por tanto con un número finito de ecuaciones (aunque en general el número de ecuaciones será elevado típicamente de miles o incluso centenares de miles). La discretización en elementos finitos ayuda a construir un algoritmo de proyección sencillo, logrando además que la solución por el método de elementos finitos sea generalmente exacta en un conjunto finito de puntos. Estos puntos coinciden usualmente con los vértices de los elementos finitos o puntos destacados de los mismos. Para la resolución concreta del enorme sistema de ecuaciones algebraicas en general pueden usarse los métodos convencionales del álgebra lineal en espacios de dimensión finita.

En lo que sigue d es la dimensión del dominio, n el número de elementos finitos y N el número de nodos total.

weak formulation

The weak formulation of a differential equation allows us to convert a differential calculus problem formulated in terms of differential equations into a linear algebra problem posed on a Banach space, generally of non-finite dimension, but which can be approximated by a finite system of algebraic equations.

Given a linear differential equation of the form:.

Where the solution is a certain function defined on a d-dimensional domain, and a set of appropriate boundary conditions have been specified, it can be assumed that the function sought is an element of a function space or Banach space V and that equation () is equivalent to:.

Where V' is the dual space of V, the weak variational form is obtained by searching for the only solution such that:.

When the linear operator is an elliptic operator, the problem can be stated as a minimization problem on the Banach space.

Domain discretization

Given a domain with a continuous boundary in the Lipschitz sense a partition into n "finite elements", is a collection of n subdomains satisfying:.

2. • Each is a compact set with a Lipschitz-continuous boundary.

Usually for practical convenience and simplicity of analysis, all "finite elements" have the same "form", that is, there is a reference domain and a collection of bijective functions:.

This reference domain is often also called isoparametric domain. In 2D analyzes (d = 2) the reference domain is typically taken to be an equilateral triangle or a square, while in 3D analyzes (d = 3), the reference domain is typically a tetrahedron or a hexahedron. Additionally, on each element, some special points will be considered, called nodes and which will generally include the vertices of the finite element and the additional condition will be required that two adjacent elements share the nodes on the subset, that is:.

Once the partition into finite elements is defined, a functional space of finite dimension is defined on each element, usually formed by polynomials. This functional space will serve to locally approximate the solution of the variational problem. The variational problem in its weak form is posed on a space of non-finite dimension, and therefore the function sought will be a function of said space. The problem in that exact form is computationally intractable, so in practice a finite-dimensional subspace of the original vector space will be considered. And instead of the exact solution of (), the projection of the original solution on said finite-dimensional vector subspace is calculated, that is, the following problem will be solved numerically:

Where:.

If the discretization is sufficiently fine and the finite functional space over each element is well chosen, the numerical solution obtained will approximate the original solution reasonably well. This will generally imply considering a very high number of finite elements and therefore a high-dimensional projection subspace. The error between the exact solution and the approximate solution can be limited thanks to Ceá's lemma"), which essentially states that the exact solution and the approximate solution satisfy:.

That is, the error will depend above all on how well the vector subspace associated with the discretization in fintio elements approximates the original vector space.

Shape and solution space functions

There are many ways to choose a set of functions that form a vector basis on which to approximate the exact solution of the problem. From a practical point of view it is useful to define a finite-dimensional vector space defined on the reference domain formed by all polynomials of degree equal to or less than a certain degree:.

Then, through the applications that apply the reference domain to each finite element, the vector space that will serve to approximate the solution is defined as:

When it is a linear function and the space is made up of polynomials then the restriction is also a polynomial. The vector space is a polynomial space in which the base of said space is formed by shape functions, which given the set of nodes of the reference domain are defined as:

This allows us to uniquely define shape functions on the real domain on which the problem is defined:

These functions can be extended to the entire domain, thanks to the fact that the set of subdomains or finite elements constitutes a partition of the entire domain:

The shape functions allow any function defined on the original domain to be projected onto the finite element space using the projector:.

Solving the equations

Given a base associated with a certain discretization of the domain, such as that given by the functions, the weak form of the problem (when the function is bilinear) can be written as a simple matrix equation:

Where N is the number of nodes. Grouping the terms and taking into account that v^h is arbitrary and that, therefore, the previous equation must be fulfilled for any value of said arbitrary vector, we have:

This is the common form of the system of equations of an element problem associated with a linear differential equation, not dependent on time. This last form is precisely the form () of the historical review. To numerically solve the system of equations (), which usually consists of thousands or even hundreds of thousands of equations, efficient algorithms are required that optimize the number of operations that must be performed and save memory.

In general, the computational complications that must be resolved in numerical resolution are:

1. • The calculation of the coefficient matrix, this generally requires approximate numerical integration which is a new source of errors in the calculation by the FEM.

2. • The use of an efficient method to solve the system of equations obtained. For example, Cramer's method is totally impractical for , a computer of about 10 GFlops would take more than 2 years to solve the system by this method, while if the Gaussian elimination method is used it would take less than one ten thousandth of a second.

To understand the need for numerical integration we need to see what form the weak form of the problem typically has, expressed in terms of the subdomains or finite elements. That weak form involves integrals of the form:.

Where:.

Error approximation

According to Ceá's lemma (), the error committed in the approximation of an exact solution using finite elements is limited by the approximation error, that is, the solution obtained through the FEM is the better the approximation. Since the approximation error depends crucially on the size of the elements, the greater their number, other factors being equal, the smaller the approximation error. Below we limit this approximation error that will limit the error of the finite element solution.

To do this we need to define the diameter of each subdomain or finite element:

h is a measure of the fineness of the discretization and is the maximum of the above values. It can be verified that the approximation error (and therefore the error of the solution using finite elements) is bounded by:.

