Finite element method overview

Finite element methods are one of several approximate numerical techniques available for the solution of engineering boundary value problems. Analysis of materials processing operations lead to equations of this type, and finite element methods have a number of advantages in modeling such processes. This document is intended as an overview of this technique, to include examples relevant to polymer processing technology. 

The examples are made with the Chemical Engineering addition to FEMLAB, version 3.1. Appendix F describes the finite element method in one dimension and two dimensions so you have some concept of the approximation going from a single differential equation to a set of algebraic equations. This appendix presents an overview of many of the choices provided by FEMLAB. Illustrations of how FEMLAB is used to solve problems are given in Chapters 9-11. Thus, you may wish to skim this appendix on a first reading, and then come back to it as you use the program to solve the examples. A more comprehensive account of FEMLAB is available in Zimmerman (2004). 

Four research teams—AECB, CLAY, KIPH and LBNL—studied the task with different computational models. The computer codes applied to the task were ROCMAS, FRACON, THAMES and ABAQUS-CLAY. All of them were based on the finite-element method (FEM). Figure 6 presents an overview of the geometry and the boundary conditions of respective models, including the nearfield rock, bentonite buffer, concrete lid, and heater. The LBNL model is the largest and explicitly includes nearby drifts as well as three main fractures... 

In this overview, three digital simulation approaches (the finite difference method (FDM), the finite element method (FEM),... 

The fuzzy finite element method (FFEM) has become a most valuable tool for the analysis of structural dynamics problems with imprecise parameters. The present paper gives an overview of the status of development of the FFEM. The paper focuses specifically on the capabilities of the method, especially in early stages of design analysis. The formulation of design criteria in the presence of parameter uncertainties is reviewed and options for early design optimisation are discussed. 

This paper gives an overview of the current state of development of the fuzzy finite element method for dynamic analysis. After a review of definitions and generic analysis... 

A differential equation that has data given at more than one value of the independent variable is a boundary-value problem (BVP). Consequently, the differential equation must be of at least second order. The solution methods for BVPs are different compared to the methods used for initial-value problems (IVPs). An overview of a few of these methods will be presented in Sections 6.2.1. 2.3. The shooting method is the first method presented. It actually allows initial-value methods to be used, in that it transforms a BVP to an IVP, and finds the solution for the IVP. The lack of boundary conditions at the beginning of the interval requires several IVPs to be solved before the solution converges with the BVP solution. Another method presented later on is the finite difference method, which solves the BVP by converting the differential equation and the boundary conditions to a system of linear or non-hnear equations. Finally, the collocation and finite element methods, which solve the BVP by approximating the solution in terms of basis functions, are presented. 

In the next sections, we give a general overview how the Density Functional Theory is applied to electronic structure calculations within the framework of the finite-element method. We show how to incorporate pseudopotentials into the equations, explaining some technical difficulties that had to be solved and sorting all the ideas out and presenting them in a fashion applicable to our problem. 

This entry is aimed to provide an overview of analytical tools that can be used to study the behavior of masOTiry-infUled RC frames under earthquake loading. It is not intended to be an exhaustive summary of the literature rather, it will focus on some of the most common and representative analysis methods developed for such stmctures including their advantages and limitations. In this respect, two types of analysis methods will be considered (i) simplified, design-oriented analysis tools, and (ii) refined tools based on the finite element method. 

By means of introduction, section 1.1.1. presents a brief overview of several of the key features of fiber-reinforced composites. Thereafter, a discussion of the multi-scale modeling methodology within the context of the finite element method is given. The Introduction concludes with an overview of the use of multi-scale modeling to determine material properties at the macro-scale. These issues will be explored in more depth in subsequent sections. 

In this section we give a brief introduction and overview of those aspects of the finite element method that are relevant to micro-macro modeling. A detailed treatment may be found, for example, in [13]. 

As the numerical simulation of flows inside a compound extruder is a complex task, engineers usually do not simulate complete units but concentrate on functional zones. Due to the modularity of a compound extruder, a wide variety of configurations is possible for any functional zone. ID simulations calculate averaged functions or determine an overview of a specific configuration on an empirical level. They are used to select a specific extruder setup to be simulated with finite element (FEM) or boundary element (BEM) methods. 

This chapter presents an overview of modeling by finite elements (FE) for analysis of part and assembly models. The theory and application of this method for problem solving in different areas of engineering in static and dynamical as well as linear and nonhnear tasks of deformation, temperature, vibration, frequency, shape, etc., analyses are not discussed in this text. This material concentrates on the less published topic of FE models and advanced modeling procedures, as they are available for advanced analyses in Computer Aided Engineering (CAE) systems. CAE emphasizes the analysis based development of products while CAD/CAM... 

The second section deals with the analysis methods that are used in fuzzy finite element analysis. An overview of different approaches is presented first the transformation method, affine analysis, and global optimisation. The application to dynamic analysis is considered. The core of this section is the presentation of a consistent analysis approach to predict the effect of parameter uncertainties on different characteristics of dynamic behaviour of structures, resonance frequencies and dynamic response levels. A hybrid approach is developed, based on modal superposition and optimisation. 

In the third part of this chapter we review numerical methods commonly used in applying the PB equation to more complicated systems than simpler one-dimensional representations. Because two major articles covering most aspects of the numerical solution of the PB equation have recently appeared,only an overview of the numerical work is presented, emphasizing those aspects of primary importance or those that have been given less coverage elsewhere. Included is a brief description of finite-difference/finite-element PB algorithms similar to those used in popular programs such as DelPhi, MEAD, and alternative approaches... 

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