Finite Element Method

10 Key excerpts on "Finite Element Method"

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  The Finite Element Method with Heat Transfer and Fluid Mechanics Applications

  • Erian A. Baskharone(Author)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  1 The Finite Element Method: Introductory Remarks The Finite Element Method is a numerical technique for obtaining approximate solu- tions to a wide spectrum of engineering problems. Although originally developed to study the stresses in complex airframe structures, it has since been extended and applied to a broad field in continuum mechanics. Because of its diversity and flex- ibility as an analysis tool, this particular technique is receiving much attention in both academia and industry fields. Although this brief comment on the Finite Element Method answers the question posed by the section heading, it does not give us the operational definition we need to apply the method to a particular problem. Such an operational definition, along with a description of the method fundamentals, requires considerably more than just one paragraph to develop. Hence, the first segment of this book is devoted to basic concepts and fundamental theory. Before discussing more aspects of the Finite Element Method, we should first consider some of the circumstances leading to its inception, and we should briefly contrast it with other numerical techniques. In more and more engineering situations today, we find that it is necessary to obtain approximate numerical solutions to problems rather than exact closed- form solutions. For example, we may want to find the load capacity of a plate that has several stiffeners and odd-shaped holes, the concentration of pollutants during nonuniform atmospheric conditions, or the rate of fluid flow through a pas- sage of arbitrary shape. Without too much effort, we can write down the governing equations and boundary conditions for these problems, but we immediately see that no simple analytical solution can be found. The difficulty in these examples lies in the fact that either the geometry or some other features of the problem are irregu- lar.

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  Mechanics of Structures

  Variational and Computational Methods

  • Walter Wunderlich, Walter D. Pilkey(Authors)
  • 2002(Publication Date)
  • CRC Press

    (Publisher)

  6 The Finite Element Method The Finite Element Method is the dominant computational tool of contemporary structural and solid mechanics. It permits difficult problems of complicated geometry to be solved with relative ease. A number of finite element-based general purpose computer programs for analysis and design are available to the engineer. These programs can be used to solve an array of difficult structural engineering problems. An introduction to finite element tech-nology from the viewpoint of the structural mechanics philosophy of this book is presented in this chapter. The Finite Element Method has its origins in the matrix methods of structural analysis (Chapter 5). Some four decades ago it was recognized to be an emerging viable com-putational method for solid mechanics with applications to many problems of structural analysis. Since then, a mathematical basis for finite elements has been developed and applications have been expanded to include fluid mechanics, heat transfer, biomechan-ics, geomechanics, and acoustics. This ever-widening applicability of the Finite Element Method is due in part to its common formulation based on the variational principles. Also, one-, two-, and three-dimensional problems are handled in like fashion with a uniform notation. Technical histories of the Finite Element Method are provided in Clough (1990), Felippa (1994), and Gupta and Meek (1996). Biographies of some of the pioneers in the development of finite element technology are available in Robinson (1985). The method can be considered as a natural evolution of the standard methods of structural mechanics for frames modeled as discrete elements, or as an approximate solution technique for continuum mechanics pro-blems utilizing a regionally discretized model with assumed strain patterns for the regions.

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  Computational Methods in Subsurface Flow

  • Peter S. Huyakorn(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  2 The Finite Element Method 2.1 General The Finite Element Method is a numerical method for solving differential equations by means of piecewise approximation. As distinct from the finite difference method, which regards the solution region as an array of grid points, the Finite Element Method envisions the region as being made up of many small interconnected subregions called finite elements. Such elements, which generally take simple shapes (e.g. triangular, quad-rilateral and rectangular) are then assembled in various ways to represent a solution domain of arbitrary geometry. The origin of the Finite Element Method can be separately traced to applied mathematicians, physicists, and engineers. The first attempt to solve elliptical partial differential equations using a piecewise approximation was made by Courant (1943). The first important application of such a technique to a problem in structural mechanics was by Turner et al. (1956). The name Finite Element Method was suggested in a later paper by Clough (1960), who used finite elements to perform plane stress analysis. Since then, there has been a rapid development and the Finite Element Method is now generally recognized as a standard tool for solving a wide range of physical problems. For a more comprehensive review of the history of finite elements, the reader is referred to Heubner (1975). We develop an understanding of the essential concepts of finite element analysis without undue mathematical complexity by the consideration of simple discrete systems. Having achieved this goal, more mathematically abstract formulations are presented later in this chapter. 2.2 Basic Concepts in Finite Element Analysis The finite element analysis of a physical problem can be described as follows: (1) The physical system is subdivided into a series of finite elements that are connected at a discrete number of nodal points; this process is 25 26 2. THE Finite Element Method often called discretization.