Where:.

being a multiindex and the partial derivative of u associated with it. The norm of the space L(Ω).

El MEF es un método numérico de resolución de ecuaciones diferenciales. La solución obtenida por MEF es solo aproximada, coincidiendo con la solución exacta solo en un número finito de puntos llamados nodos. En el resto de puntos que no son nodos, la solución aproximada se obtiene interpolando a partir de los resultados obtenidos para los nodos, lo cual hace que la solución sea solo aproximada debido a ese último paso.

El MEF convierte un problema definido en términos de ecuaciones diferenciales en un problema en forma matricial que proporciona el resultado correcto para un número finito de puntos e interpola posteriormente la solución al resto del dominio, resultando finalmente solo una solución aproximada. El conjunto de puntos donde la solución es exacta se denomina conjunto nodos. Dicho conjunto de nodos forma una red, denominada malla formada por retículos. Cada uno de los retículos contenidos en dicha malla es un «elemento finito». El conjunto de nodos se obtiene dividiendo o discretizando la estructura en elementos de forma variada (pueden ser superficies, volúmenes y barras).

Desde el punto de vista de la programación algorítmica modular las tareas necesarias para llevar a cabo un cálculo mediante un programa MEF se dividen en:.

• - Preproceso, que consiste en la definición de geometría, generación de la malla, las condiciones de contorno y asignación de propiedades a los materiales y otras propiedades. En ocasiones existen operaciones cosméticas de regularización de la malla y precondicionamiento para garantizar una mejor aproximación o una mejor convergencia del cálculo.

• - Cálculo, el resultado del preproceso, en un problema simple no-dependiente del tiempo, permite generar un conjunto de N ecuaciones y N incógnitas, que puede ser resuelto con cualquier algoritmo para la resolución de sistemas de ecuaciones lineales. Cuando el problema a tratar es un problema no lineal o un problema dependiente del tiempo a veces el cálculo consiste en una sucesión finita de sistemas de N ecuaciones y N incógnitas que deben resolverse uno a continuación de otro, y cuya entrada depende del resultado del paso anterior.

• - Postproceso, el cálculo proporciona valores de cierto conjunto de funciones en los nodos de la malla que define la discretización, en el postproceso se calculan magnitudes derivadas de los valores obtenidos para los nodos, y en ocasiones se aplican operaciones de suavizado, interpolación e incluso determinación de errores de aproximación.

Preprocessing and mesh generation

The mesh is generated and generally consists of thousands (and even hundreds of thousands) of points. Information about the material properties and other characteristics of the problem is stored along with information that describes the mesh. On the other hand, the forces, thermal flows or temperatures are reassigned to the points of the mesh. Mesh nodes are assigned a density throughout the material depending on the level of mechanical stress or other property. Regions that will receive a lot of stress typically have a higher density of nodes (mesh density) than those that experience little or none. Points of interest consist of: previously tested material fracture points, recesses, corners, complex details, and areas of high stress. The mesh acts like a spider's web in that from each node a mesh element extends to each adjacent node. This type of vector network is what brings the properties of the material to the object, creating various elements.

The tasks assigned to the preprocess are:

1. • The continuum is divided, by means of imaginary lines or surfaces, into a number of finite elements. This part of the process is usually developed using algorithms incorporated into meshing computer programs during the preprocessing stage.

2. • It is assumed that the elements are connected to each other by a discrete number of points or "nodes", located on their contours. The displacements of these nodes will be the fundamental unknowns of the problem, as occurs in the simple analysis of structures by the matrix method.

3. • A set of functions is taken that uniquely define the displacement field within each finite element based on the nodal displacements of said element. For example, the displacement field within a two-node linear element could be defined by: u = N**u + N**u, with N and N being the commented functions (shape functions) and u1 and u2 being the displacements in node 1 and node 2.

4. • These displacement functions will then uniquely define the deformation state of the element based on the nodal displacements. These deformations, together with the constitutive properties of the material, will in turn define the state of stress throughout the element, and consequently in its contours.

5. • A system of forces concentrated in the nodes is determined, such that it balances the tensions in the contour and any distributed loads, thus resulting in a relationship between forces and displacements of the form F = K·u, which as we see is similar to that of the matrix calculation.

Calculation and resolution of systems of equations

In a non-time-dependent linear mechanical problem, such as a static structural analysis problem or an elastic problem, the calculation is generally reduced to obtaining the displacements in the nodes and with them approximately defining the displacement field in the finite element.

When the problem is nonlinear, in general, the application of forces requires the incremental application of forces and considering numerical increments, and calculating in each increment some magnitudes referred to the nodes. Something similar happens with time-dependent problems, for which a succession of instants is considered, generally quite close in time, and the instantaneous equilibrium at each instant is considered. In general, these last two types of problems require a substantially higher calculation time than in a stationary and linear problem.

Post-processing

Currently, the MEF is used to calculate such complex problems that the files generated as a result of the MEF have such an amount of data that it is convenient to process them in some additional way to make them more understandable and illustrate different aspects of the problem.

In the post-processing stage, the results obtained from the resolution of the system are treated to obtain graphic representations and obtain derived magnitudes that allow drawing conclusions from the problem.

FEM post-processing generally requires additional software to organize the output data in such a way that the result is more easily understood and allows deciding whether certain consequences of the problem are acceptable or not. In the calculation of structures for example, post-processing may include additional checks of whether a structure meets the requirements of the relevant standards, calculating whether allowable stresses are exceeded, or whether there is a possibility of buckling in the structure.

