Finite Element Method
The Finite Element Method is a numerical technique used to solve complex engineering problems by dividing a structure into smaller, simpler elements. It involves representing the behavior of each element using mathematical models and then assembling these models to analyze the overall behavior of the structure. This method is widely used in various engineering disciplines to simulate and optimize the performance of designs.
8 Key excerpts on "Finite Element Method"
- Andy Choi(Author)
- 2018(Publication Date)
- Bentham Science Publishers(Publisher)
...The Basics The Finite Element Method is a mathematical approach used to examine continua and structures. Typically, the problem at hand is too difficult to be resolved in a satisfactory manner using classical analytical means. The finite element process generates a lot of simultaneous algebraic equations, which are created and calculated on a digital computer. Finite element calculations are carried out on laptops, mainframes, and personal computers. Results are seldom precise. However, errors are reduced by processing more equations, and results accurate enough for engineering applications are attainable at reasonable costs. Finite element analysis examines a complex problem by redefining it as the summation of the solutions of a series of interrelated simpler problems. The first stage involves subdividing (that is, discretize) the complex geometry into a suitable set of smaller “elements” of “finite” dimensions. This forms the “mesh” model of the investigated structure when combined (Fig. 29). Fig. (29)) Schematic illustration of a dental implant. (A) Solid model; (B) Finite element mesh. Each element can undertake a specific geometric shape (that is, square, cube, triangle, tetrahedron, for example) with a specific internal strain function. The equilibrium equations between the displacements taking place at its corner points or “nodes” and the external forces acting on the element can be written using these functions, the actual geometry of the element, and a suite of boundary conditions such as constrain points. One equation for each degree of freedom will be created for each node of the element. In general, these equations are appropriately written in matrix form for utilization in a computer algorithm. From the above example, and as a whole, the Finite Element Method models a structure as an assembly of small parts (elements). A simple geometry is used to define each element and therefore is much easier to examine than the actual structure...
- Matthew N.O. Sadiku(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
...6 Finite Element Method Prayer without action is hypocrisy and action without prayer is arrogance. —Unknown 6.1 Introduction The Finite Element Method (FEM) has its origin in the field of structural analysis. Although the earlier mathematical treatment of the method was provided by Courant [ 1 ] in 1943, the method was not applied to electromagnetic (EM) problems until 1968. Since then the method has been employed in diverse areas such as waveguide problems, electric machines, semiconductor devices, microstrips, and absorption of EM radiation by biological bodies. Although the finite difference method (FDM) and the method of moments (MoM) are conceptually simpler and easier to program than the FEM, FEM is a more powerful and versatile numerical technique for handling problems involving complex geometries and inhomogeneous media. The systematic generality of the method makes it possible to construct general-purpose computer programs for solving a wide range of problems. Consequently, programs developed for a particular discipline have been applied successfully to solve problems in a different field with little or no modification. A brief history of the beginning of FEM is provided in Reference 2. The finite element analysis of any problem involves basically four steps [ 3 ]: • Discretizing the solution region into a finite number of nonoverlap subregions or elements, • Deriving governing equations for a typical element, • Assembling of all elements in the solution region, and • Solving the system of equations obtained. Discretization of the continuum involves dividing the solution region into sub-domains, called finite elements. Figure 6.1 shows some typical elements for one-, two-, and three-dimensional problems. The problem of discretization will be fully treated in Sections 6.5 and 6.6...