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  Nonlinear Problems in Machine Design

  • Eliahu Zahavi, David M. Barlam(Authors)
  • 2000(Publication Date)
  • CRC Press

    (Publisher)

  49 2 Finite Element Method Much like the methods described in Section 1.4, the Finite Element Method (FEM) provides an approximate solution to the boundary value problem, which determines the relationship between stresses and strains and the acting forces in a loaded body. There are a number of ways to derive the main equations of the FEM that depend on its usage: in application to conservative systems, one may use either the principle of minimum of potential energy, or the principle of virtual work, in nonconservative systems, e.g., involving inelastic materials or friction, one must use the principle of virtual work or its equivalent—the Galerkin method. 1 2.1 INTRODUCTION TO FINITE ELEMENT THEORY In the following derivation of FEM equations, we use the Galerkin method in a weak form (see Section 1.5.1). The body that is subjected to analysis is divided into subdomains, called elements (see Figure 2.1). The elements usually have simple geometric forms such as triangles, quadrilaterals, hexahedrons, tetrahedrons, prisms, FIGURE 2.1 Body subject to analysis, divided into elements. 50 Nonlinear Problems in Machine Design etc. (Figure 2.2). The elements are interconnected at so-called nodes . The FEM model must meet two fundamental requirements. 1. The equilibrium within and among the elements 2. A compatibility—continuos displacements within and across element boundaries 2.1.1 T HE B OUNDARY V ALUE P ROBLEM To simplify the presentation, we consider a two-dimensional body of unity thickness. See Figure 1.7. Body forces (X,Y) act upon the volume, while surface tractions (p x ,p y ) are applied to part S 1 of the boundary surface. Displacements (u 0 ,v 0 ) are applied to part S 2 of the boundary surface. The problem pertains to a derivation of approximate expressions of strains and stresses in an explicit form.

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  The Finite Element Method in Engineering

  Pergamon International Library of Science, Technology, Engineering and Social Studies

  • S. S. Rao(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER 1 INTRODUCTION TO Finite Element Method 1.1 BASIC CONCEPT The basic idea in the Finite Element Method is to find the solution of a complicated problem by replacing it by a simpler one. Since the actual problem is replaced by a simpler one in finding the solution, we will be able to find only an approximate solution rather than the exact solution. The existing mathematical tools will not be sufficient to find the exact solution (and sometimes, even an approximate sol-ution) of most of the practical problems. Thus in the absence of any other conven-ient method to find even the approximate solution of a given problem, we have to prefer the Finite Element Method. Moreover, in the Finite Element Method, it will often be possible to improve or refine the approximate solution by spending more computational effort. In the Finite Element Method, the solution region is considered as built up of many small, interconnected subregions called finite elements. As an example of how a finite element model might be used to represent a complex geometrical shape, consider the milling machine structure shown in Figure 1.1(a). Since it is very difficult to find the exact response (like stresses and displacements) of the machine under any specified cutting (loading) condition, this structure is approx-imated as composed of several pieces as shown in Figure 1.1(b) in the finite elem-ent method. In each piece or element, a convenient approximate solution is assumed and the conditions of overall equilibrium of the structure are derived. The satisfaction of these conditions will yield an approximate solution for the displacements and stresses. 1.2 HISTORICAL BACKGROUND Although the name of the Finite Element Method was given recently, the concept has been used several centuries back. For example, ancient mathematicians found the circumference of a circle by approximating it as a polygon as shown in Figure 1.2.