Thermomechanical problems

A wide range of objective functions (variables with the system) are available for minimization or maximization:

• - Mass, volume, temperature.

• - Tension energy, tension effort.

• - Force, displacement, speed, acceleration.

• - Synthetic (defined by the user).

There are multiple loading conditions that can be applied to the system. Some examples are:

• - Punctual, pressure, thermal, gravity, and static centrifugal loads.

• - Thermal loads of heat transmission analysis solutions.

• - Forced displacements.

• - Heat flow and convection.

• - Point, pressure, and dynamic gravity loads.

Each MEF program can come with a library of elements, or one that is built over time. Some examples of elements are:

• - Bar type elements.

• - Beam type elements.

• - Plate/Shell/Composite elements.

• - Sandwich panel.

• - Solid elements.

• - Spring type elements.

• - Mass elements.

• - Rigid elements.

• - Viscous damping elements.

Many MEF programs are also equipped with the ability to use multiple materials in the structure, such as:.

• - General isotropic / orthotropic / anisotropic elastic models.

• - Homogeneous / heterogeneous materials.

• - Plasticity models.

• - Viscous models.

El programador puede insertar numerosos algoritmos o funciones que pueden hacer al sistema comportarse de manera lineal o no lineal. Los sistemas lineales son menos complejos y normalmente no tienen en cuenta deformaciones plásticas. Los sistemas no lineales") toman en cuenta las deformaciones plásticas, y algunos incluso son capaces de verificar si se presentaría fractura en el material.

Algunos tipos de análisis ingenieriles comunes que usan el método de los elementos finitos son:.

• - Análisis estático se emplea cuando la estructura está sometida a acciones estáticas, es decir, no dependientes del tiempo.

• - Análisis vibracional es usado para analizar la estructura sometido a vibraciones aleatorias, choques e impactos. Cada uno de estas acciones puede actuar en la frecuencia natural de la estructura y causar resonancia y el consecuente fallo.

• - Análisis de fatiga ayuda a los diseñadores a predecir la vida del material o de la estructura, prediciendo el efecto de los ciclos de carga sobre el espécimen. Este análisis puede mostrar las áreas donde es más probable que se presente una grieta. El análisis por fatiga puede también predecir la tolerancia al fallo del material.

Los modelos de análisis de transferencia de calor por conductividad o por dinámicas térmicas de flujo del material o la estructura. El estado continuo de transferencia se refiere a las propiedades térmicas en el material que tiene una difusión lineal de calor.

MEF results

FEM has become a solution to the task of predicting failures due to unknown stresses by teaching the problems of stress distribution in the material and allowing designers to see all the stresses involved. This method of product design and testing is better than trial and error where manufacturing costs associated with the construction of each specimen for testing must be maintained.

The great advantages of computer calculation can be summarized as:

• - It makes possible the calculation of structures that, either due to the large number of operations that their resolution presents (multi-story frameworks, for example) or because of their tediousness (spatial frameworks, for example) which were, in practice, unapproachable through manual calculation.

• - In most cases it reduces the risk of operational errors to negligible limits.

Higher Order MEF

The latest advances in this field indicate that its future lies in higher-order adaptation methods, which satisfactorily respond to the increasing complexity of engineering simulations and satisfy the general trend of simultaneous resolution of phenomena with multiple scales. Among the various adaptation strategies for finite elements, the best results can be achieved with hp-adaptability. Goal-oriented adaptivity is based on the adaptation of the finite element mesh, with the goal of improving the resolution by a specific amount of interest (rather than minimizing the error

of the approximation in some global norm), and hp-adaptability is based on the combination of spatial refinements (h-adaptability), with a simultaneous variation of the order of the approximation polynomial (p-adaptability). There are examples where 'hp-adaptability' turned out to be the only way to solve the problem at a required level of accuracy.

Limitations

In general, the FEM as currently used has some limitations:

• - The FEM calculates specific numerical solutions adapted to particular input data; a simple sensitivity analysis cannot be performed to determine how the solution will vary if any of the parameters are slightly altered.[8] That is, it provides only specific quantitative numerical responses, not general qualitative relationships.

• - The FEM provides an approximate solution whose margin of error is generally unknown. Although some types of problems allow the error of the solution to be limited, due to the various types of approximations used by the method, nonlinear or time-dependent problems in general do not allow us to know the error.

• - In the FEM, most practical applications require a lot of time to adjust details of the geometry, with frequent problems of poor conditioning of the meshes, unequal degree of convergence of the approximate solution towards the exact solution at different points, etc. In general, a simulation requires the use of numerous tests and trials with simplified geometries or less general cases than the one that is finally intended to be simulated, before beginning to achieve satisfactory results.

En problemas dinámicos, donde las magnitudes cambian a lo largo del tiempo, existen diversos métodos para integrar en el tiempo. En ambos métodos se discretiza el tiempo, por lo que se considera la solución solo para un cierto número de instantes (para el resto de valores del tiempo se puede interpolar la solución por intervalos). La diferencia entre un instante en el que se busca la solución y el siguiente se denomina, paso de tiempo. Las dos principales variantes del cálculo por FEM son:.

• - Método implícito, que requieren resolver a cada paso de tiempo un sistema de ecuaciones, aunque pueden usarse pasos de tiempo más largos.

• - Método explícito, que no requieren resolver un sistema de ecuaciones a cada paso de tiempo, aunque debido a que la convergencia no siempre está asegurada el paso de tiempo debe escogerse convenientemente pequeño.

The implicit method

These calculations are usually used for stiffness calculations (although sometimes they can also be calculated dynamically). Among implicit methods some are unconditionally convergent (they do not diverge exponentially from the exact solution) only for a certain fixed choice of method parameters.

