有限元方法及其应用

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

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.