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  Computational Techniques for Differential Equations

  • J. Noye(Author)
  • 2000(Publication Date)
  • North Holland

    (Publisher)

  520 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 Finite EZement Method 4 79 ABSTRACT The Finite Element Method i s a numerical method originally invented by engineers o u t of necessity t o solve complex practical problems. I t later attracted the interest of mathematicians who recognised i t s place amongst known mathematical procedures. This allowed t o generalise the method and to use i t w i t h confidence in other fields. Fast and large computers made possible applications to more and more complicated practical problems a t reasonable cost. General purpose computer codes written by engineers appeared allowing the solution of a wide range of problems by one code and with minimum theoretical knowledge required by the user. The Finite Element Method became a tool for an engineer and scientist comparable with other experimental procedures used in the practice. The success unfortunately ( b u t naturally) often led to an overconfidence of the users who forgot t h a t the computer cannot replace clear understanding of the fundamentals of the method, experience and skill i n modelling real structures and t o coupling techniques and theory correctly and efficiently together. The correct usage for the future progress of the method i s in close cooperation of mathematicians, scientists, computer programmers and users. This paper tries t o de- monstrate how an engineer understands the Finite Element Method and i t s application. INTRODUCTION I t i s essential t o distinguish four areas of interests in the Finite Element Methods, each on i t s own very important for the future development and usage of the technique. Firstly, mathematical theory of the method, which has t o work with idealised models of reality in order t o achieve generalisation, i s essential for mathematical validity of specific approaches and for future direction of the technique as a whole.

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  Finite Element Method for Solids and Structures

  A Concise Approach

  • Sung W. Lee, Peter W. Chung(Authors)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  1 Introduction to the Finite Element Method The finite element (FE) method is a powerful technique that can be used to transform any continuous body into a set of governing equations with a finite number of unknowns, called degrees of freedom (DOF). For solids and structures, DOF are displacements or rotational angles at discrete points in the body called nodes or nodal points. The simplest model for any structural system is a single degree of freedom (SDOF) system with a spring and a concen- trated mass. For a SDOF system, the spring constant and the mass can be determined via experiments or analytical means. However, for complicated structures such as aircraft, power plants, launch vehicles, buildings, bridges, machinery, or electronic devices, we need systematic and rational methods based on the fundamental principles of solid and structural mechanics in order to construct the governing equations. In the FE formulation, the domain of interest is first divided into a mesh of subdomains called elements. An assumed displacement is then introduced into each element to construct the element stiffness matrix and the load vector in terms of the element nodal DOF to form the equilibrium equations at the element level. Subsequently, they are assembled into the global stiffness matrix and the global load vector to construct the equilibrium equation at the global level, which can be solved for the unknown global nodal DOF. In this chapter, we will introduce the fundamentals of the FE method using a system of linear springs and a slender linear elastic body undergoing axial deformation as examples. These simple problems are chosen to describe the essential features of the FE method which are common to analysis of more complicated structural systems such as three-dimensional (3D) bodies. In the example problems given in this chapter, we will try to keep the solutions in non- dimensional form.

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  Finite Element and Boundary Methods in Structural Acoustics and Vibration

  • Noureddine Atalla, Franck Sgard(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  63 Chapter 4 The Finite Element Method An introduction 4.1 INTRODUCTION This chapter introduces the fundamental concepts of the FEM through the study of a one-dimensional acoustic problem. The finite element dis-cretization is based on a representation of the geometry and the unknown variables in terms of shape functions and nodal variables. This choice is mainly motivated by the requirements of generality and flexibility: to ana-lyze, using a unified and systematic approach, boundary value problems related to domains of various shapes subjected to miscellaneous boundary conditions. The analysis of a problem by FEM consists of seven basic steps. They are (see Figure 4.1) Step 1 : Writing the weak (or variational) integral form of the equations governing the problem. Step 2 : Meshing or geometry discretization. It consists in breaking up the geometry of the domain Ω into a set of subdomains Ω e called “the elements.” The subdomains are usually simple geometrical primitives (e.g., lines, triangles, quadrangles, tetrahedra …). Step 3 : Approximation of variables and calculation of elementary matri-ces. It consists in defining on each subdomain Ω e an approximate function p e of the exact function p . This so-called nodal approxima-tion function must have the following features: • It must only involve the nodal variables attached to nodes located on Ω e and its boundary; • It must be continuous on Ω e . In addition, all nodal approxima-tion functions must satisfy the continuity conditions between subdomains. The calculation of the elementary matrices consist then in evaluating, on each subdomain Ω e , the nodal approximation of the integral form, determined in step 1. 64 Finite element and boundary methods in structural acoustics and vibration Step 4 : Assembling. In this step, the elementary matrices are combined to form global matrices. In other words, the integral form is evaluated over the entire domain Ω .

