Finite Element Analysis with Error Estimators

An Introduction to the FEM and Adaptive Error Analysis for Engineering Students

J. E. Akin

• Publisher: Butterworth-Heinemann
• Year: 2005
• Print ISBN: 9780750667227
• eBook ISBN: 9780080472751
• Topic: Technik & Maschinenbau
• Subtopic: Angewandte Mathematik

This key text is written for senior undergraduate and graduate engineering students. It delivers a complete introduction to finite element methods and to automatic adaptation (error estimation) that will enable students to understand and use FEA as a true engineering tool. It has been specifically developed to be accessible to non-mathematics students and provides the only complete text for FEA with error estimators for non-mathematicians. Error estimation is taught on nearly half of all FEM courses for engineers at senior undergraduate and postgraduate level; no other existing textbook for this market covers this topic. - The only introductory FEA text with error estimation for students of engineering, scientific computing and applied mathematics - Includes source code for creating and proving FEA error estimators

Chapter 1

Introduction

1.1 Finite element methods

The goal of this text is to introduce finite element methods from a rather broad perspective. We will consider the basic theory of finite element methods as utilized as an engineering tool. Likewise, example engineering applications will be presented to illustrate practical concepts of heat transfer, stress analysis, and other fields. Today the subject of error analysis for adaptivity of finite element methods has reached the point that it is both economical and reliable and should be considered in an engineering analysis. Finally, we will consider in some detail the typical computational procedures required to apply modern finite element analysis, and the associated error analysis. In this chapter we will begin with an overview of the finite element method. We close it with consideration of modern programming approaches and a discussion of how the software provided differs from the author’s previous implementations of finite element computational procedures.

In modern engineering analysis it is rare to find a project that does not require some type of finite element analysis (FEA). The practical advantages of FEA in stress analysis and structural dynamics have made it the accepted tool for the last two decades. It is also heavily employed in thermal analysis, especially for thermal stress analysis.

Clearly, the greatest advantage of FEA is its ability to handle truly arbitrary geometry. Probably its next most important features are the ability to deal with general boundary conditions and to include nonhomogeneous and anisotropic materials. These features alone mean that we can treat systems of arbitrary shape that are made up of numerous different material regions. Each material could have constant properties or the properties could vary with spatial location. To these very desirable features we can add a large amount of freedom in prescribing the loading conditions and in the post-processing of items such as the stresses and strains. For elliptical boundary value problems the FEA procedures offer significant computational and storage efficiencies that further enhance its use. That class of problems include stress analysis, heat conduction, electrical fields, magnetic fields, ideal fluid flow, etc. FEA also gives us an important solution technique for other problem classes such as the nonlinear Navier – Stokes equations for fluid dynamics, and for plasticity in nonlinear solids.

Here we will show what FEA has to offer and illustrate some of its theoretical formulations and practical applications. A design engineer should study finite element methods in more detail than we can consider here. It is still an active area of research. The current trends are toward the use of error estimators and automatic adaptive FEA procedures that give the maximum accuracy for the minimum computational cost. This is also closely tied to shape modification and optimization procedures.

1.2 Capabilities of FEA

There are many commercial and public-domain finite element systems that are available today. To summarize the typical capabilities, several of the most widely used software systems have been compared to identify what they have in common. Often we find about 90 percent of the options are available in all the systems. Some offer very specialized capabilities such as aeroelastic flutter or hydroelastic lubrication. The mainstream capabilities to be listed here are found to be included in the majority of the commercial systems. The newer adaptive systems may have fewer options installed but they are rapidly adding features common to those given above. Most of these systems are available on engineering workstations and personal computers as well as mainframes and supercomputers. The extent of the usefulness of an FEA system is directly related to the extent of its element library. The typical elements found within a single system usually include membrane, solid, and axisymmetric elements that offer linear, quadratic, and cubic approximations with a fixed number of unknowns per node. The new hierarchical elements have relatively few basic shapes but they do offer a potentially large number of unknowns per node (more than 80). Thus, the actual effective element library size is extremely large.

In the finite element method, the boundary and interior of the region are subdivided by lines (or surfaces) into a finite number of discrete sized subregions or finite elements. A number of nodal points are established with the mesh. The size of an element is usually associated with a reference length denoted by h. It, for example, may be the diameter of the smallest sphere that can enclose the element. These nodal points can lie anywhere along, or inside, the subdividing mesh, but they are usually located at intersecting mesh lines (or surfaces). The elements may have straight boundaries and thus, some geometric approximations will be introduced in the geometric idealization if the actual region of interest has curvilinear boundaries. These concepts are graphically represented in Fig. 1.1.

Figure 1.1 Piecewise approximation of a scalar function

The nodal points and elements are assigned identifying integer numbers beginning with unity and ranging to some maximum value. The assignment of the nodal numbers and element numbers will have a significant effect on the solution time and storage requirements. The analyst assigns a number of generalized degrees of freedom to each and every node. These are the unknown nodal parameters that have been chosen by the analyst to govern the formulation of the problem of interest. Common nodal parameters are displacement components, temperatures...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Preface
• Features of the text and accompanying resources
• Notation
• Chapter 1: Introduction
• Chapter 2: Mathematical preliminaries
• Chapter 3: Element interpolation and local coordinates
• Chapter 4: One-dimensional integration
• Chapter 5: Error estimates for elliptic problems
• Chapter 6: Super-convergent patch recovery
• Chapter 7: Variational methods
• Chapter 8: Cylindrical analysis problems
• Chapter 9: General interpolation
• Chapter 10: Integration methods
• Chapter 11: Scalar fields
• Chapter 12: Vector fields
• Index