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內容簡介

　　Finite Element Method and its Applications discusses the methods in a general frame and the performance on the computer, the variational formulations for elliptic boundary value problems, the error estimates and convergence for finite element approximate solutions and nonstandard finite element. In particular, presentations of the subject include the applications of finite element method to various scientific and engineering problems, for example, three dimensional elastic beam, elastic mechanics, three dimensional neutron diffusion problems, magneto hydrodynamics, three dimensional turbomachinery flows, Navier-Stokes equations and bifurcation phenomena for nonlinear problem, etc. Most applications results were established by the authors in the past three decades. This book was written by Kaitai Li, Aixiang Huang, Qinghuai Huang.

目錄

Chapter 1 The Structure of Finite Element Method1.1 Galerkin Variational Principle and Ritz Variational Principle1.2 Galerkin Approximation Solution1.3 Finite Element Subspace1.4 Element Stiffness and Total Stiffness
Chapter 2 Elements and Shape Functions2.1 Rectangular Shape Function2.1.1 Lagrange Type Shape Function of Rectangular2.1.2 Hermite Type Shape Function of Rectangular2.2 Triangular Element2.2.1 Area Coordinate and Volume Coordinate2.2.2 Lagrange Type Shape Function of Triangular Element2.2.3 Hermite Type Shape Function of Triangular Element2.3 Shape Function of Three Dimensional Element2.3.1 Lagrange Type Shape Function of Hexahedron Element2.3.2 Lagrange Type Shape Function of Tetrahedron Element2.3.3 Shape Function of The Three Prism Element2.3.4 Hermite-Type Shape Function of Tetrahedron Element2.4 Iso-parametric Finite Element2.5 Curve Element
Chapter 3 Procedure and Performance of Computation of Finite Element Method3.1 The Procedure of Finite Element Computation3.2 One dimensional Store of Symmetric and Band Matrix3.3 Numerical Integration3.4 Computation of Element Stiffness Matrix and Synthesis of Total Stiffness Matrix3.4.1 Computation of Shape Function3.4.2 The Computation of Element Stiffness Matrix and Element Array3.4.3 Superposition of Elements of Total Stiffness Matrix3.5 Direct Solution Method for Finite Element Algebraic Equations3.5.1 Decomposition for Symmetric and Positive Definition Matrix3.5.2 Direct Solution for Algebraic equations3.6 Other Solution Method for Finite Element Algebraic Equations3.6.1 The Steepest Descent Method3.6.2 Conjugate Gradient Method3.7 Treatment of Constraint Conditions3.7.1 Treatment of Imposed Constraint Conditions3.7.2 Treatment of Periodic Constrain Condition3.7.3 Remove Periodic Constrain and Matrix Transformation3.7.4 Performance of the Method on Computer3.8 Calculation of Derivatives of Function3.9 Automatic Generation of Finite Element Mesh
Chapter 4 Sobolev Space4.1 Some Notations and Assumptions on Domain4.2 Classical Function Spaces4.3 LP（Ω） Space4.4 Spaces of Distribution4.5 Sobolev Spaces with Integer Index4.6 Sobolev Space with a Real Index HσP（Ω）4.7 Embedding Theorem and Interpolate Inequalities4.8 The Trace Spaces
Chapter 5 The Variational Principle for Elliptic Boundary Value Problem and Error Estimate of Finite Element Approximation Solution.5.1 Elliptic Boundary Value Problem5.1.1 Regularity5.1.2 The Existence and Uniqueness of the Solution5.1.3 Maximum Principle5.2 Variational Formulations5.3 Finite Element Approximation Solutions5.4 Coordinate Transformation and Equivalent Finite Element5.4.1 Affine Transformation and Affine Equivalent Finite Element5.4.2 Isoparametric Transformation and Isopavametric Finite Element5.5 The Theory of Finite Element Interpolation5.5.1 Some Lemma……Chapter 6 Nonstandard Finite Element MethodsChapter 7 Applications of Finite Element Method in the EngineeringChapter 8 Finite Element Analysis for Internal Flow in TurbomachineChapter 9 Finite Element Approximation for the Navier-Stokes EquationsReferences