Calculations by the implicit method (or semi-implicit to the most rigid part of the system) require much more computing time to take a step in time, since they must invert a matrix of very large size, for this reason, iterative methods are usually used, instead of direct methods, such as those associated with the Krylov subspace. In compensation, much larger time steps can be used as they are stable.

The explicit method

An explicit method is one that does not require the solution of a non-trivial system of equations at each time step. In these calculations, a simulation is carried out with modification of the mesh over time. In general, explicit methods require less computing time than implicit methods, although they frequently present the problem of not being unconditionally convergent, and require first evaluating the maximum time step for the computation to be numerically stable. Explicit methods are usually conditionally convergent but not unconditionally convergent, so the time step used in the finite difference scheme must be less than a certain value:.

In explicit finite elements, the use of simple elements, such as quadrilaterals with an integration point and stabilization against zero energy modes, is preferable to higher order elements.

Explicit methods find their optimal field of application in fast dynamics problems, in which strong nonlinearities occur and the use of small time intervals becomes a necessity.

An important advantage of the explicit method is the resolution of the equations at an exclusively local level, without ever posing systems of coupled global equations. This allows the use of element-by-element algorithms, which facilitate parallel calculation. Posed as dynamic relaxation or viscous relaxation methods, they are framed together with iterative methods for solving nonlinear equations, such as Gauss-Seidel relaxation methods, or preconditioned conjugate gradient with element-by-element techniques. Being very interesting for parallel calculation.

• - Method of finite volumes.

• - Method of finite differences.

• - K. J. Bathe (1995): Finite Element Procedures, Prentice Hall, 2nd edition.

• - P. G. Ciarlet (1978): The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.

• - P. G. Ciarlet (1991): Basic error estimates for elliptic problems in Handbook of Numerical Analysis (Vol II) J.L. Lions and P. G. Ciarlet (ed.), North-Holland, Amsterdam, 1991, p. 17-351.

• - O. C. Zienkiewicz, R. L. Taylor (1994): The Finite Element Method, volumes 1 and 2. CIMNE-Mc Graw Hill, 1994.

• - E. Oñate (1995): Calculation of Structures by the Finite Element Method, CIMNE, Barcelona, ​​1995.

• - M. Cerrolaza (2007): The Finite Element Method in Engineering and Applied Sciences: Theory and Programs, CDCH-UCV, Caracas, 2007.

• - J. J. Benito Muñoz, R. Álvarez Cabal, F. Ureña Prieto, E. Salete Casino, E. Aranda Ortega (2014): Introduction to the Finite Element Method, UNED, Madrid, 2014.

• - Finite Element Analysis.

• - Finite element method in structural mechanics.

• - Quickfield.

• - Analysis of a Railway Detour by the Finite Element Method (Final Year Project in Mechanical Engineering at the University of Oviedo), Cover, Annex.

Contenido

El MEF fue al principio desarrollado en 1943 por Richard Courant, quien utilizó el método de Ritz de análisis numérico y minimización de las variables de cálculo para obtener soluciones aproximadas a un sistema de vibración. Poco después, un documento publicado en 1956 por M. J. Turner, R. W. Clough, H. C. Martin, y L. J. Topp estableció una definición más amplia del análisis numérico.[7]​ El documento se centró en «la rigidez y deformación de estructuras complejas».

Con la llegada de los primeros ordenadores instaura el cálculo matricial de estructuras. Éste parte de la discretización de la estructura en elementos lineales tipo barra de los que se conoce su rigidez frente a los desplazamientos de sus nodos. Se plantea entonces un sistema de ecuaciones resultado de aplicar las ecuaciones de equilibrio a los nodos de la estructura. Este sistema de ecuaciones se esquematiza de la siguiente manera:.

Donde las incógnitas son los desplazamientos en los nodos (vector u) que se hallan a partir de las "fuerzas" o "solicitaciones" en los nodos (vector ) y de la rigidez de las barras (matriz de rigidez ). Conocidos dichos desplazamientos es posible determinar los esfuerzos en las barras. La solución obtenida es bastante precisa si se cumplen las hipótesis implícitas en la formulación.

Practical use of the method around 1950

When the first computer equipment arrived in the 1950s, the calculation of structures was at a point where the predominant calculation methods consisted of iterative methods (Cross and Kani methods) that were carried out manually and, therefore, were quite tedious. The calculation of a multi-story building structure, for example, could take several weeks, which represented a substantial cost of time to the detriment of the possibility of investing time in optimizing the structure.

The arrival of the computer allowed the resurgence of the method of displacements already known in previous centuries (Navier, Lagrange, Cauchy), but which were difficult to apply since in the end they led to the resolution of enormous systems of equations that were unapproachable from a manual point of view.

From 1960 to 1970

As practical applications of finite elements grew in size, the computing time and memory requirements of computers grew. At that point the development of more efficient algorithms became important. To solve the systems of equations, the study of the adaptability of already known algorithms (Gauss, Cholesky, Crout, Conjugate Gradient, etc.) is promoted. Saving time is unthinkable and with it the use of the matrix method spreads. This development is especially notable in building structures where the discretization of the frames into bars is practically immediate from the beams and columns.

However, and despite developing modeling of surface elements using bars (slabs with grids, curved elements using approximations of straight elements, etc.), great difficulties arise in the face of continuous structures (surfaces and volumes) and with complex geometries. Hence, it is precisely within the aerospace field where new FEM techniques begin to be developed.

Given its generality, the method was extended to other non-structural fields such as heat conduction, fluid mechanics, etc., where it competed with other numerical methods such as the finite difference method, which, although more intuitive, once again had difficulties in planning for complex geometries.