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  Applied Metal Forming

  Including FEM Analysis

  • Henry S. Valberg(Author)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  Process simulation using FEA has yielded considerable cost and quality improvements in the industry. FEM simulation has become a practical and efficient tool for prediction of performance in industrial forming operations before dies are manufactured. Today, FEA is an extremely effective tool for research and also in industrial applications. FEM programs can predict large deformations and thermally influenced mate- rial flow in metal forming. Thus, they can provide detailed information about the forming mechanics. FEM simulation of metal forming is highly complex and compu- tation intensive. This is due to extreme nonlinearities because of large strains, plas- tic flow of anisotropic materials, with interfacial friction between irregular-shaped surfaces under changing contact. Because of this, it is difficult to obtain analytical solutions to such problems for quantitative evaluation of temperature, stress, and strain distributions, within the deforming body. It is therefore common to use the FEM for this purpose. 14.3.1 FEM Modeling Approach In FEM modeling, 1 the workpiece is represented by a collection of subdomains called finite elements; see Fig. 14.1. The elements are bounded by sets of nodes. They define localized mass and stiffness properties of the model. Equations of equi- librium, in conjunction with applicable physical considerations such as compatibility and constitutive relations, are applied to each element to construct a system of equa- tions. The system is then solved to find unknown values using advanced numerical techniques. FEA is an approximate method, and its accuracy can be improved by increasing the number of elements used.

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  Computational Finite Element Methods in Nanotechnology

  • Sarhan M. Musa(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  15 Finite Element Method (FEM) for Nanotechnology Application in Engineering: Integrated Use of Macro-, Micro-, and Nano-Systems Radostina Petrova

  Technical University of Sofia, Sofia, Bulgaria

  P. Genova

  Bulgarian Academy of Science, Sofia, Bulgaria

  M. Tzoneva

  Technical University of Sofia, Sofia, Bulgaria

  CONTENTS 15.1  Introduction 15.2  Basic Idea and Applied Methods 15.3  Examples 15.3.1  Clamps for Operating with Micro or Nano Objects 15.3.1.1  Background of the Problem 15.3.1.2  Aim of the Example 15.3.1.3  Description of the Finite Element Model 15.3.1.4  Solution for a Set of Materials 15.3.1.5  Numerical Experiments and Results 15.3.1.6  Conclusions on the Cited Numerical Example 15.3.2  Analysis and Design of Parallel Micro-Robot Structures 15.3.2.1  Why Tripod Structure? 15.3.2.2  Aim of the Provided Example 15.3.2.3  Description of the Established Mathematical Model and of the Developed Solution Strategy 15.3.2.4  Numerical Examples for Outlining the Accessible Zone of the Vertex of the Controlled Link 15.3.2.5  FEM of a Spatial Three-Pod Mechanism with Three Built-in Drives 15.3.2.6  Conclusions on the Preceding Numerical Example 15.3.3  Kinematics of a Macro-Mechatronic System 15.3.3.1  Analytical Solution 15.3.3.2  Finite Element Solution 15.3.3.3  Conclusions on the Preceding Example 15.3.4  Analysis and Synthesis of Mechanical Bioreactors for Bio-Nanotechnologies 15.3.4.1  Short Review of Existing Devices for Bioreactor Systems 15.3.4.2  Kinematics of Spatial Mechanisms in Biotechnologies 15.3.4.3  Conclusions on the Preceding Example References 15.1  Introduction

  “The step from micro-technology to nano-technology requires more than a reduction of the size by a factor of a thousand. If you want to move precisely in the nano-world, you don’t succeed by perfecting proven techniques.” This statement of Handelsblat is cited by Klocke and Gesang (2002) and describes briefly the innovative approach in nanoscience.

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