精彩書摘

　　Chapter 1
　　The Structure of Finite Element Method
　　The finite element method is a numerical computational method for differential equations and partial differential equations. In order to solve the general field problem by using finite element method， it must pass through the following processes：
　　1） Find the variational formulation associated with original field problem.
　　2） Establish finite element subspace. For example， select the element type and associated phase functions.
　　3） Establish element stiffness matrix， element column and assemble global stiffness matrixfull column.
　　4） Treatment of the boundary conditions and solving of the system of finite element equations.
　　5） Come back to the real world. In this book， the first four processes will be systematic formulations in the first chapter till third chapter.
　　1.1 Galerkin Variational Principle and Ritz Variational Principle As an example， we consider the linear elliptic boundary value problem of two dimension，
　　（1.1.1） where， Ω is a connected domain in R2， .Ω = Γ1 ∪ Γ2 is a piecewise smooth boundary. Letn denote the unit outward normal vector to .Ω defined almost everywhere on .Ω. p（x， y） ∈C1（Ω）， p（x， y） ≥ p0 > 0， σ（x， y） ∈ C0（Ω） and σ（x， y） ≥ 0.
　　Throughout this chapter we make notation： C0（Ω） = the set of all continuous function in an open subset in Rn. Ck（Ω） = the set of functions v ∈ C0（Ω）， whose derivatives of order k，exist and are continuous；
　　where α = （α1， ？ ？ ？ ， αn）， |α| = α1 + ？ ？ ？ + αn.
　　Assume that u（x， y） ∈ C2（Ω） satisfies （1.1.1） in Ω and on .Ω， the function u（x， y） is called classical solution of problem （1.1.1）.Next， we consider weak solution of （1.1.1）. Define the norm
　　（1.1.2） Sobolev space H1（Ω） is a closure of C∞（Ω）， under the norm （1.1.2） with the inner product
　　（1.1.3） H1（Ω） is a Hilbert space which is called one order Sobolev space. Let C∞0 （Ω） = {v ： v is an infinite differentiable function and support of v . Ω}， H10 （Ω） = the closure of C∞0 （Ω） under the norm（1.1.2），it is equivalent to H10 （Ω） = {v ： v ∈ H1（Ω）， v|.Ω = 0}.In addition， let C∞# （Ω） = {v ： v ∈ C∞（Ω）， v|Γ1 = 0}，V （Ω） = closure ofC∞# （Ω） under the norm（1.1.2），which is equivalent toV = {v ： v ∈ H1（Ω）， v|Γ1 = 0}.
　　It is clear that V is a Hilbert space with inner product （1.1.3）. Furthermore，H10 （Ω） . V . H1（Ω）.Let us introduce bilinear functional
　　（1.1.4） In （1.1.4）， fixed u， then B（u， v） is a linear functional of v， while v is fixed， it is a linear functional of u. In other words， suppose α1， α2， β1， β2 are arbitrary constants， then B（α1u1 + α2u2， β1v1 + β2v2） =α1β1B（u1， v1） + α1β2B（u1， v2） + α2β1B（u2， v1） + α2β2B（u2， v2）， .u1， u2， v1， v2 ∈ H1（Ω）. It is clear that （1.1.4） satisfies
　　（1） Symmetry，B（u， v） = B（v， u）. （1.1.5）
　　（2） The continuity in V × V ， i.e.， there exists a constant M >0， such that|B（u， v）| M u 1，Ω v 1，Ω， .u， v ∈ V. （1.1.6）
　　（3） Coerciveness in V ， i.e.， there exists constant γ > 0， such that B（u， u） γ u 2 1，Ω， .u ∈ V. （1.1.7） Of course， is a continuous linear functional in v.
　　The Galerkin Variational Formulation for （1.1.1）： Find u ∈ V ， such thatB（u， v） = f（v）， .v ∈ V. （1.1.8）A solution u satisfying （1.1.8） is called a weak solution of （1.1.1）. The space V is calledadmissible space or trial space. On the other hand， （1.1.8） must be satisfied for every v ∈ V ，therefore， V is called test function space. If trial and test space for the variational problem arethe same Hilbert V ， in this case， V is called energy space.
　　Owing to the boundary condition on Γ2 is contained in the variational problem （1.1.8）， theboundary condition on Γ2 is called nature boundary condition， while the boundary conditiononΓ1 is called essential boundary condition.
　　The following proposition gives the relationship between classical solution and weak solution of （1.1.1）.
　　Proposition 1.1 Suppose u ∈ C2（Ω）. If u is a classical solution of （1.1.1）， then， u isthe weak solution of （1.1.1）. Otherwise， if u is a weak solution of （1.1.1）， then u is a classicalsolution of （1.1.1）.
　　……

前言/序言

有限元方法及其應用（英文版） [Finite Element Method and its Applications] 下載 mobi epub pdf txt 電子書

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