Nizeh Botros for reviewing the materials. I would like to expecially thank Dr. Budzban for his tireless efforts in making sure that I had all that was needed to be successful and ensuring the educational and mathematical integrity of this research paper.
I would also like to thank Mary Ann Budzban for being so warm and caring to me. Both of them have went above and beyond to care for my education and my well being. I would like to especially thank Dr. I would like to thank Dr. Botros, a fellow engineer, who did not hesitate to offer guidance or serve on my research paper committee. There are a host of others who have been everything from a study buddy to a best friend, I would like to thank them all.
I would not have survived the emotional stress doing rigorous mathematics can bring, had you all not been there. Of this host, I would like to give a special thanks to Lochana Siriwardena and Yasanthi Kotegoda, you two have become my family. Many Thanks. The sophistication of the method, its accuracy, simplicity, and computability all make it a widely used tool in the engineering modeling and design process.
This paper will discuss finite element analysis from mathematical theory to applications. For purposes of analysis of the method, it is easier to study theory along side applications. This hopefully gives the reader an opportunity to draw direct connections between application and theory, putting the mathematics into context. For the basis of understanding the mathematical theory, we will utilize a one dimensional problem.
However, all the concepts and proofs can be easily transformed to multidimensional situations with a few adjustments. In many cases, the solution to even second order differential equations can be quite complicated and an alternative method to computing an exact answer would be needed. Finite element analysis provides the tools necessary to approximate the solution. We will consider a simple example to help illustrate the theory. In this case an exact solution using traditional methods for solving differential equations can be found.
However, we will utilize this example for explanation purposes and as a comparison for the accuracy of the method. We will return to this equation in section chapter 4. Before we begin, let us build the mathematical framework and key ideas needed for the theoretical foundation of finite element anlaysis.
Vector Space Let V be a set. Let F be a field of scalars. Inner Product Let V be a vector space. Norm Let V be a vector space. Example 1. An example of a norm is the absolute value function on the reals. R1 Example 1. Here we use the Riemann Integral to define an inner product on a function space. Proposition 1. Let H, h, i be an inner product space. We will call this the norm induced by the inner product. In what follows C will denote the set of all continuous functions and L will denote the set of all polynomials of degree 1.
Superscripts will denote the order of the smoothness of the function. For example C 2 [0, 1] is the set of all second order differentiable functions defined on the closed interval [0, 1]. When we discuss the weak or variational form of an object in mathematics it generalizes the standard form and exists in context where the standard form may not.
For example the weak derivative of a function may exist where the standard form does not. To restate our problem in its weak or variational form we will utilize the prop- erties of the inner product. Suppose there is a linear differential relationship between the two functions y and f which exist in the function space H. We will refer to this inner product as the energy inner product. Note: The inner product described here is only valid for a specific set of func- tions. The Neumann and Dirichlet boundary conditions become important assump- tions for our analysis.
To do this we need a measure of distance. In mathematics one way to generate a measure of distance is the norm. We will utilize this norm within our calculations. Let us illustrate this theory in the context of an example. Suppose h meets all bound- R1 R1 ary requirements.
Now that we have established that the weak form solves the original equation, we can utilize the weak form of the differential equation to estimate local solutions using peice-wise polynomial approximations in the finite element method. However, before we do this we must verify the uniqueness of our proposed solution.
Previ- ously, we have considered infinite dimensional spaces. For purposes of finite element analysis we will truncate our dimensions. We can consider a finite degree polymonial approximation to each of our involved quantities. As we build on our analysis we can begin to specify characteristics of our subspaces, keeping in mind that all subspaces must contain the same properties of the original space.
These include general properties of eigenvalues, the Perron-Frobenius theory for nonnegative matrices, Gerschgorin analysis, and Schur complements. Chapters 7 through 10 present preconditioners derived from algebraic considerations, such as incomplete factorizations, and tools for analyzing their performance. Chapters 11 through 13 discuss the conjugate gradient method and some variants for nonsymmetric sys- tems. There are three appendices on additional matrix theory and Chebyshev polynomials.
The book should be valuable as a reference volume, updating the classic books of Varga [1] and Young [2], and it also has potential as a text for an advanced graduate course. The chapters on matrix theory give a good concise overview of material for analyzing iterative methods, and many of the topics, such as generalizations of regular splittings and uses of the field of values, are not seen in standard texts.
The chapters on classical and conjugate gradient-like methods comprise an excellent summary of what is known about convergence rates of such methods, including important results for special cases of eigenvalue distributions and for nonsymmetric systems. The chapters on preconditioners are somewhat more difficult to read than the rest of the book, in part because there is little background on the discrete partial differential equations from which most of the ideas derive.
For example, many results in Chapter 9 were designed for multilevel methods for elliptic problems, which is hard to appreciate from the text. However, the large amount of analysis in these chapters makes them of potential long-term use for reference. For use as a text, the first seven chapters contain a very good collection of exercises as well as the nice feature through Chapter 6 of the presentation of definitions in the introduction to the chapters.
