Partial Differential Equations and the Finite Element MethodJohn Wiley & Sons, 16 dic 2005 - 512 pagine A systematic introduction to partial differential equations and modern finite element methods for their efficient numerical solution Partial Differential Equations and the Finite Element Method provides a much-needed, clear, and systematic introduction to modern theory of partial differential equations (PDEs) and finite element methods (FEM). Both nodal and hierachic concepts of the FEM are examined. Reflecting the growing complexity and multiscale nature of current engineering and scientific problems, the author emphasizes higher-order finite element methods such as the spectral or hp-FEM. A solid introduction to the theory of PDEs and FEM contained in Chapters 1-4 serves as the core and foundation of the publication. Chapter 5 is devoted to modern higher-order methods for the numerical solution of ordinary differential equations (ODEs) that arise in the semidiscretization of time-dependent PDEs by the Method of Lines (MOL). Chapter 6 discusses fourth-order PDEs rooted in the bending of elastic beams and plates and approximates their solution by means of higher-order Hermite and Argyris elements. Finally, Chapter 7 introduces the reader to various PDEs governing computational electromagnetics and describes their finite element approximation, including modern higher-order edge elements for Maxwell's equations. The understanding of many theoretical and practical aspects of both PDEs and FEM requires a solid knowledge of linear algebra and elementary functional analysis, such as functions and linear operators in the Lebesgue, Hilbert, and Sobolev spaces. These topics are discussed with the help of many illustrative examples in Appendix A, which is provided as a service for those readers who need to gain the necessary background or require a refresher tutorial. Appendix B presents several finite element computations rooted in practical engineering problems and demonstrates the benefits of using higher-order FEM. Numerous finite element algorithms are written out in detail alongside implementation discussions. Exercises, including many that involve programming the FEM, are designed to assist the reader in solving typical problems in engineering and science. Specifically designed as a coursebook, this student-tested publication is geared to upper-level undergraduates and graduate students in all disciplines of computational engineeringand science. It is also a practical problem-solving reference for researchers, engineers, and physicists. |
Dall'interno del libro
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Sommario
1 | |
2 Continuous Elements for 1D Problems | 45 |
3 General Concept of Nodal Elements | 103 |
4 Continuous Elements for 2D Problems | 125 |
5 Transient Problems and ODE Solvers | 167 |
6 Beam and Plate Bending Problems | 209 |
7 Equations of Electromagnetics | 269 |
Appendix A Basics of Functional Analysis | 319 |
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Partial Differential Equations and the Finite Element Method Pavel Ŝolín Anteprima non disponibile - 2005 |
Partial Differential Equations and the Finite Element Method Pavel Ŝolín Anteprima non disponibile - 2005 |
Parole e frasi comuni
a₁ Algorithm analogously basis functions bilinear bounded bubble functions coefficients Consider constant convergence defined Definition degrees of freedom Dirichlet boundary conditions discrete edge elements EdgeList Elem element Km elliptic exact solution example Exercise existence and uniqueness Fekete points Figure finite element mesh finite element methods Gaussian quadrature H¹(N Hermite elements hierarchic shape functions higher-order Hilbert space inequality inner product integral interface Lagrange nodal Lax-Milgram lemma Lemma linear forms linear operator linear space Lobatto lowest-order maximum norm Maxwell's equations nodal basis nodal points nodal shape functions normed space obtain orthogonal Paragraph PDEs piecewise-affine polynomial degree polynomial space problem Proof reference domain reference map right-hand side RK methods second-order sequence Sobolev space solver stiffness matrix subspace Th,p Theorem triangular unisolvent V-elliptic V₁ vector vert_dof vertex basis functions vertex functions Vh.p W₁ weak formulation zero ди
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Pagina 186 - X if and only if, for every e > 0, there exists a 8 > 0 such that c*'(/(*>, /(P...
Pagina 2 - We recall that an nxn matrix A is said to be positive definite if vTAv > 0 for all non-zero v € R".
Pagina 341 - A is diagonalizable if and only if it has n linearly independent eigenvectors. In that case, the diagonal matrix D, similar to A is given by /X, 0 0 0 X2 0 0 0 X, D = \ \0 0 0 ••• A.,,/ where X, , X2, ..., Xn are the eigenvalues of A.
Pagina 66 - It is left to the reader as an exercise to verify that the module generated by a,b,c,d with the above relations give A/(2t - l,t — 2) ® A.
Pagina 84 - S consists of three arrays: 1. Array A of length NNZ: This is a real-valued array containing all nonzero entries of the matrix S listed from the left to the right, starting with the first and ending with the last row. 2. Array IA of length N + 1: This is an integer array, IA[l] = 1.