eBook - ePubBiomaterials Science
An Introduction to Materials in Medicine
- Buddy D. Ratner, Allan S. Hoffman, Frederick J. Schoen, Jack E. Lemons(Authors)
- 2012(Publication Date)
- Academic Press(Publisher)
...Although the approach was originally formulated to perform structural analyses, it has become increasingly adopted in biomechanics and biomaterial sciences to solve complex multi-physics problems. In the following, we present a brief introduction of the mathematical formulation of FEM for the biomaterials audience. We utilize the infinitesimal strain linear elasticity which, while not realistic for most biological materials, allows the basic theory to be presented with maximum clarity. Overview of the Finite Element Method In FEM, a real structure is replaced by a discrete model obtained by subdivision into a number of finite elements (Figure I.1.4.1). The discretized model is composed of appropriately shaped elements defined by a series of interconnected points known as nodes. The continuum problem with infinite degrees of freedom can thus be reduced to a discrete problem with finite degrees of freedom, and solved computationally with a series of simultaneous algebraic equations. In the ordinary formulation, the displacement field within each finite element is strictly related to nodal displacement by shape functions that can be derived from the interpolation of nodal displacements. Under this assumption, the initial problem can be reduced to a discrete problem where the unknowns are the Cartesian components of the nodal displacements, in effect reducing the initial three-dimensional problem to one with only three degrees of freedom per node (Tottenham and Brebbia, 1970 ; Zienkiewicz, 1971 ; Middleton et al., 1996). FIGURE I.1.4.1 An arbitrary area discretized with tetrahedral elements. Problem Consider first the simple case of a triangular plane element defined by three nodes (Figure I.1.4.2 a) (Tottenham and Brebbia, 1970 ; Zienkiewicz, 1971). Two displacement components u (x, y) and v (x, y) of an arbitrary point P (x, y) within the element can be expressed by the relationship (Eq. 1): (1) which can be expressed in matrix form as (Eq...
- João Pedro A. Bastos, Nelson Sadowski(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...5 Finite Element Method Brief Presentation 5.1 INTRODUCTION The evolution of the FEM is intimately linked to developments in engineering and computer sciences. Its application in a variety of areas, especially in the nuclear, aeronautics, and transportation industries, is a testimony to the high degree of accuracy the method is capable of, as well as to its ability to model complex problems. There are different ways to define, present, and use FEM. Our choice here is based on our implementation experience on which we adopted the approaches using the FE real coordinates as well as the concept of”reference element We intend to present, in a relatively direct and short manner, the FEM by using very classical algebra, and it is not our intention to present the FEM in deeply theoretical detail. We believe that it can be followed by most of the readers with an engineering background. We have already published scientific papers and two books [ 1, 2 ]. The latter is dedicated to 2 D problems. Our interest here is to present 3 D formulations applied to low‐frequency cases. Here we will focus mainly on the concepts and practical aspects that will be directly applied to the cases treated in this work. Generally, in EM, the FEM is associated with variational methods or residual methods [ 1, 2 ]. In the first case, the numerical procedure is established using a functional to be minimized. For each problem a particular functional has to be defined. It is worth mentiomng that for the classica12D problems, the functionals are well known, but for less usual phenomena a search for a functional is necessary, which can be a difficult task in some cases. Moreover, we do not work directly with the physical equation related to our problem, but with the corresponding functional. Contrarily, residual methods are established directly from the physical equation that has to be solved...
- Saad A. Ragab, Hassan E. Fayed(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...Preface Computational approaches to solving engineering problems have become essential analysis and design tools for engineers. The finite-element method is at the center of modern computer analysis techniques. Before embarking on using massive finite-element commercial software, engineers need to know how finite- element models are derived for the basic principles that are usually expressed as differential or integral statements. Having strong mathematical foundation of the finite-element method, engineering students will be better prepared to tackle complex problems. This textbook Introduction to Finite Element Analysis has evolved from the first author’s lecture notes for finite-element courses that were taught in the department of engineering science and mechanics (now biomedical engineering and mechanics) at Virginia Tech for the past 17 years. The book serves as an introduction to the finite-element method, and presents it as a numerical technique for solving differential equations that describe problems in civil, mechanical, aerospace, and biomedical engineering. It enables engineering students to formulate and solve finite-element models of practical problems and analyze the results. Although commercial finite-element software are not used in this book, it explains the mathematical foundation underpinning such software. Mastering the techniques presented in this book, students will be better prepared to use commercial software as practicing engineers. The book is intended for senior or first-year graduate students in engineering or related disciplines. Thus, mathematical rigor is not compromised but presented at a level consistent with the anticipated mathematics background required in most engineering curricula. The power and versatility of the finite-element method is demonstrated by a large number of examples and exercises of practical engineering problems...