With the arrival of computer centers and the first commercial programs in the 60s, the MEF, while becoming popular in the industry, reinforced its theoretical bases in university centers.

In the 70s there was a great growth in the bibliography as well as the extension of the method to other problems such as non-linear ones. In this decade, the MEF was limited to expensive mainframe computers generally owned by the aeronautical, automotive, defense and nuclear industries. New types of elements are studied and the rigorous mathematical foundations of the method are laid, which had previously appeared more as an engineering technique than as a numerical method of mathematics.

Since 1980

Finally, starting in the 1980s, with the widespread use of personal computers, the use of commercial programs that specialize in various fields spread, establishing the use of graphic pre- and post-processors that perform meshing and graphical representation of the results. The study of the application of the method to new behavioral models (plasticity, fracture, continuous damage, etc.) and the analysis of errors continue.

Currently, within the structural field, the FEM shares prominence with the matrix method, with many programs that mix analysis by both methods, mainly due to the greater need for memory required by finite element analysis. Thus, the application of the FEM has been left to the analysis of continuous slab or screen type elements, while the frames are still being discretized into bars and using the matrix method. And since the rapid decline in the cost of computers and the phenomenal increase in computing power, the MEF has developed incredible precision. Today, supercomputers are capable of giving quite precise results for all types of parameters.

El desarrollo de un algoritmo de elementos finitos para resolver un problema definido mediante ecuaciones diferenciales y condiciones de contorno requiere en general cuatro etapas:.

1. • El problema debe reformularse en forma variacional.

2. • El dominio de variables independientes (usualmente un dominio espacial) debe dividirse mediante una partición "Partición (matemática)") en subdominios, llamados elementos finitos. Asociada a la partición anterior se construye un espacio vectorial de dimensión finita, llamado espacio de elementos finitos. Siendo la solución numérica aproximada obtenida por elementos finitos una combinación lineal en dicho espacio vectorial.

3. • Se obtiene la proyección del problema variacional original sobre el espacio de elementos finitos obtenido de la partición. Esto da lugar a un sistema con un número de ecuaciones finito, aunque en general con un número elevado de ecuaciones incógnitas. El número de incógnitas será igual a la dimensión del espacio vectorial de elementos finitos obtenido y, en general, cuanto mayor sea dicha dimensión tanto mejor será la aproximación numérica obtenida.

4. • El último paso es el cálculo numérico de la solución del sistema de ecuaciones.

Los pasos anteriores permiten construir un problema de cálculo diferencial en un problema de álgebra lineal. Dicho problema en general se plantea sobre un espacio vectorial de dimensión no-finita, pero que puede resolverse aproximadamente encontrando una proyección sobre un subespacio de dimensión finita, y por tanto con un número finito de ecuaciones (aunque en general el número de ecuaciones será elevado típicamente de miles o incluso centenares de miles). La discretización en elementos finitos ayuda a construir un algoritmo de proyección sencillo, logrando además que la solución por el método de elementos finitos sea generalmente exacta en un conjunto finito de puntos. Estos puntos coinciden usualmente con los vértices de los elementos finitos o puntos destacados de los mismos. Para la resolución concreta del enorme sistema de ecuaciones algebraicas en general pueden usarse los métodos convencionales del álgebra lineal en espacios de dimensión finita.

En lo que sigue d es la dimensión del dominio, n el número de elementos finitos y N el número de nodos total.

weak formulation

The weak formulation of a differential equation allows us to convert a differential calculus problem formulated in terms of differential equations into a linear algebra problem posed on a Banach space, generally of non-finite dimension, but which can be approximated by a finite system of algebraic equations.

Given a linear differential equation of the form:.

Where the solution is a certain function defined on a d-dimensional domain, and a set of appropriate boundary conditions have been specified, it can be assumed that the function sought is an element of a function space or Banach space V and that equation () is equivalent to:.

Where V' is the dual space of V, the weak variational form is obtained by searching for the only solution such that:.

When the linear operator is an elliptic operator, the problem can be stated as a minimization problem on the Banach space.

Domain discretization

Given a domain with a continuous boundary in the Lipschitz sense a partition into n "finite elements", is a collection of n subdomains satisfying:.

2. • Each is a compact set with a Lipschitz-continuous boundary.

Usually for practical convenience and simplicity of analysis, all "finite elements" have the same "form", that is, there is a reference domain and a collection of bijective functions:.

This reference domain is often also called isoparametric domain. In 2D analyzes (d = 2) the reference domain is typically taken to be an equilateral triangle or a square, while in 3D analyzes (d = 3), the reference domain is typically a tetrahedron or a hexahedron. Additionally, on each element, some special points will be considered, called nodes and which will generally include the vertices of the finite element and the additional condition will be required that two adjacent elements share the nodes on the subset, that is:.

Once the partition into finite elements is defined, a functional space of finite dimension is defined on each element, usually formed by polynomials. This functional space will serve to locally approximate the solution of the variational problem. The variational problem in its weak form is posed on a space of non-finite dimension, and therefore the function sought will be a function of said space. The problem in that exact form is computationally intractable, so in practice a finite-dimensional subspace of the original vector space will be considered. And instead of the exact solution of (), the projection of the original solution on said finite-dimensional vector subspace is calculated, that is, the following problem will be solved numerically:

Where:.

If the discretization is sufficiently fine and the finite functional space over each element is well chosen, the numerical solution obtained will approximate the original solution reasonably well. This will generally imply considering a very high number of finite elements and therefore a high-dimensional projection subspace. The error between the exact solution and the approximate solution can be limited thanks to Ceá's lemma"), which essentially states that the exact solution and the approximate solution satisfy:.

