Finite Element Modeling
Finite Element Modeling is a computational technique used to analyze and simulate complex physical systems. It involves dividing a structure or system into smaller elements to accurately predict its behavior under various conditions. By applying mathematical equations and algorithms, Finite Element Modeling helps engineers and researchers understand stress, strain, and other physical phenomena, aiding in the design and optimization of structures and materials.
8 Key excerpts on "Finite Element Modeling"
eBook - ePubParticle Technology and Engineering
An Engineer's Guide to Particles and Powders: Fundamentals and Computational Approaches
- Jonathan P.K. Seville, Chuan-Yu Wu(Authors)
- 2016(Publication Date)
- Butterworth-Heinemann(Publisher)
...Chapter 10 Finite Element Modeling Abstract The finite element method (FEM) is used for modeling of physical systems in a wide variety of engineering disciplines, including structural dynamics, heat transfer, fluid dynamics, and aerodynamics. In this chapter, the application of FEM in modeling particle systems is introduced, which covers three aspects: (1) modeling of particle–particle interactions, (2) multiple particle Finite Element Modeling (MPFEM), and (3) continuum modeling of powder compaction. A detailed discussion of the constitutive models used for powder compaction is presented, with particular reference to the Drucker–Prager-cap (DPC) model. In addition, methods for determination of material properties for Finite Element Modeling with the DPC model are discussed in detail. Typical applications of FEM in modeling particle systems are illustrated. Keywords Contact modeling; Drucker–Prager-cap model; Finite discrete element method; Finite Element Modeling; Model calibration; Multiple particle finite element method; Particle interaction; Powder compaction The finite element method (FEM) is one of the computational techniques for finding approximate solutions to differential and integral equations. Although FEM was developed in the first half of the twentieth century, with its initial application in solid mechanics, civil, and structural engineering, it has since advanced significantly into modeling of physical systems in a wide variety of engineering disciplines, including structural dynamics, heat transfer, fluid dynamics, and aerodynamics. In these and other fields of engineering and applied sciences, most real problems are complex in both geometry and boundary conditions. It is generally impossible to obtain analytical solutions for such problems, so approximate methods are needed. In FEM, a computational domain is generally set up within which the underlying physics can be defined using partial differential equations or integral equations...
- 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...
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...
eBook - ePubMechanics of Solid Polymers
Theory and Computational Modeling
- Jorgen S Bergstrom(Author)
- 2015(Publication Date)
- William Andrew(Publisher)
...In practice, finite element analysis (FEA) can be divided into two different categories: implicit and explicit simulations, see Table 3.3. FEA can also be used to study eigenfrequencies and eigenmodes of deformation for a component or system. Table 3.3 Comparison Between Implicit and Explicit FEA Implicit Analysis Explicit Analysis Solves the equilibrium equations at each time step Solves the problem using Newton’s law of motion Good for static problems Good for short duration dynamic problems Is numerically stable Is only numerically stable for small time increments If the FE software finds a solution, that solution is likely to have small numerical errors Often easy to find a solution, but care is needed to find a solution with small numerical errors Contact problems are sometimes difficult to handle Good at handling problems with contact As mentioned, one of the most difficult steps in an FE simulation is to specify the material model. A material model is here defined as a constitutive equation and a corresponding set of material parameters: FE software contains a library of different constitutive equations that can be chosen, but the material parameters are typically not provided and the selection of constitutive models that are available is typically targeted to metals. There are generally only a limited set of constitutive models that are suitable for predicting the deformation behavior of polymers. One way to get around this is to use an external user material subroutine (UMAT) to define the material behavior, see Chapter 10 for more details. 3.3 Review of Modeling Techniques Most polymer mechanics problems can be divided into two main categories: predictions of deformation behavior and predictions of failure events. A common approach is to start with a deformation analysis to determine the magnitudes and distributions of stress and strain, and then, if needed, use this information as part of a predictive failure analysis...
- 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...
- Ulrich Häußler-Combe(Author)
- 2014(Publication Date)
- Ernst & Sohn(Publisher)
...Some benefit is desirable finally. Thus, a model which passed validations is usable for predictions. Structures created along such predictions hopefully prove their worth in the reality of interest. This textbook covers the range of conceptual models, mathematical models, and numerical models with special attention to reinforced concrete structures. Notes regarding the computational model including available programs and example data are given in Appendix F. A major aspect of the following is modeling of ultimate limit states : states with maximum bearable loading or acceptable deformations and displacements in relation to failure. Another aspect is given with serviceability : Deformations and in some cases oscillations of structures have to be limited to allow their proper usage and fulfillment of intended services. Durability is a third important aspect for building structures: deterioration of materials through, e.g., corrosion, has to be controlled. This is strongly connected to cracking and crack width in the case of reinforced concrete structures. Both topics are also treated in the following. 1.2 Discretization Outline The finite element method (FEM) is a predominant method to derive numerical models from mathematical models. Its basic theory is described in the remaining sections of this chapter insofar as it is needed for its application to different types of structures with reinforced concrete in the following chapters. The underlying mathematical model is defined in one-, two-, or three-dimensional fields of space related to a body and one-dimensional space of time. A body undergoes deformations during time due to loading. We consider a simple example with a plate defined in 2D space, see Fig. 1.2. Loading is generally defined depending on time whereby time may be replaced by a loading factor in the case of quasistatic problems...
- Ehab Ellobody, Ran Feng, Ben Young(Authors)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
...Chapter 3 Finite Element Modeling Chapter 3 focuses on Finite Element Modeling of metal structures and details the choice of element type and mesh size that can accurately simulate the complicated behavior of different metal structural elements. The chapter details how the nonlinear material behavior can be efficiently modeled and how the initial local and overall geometric imperfections were incorporated in the finite element analysis. The chapter also details modeling of different loading and boundary conditions commonly applied to metal structures. The chapter focuses on the Finite Element Modeling using any software or finite element package, e.g., the use of ABAQUS software in Finite Element Modeling. Keywords Boundary conditions; finite element mesh; initial imperfections; load application; material modeling; residual stresses 3.1 General Remarks The brief revision of the finite element method is presented in Chapter 2. It is now possible to detail the main parameters affecting Finite Element Modeling and simulation of different metal structural members, which is highlighted in this chapter. The chapter provides useful guidelines on how to choose the best finite element type and mesh to represent metal columns, beams and beam columns, and connections. The behavior of different finite elements, briefed in Chapter 2, is analyzed in this chapter to assess their suitability for simulating the structural member. There are many parameters that control the choice of finite element type and mesh such as the geometry, cross section classification, loading, and boundary conditions of the structural member. The aforementioned issues are also covered in this chapter. Accurate Finite Element Modeling depends on the efficiency in simulating the nonlinear material behavior of metal structural members. This chapter shows how to correctly represent different linear and nonlinear regions in the stress–strain curves of metal structures...