In addition, such a course would almost certainly have to be supplemented with material on discrete partial differential equations. In general, the organization and presentation of the book is good, although there are a few technical concepts that are used before they are defined. The most no- table weakness concerns the references. Citations appear at the end of individual chapters, but there is no index of references, and it is difficult to find individual ref- erences.
As a consequence, I believe it will be somewhat difficult to follow up in the published literature. Despite this flaw, I believe the book will serve as an excellent reference on the subject of iterative methods. It is a good introduction to the topic albeit at a fairly advanced level, and it is also a potential source of new ideas. References 1. MR 2. A great deal of research has been done since the s, and the authors state that most of the material they present has not previously appeared in textbooks.
Both serial and parallel algorithms are described, the latter in the fourth and final chapter. There is very little treatment of computation in floating-point arithmetic, and the emphasis is on the computational complexity of algorithms rather than their actual cost in a com- puter implementation.
The book can be described as being more theoretical than a numerical analysis textbook, but more practical than a textbook in computational complexity, and it makes contributions to both areas. A strength is the treatment of structured matrices. The organization of the book is a little unusual in that important topics are often relegated to appendices and exercises.
A full rounding error analysis is given for the FFT Proposition 4. The presentation is good, the book having been prepared with TEX, but it could be improved. The numbering of equations, theorems and so on does not reflect the chapter number, which makes cross-referencing difficult. Citations are given using an ugly alphanumeric scheme, under which the book under review would be cited as [BP94]; this makes finding a reference in the extensive bibliography more difficult than if numbers were used.
It appears that Volume 2 will include a treatment of fast matrix multiplication, which is only briefly considered in Volume 1. Aho, J.
Hopcroft, and J. Borodin and I. Munro, The computational complexity of algebraic and numeric problems, American Elsevier, New York, MR 3. MR 93g 4. Charles F. MR 93a Nicholas J. The book, however, concentrates on the content of only a portion of the lectures, and enters into considerable detail on the chosen topics.
These topics are quickly enumerated by giving the chapter headings: 1. Matrix Subdivision, 2. Stationary Subdivision, 3. Piecewise Polynomial Curves, 4. Geo- metric Methods for Piecewise Polynomial Surfaces, and 5. Recursive Algorithms for Polynomial Evaluation. This leads naturally into the subject of general matrix subdivision schemes. Chapter 2 discusses the de Rham-Chaikin algorithm and the Lane-Reisenfeld algorithm. Here we encounter the refinement equation that plays an important role in wavelets, and one section is devoted to applications in that subject.
Chapter 3 concerns representation of curves parametrically by spline func- tions. Many subtopics are dealt with, such as knot insertion, variation-diminishing properties of the B-spline basis, and connection matrices which relate adjacent parts of the piecewise polynomial. In Chapter 4 the emphasis is on multivari- ate splines and their use in surface representation.
The geometric interpretation of higher-dimensional spline functions as volumes of slices of polyhedra is central. This chapter closes with a historical vignette in the form of letters by I. Schoenberg and H.
Chapter 5 discusses, among other topics, blossoming, pyramid schemes, and subdivision for multivariate polynomials. This book of pages of- fers a concentrated mathematical development of the representation of curves and surfaces by one of the most authoritative experts in the field. It certainly establishes the current status of this rapidly unfolding area. It contains twenty-eight papers, broadly sorted into seven categories. It records the transactions of a conference held in October at Taormina, Italy.
Thirteen of the articles were invited specifically for the volume and fifteen were selected from a pool of contributions. There are four papers in the first group on multiresolution analysis. In the fifth group there are two papers on wavelets and fractals, by Jaffard and Holschneider. The sixth group addresses numerical methods. This volume provides an excellent snapshot of the broad activity in wavelet the- ory. For readers of Mathematics of Computation, the papers in the sixth group may be of particular interest.
NURBS are basic objects that are the building blocks for representing curves and surfaces. Such representations, in turn, are essential in computerized design, drafting, modeling, and so on. NURBS are suf- ficiently versatile to fit several distinct systems for computerized design. In the s, such systems grew up independently in different companies mainly in the automobile and aircraft industries and even in different branches of the same com- pany. NURBS eventually made it possible to avoid the chaos in this field that the industry was apparently facing.
The author gives a little of this history in his preface. The book is intended as a textbook for a course in computer-aided-design at the beginning graduate level. Prerequisites are knowledge of linear algebra, calculus, and basic computer graphics. Since formal geometry is NOT a prerequisite, the author begins with a snappy account of projective geometry. I particularly like his definition of the projective plane, which requires just three simple sentences.
Chapter 2 is devoted to projective maps, affine maps, Moebius transformations, perspectivities and collineations. In Chapter 3, conics are introduced in a manner going back to Steiner.
In Chapter 4, more concrete representations of conics are considered, in parametric form. Here we meet the Bernstein form of a conic and the de Casteljau algorithm for computing points on it. The notion of a control polygon is introduced in this context. Interpolating conics, blossoms, and polars make their entrance. In Chapter 5, emphasis shifts from projective geometry to affine geometry, which is closer to the environment of most applications.