That is, the error will depend above all on how well the vector subspace associated with the discretization in fintio elements approximates the original vector space.

Shape and solution space functions

There are many ways to choose a set of functions that form a vector basis on which to approximate the exact solution of the problem. From a practical point of view it is useful to define a finite-dimensional vector space defined on the reference domain formed by all polynomials of degree equal to or less than a certain degree:.

Then, through the applications that apply the reference domain to each finite element, the vector space that will serve to approximate the solution is defined as:

When it is a linear function and the space is made up of polynomials then the restriction is also a polynomial. The vector space is a polynomial space in which the base of said space is formed by shape functions, which given the set of nodes of the reference domain are defined as:

This allows us to uniquely define shape functions on the real domain on which the problem is defined:

These functions can be extended to the entire domain, thanks to the fact that the set of subdomains or finite elements constitutes a partition of the entire domain:

The shape functions allow any function defined on the original domain to be projected onto the finite element space using the projector:.

Solving the equations

Given a base associated with a certain discretization of the domain, such as that given by the functions, the weak form of the problem (when the function is bilinear) can be written as a simple matrix equation:

Where N is the number of nodes. Grouping the terms and taking into account that v^h is arbitrary and that, therefore, the previous equation must be fulfilled for any value of said arbitrary vector, we have:

This is the common form of the system of equations of an element problem associated with a linear differential equation, not dependent on time. This last form is precisely the form () of the historical review. To numerically solve the system of equations (), which usually consists of thousands or even hundreds of thousands of equations, efficient algorithms are required that optimize the number of operations that must be performed and save memory.

In general, the computational complications that must be resolved in numerical resolution are:

1. • The calculation of the coefficient matrix, this generally requires approximate numerical integration which is a new source of errors in the calculation by the FEM.

2. • The use of an efficient method to solve the system of equations obtained. For example, Cramer's method is totally impractical for , a computer of about 10 GFlops would take more than 2 years to solve the system by this method, while if the Gaussian elimination method is used it would take less than one ten thousandth of a second.

To understand the need for numerical integration we need to see what form the weak form of the problem typically has, expressed in terms of the subdomains or finite elements. That weak form involves integrals of the form:.

Where:.

Error approximation

According to Ceá's lemma (), the error committed in the approximation of an exact solution using finite elements is limited by the approximation error, that is, the solution obtained through the FEM is the better the approximation. Since the approximation error depends crucially on the size of the elements, the greater their number, other factors being equal, the smaller the approximation error. Below we limit this approximation error that will limit the error of the finite element solution.

To do this we need to define the diameter of each subdomain or finite element:

h is a measure of the fineness of the discretization and is the maximum of the above values. It can be verified that the approximation error (and therefore the error of the solution using finite elements) is bounded by:.

Where:.

being a multiindex and the partial derivative of u associated with it. The norm of the space L(Ω).

El MEF es un método numérico de resolución de ecuaciones diferenciales. La solución obtenida por MEF es solo aproximada, coincidiendo con la solución exacta solo en un número finito de puntos llamados nodos. En el resto de puntos que no son nodos, la solución aproximada se obtiene interpolando a partir de los resultados obtenidos para los nodos, lo cual hace que la solución sea solo aproximada debido a ese último paso.

El MEF convierte un problema definido en términos de ecuaciones diferenciales en un problema en forma matricial que proporciona el resultado correcto para un número finito de puntos e interpola posteriormente la solución al resto del dominio, resultando finalmente solo una solución aproximada. El conjunto de puntos donde la solución es exacta se denomina conjunto nodos. Dicho conjunto de nodos forma una red, denominada malla formada por retículos. Cada uno de los retículos contenidos en dicha malla es un «elemento finito». El conjunto de nodos se obtiene dividiendo o discretizando la estructura en elementos de forma variada (pueden ser superficies, volúmenes y barras).

Desde el punto de vista de la programación algorítmica modular las tareas necesarias para llevar a cabo un cálculo mediante un programa MEF se dividen en:.

• - Preproceso, que consiste en la definición de geometría, generación de la malla, las condiciones de contorno y asignación de propiedades a los materiales y otras propiedades. En ocasiones existen operaciones cosméticas de regularización de la malla y precondicionamiento para garantizar una mejor aproximación o una mejor convergencia del cálculo.

• - Cálculo, el resultado del preproceso, en un problema simple no-dependiente del tiempo, permite generar un conjunto de N ecuaciones y N incógnitas, que puede ser resuelto con cualquier algoritmo para la resolución de sistemas de ecuaciones lineales. Cuando el problema a tratar es un problema no lineal o un problema dependiente del tiempo a veces el cálculo consiste en una sucesión finita de sistemas de N ecuaciones y N incógnitas que deben resolverse uno a continuación de otro, y cuya entrada depende del resultado del paso anterior.

• - Postproceso, el cálculo proporciona valores de cierto conjunto de funciones en los nodos de la malla que define la discretización, en el postproceso se calculan magnitudes derivadas de los valores obtenidos para los nodos, y en ocasiones se aplican operaciones de suavizado, interpolación e incluso determinación de errores de aproximación.

Preprocessing and mesh generation

The mesh is generated and generally consists of thousands (and even hundreds of thousands) of points. Information about the material properties and other characteristics of the problem is stored along with information that describes the mesh. On the other hand, the forces, thermal flows or temperatures are reassigned to the points of the mesh. Mesh nodes are assigned a density throughout the material depending on the level of mechanical stress or other property. Regions that will receive a lot of stress typically have a higher density of nodes (mesh density) than those that experience little or none. Points of interest consist of: previously tested material fracture points, recesses, corners, complex details, and areas of high stress. The mesh acts like a spider's web in that from each node a mesh element extends to each adjacent node. This type of vector network is what brings the properties of the material to the object, creating various elements.