eBook - ePub- Chandrakant S. Desai, Tribikram Kundu(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...2 Steps in the Finite Element Method Introduction Formulation and application of the Finite Element Method are considered to consist of eight basic steps. These steps are stated in this chapter in a general sense. The main aim of this general description is to prepare for complete and detailed consideration of each of these steps in this and subsequent chapters. At this stage, the reader may find the general description of the basic steps in this chapter somewhat overwhelming. However, when these steps are followed in detail with simple illustrations in the subsequent chapters, the ideas and concepts will become clear. Mathematical foundations of the variational formulation and the residual formulation (Galerkin’s method) are given in more detail after a brief description of the eight steps in the Finite Element Method. A good comprehension of these procedures is necessary for a thorough understanding of the derivation of the element equations. General Idea Engineers are interested in evaluating effects such as deformations, stresses, temperature, fluid pressure, and fluid velocities caused by forces such as applied loads or pressures and thermal and fluid fluxes. The nature of distribution of the effects (deformations) in a body depends on the characteristics of the force system and of the body itself. Our aim is to find this distribution of the effects. For convenience, we shall often use displacements or deformations u (Figure 2.1) in place of effects. Subsequently, when other problems such as heat and fluid flow are discussed they will involve distribution of temperature and fluid heads and their gradients. We assume that it is difficult to find the distribution of u by using conventional methods and decide to use the Finite Element Method, which is based on the concept of discretization, as explained in Chapter 1...
eBook - ePubMechanics of Solid Polymers
Theory and Computational Modeling
- Jorgen S Bergstrom(Author)
- 2015(Publication Date)
- William Andrew(Publisher)
...For example, using this technique it is possible to determine not only the stress state for a given imposed deformation state but also the mathematical dependence of how geometry and load history directly influence the stress state. FE tools have reached a high level of maturity and are widely used in both academia and industry. The easy access to commercial FE codes has created both great possibilities to solve advanced problems, and to some extent reduced the need for costly experimental tests. However, this computational modeling approach also presents serious challenges since the FE programs, albeit being easy to use, can provide inaccurate and misleading results if not used properly. One of the overall aims of this text is to assist the creation and selection of FE models and analysis techniques for polymer problems. 3.1.1 Required Inputs for FEA To perform an FE simulation, or indeed any stress analysis calculation, there are three different types of inputs that need to be specified, see Figure 3.1. Note that these inputs are also needed for traditional closed-form solution methods. Figure 3.1 Inputs needed for FE analysis. The fundamental problem of polymer mechanics can be written in mathematical form as a boundary value problem (BVP), with governing equations for: compatibility, constitutive response, and equilibrium. More details of these equations are given in Chapter 4. One of the overall themes of this book is that of the three different types of input to the FE models: (1) geometry; (2) loading and boundary conditions (BC); and (3) material behavior; it is the specification of the material behavior that is typically most challenging. 3.2 Types of FEA The FE method can be described as a numerical tool for solving ordinary and partial differential equations over nontrivial geometric domains. In practice, finite element analysis (FEA) can be divided into two different categories: implicit and explicit simulations, see Table 3.3...
eBook - ePubFinite Element Analysis
Method, Verification and Validation
- Barna Szabó, Ivo Babuška(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
...1 Introduction to the Finite Element Method This book covers the fundamentals of the Finite Element Method in the context of numerical simulation with specific reference to the simulation of the response of structural and mechanical components to mechanical and thermal loads. We begin with the question: what is the meaning of the term “simulation”? By its dictionary definition, simulation is the imitative representation of the functioning of one system or process by means of the functioning of another. For instance, the membrane analogy introduced by Prandtl 1 in 1903 made it possible to find the shearing stresses in bars of arbitrary cross‐section, loaded by a twisting moment, through mapping the deflected shape of a thin elastic membrane. In other words, the distribution and magnitude of shearing stress in a twisted bar can be simulated by the deflected shape of an elastic membrane. The membrane analogy exists because two unrelated phenomena can be modeled by the same partial differential equation. The physical meaning associated with the coefficients of the differential equation depends on which problem is being solved. However, the solution of one is proportional to the solution of the other: At corresponding points the shearing stress in a bar, subjected to a twisting moment, is oriented in the direction of the tangent to the contour lines of a deflected thin membrane and its magnitude is proportional to the slope of the membrane. Furthermore, the volume enclosed by the deflected membrane is proportional to the twisting moment. In the pre‐computer years the membrane analogy provided practical means for estimating shearing stresses in prismatic bars. This involved cutting the shape of the cross‐section out of sheet metal or a wood panel, covering the hole with a thin elastic membrane, applying pressure to the membrane and mapping the contours of the deflected membrane...
