Concepts and Applications of Finite Element Analysis, 4th Edition

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To explain the approximation in this process, FEM is commonly introduced as a special case of Galerkin method. The process, in mathematical language, is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE.

The residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual. These equation sets are the element equations. They are linear if the underlying PDE is linear, and vice versa. Algebraic equation sets that arise in the steady state problems are solved using numerical linear algebra methods, while ordinary differential equation sets that arise in the transient problems are solved by numerical integration using standard techniques such as Euler's method or the Runge-Kutta method.

In step 2 above, a global system of equations is generated from the element equations through a transformation of coordinates from the subdomains' local nodes to the domain's global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system. The process is often carried out by FEM software using coordinate data generated from the subdomains. FEA as applied in engineering is a computational tool for performing engineering analysis.

It includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software program coded with FEM algorithm. In applying FEA, the complex problem is usually a physical system with the underlying physics such as the Euler-Bernoulli beam equation , the heat equation , or the Navier-Stokes equations expressed in either PDE or integral equations , while the divided small elements of the complex problem represent different areas in the physical system.

FEA is a good choice for analyzing problems over complicated domains like cars and oil pipelines , when the domain changes as during a solid state reaction with a moving boundary , when the desired precision varies over the entire domain, or when the solution lacks smoothness. FEA simulations provide a valuable resource as they remove multiple instances of creation and testing of hard prototypes for various high fidelity situations. Another example would be in numerical weather prediction , where it is more important to have accurate predictions over developing highly nonlinear phenomena such as tropical cyclones in the atmosphere, or eddies in the ocean rather than relatively calm areas.

While it is difficult to quote a date of the invention of the finite element method, the method originated from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering. Its development can be traced back to the work by A. Hrennikoff [4] and R. Courant [5] in the early s. Another pioneer was Ioannis Argyris.

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In the USSR, the introduction of the practical application of the method is usually connected with name of Leonard Oganesyan. Feng proposed a systematic numerical method for solving partial differential equations. The method was called the finite difference method based on variation principle, which was another independent invention of the finite element method.

Although the approaches used by these pioneers are different, they share one essential characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains, usually called elements. Hrennikoff's work discretizes the domain by using a lattice analogy, while Courant's approach divides the domain into finite triangular subregions to solve second order elliptic partial differential equations PDEs that arise from the problem of torsion of a cylinder.

Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by Rayleigh , Ritz , and Galerkin. The finite element method obtained its real impetus in the s and s by the developments of J.

Concepts and Applications of Finite Element Analysis

Argyris with co-workers at the University of Stuttgart , R. Clough with co-workers at UC Berkeley , O. Further impetus was provided in these years by available open source finite element software programs. A finite element method is characterized by a variational formulation , a discretization strategy, one or more solution algorithms and post-processing procedures. Examples of variational formulation are the Galerkin method , the discontinuous Galerkin method, mixed methods, etc.

A discretization strategy is understood to mean a clearly defined set of procedures that cover a the creation of finite element meshes, b the definition of basis function on reference elements also called shape functions and c the mapping of reference elements onto the elements of the mesh. Examples of discretization strategies are the h-version, p-version , hp-version , x-FEM , isogeometric analysis , etc. Each discretization strategy has certain advantages and disadvantages. A reasonable criterion in selecting a discretization strategy is to realize nearly optimal performance for the broadest set of mathematical models in a particular model class.

There are various numerical solution algorithms that can be classified into two broad categories; direct and iterative solvers. These algorithms are designed to exploit the sparsity of matrices that depend on the choices of variational formulation and discretization strategy.

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Postprocessing procedures are designed for the extraction of the data of interest from a finite element solution. In order to meet the requirements of solution verification, postprocessors need to provide for a posteriori error estimation in terms of the quantities of interest.

When the errors of approximation are larger than what is considered acceptable then the discretization has to be changed either by an automated adaptive process or by action of the analyst.

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There are some very efficient postprocessors that provide for the realization of superconvergence. We will demonstrate the finite element method using two sample problems from which the general method can be extrapolated. It is assumed that the reader is familiar with calculus and linear algebra. P2 is a two-dimensional problem Dirichlet problem. The problem P1 can be solved directly by computing antiderivatives.

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For this reason, we will develop the finite element method for P1 and outline its generalization to P2. Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem BVP using the FEM. After this second step, we have concrete formulae for a large but finite-dimensional linear problem whose solution will approximately solve the original BVP. This finite-dimensional problem is then implemented on a computer. The first step is to convert P1 and P2 into their equivalent weak formulations.

Existence and uniqueness of the solution can also be shown.

P1 and P2 are ready to be discretized which leads to a common sub-problem 3. The basic idea is to replace the infinite-dimensional linear problem:. One hopes that as the underlying triangular mesh becomes finer and finer, the solution of the discrete problem 3 will in some sense converge to the solution of the original boundary value problem P2.

This parameter will be related to the size of the largest or average triangle in the triangulation.


Since we do not perform such an analysis, we will not use this notation. Depending on the author, the word "element" in "finite element method" refers either to the triangles in the domain, the piecewise linear basis function, or both. So for instance, an author interested in curved domains might replace the triangles with curved primitives, and so might describe the elements as being curvilinear. On the other hand, some authors replace "piecewise linear" by "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". Finite element method is not restricted to triangles or tetrahedra in 3-d, or higher order simplexes in multidimensional spaces , but can be defined on quadrilateral subdomains hexahedra, prisms, or pyramids in 3-d, and so on.

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Ask the provider about this item. Most renters respond to questions in 48 hours or less. The response will be emailed to you. Cancel Send message. There is a remarkable This text, originally published in , had as its amount of new material spread throughout the objective at that time the description of the finite book, however. This is especially true in connec- element method as a procedure for stress analysis. The author has kept pace wanted to understand why the method behaves as with the field and is perceptive in his inter- it does, but was not greatly concerned with pretation of developments since the prior edition.

The first Moreover, he presents this material in a clear and edition succeeded admirably in its attempt to concise manner. He has also added an impressive achieve those goals and for that reason it became number of new problems for assignment to one of the most popular and widely-cited texts. This is a welcome addition to what has We are pleased to report that this second edition otherwise become a rather crowded shelf of books meets an even higher standard.

Superficially, there are not many changes from the first edition. The key to the successful solutions problems now lies in the choice of appropriate numerical models and their zyxw a Monotonic b Cyclic and c Transient loading Comparison of the results from numerical models with the results of standard laboratory associated parameters for geological media.

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Much tests. Comparison of field measurements and numer- The main objective of the symposium is to ical predictions. Final manuscripts will be due before engineers. A special emphasis will be given to the 30 April