The tasks assigned to the preprocess are:

1. • The continuum is divided, by means of imaginary lines or surfaces, into a number of finite elements. This part of the process is usually developed using algorithms incorporated into meshing computer programs during the preprocessing stage.

2. • It is assumed that the elements are connected to each other by a discrete number of points or "nodes", located on their contours. The displacements of these nodes will be the fundamental unknowns of the problem, as occurs in the simple analysis of structures by the matrix method.

3. • A set of functions is taken that uniquely define the displacement field within each finite element based on the nodal displacements of said element. For example, the displacement field within a two-node linear element could be defined by: u = N**u + N**u, with N and N being the commented functions (shape functions) and u1 and u2 being the displacements in node 1 and node 2.

4. • These displacement functions will then uniquely define the deformation state of the element based on the nodal displacements. These deformations, together with the constitutive properties of the material, will in turn define the state of stress throughout the element, and consequently in its contours.

5. • A system of forces concentrated in the nodes is determined, such that it balances the tensions in the contour and any distributed loads, thus resulting in a relationship between forces and displacements of the form F = K·u, which as we see is similar to that of the matrix calculation.

Calculation and resolution of systems of equations

In a non-time-dependent linear mechanical problem, such as a static structural analysis problem or an elastic problem, the calculation is generally reduced to obtaining the displacements in the nodes and with them approximately defining the displacement field in the finite element.

When the problem is nonlinear, in general, the application of forces requires the incremental application of forces and considering numerical increments, and calculating in each increment some magnitudes referred to the nodes. Something similar happens with time-dependent problems, for which a succession of instants is considered, generally quite close in time, and the instantaneous equilibrium at each instant is considered. In general, these last two types of problems require a substantially higher calculation time than in a stationary and linear problem.

Post-processing

Currently, the MEF is used to calculate such complex problems that the files generated as a result of the MEF have such an amount of data that it is convenient to process them in some additional way to make them more understandable and illustrate different aspects of the problem.

In the post-processing stage, the results obtained from the resolution of the system are treated to obtain graphic representations and obtain derived magnitudes that allow drawing conclusions from the problem.

FEM post-processing generally requires additional software to organize the output data in such a way that the result is more easily understood and allows deciding whether certain consequences of the problem are acceptable or not. In the calculation of structures for example, post-processing may include additional checks of whether a structure meets the requirements of the relevant standards, calculating whether allowable stresses are exceeded, or whether there is a possibility of buckling in the structure.

Thermomechanical problems

A wide range of objective functions (variables with the system) are available for minimization or maximization:

• - Mass, volume, temperature.

• - Tension energy, tension effort.

• - Force, displacement, speed, acceleration.

• - Synthetic (defined by the user).

There are multiple loading conditions that can be applied to the system. Some examples are:

• - Punctual, pressure, thermal, gravity, and static centrifugal loads.

• - Thermal loads of heat transmission analysis solutions.

• - Forced displacements.

• - Heat flow and convection.

• - Point, pressure, and dynamic gravity loads.

Each MEF program can come with a library of elements, or one that is built over time. Some examples of elements are:

• - Bar type elements.

• - Beam type elements.

• - Plate/Shell/Composite elements.

• - Sandwich panel.

• - Solid elements.

• - Spring type elements.

• - Mass elements.

• - Rigid elements.

• - Viscous damping elements.

Many MEF programs are also equipped with the ability to use multiple materials in the structure, such as:.

• - General isotropic / orthotropic / anisotropic elastic models.

• - Homogeneous / heterogeneous materials.

• - Plasticity models.

• - Viscous models.

El programador puede insertar numerosos algoritmos o funciones que pueden hacer al sistema comportarse de manera lineal o no lineal. Los sistemas lineales son menos complejos y normalmente no tienen en cuenta deformaciones plásticas. Los sistemas no lineales") toman en cuenta las deformaciones plásticas, y algunos incluso son capaces de verificar si se presentaría fractura en el material.

Algunos tipos de análisis ingenieriles comunes que usan el método de los elementos finitos son:.

• - Análisis estático se emplea cuando la estructura está sometida a acciones estáticas, es decir, no dependientes del tiempo.

• - Análisis vibracional es usado para analizar la estructura sometido a vibraciones aleatorias, choques e impactos. Cada uno de estas acciones puede actuar en la frecuencia natural de la estructura y causar resonancia y el consecuente fallo.

• - Análisis de fatiga ayuda a los diseñadores a predecir la vida del material o de la estructura, prediciendo el efecto de los ciclos de carga sobre el espécimen. Este análisis puede mostrar las áreas donde es más probable que se presente una grieta. El análisis por fatiga puede también predecir la tolerancia al fallo del material.

Los modelos de análisis de transferencia de calor por conductividad o por dinámicas térmicas de flujo del material o la estructura. El estado continuo de transferencia se refiere a las propiedades térmicas en el material que tiene una difusión lineal de calor.

MEF results

FEM has become a solution to the task of predicting failures due to unknown stresses by teaching the problems of stress distribution in the material and allowing designers to see all the stresses involved. This method of product design and testing is better than trial and error where manufacturing costs associated with the construction of each specimen for testing must be maintained.

The great advantages of computer calculation can be summarized as:

• - It makes possible the calculation of structures that, either due to the large number of operations that their resolution presents (multi-story frameworks, for example) or because of their tediousness (spatial frameworks, for example) which were, in practice, unapproachable through manual calculation.