These are curves made up piecewise from conics, with certain smoothness imposed at the junctions. We have a Bernstein representation, again with control points and weights, either of which can be manipulated to affect the shape of the curve. There is a projective form of the de Casteljau algorithm, due to the author Rational cubics are the subject of Chapter 8.
Rational cubic splines are treated from the projective viewpoint in Chapter 9, with second-order smoothness imposed by use of the osculants. Thus, rational B-splines of arbitrary degree are permitted.
The basic operation of knot insertion is described. The bilinear and the bicubic cases are singled out. Surfaces of revolution and developable surfaces are considered specially. Triangular patches, quadric surfaces, and Gregory patches are topics considered in the later chapters.
There is a good bibliography and a good index. Copious references to the literature are made throughout the book. All-in-all, this is a very appealing book that should have a stimulating effect on the teaching of this important subject. It can certainly be recommended for solo study because of the gentle expository style of the writing. A common thread is the use of differential geometry constructs to express fair- ness.
Typically, not these quantities alone, but also their arcwise derivatives or divided differences , are the determinants of shape quality. Sur- faces are to be treated similarly, with at least five double-integral fairness metrics to choose from. Other chapters present comparable schemes. Eck and Jaspert work with point sets only.
Burchard et al. Ginnis et al. Skillful handling leads to an optimization problem wherein both objective and constraint functions are polynomial in the unknown spline coefficients.
Rounding out the volume are various papers that speak to the issue of shape control without explicitly defining, or attempting to measure, fairness: Gallagher and Piper on convexity-preserving surface interpolation, Bloor and Wilson on in- teractive design using PDEs, J.
Peters on surfaces of arbitrary topology using bi- quadratic and bicubic splines, Zhao and Rockwood on a convolution approach to N -sided patches, Beier and Chen on a simplified reflection model for interactive smoothness evaluation.
The relevant parts of the book are written in the same spirit as the classical monograph of McClellan and Rader [2]. The underlying algebraic structures are the residue class rings Z M of the integers modulo M as well as polynomial rings over Z M.
The third main topic of the book is error detection and correction by linear codes over Z M , with special reference to fault tolerance in modular arithmetic. The relevant mathematics are expressed in non-technical terms whenever possible, in the interests of keeping the treatment accessible to a majority of students.
This symposium relates considerable numerical analysis involved in research in both theoretical and practical aspects of the finite element method. This text is organized into three parts encompassing 34 chapters. Part I focuses on the mathematical foundations of the finite element method, including papers on theory of approximation, variational principles, the problems of perturbations, and the eigenvalue problem.
Part II covers a large number of important results of both a theoretical and a practical nature. This part discusses the piecewise analytic interpolation and approximation of triangulated polygons; the Patch test for convergence of finite elements; solutions for Dirichlet problems; variational crimes in the field; and superconvergence result for the approximate solution of the heat equation by a collocation method.
Part III explores the many practical aspects of finite element method. This book will be of great value to mathematicians, engineers, and physicists. Understanding and Implementing the Finite Element Method is essential reading for those interested in understanding both the theory and the implementation of the? This book contains a thorough derivation of the finite element equations as well as sections on programming the necessary calculations, solving the finite element equations, and using a posteriori error estimates to produce validated solutions.
Accessible introductions to advanced topics, such as multigrid solvers, the hierarchical basis conjugate gradient method, and adaptive mesh generation, are provided.
Each chapter ends with exercises to help readers master these topics. Readers will bene? Students can use the MATLAB codes to experiment with the method and extend them in various ways to learn more about programming?
This practical book should provide an excellent foundation for those who wish to delve into advanced texts on the subject, including advanced undergraduates and beginning graduate students in mathematics, engineering, and the physical sciences.
Chapter 6: The mesh data structure; Chapter 7: Programming the finite element method: Linear Lagrange triangles; Chapter 8: Lagrange tri. Written by the pre-eminent professors in their fields, this new edition of the Finite Element Method maintains the comprehensive style of the earlier editions and authoritatively incorporates the latest developments of this dynamic field.
Expanded to three volumes the book now covers the basis of the method and its application to advanced solid mechanics and also advanced fluid dynamics.
Volume Two: Solid and Structural Mechanics is intended for readers studying structural mechanics at a higher level. Although it is an ideal companion volume to Volume One: The Basis, this advanced text also functions as a "stand-alone" volume, accessible to those who have been introduced to the Finite Element Method through a different route.
Volume 1 of the Finite Element Method provides a complete introduction to the method and is essential reading for undergraduates, postgraduates and professional engineers. Volume 3 covers the whole range of fluid dynamics and is ideal reading for postgraduate students and professional engineers working in this discipline. Coverage of the concepts necessary to model behaviour, such as viscoelasticity, plasticity and creep, as well as shells and plates. Up-to-date coverage of new linked interpolation methods for shell and plate formations.
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