• - In most cases it reduces the risk of operational errors to negligible limits.

Higher Order MEF

The latest advances in this field indicate that its future lies in higher-order adaptation methods, which satisfactorily respond to the increasing complexity of engineering simulations and satisfy the general trend of simultaneous resolution of phenomena with multiple scales. Among the various adaptation strategies for finite elements, the best results can be achieved with hp-adaptability. Goal-oriented adaptivity is based on the adaptation of the finite element mesh, with the goal of improving the resolution by a specific amount of interest (rather than minimizing the error

of the approximation in some global norm), and hp-adaptability is based on the combination of spatial refinements (h-adaptability), with a simultaneous variation of the order of the approximation polynomial (p-adaptability). There are examples where 'hp-adaptability' turned out to be the only way to solve the problem at a required level of accuracy.

Limitations

In general, the FEM as currently used has some limitations:

• - The FEM calculates specific numerical solutions adapted to particular input data; a simple sensitivity analysis cannot be performed to determine how the solution will vary if any of the parameters are slightly altered.[8] That is, it provides only specific quantitative numerical responses, not general qualitative relationships.

• - The FEM provides an approximate solution whose margin of error is generally unknown. Although some types of problems allow the error of the solution to be limited, due to the various types of approximations used by the method, nonlinear or time-dependent problems in general do not allow us to know the error.

• - In the FEM, most practical applications require a lot of time to adjust details of the geometry, with frequent problems of poor conditioning of the meshes, unequal degree of convergence of the approximate solution towards the exact solution at different points, etc. In general, a simulation requires the use of numerous tests and trials with simplified geometries or less general cases than the one that is finally intended to be simulated, before beginning to achieve satisfactory results.

En problemas dinámicos, donde las magnitudes cambian a lo largo del tiempo, existen diversos métodos para integrar en el tiempo. En ambos métodos se discretiza el tiempo, por lo que se considera la solución solo para un cierto número de instantes (para el resto de valores del tiempo se puede interpolar la solución por intervalos). La diferencia entre un instante en el que se busca la solución y el siguiente se denomina, paso de tiempo. Las dos principales variantes del cálculo por FEM son:.

• - Método implícito, que requieren resolver a cada paso de tiempo un sistema de ecuaciones, aunque pueden usarse pasos de tiempo más largos.

• - Método explícito, que no requieren resolver un sistema de ecuaciones a cada paso de tiempo, aunque debido a que la convergencia no siempre está asegurada el paso de tiempo debe escogerse convenientemente pequeño.

The implicit method

These calculations are usually used for stiffness calculations (although sometimes they can also be calculated dynamically). Among implicit methods some are unconditionally convergent (they do not diverge exponentially from the exact solution) only for a certain fixed choice of method parameters.

Calculations by the implicit method (or semi-implicit to the most rigid part of the system) require much more computing time to take a step in time, since they must invert a matrix of very large size, for this reason, iterative methods are usually used, instead of direct methods, such as those associated with the Krylov subspace. In compensation, much larger time steps can be used as they are stable.

The explicit method

An explicit method is one that does not require the solution of a non-trivial system of equations at each time step. In these calculations, a simulation is carried out with modification of the mesh over time. In general, explicit methods require less computing time than implicit methods, although they frequently present the problem of not being unconditionally convergent, and require first evaluating the maximum time step for the computation to be numerically stable. Explicit methods are usually conditionally convergent but not unconditionally convergent, so the time step used in the finite difference scheme must be less than a certain value:.

In explicit finite elements, the use of simple elements, such as quadrilaterals with an integration point and stabilization against zero energy modes, is preferable to higher order elements.

Explicit methods find their optimal field of application in fast dynamics problems, in which strong nonlinearities occur and the use of small time intervals becomes a necessity.

An important advantage of the explicit method is the resolution of the equations at an exclusively local level, without ever posing systems of coupled global equations. This allows the use of element-by-element algorithms, which facilitate parallel calculation. Posed as dynamic relaxation or viscous relaxation methods, they are framed together with iterative methods for solving nonlinear equations, such as Gauss-Seidel relaxation methods, or preconditioned conjugate gradient with element-by-element techniques. Being very interesting for parallel calculation.

• - Method of finite volumes.

• - Method of finite differences.

• - K. J. Bathe (1995): Finite Element Procedures, Prentice Hall, 2nd edition.

• - P. G. Ciarlet (1978): The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.

• - P. G. Ciarlet (1991): Basic error estimates for elliptic problems in Handbook of Numerical Analysis (Vol II) J.L. Lions and P. G. Ciarlet (ed.), North-Holland, Amsterdam, 1991, p. 17-351.

• - O. C. Zienkiewicz, R. L. Taylor (1994): The Finite Element Method, volumes 1 and 2. CIMNE-Mc Graw Hill, 1994.

• - E. Oñate (1995): Calculation of Structures by the Finite Element Method, CIMNE, Barcelona, ​​1995.

• - M. Cerrolaza (2007): The Finite Element Method in Engineering and Applied Sciences: Theory and Programs, CDCH-UCV, Caracas, 2007.

• - J. J. Benito Muñoz, R. Álvarez Cabal, F. Ureña Prieto, E. Salete Casino, E. Aranda Ortega (2014): Introduction to the Finite Element Method, UNED, Madrid, 2014.

• - Finite Element Analysis.

• - Finite element method in structural mechanics.

• - Quickfield.

• - Analysis of a Railway Detour by the Finite Element Method (Final Year Project in Mechanical Engineering at the University of Oviedo), Cover, Annex.